y ( m) y ( m) A new tool for miscibility control: Linear coupling 0 4 8 12 0 4 8 12 0 |2 |1 50 100 x ( m) Eike Nicklas 2013 150 200 Dissertation submied to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg. Germany for the degree of Doctor of Natural Sciences Put forward by Eike Nicklas born in: Heidelberg, Germany Oral examination: 23. July, 2013 A new tool for miscibility control: Linear coupling Referees: Prof. Dr. Markus K. Oberthaler Prof. Dr. omas Gasenzer Abstract In this work we experimentally study the rich interplay of a linear coupling and non-linear interactions between the two components of an elongated Bose-Einstein condensate of 87 Rb. In the limit of strong linear coupling we generate dressed states and explore the eﬀective interactions between them. We ﬁnd that the miscibility of dressed states is opposite to that of the atomic states. If the characteristic energies of interactions and linear coupling are equal they give rise to a miscible-immiscible quantum phase transition. We study the linear response of the system to sudden quenches in the vicinity of the critical point by analyzing spin correlations in the system. A power law scaling of the characteristic length scales is observed on both sides of the phase transition and the scaling exponents agree with the mean ﬁeld prediction. Temporal scaling is found on the miscible side in agreement with a prediction based on Bogoliubov theory. In addition, experimental results for ﬁnite-time quenches through the critical point are presented. e good control over amplitude and phase of the linear coupling ﬁeld oﬀers new possibilities for the study of both equilibrium and dynamical properties of phase transitions. Zusammenfassung Diese Arbeit behandelt das vielältige Zusammenspiel von linearer Kopplung und nichtlinearen Wechselwirkungen zwischen zwei Komponenten eines elongierten Bose-Einstein Kondensats von 87 Rb. Im Grenzfall starker linearer Kopplung erzeugen wir sogenannte ’dressed states’ und untersuchen die Wechselwirkungen zwischen ihnen. Dabei zeigen wir, dass die Mischbarkeit von ’dressed states’ entgegengesetzt der von atomaren Zuständen ist. Im Falle gleich großer charakteristischer Energien von Wechselwirkung und Kopplung ﬁndet ein antenphasenübergang von mischbar zu nicht mischbar sta. Wir untersuchen die Antwort des Systems auf plötzliche Parameteränderungen in der Nähe des kritischen Punktes anhand des Verhaltens von Spinkorrelationen. Dabei zeigt sich ein Potenzgesetz im Skalieren der charakteristischen Längenskalen auf beiden Seiten des Phasenübergangs, wobei die Exponenten mit der Vorhersage der Molekularfeldnäherung übereinstimmen. Auf der mischbaren Seite wird ein zeitliches Skalierungsverhalten in Übereinstimmung mit der Vorhersage der Bogoliubov-eorie beobachtet. Zusätzlich präsentieren wir experimentelle Ergebnisse zu langsamen Rampen in der Nähe des kritischen Punktes. Die gute Kontrolle über die Amplitude und Phase des linearen Kopplungsfeldes scha neue Möglichkeiten, die Eigenschaen von Phasenübergängen sowohl im dynamischen Fall als auch unter Gleichgewichtsbedigungen zu untersuchen. Contents 1. Introduction 2. Theory of linearly coupled interacting Bose-Einstein condensates 2.1. Hamiltonian and equations of motion . . . . . . . . . . . . . . . . . . . 2.2. Single spatial mode approximation . . . . . . . . . . . . . . . . . . . . . 2.2.1. Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Internal Josephson junction . . . . . . . . . . . . . . . . . . . . 2.3. Ground state properties of elongated binary condensates . . . . . . . . 2.3.1. Homogeneous system without dressing . . . . . . . . . . . . . 2.3.2. Homogeneous system with dressing . . . . . . . . . . . . . . . 2.3.3. Eﬀects of a trapping potential . . . . . . . . . . . . . . . . . . . 2.4. Bogoliubov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Linearization of the equations of motion . . . . . . . . . . . . . 2.4.2. Bogoliubov spectrum of a homogeneous system . . . . . . . . . 2.4.3. Bogoliubov spectrum of a dressed system . . . . . . . . . . . . 2.4.4. Eﬀects of a trap and numerical Bogoliubov-de Gennes analysis 2.5. Extension to negative coupling strengths . . . . . . . . . . . . . . . . . 2.5.1. Phase and stability diagrams and summary . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 5 7 8 8 10 11 12 13 14 15 19 20 23 3. Experimental system and analysis methods 3.1. Experimental system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Optical dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Linear coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Employed atomic states and Feshbach resonance . . . . . . . . . . . . . . 3.1.4. Detection of the atomic cloud . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Free evolution experiments and their analysis . . . . . . . . . . . . . . . . . . . . 3.2.1. Free evolution far from the Feshbach resonance in the charger . . . . . . . 3.2.2. Formation of spin domains near the Feshbach resonance in the waveguide 3.2.3. Analysis methods: Counting, Fourier spectra and correlations . . . . . . . 3.2.4. Mapping out the Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 25 28 30 31 31 31 33 34 37 . . . . 41 41 42 44 45 5. A miscible-immiscible phase transition 5.1. Non-adiabatic generation of dressed states . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 51 52 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Interacting dressed states 4.1. Rabi oscillations in the presence of interactions . . . . . . . . . . . . . . . . . . 4.1.1. Amplitude of long Rabi oscillations . . . . . . . . . . . . . . . . . . . . . 4.2. Interacting dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Reconstruction of dressed states from spatially resolved Rabi oscillations . . . . i 5.2. Dynamics beyond the strong coupling limit . . . . . 5.3. Linear response to quenches near the critical point 5.3.1. Scaling on the miscible side of the transition 5.3.2. Scaling in immiscible regime . . . . . . . . 5.4. Summary, outlook and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 54 58 63 68 6. Dynamics of phase transitions and the Kibble-Zurek mechanism 6.1. Proposed implementation in binary Bose-Einstein condensates . . . . 6.1.1. A criterion for adiabatic quenches . . . . . . . . . . . . . . . 6.1.2. Numerical simulations and inhomogeneity eﬀects . . . . . . . 6.1.3. Experimental feasibility . . . . . . . . . . . . . . . . . . . . . 6.2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Adiabatic and non-adiabatic ramps towards the critical point 6.2.2. Non-adiabatic ramps through the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 74 74 77 78 79 79 81 7. Conclusion and Outlook A. Summary of atomic and experimental parameters A.1. Properties of 87 Rb . . . . . . . . . . . . . . . . . . . . . A.1.1. Scaering lengths . . . . . . . . . . . . . . . . . A.1.2. Loss coeﬃcients . . . . . . . . . . . . . . . . . . A.1.3. Scaering lengths near the Feshbach resonance A.2. Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Optical dipole traps . . . . . . . . . . . . . . . . . . . . A.3.1. Charger . . . . . . . . . . . . . . . . . . . . . . A.3.2. Waveguide . . . . . . . . . . . . . . . . . . . . 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 87 87 87 88 88 88 88 88 B. Numerical methods for simulating Bose-Einstein condensates B.1. Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . B.1.1. One-dimensional Gross-Pitaevskii equation . . . . . . B.1.2. Nonpolynomial nonlinear Schrödinger equation . . . . B.2. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . B.2.1. Computing the ground state . . . . . . . . . . . . . . . B.2.2. Bogoliubov - de Gennes stability analysis . . . . . . . B.2.3. Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 89 89 90 91 91 94 96 . . . . . . . . . . . . . . . . . . . . . . . . C. Calibration of in-situ imaging near the Feshbach resonance 99 C.1. Adjusting the imaging frequency for maximum detectivity . . . . . . . . . . . . . . 99 C.2. Absolute atom number calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 C.3. Imaging in the presence of a linear coupling ﬁeld . . . . . . . . . . . . . . . . . . . 100 Bibliography ii 103 1. Introduction Phase transitions are ubiquitous in nature. One everyday example is the boiling of water, which marks the transition from a liquid to a gaseous state at a temperature of 100◦ C under standard pressure conditions of 1 bar. Further examples of phase transitions range from Bose-Einstein condensation to the high temperature plasma in the early stages of the universe aer the Big Bang. ese examples demonstrate that phase transitions occur on all energy scales from the nanokelvin regime up to beyond 1020 Kelvin. e boiling of water is a paradigm example of a ﬁrst order phase transition as the two phases, water and steam, can coexist at the transition temperature due to latent heat in the process. As heat is added to boiling water its temperature stays constant, but the fraction of particles in gaseous form is increased. In second order phase transitions the two phases cannot coexist at the transition point. A well known example is the Curie point separating ferromagnetic from paramagnetic behavior in magnetic materials. Above the Curie temperature and in the absence of external magnetic ﬁelds the magnetic moments of the atoms in the material point in random directions. When cooling below the transition point the magnetic moments align along a randomly chosen axis of magnetization. is is an illustrative example of a symmetry breaking in phase transitions, as the system spontaneously chooses one out of a set of equivalent conﬁgurations. A phase transition is caused by the competition of two energy scales, one of them favoring an ordered state and the other a disordered conﬁguration. ese energy scales are equal at the transition point. In thermodynamic or classical phase transitions one of these competing quantities is given by temperature or pressure and the loss of order is driven by entropy due to thermal ﬂuctuations. In contrast, quantum phase transitions occur at zero temperature and the symmetry breaking is caused by quantum ﬂuctuations [1]. e study of critical phenomena at second order phase transitions has triggered many new concepts in theoretical physics such as the notion of universality or renormalization group methods. For example, the characteristic length and time scales of the system diverge at the critical point with a power law scaling. e corresponding critical exponents depend only on the universality class of the system given for example by its dimension or the range of interactions, but not on the microscopic details. Mean ﬁeld theories can predict values for the critical exponents, but their validity breaks down close to the critical point. Ultracold quantum gases oﬀer new prospects for experimental studies of phase transitions and criticality as they are well isolated from the environment and oﬀer a high level of control over the relevant system parameters such as interactions or the dimensionality of the trapping potential. ey have been employed to study thermodynamic phase transitions such as Bose-Einstein condensation [2] or the Berezinskii-Kosterlitz-ouless (BKT) transition in two dimensions [3] as well as quantum phase transitions. Prominent examples of the laer are the Mo insulator to superﬂuid transition [4, 5] or the Dicke phase transition [6]. Symmetry breaking has been observed in one-[7] and multi-component Bose-Einstein condensates [8, 9] as well as in the context of the aforementioned quantum phase transitions [10, 11]. Universality and scale invariance have been measured in the density ﬂuctuations near the BKT transition [12] and quantum critical behavior was observed at the vacuum-to-superﬂuid transition in two-dimensional optical laices [13]. Some of the experiments mentioned above employ sudden or ﬁnite-time quenches through or 1 1. Introduction towards the critical point. e relaxation dynamics following quenches is an interesting subject on its own and has been studied in the context of the light-cone-like spreading of correlations [14] and prethermalization [15]. In the context of phase transitions quenches have been proposed as a tool to probe both equilibrium and dynamical scaling properties near critical points [16]. In this thesis we report on the experimental realization of a quantum phase transition in a quasi one-dimensional two-component Bose-Einstein condensate as proposed in [17, 18]. e inter-atomic interactions are chosen such that the two components are immiscible, i.e. in the ground state their overlap is minimized. us, the ground state of the system consists of two separate domains each containing atoms of one component. In a homogeneous one-dimensional geometry this state breaks translational symmetry as the position of the boundary between the domains is chosen randomly [19]. e symmetry restoring component is realized by a linear coupling of the involved states employing an electromagnetic ﬁeld, which favors spatial overlap of the two atomic clouds. A critical point occurs when the energy scales of the symmetry-breaking interactions and the symmetryrestoring linear coupling are equal. In our experiments we characterize both ingredients for the phase transition, namely atomic interactions and the linear coupling. In the absence of a linear coupling the dynamics of the system is determined by inter-atomic interactions. A Feshbach resonance [20] allows us to tune the relevant interaction strength and we systematically analyze its eﬀect on the spatial dynamics of the system. If the linear coupling is much stronger than the interactions, the laer can be neglected and the system is dominated by the coupling ﬁeld. In analogy to optical dressed states [21] the system can be described by eigenstates of the linear coupling Hamiltonian. We experimentally generate dressed condensates using a novel non-adiabatic preparation scheme. In addition the eﬀective interactions among dressed states are examined and contrasted to the interactions of the involved bare states. Aer having independently studied the relevant ingredients for the miscible-immiscible transition we characterize the phase transition and measure the linear response of the system aer sudden quenches to the vicinity of the transition point. A power law divergence of the characteristic length scales at the critical point is observed both on the miscible and on the immiscible side. In addition the extracted relaxation time in the miscible regime is compared to the energy gap of the excitation spectrum of the system. Our ﬁndings agree with theoretical mean ﬁeld predictions based on Bogoliubov theory. is thesis is organized as follows: Aer this introduction, we discuss the theoretical description of linearly coupled binary Bose-Einstein condensates and summarize the results necessary for the interpretation of the experimental observations. e third chapter introduces our experimental system and presents the employed methods for analyzing the obtained data. Chapter 4 deals with the interaction properties of dressed states, which are obtained from the amplitude of Rabi oscillations aer several hundred oscillation cycles. ese results are published in [22]. e implementation of the miscible-immiscible quantum phase transition is presented in chapter 5 and the non-equilibrium dynamics of the system following a sudden quench is discussed. e ﬁnal chapter discusses dynamical scaling near phase transitions in the context of the Kibble-Zurek mechanism. A compact overview of the relevant experimental parameters and numerical methods for the simulation of Bose-Einstein condensates is provided in the appendices. e following experiments are not discussed in this thesis, but were performed during the same time. • Nonlinear atom interferometry beats classical precision limit [23] • Classical Bifurcation at the Transition from Rabi to Josephson Dynamics [24] • Atomic homodyne detection of continuous variable entangled twin-atom states [25] • Optimized absorption imaging of mesoscopic atomic clouds [26] 2 2. Theory of linearly coupled interacting Bose-Einstein condensates Bose-Einstein condensates are versatile tools for the study of interacting macroscopic quantum systems. e theoretical description of such degenerate quantum gases is signiﬁcantly simpliﬁed by their good isolation from the environment and their low temperature. As the ground state of the system is macroscopically occupied it can be described as a classical ﬁeld in the so-called mean ﬁeld approximation [27]. In addition, due to the small kinetic energies of the atoms, interactions are possible only via s-wave scaering. us, atomic interactions are characterized by a single parameter, the s-wave scaering length as . A positive scaering length as > 0 denotes repulsive interactions, while as < 0 implies araction between the atoms. Compared to the situation with a single atomic species, the interaction properties of two-component condensates are much richer as each component interacts with atoms both of the same and of the other species. e relative values of the diﬀerent scaering lengths determine fundamental properties of the system such as its miscibility or stability. ese characteristics are fundamentally modiﬁed in the presence of a radiation ﬁeld that linearly couples the two components. For example, an immiscible system can be tuned miscible by the radiation ﬁeld [17]. In this chapter we introduce the theoretical description of linearly coupled interacting two-component Bose-Einstein condensates in the mean ﬁeld approximation. We will discuss their ground state properties and excitation spectra along with the resulting dynamics. In our experimental system the atomic clouds are conﬁned in an elongated trapping potential. us, we will constrain the discussion to one dimension. We will simplify the description by assuming a homogeneous system without longitudinal conﬁnement as it allows to derive many results analytically. e deviations caused by a trapping potential will be discussed where necessary and quantiﬁed by numerical simulations. 2.1. Hamiltonian and equations of motion We begin our discussion with the Hamiltonian of two interacting atomic clouds in the presence of a linear coupling ﬁeld. In addition, this section introduces the relevant quantities and the mean ﬁeld equations of motion. Hamiltonian Our experimental system consists of a Bose-Einstein condensate of 87 Rb. e two components are realized by diﬀerent hyperﬁne states of the electronic ground state, which are coupled via microwave and radio frequency radiation. e Hamiltonian of the system consists of three terms [28, 18] Ĥ = Ĥ0 + Ĥint + Ĥcpl (2.1) Ĥ0 describes the single particle eﬀects, Ĥint the interactions among atoms in the same as well as in diﬀerent states, while Ĥcpl summarizes the eﬀects of a possibly detuned linear coupling of the two 3 2. eory of linearly coupled interacting Bose-Einstein condensates components. In second quantization these term are given by ) ( ∑∫ ℏ2 2 † Ĥ0 = dx Ψ̂i − ∇ + V Ψ̂i 2m i=1,2 ∫ 1 ∑ † † Ĥint = gij dx Ψ̂i Ψ̂j Ψ̂j Ψ̂i 2 i,j=1,2 ∫ [ ] 1 ∫ [ † ] 1 † † ∗ † Ĥcpl = − dx ℏΩ̃Ψ̂1 Ψ̂2 + ℏΩ̃ Ψ̂2 Ψ̂1 + ℏδ dx Ψ̂2 Ψ̂2 − Ψ̂1 Ψ̂1 2 2 where Ψ̂i = Ψ̂i (x, t) denote bosonic ﬁeld annihilation operators obeying the bosonic commutation relations, 2πℏ Planck’s constant, m the atomic mass, V = V(x) an external potential identically 4πℏ2 a acting on the two components and gij = m ij the interaction strength parametrized by the s-wave scaering length aij .¹ e linear coupling between the components is characterized by the Rabi frequency Ω̃ = Ωeiφ and the detuning δ from the atomic resonance. Ĥ0 and Ĥcpl are single particle Hamiltonians, whereas Ĥint introduces nonlinear eﬀects via two-body collisions. Mean field description and classical Hamiltonian At low temperatures and macroscopic atom numbers the ﬁeld operators Ψ̂i for the Bose-Einstein condensates can be replaced by classical complex-valued functions ψ i (x, t), which we call the order parameter or wave function of the condensate [27]. In this mean ﬁeld description, the linear atom density is given by ni (x, t) = ψ ∗i (x, t)ψ i (x, t) and the classical Hamiltonian takes the form (2.2) H = H0 + Hint + Hcpl with ( ) ℏ2 ∗ 2 H0 = dx − ψ i ∇ ψ i + Vni 2m i=1,2 ∫ 1 ∑ Hint = gij dx ni nj 2 i,j=1,2 ∫ [ ] 1 ∫ 1 ∗ ∗ ∗ Hcpl = − dx ℏΩ̃ψ 1 ψ 2 + ℏΩ̃ ψ 2 ψ 1 + ℏδ dx [n2 − n1 ] 2 2 ∑∫ In the following we denote the number of atoms in the i-th component by Ni = 1 their normalized diﬀerence by the imbalance z = NN21 −N +N2 . ∫ dx ni (x) and Equations of motion e equations of motion governing the dynamics of a two-component condensate can be obtained from the Hamiltonian given by Equation 2.1 using the Heisenberg equation iℏ∂t Ψ̂i = [Ψ̂i , Ĥ], which yields the coupled pair of equations ¹e interaction parameters gij are modiﬁed for (quasi-)one-dimensional systems. See subsection B.1.1 for details. 4 2.2. Single spatial mode approximation ] [ ∂ ℏ2 2 2 2 iℏ ψ 1 = − ∇ + V + g11 |ψ 1 | + g12 |ψ 2 | ψ 1 − ∂t 2m [ ] ∂ ℏ2 iℏ ψ 2 = − ∇2 + V + g22 |ψ 2 |2 + g12 |ψ 1 |2 ψ 2 − ∂t 2m ℏΩ̃ ℏδ ψ2 − ψ 2 2 1 ∗ ℏΩ̃ ℏδ ψ + ψ 2 1 2 2 (2.3) ese equations are the main tool for modeling the dynamics of the two-component condensate in the mean ﬁeld regime. eir single-component version was derived independently by Gross [29] and Pitaevskii [30] and is referred to as the Gross-Pitaevskii equation. 2.2. Single spatial mode approximation e dynamics of the system is governed by the interplay of several ingredients: A linear coupling of two states, non-linear interactions among the atoms, as well as the spatial degree of freedom. Before discussing the properties of this complex system we will ﬁrst focus on each of these contributions separately in order to understand their physical characteristics. As a ﬁrst simpliﬁcation we neglect the spatial degree of freedom and we reduce the one-dimensional system to zero dimensions such that only the internal degree of freedom remains, i.e. dynamics in the relative population of the two states. In this case, the spatial part of wave function ϕ 1 (x) = ϕ 2 (x) = ϕ(x) is the same for both components and we assume ϕ(x) to be normalized to 1. Experimentally the single spatial mode approximation is applicable when the atomic cloud is conﬁned in a tight trap, such that the extent of the cloud is smaller than the typical length scale of density variations due to interactions. In the single mode approximation the system reduces to well-known models: In the absence of interactions it is described by the Rabi Hamiltonian whose eigenstates are the dressed states. e inclusion of interactions leads to the Josephson Hamiltonian. In this section, we will discuss these two systems. Bloch sphere representation In the absence of spatial degrees of freedom the system is fully described by N atoms, each having two internal states |1⟩ and |2⟩. Such a two-level system can be mapped onto a spin J = 1/2 system by assigning the atomic state |1⟩ (|2⟩) to the eigenstate of the spin operator Ĵz with the eigenvalue jz = −1/2 (+1/2). A general pure quantum state of the two-level system is characterized by the normalized probability amplitude of being in either state and the relative quantum mechanical phase φ. us it can be represented in spherical coordinates as |θ, φ⟩ = sin(θ/2)|1⟩ + e−iφ cos(θ/2)|2⟩. is expression describes the point on the surface of a sphere with the polar angle θ and the azimuthal angle φ. In this Bloch sphere representation the axes were chosen such that the south pole corresponds to state |1⟩ and the north pole to state |2⟩. e population diﬀerence (|2⟩⟨2| − |1⟩⟨1|)/2 is mapped onto Ĵz and the coherences are represented by Ĵx = (|2⟩⟨1| + |1⟩⟨2|)/2 and Ĵy = (|2⟩⟨1| − |1⟩⟨2|)/2i [31, 32]. 2.2.1. Dressed states In the limit of strong linear coupling, ℏΩ ≫ ngij , interactions can be neglected and the dynamics of the system is governed by Ĥcpl . Identifying the two modes |1⟩ = ψ 1 and |2⟩ = ψ 2 the Hamiltonian 5 2. eory of linearly coupled interacting Bose-Einstein condensates z x y Figure 2.1.: Schematic representation of a quantum state |θ, φ⟩ of a two-level system on the Bloch sphere. e shaded regions illustrate the deﬁnition of θ and φ. can be wrien in matrix notation as Ĥcpl ) ( ) ( 1 1 δ Ω̃ δ Ωeiφ =− ℏ =− ℏ ∗ Ωe−iφ −δ 2 2 Ω̃ −δ (2.4) In analogy to quantum optics [21] the eigenstates of this Hamiltonian are the dressed states |+⟩ = eiφ/2 sin(θ/2)|1⟩ + e−iφ/2 cos(θ/2)|2⟩ |−⟩ = eiφ/2 cos(θ/2)|1⟩ − e−iφ/2 sin(θ/2)|2⟩ (2.5) with the mixing angle tan θ = −Ω/δ. e corresponding eigenenergies are E± = ℏ2 (∓Ωeﬀ − δ), √ where Ωeﬀ = Ω2 + δ 2 denotes the oscillation (Rabi) frequency in the presence of a detuning [31]. In the limit of a large detuning |δ| ≫ Ω, θ = 0 and the dressed states coincide with the atomic states |1⟩ and |2⟩ [33]. On resonance, δ = 0, the dressed states are equal superpositions of the atomic states |+⟩ = √12 (|1⟩ + |2⟩) and |−⟩ = √12 (|1⟩ − |2⟩) with eigenenergies − 12 ℏΩ and + 12 ℏΩ, respectively. Note that the |+⟩ state is lower in energy and corresponds to the ground state while |−⟩ is an excited state.² In terms of the spin representation introduced above the linear coupling Hamiltonian is wrien as Ĥcpl = ℏΩ(cos φ Ĵx + sin φ Ĵy ) + ℏδ Ĵz [34]. us, in the Bloch sphere picture, the action of the Hamiltonian Equation 2.4 corresponds to a rotation around the axis (θ, φ). For resonant coupling θ = π/2 and the rotation axis passes through the equator, whereas it is tilted towards one of the poles by a non-zero detuning. e |+⟩ dressed state is parallel and the |−⟩ state antiparallel to this rotation axis and thus they are stationary [31]. A generic pure quantum state |θ ′ , φ ′ ⟩ which is not aligned with the rotation axis oscillates with the frequency Ωeﬀ and a non-zero amplitude around (θ, φ). e amplitude of these Rabi oscillations depends on the relative phase Δφ = φ ′ − φ of the atomic state and the linear coupling as well as Δθ = θ ′ −θ. e angle θ ′ is determined by the imbalance of the initial state and θ by the detuning δ. In particular if the system is prepared in an atomic state, e.g. |1⟩, and the linear coupling is resonant, then all atoms oscillate coherently between the two levels. In the following, we will restrict the discussion to the case of resonant coupling, δ = 0, where the dressed states |+⟩ and |−⟩ lie on the equator of the Bloch sphere and correspond to an equal probability of being in state |1⟩ or |2⟩. A possible experimental scheme for preparing a dressed state is to generate an equal superposition of the atom states and to adjust the relative phase Δφ = 0 ²is deﬁnition is opposite to the commonly used convention due to the leading minus sign in Equation 2.4. is sign is included in order to conform with Equation 2.1. 6 2.2. Single spatial mode approximation in order to obtain |+⟩ and Δφ = π for |−⟩. is scheme aligns the rotation axis to be (anti-) parallel with the atomic state. e experimental implementation of this method will be discussed in subsection 3.1.2 and section 5.1. 2.2.2. Internal Josephson junction In the presence of interactions the oscillation dynamics on the Bloch sphere is modiﬁed by additional non-linear terms. e eﬀect of interactions can be wrien in the spin representation as Ĥint = ℏχ Ĵ2z ∫ with the eﬀective nonlinearity ℏχ = 21 (g11 +g22 −2g12 ) d3 x |ϕ(x)|4 [34]. is term corresponds to a rotation around the z-axis with an angular velocity that depends on the distance from the equator, i.e. the mean value of z. is eﬀect is referred to as ’one-axis twisting’ [35] and has been used to generate squeezed states which allow to improve interferometry beyond the statistical limit of independent measurements known as the standard quantum limit [23, 36]. e interesting quantum properties of a two-mode system with both on-site interaction and a linear coupling (equivalent to a tunneling link) between the sites were ﬁrst discovered by Brian D. Josephson in 1962, when he considered two superconductors that are coupled via a weak insulating layer [37]. For example, he predicted an oscillating current when applying a constant external voltage, a phenomenon known as the ac Josephson eﬀect. ese characteristics have been observed in various experimental systems, in particular using interacting atomic states in a Bose-Einstein condensate coupled via Rabi coupling [24]. is realization is referred to as an internal Josephson junction [38]. For an in-depth derivation and discussion of the dynamics we refer to [39, 40]. In this section, we will only summarize the results that are relevant in the context of the quantum phase transition discussed in this thesis. Plasma and π oscillations We consider a resonantly coupled system leading to θ = π/2 and without loss of generality assume the phase of the linear coupling to be φ = 0. In the absence of interactions a quantum state that is prepared close to the rotation axis, e.g. |θ = π/2, φ = ε⟩ with ε ≪ 1, rotates on a circular path around (θ = π/2, φ = 0) with the frequency Ω. e presence of interactions Ĥint modiﬁes both the shape of the trajectory in the classical phase space spanned by the relative population imbalance and phase of the two states and the oscillation frequency depending on the sign of χ and the relative strength of interactions and linear coupling Λ = N|χ| Ω [24]. In the Rabi regime of weak interactions (Λ < 1) and assuming χ > 0, the velocity ﬁelds on the Bloch sphere due to linear√coupling and interactions co-propagate resulting in an increased oscillation frequency ω pl = Ω 1 + Λ. ese oscillations are oen referred to as plasma oscillations. A relative phase of φ = π leads √ to counter-propagating velocity ﬁelds resulting in slower π oscillations with a frequency ω π = Ω 1 − Λ. Graphical illustrations of the velocity ﬁelds are found in [39]. ese oscillation frequencies are given by the gap between the ground state and the ﬁrst excited state when mapping this internal Josephson junction onto an eﬀective potential in Fock space [39, 40]. Bifurcation and symmetry breaking phase transition e symmetry points of the plasma and π oscillations are stable ﬁxed points in the phase space of the system. However, for Λ > 1 the value of ω π becomes imaginary which indicates a critical point. e formerly stable ﬁxed point on the equator of the Bloch √ sphere is replaced by an unstable ﬁxed point and two new stable ﬁxed points emerge at ±z0 = ± 1 − (1/Λ2 ). is phenomenon is called a pitchfork bifurcation and corresponds to a symmetry breaking phase transition [28]. For χ < 0 7 2. eory of linearly coupled interacting Bose-Einstein condensates the bifurcating ﬁxed point correspond to the ground state of the system. At the bifurcation point the system switches over from a single ground state conﬁguration to two equivalent degenerate lowest energy eigenstates. In terms of symmetry groups the Hamiltonian of the system has a U(1) × Z2 symmetry. e U(1) means that the Hamiltonian is not aﬀected by a global phase in the wave function while the Z2 indicates an invariance under the transformation z → −z as the Hamiltonian is quadratic in z [41]. is symmetry is broken when the system randomly chooses the ground state at the positive or the negative value of z0 and the symmetry reduces to U(1) × I. In elongated condensates the single mode approximation is not valid and this symmetry breaking is the origin of a miscible-immiscible phase transition. For χ < 0 and Λ < 1 the two components will overlap, i.e. they are miscible, because the ground state of the system corresponds to a state prepared on the isolated stable ﬁxed point. When Λ is increased to Λ > 1 the symmetry is broken and one of the equivalent ground states at ±z0 is chosen spontaneously. As the single spatial mode approximation is not valid in elongated condensates diﬀerent spatial regions will independently choose the +z0 or −z0 conﬁguration and alternating domains each predominantly containing atoms of one or the other species will form. We now turn to the discussion of the ground state properties and excitation spectra properties of elongated binary Bose-Einstein condensates. 2.3. Ground state properties of elongated binary condensates Aer having discussed the interplay of atomic interactions and a linear coupling ﬁeld in the single spatial mode approximation we will now focus on two interacting species in a spatially extended system, in particular a one-dimensional waveguide. First, we will discuss the ground state properties of two-component Bose-Einstein condensates in the absence of the linear coupling, Ω = 0. We will restrict ourselves to the case of equal atom numbers in the two components N1 = N2 . 2.3.1. Homogeneous system without dressing In a one-dimensional waveguide the ground state can either be a spatially uniform superposition of the two components or a phase separated one, where the two components occupy diﬀerent regions and their overlap is minimized. e following energetic consideration allows to derive a criterion determining which conﬁguration is energetically favorable. For simplicity we ignore the kinetic energy contribution to the classical Hamiltonian Equation 2.2 and use a box potential of length L. e energy of the uniform superposition state with N1 and N2 atoms in the two components is given by [27, 42] Eunif = g11 N21 g22 N22 N1 N2 + + g12 2 L 2 L L (2.6) e corresponding expression for the phase-separated state reads Esep = g11 N21 g22 N22 + 2 L1 2 L2 (2.7) e conditions of a ﬁxed system size L = L1 + L2 and equal pressures ∂Esep /∂L1 = ∂Esep /∂L2 lead to g11 N21 g22 N22 √ N1 N2 Esep = + + g11 g22 (2.8) 2 L 2 L L 8 2.3. Ground state properties of elongated binary condensates us the phase separated state is energetically favorable if Esep < Eunif ⇔ √ g11 g22 < g12 ⇔ Δ= g11 g22 <1 g212 (2.9) is is the condition for (im-)miscibility [27]: e ground state of two components consists of two separate phases, if their inter-species repulsion is stronger than the geometric mean of the intra-species repulsion strengths. Note that Δ < 1 corresponds to χ < 0 for the nonlinearity χ introduced in subsection 2.2.2. For the remainder of this section we will focus on the immiscible regime Δ < 1, where the ground state consists of two separate domains. e kinetic energy was neglected in the discussion above. Its inclusion leads to a non-zero width of the domain wall separating the two components. e characteristic width of the domain wall, the spin healing length ξ s , can be determined in analogy to the single component healing length [27] by equating the quantum pressure due to kinetic energy with the interaction pressure at the domain wall √ √ √ ℏ2 ℏ2 (2.10) √ ∇2 n1 + √ ∇2 n2 = g11 n1 + g22 n2 − 2g12 n1 n2 2m n1 2m n2 Assuming equal densities n1 = n2 = n/2 and identifying ξ s with the characteristic length scale of density variations at the domain wall we yield³ n ℏ2 = (g11 + g22 − 2g12 ) 2 2 mξ s ⇔ ℏ ξs = √ mngs (2.11) with gs = 12 (g11 + g22 − 2g12 ).⁴ Note that ξ s takes on imaginary values in the immiscible regime, gs < 0, where domain walls are stable. In the following we will always refer to |ξ s | when giving numerical values of the spin healing length. e term ngs corresponds to the energy contained in one domain wall due to the overlap of the two components. It sets an important energy scale of binary immiscible atomic clouds. As we will see in the remainder of this chapter many experimental observables in the context of the miscibility phase transition are directly related to this energy. In summary, the ground state of an immiscible binary condensate consists of two domains each containing one component, separated by a domain wall. Although the Hamiltonian is translationally invariant, the ground state is not: e existence of two atomic components in separated regions breaks translational symmetry and the choice of which component populates which side of the domain wall is made spontaneously as both conﬁgurations are energetically equivalent [43]. e number of domain walls is minimized to one by the additional energy cost of each boundary. e width of the boundary is given by the spin healing length and depends on the value of the relevant interaction energy ngs . e mean ﬁeld ground states for both miscible and immiscible conﬁgurations are illustrated in Figure 2.2. ³In the limit g11 ≈ g22 , this deﬁnition of the spin healing length is equivalent to the ’penetration depth’ deﬁned in [42]. e interaction parameters accessible in our experiments are all in the ’weakly segregated phase’ discussed the same publication. ⁴gs is (except for a factor ℏ) equivalent to the previous deﬁnition of χ which was used in order to obey the nomenclature conventions for internal Josephson junctions. In the context of miscibility in an elongated atomic cloud we will from now on use gs . 9 200 linear density (atoms/ m) linear density (atoms/ m) 2. eory of linearly coupled interacting Bose-Einstein condensates 150 100 50 0 -10 -5 0 x ( m) 5 10 200 s 150 100 50 0 -10 -5 0 x ( m) 5 10 Figure 2.2.: Ground state density proﬁles of a miscible (le, Δ = 1.20) and immiscible (right, Δ = 0.80) homogeneous binary condensate. e width of the domain wall is given by the spin healing length ξ s as illustrated in the right panel. e value of the spin healing lengths is ξ s = 1.91 μm for the parameters used in this ﬁgure . 2.3.2. Homogeneous system with dressing We will now include the eﬀects of a linear coupling ﬁeld on the ground state of the one-dimensional system. e linear coupling and its eigenstates, the dressed states, have been discussed in the single spatial mode approximation in subsection 2.2.1. e concept of dressed states has been generalized to dressed condensates in [44]. As we analyze the ground states properties, we choose the relative phase between linear coupling and the atoms such that the lower energy dressed state |+⟩ is prepared. In the miscible regime Δ > 1 the wave functions of the two components overlap and the previously discussed single mode case remains valid. For immiscible parameters, Δ < 1, the single mode approximation cannot be applied as the ground state consists of two separate domains in the absence of a linear coupling. It was shown theoretically that a linear coupling can tune the system to miscibility [17]. Above a critical coupling strength Ωc = −ngs the mean ﬁeld ground state density proﬁles of the two components overlap and cannot be distinguished from a miscible system. e numerically computed ground state density proﬁles for various values of Ω are illustrated in the le panel of Figure 2.3. As Ω increases the domain wall widens and the background density far from the domain wall increases. When crossing the critical point the density proﬁles of the two components become ﬂat and the system becomes miscible. In order to understand these numerical observations we recall the bifurcation occurring in the single spatial mode approximation (see subsection 2.2.2). As the stable ﬁxed points correspond to energy minima they predict the imbalance of the ground state [45]. When assuming an equal population of the two components N1 = N2 translational symmetry is broken and one component will predominate on the le side of the domain wall, while the right side is occupied by the other component.⁵ e imbalance far from the domain wall is given by the prediction of the bifurcation √ √ 2 z0 = ± 1 − (1/Λ) = ± 1 − (Ω/Ωc )2 (2.12) is prediction is in excellent agreement with the numerical ﬁndings as shown in the right panel of ⁵Due to the linear coupling only the total atom number is conserved but not the individual populations N1 and N2 (see [45] for an in-depth discussion). However, as we are interested in the width of the domain wall in the presence of a linear coupling we ﬁx the atom numbers to be equal. In addition we assume equal intra-species scaering lengths a11 = a22 . 10 1.0 =0Hz =5Hz =10Hz =15Hz =20Hz =25Hz =30Hz 200 150 0.5 imbalance linear density (atoms/ m) 2.3. Ground state properties of elongated binary condensates 100 0.0 -0.5 50 0 -10 -5 0 x ( m) 5 10 -1.0 0 5 10 15 20 25 (Hz) 30 35 40 Figure 2.3.: (le) Numerically computed ground state density proﬁles near the domain wall for various linear coupling strengths. For clarity only the density of one component is drawn. e proﬁle of the other component is obtained by reﬂection along x = 0. e width of the domain wall increases as Ω approaches Ωc , which is 27.3 Hz for the chosen parameters. When exceeding the critical coupling strength the system becomes miscible. (right) e imbalance of the computed ground state proﬁles far from the domain wall (black circles) is compared to the analytical single mode prediction (solid line). e excellent agreement conﬁrms that the ground state imbalance is given by the ﬁxed points of the bifurcation. Figure 2.3. e presence of the linear coupling ﬁeld reduces the energy of the uniform superposition state by ℏΩ [46] and thus reduces the energy contained in a domain wall by this value. In analogy to the previous subsection 2.3.1 the resulting width of the domain wall can be calculated as ℏ2 = ngs + ℏΩ mξ 2s ⇔ ξs = √ ℏ m(ngs + ℏΩ) =√ ℏ m(ℏΩ − ℏΩc ) (2.13) e eﬀective spin healing length increases with Ω and diverges as Ω → Ωc . At this point the system transitions to miscibility and the two components overlap spatially. 2.3.3. Eﬀects of a trapping potential In the presence of a longitudinal trapping potential translational symmetry is broken as the sum density of the two components is not homogeneous anymore. is symmetry breaking changes the ground state and it takes on the symmetry of the trap due to the density dependence of the interaction energy gii ni . Assuming a11 > a22 , component 1 will prefer lower densities than component 2 and occupies the edges of the trap while component 2 populates the trap center [47, 48]. In the immiscible regime this asymmetry in the intra-species scaering lengths can compensate the energy cost of a second domain wall. is situation is depicted in the right panel of Figure 2.4. is three-domain ground state has the same symmetry as the trap in contrast to the symmetry broken two-domain state, in which one component occupies the le half of the trap and the other component the right half. For most conﬁgurations of the relative interaction strengths and atom numbers the three-domain ground is lower in energy than the two-domain state. A detailed discussion and categorization of the possible ground state conﬁgurations is found in [49]. e presence of the trap does not aﬀect the width of the domain walls as long as ξ s is a lot smaller 11 2. eory of linearly coupled interacting Bose-Einstein condensates 250 linear density (atoms/ m) linear density (atoms/ m) 250 200 150 100 50 0 -150 -100 -50 0 50 x ( m) 100 150 200 150 100 50 0 -150 -100 -50 0 50 x ( m) 100 150 Figure 2.4.: Ground state density proﬁles of a miscible (le, Δ = 1.20) and immiscible (right, Δ = 0.80) binary condensate in the presence of a harmonic trapping potential. Due to a11 > a22 the ﬁrst component (solid lines) is pushed to the outer regions of the trap with a lower density while the other component (dashed lines) prevails in the trap center. is eﬀect reduces the overlap of the two components for miscible parameters (le), a phenomenon that does not occur in a homogeneous system. In the immiscible case the asymmetry in intra-species scaering lengths leads to a ground state consisting of three-domains. than the extent of the atomic clouds in the trap [42]. However, due to the inhomogeneous density proﬁle ξ s depends on the position of the domain wall relative to the trap center. For miscible scaering parameters and in the absence of a trapping potential the overlap of the two components is perfect also for an asymmetry in the relative values of a11 and a22 . However, in the presence of a trap such an asymmetry causes diﬀerent density proﬁles of the atomic clouds and their overlap is reduced as shown in the le panel of Figure 2.4. In contrast to phase separation, which requires immiscible interaction parameters, this eﬀect is referred to as potential separation [43] as it occurs only in the presence of an external potential. 2.4. Bogoliubov theory Aer the discussion of the ground state properties of binary interacting condensates we will now focus on their excitation spectra. e excitation spectrum can be calculated using a linear response analysis of the Gross-Pitaevskii equation and diagonalizing the resulting set of equations by employing the Bogoliubov transformation. is approach is the classical counterpart to Bogoliubov theory, which describes elementary excitations as bosonic quasiparticles whose vacuum is given by the unperturbed condensates [27]. We stress that this analysis is based on the mean ﬁeld approximation and valid only in the linear response of the system, i.e. small deviations from the unperturbed condensates. e notion of immiscibility of the two components in the ground state is related to a modulational instability of a uniform superposition of the two components. Unstable modes in the excitation spectrum have imaginary eigenenergies and thus grow exponentially in time. Initial work on single component condensates associated the appearance of unstable modes with aractive atomic interactions, which manifest themselves for example in the formation of a train of solitons [50]. e Bogoliubov approach has been generalized to two-component condensates, where eﬀective arac- 12 2.4. Bogoliubov theory tive interactions can appear although all atomic interactions are repulsive [51, 52, 43].⁶ is section summarizes the Bogoliubov spectra of binary condensates, which can be derived analytically for homogeneous one-dimensional systems. We begin with a simple linearization of the equations of motion yielding the characteristic length scales of the excitation modes. en, we discuss the full spectrum of elementary excitations in the absence of a linear coupling ﬁeld and include its eﬀects later on. As previously, we will restrict the discussion to the case of repulsive interactions, gij > 0. 2.4.1. Linearization of the equations of motion Following the ansatz presented in the appendices of [55, 56] many properties such as stability or characteristic length and energy scales can be deduced by linearizing the equations of motion around √ a stationary state. We begin the analysis by inserting ψ i = ψ 0i + nδψ i into Equation 2.3, where ψ 0i denotes the real valued background wave function of component i and δψ i is a small perturbation ∂ψ 0 around it. Using the stationary character of the background state⁷ iℏ ∂ti = μ i ψ 0i and ignoring all terms of second or higher order in δψ i we obtain the equations of motion for the perturbations ∑ ℏ2 2 ∇ δψ i = Sik δψ k 2m (2.14) k with S= ( ) ng11 + ℏΩ/2 ng12 − ℏΩ/2 ng12 − ℏΩ/2 ng22 + ℏΩ/2 e matrix S can be diagonalized by CSC−1 with the eigenvalues ( ) √ 1 2 2 Γ± = ℏΩ + ng11 + ng22 ± (ng11 − ng22 ) + (2ng12 − ℏΩ) 2 (2.15) (2.16) e ﬁrst term in the square root is a lot smaller⁸ than the second one and can be neglected leading to 1 1 Γ± = n(g11 + g22 ± 2g12 ) + (ℏΩ ∓ ℏΩ) (2.17) 2 2 e signs of the eigenvalues Γ± act as a stability signature of the system [52]. If all eigenvalues are positive the system is stable, while a negative eigenvalue indicates an instability. e physical reason for this correspondence is that small ﬂuctuations δψ i can decrease the energy of the system and thus grow exponentially if S has a negative eigenvalue [52]. Furthermore the nature of the excitations can be deduced from a similar energetic consideration: If all atomic interactions are repulsive but fulﬁll the immiscibility condition Equation 2.9, the unstable modes locally increase the atom number diﬀerence with constant sum density (out-of-phase mode), while an instability due to aractive interactions, g12 < 0, tends to increase the local sum density (in-phase mode) [52]. We consider ∑ −1 the characteristic length scale ξ of the perturbations in the eigenbasis of S given by δ ψ̄ i = k Cik δψ k . is ansatz leads to ξ 2 ∇2 δ ψ̄ i = δ ψ̄ i (2.18) ⁶e (im-)miscibility condition Equation 2.9 was ﬁrst derived in this context [53, 54]. ⁷We assume an equal superposition of the two components as the background state, which is stationary but unstable for a homogeneous system with immiscible scaering parameters. In the presence of a trapping potential and a11 ̸= a22 a stationary background state is diﬃcult to realize. ⁸About a factor 1000 for 87 Rb and taking into account the values of g12 and Ω we can realize in our experiments. 13 2. eory of linearly coupled interacting Bose-Einstein condensates with ξ2 = ℏ2 2mΓ± (2.19) In the following we focus on the smallest eigenvalue Γ− resulting in ξ = ξ(Ω) = √ ℏ 2m(ngs + ℏΩ) (2.20) In the absence √ of linear coupling ξ recovers the spin healing length deﬁned in Equation 2.11, ξ(Ω = 0) = ξ s / 2. Furthermore ξ diverges at the critical coupling strength ℏΩc = −ngs . For Ω < Ωc , ξ is imaginary and the unstable solutions of Equation 2.18 are sinusoidal with a wave vector k = 1/ξ, i.e. a wavelength of λ = 2πξ. In the case Ω > Ωc , ξ is real and the stable solutions of Equation 2.18 are exponential functions with length scales of ξ. us ξ plays the role of a correlation length. Aer this simpliﬁed linearization approach we will discuss the full dispersion of the excitations resulting from a mean ﬁeld Bogoliubov analysis in the following section. 2.4.2. Bogoliubov spectrum of a homogeneous system In the absence of an external trapping potential the excitation spectrum of a single condensate of atoms in state i is given by [27] ( 2 2 ℏ ω (k) = c̃2i ℏ2 k2 + ℏ2 k2 2m )2 (2.21) √ where c̃i = ni gii /m is the sound velocity, k the wave vector of the excited mode and ℏω the corresponding excitation energy. e dispersion starts linearly for small k and becomes quadratic as k increases, respectively corresponding to the phonon and the free particle regimes of the quasiparticle excitations [27]. e dispersion of two interacting condensates can be wrien in the same form [43, 57, 58] ( 2 ℏ ω 2± (k) = c2± ℏ2 k2 + ℏ2 k2 2m )2 (2.22) where the sound velocities c± are calculated from the single condensate sound velocities c̃i by [ ] √ 1 2 2 2 2 2 2 2 2 2 c± = (c̃1 + c̃2 ) ± (c̃1 − c̃2 ) + 4(g12 /g11 g22 )c̃1 c̃2 (2.23) 2 e spectrum consists of two branches ω ± depicted in Figure 2.5. ω + is higher in energy because > c2− . In the absence of inter-species interactions, g12 = 0, the two atomic clouds are independent of each other and the two branches reduce to the dispersions of each condensate given by Equation 2.21. In the case g12 ̸= 0 excitations on top of the two atomic species are not independent of each other and become coupled. e upper branch ω + describes in-phase excitations on the two components, i.e. excitations on the sum density such as breathing where both components move in unison. It is also called the ’stiﬀ mode’. e lower branch ω − represents the dispersion of out-ofphase modes, i.e. excitations on the diﬀerence density of the two components. ey are referred to as spin excitations or ’so modes’ [59, 41]. In consequence the spectrum of two interacting condensates does not describe excitations on top of the single components but rather on their sum or diﬀerence density. c2+ 14 2.4. Bogoliubov theory 3000 3000 2000 1000 (Hz2 ) 4000 2000 2 4000 (Hz2 ) 5000 2 5000 1000 0 0 -1000 0.00 -1000 0.00 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 kc (kf , - -2f ) 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 Figure 2.5.: Bogoliubov spectra in the miscible (Δ > 1, le panel) and immiscible (Δ < 1, right panel) regimes. Dashed lines represent sum density excitations ω 2+ and solid lines correspond to spin excitations described by ω 2− . In the immiscible case the eigenenergies of long wavelength spin excitations with k < kc become imaginary indicating a modulational instability. e fastest growing mode kf and its growth rate τ f are depicted. If the condition for immiscibility Equation 2.9 is fulﬁlled, c2− becomes negative resulting in imaginary excitation energies. us the amplitudes of long wavelength modes 0 < k < kc = 2m|c− |/ℏ grow exponentially [43, 58]. is phenomenon is referred to as modulational instability, because weak perturbations on the background state cause the subsequent exponential growth of excitation modes in a given range of wavelengths √ [58]. √ e fastest growing mode is determined from the 2 minimum of ω − resulting in kf = kc / 2 = 2m|c− |/ℏ and a growth rate of 1/τ f = |ω − (kf )| = m|c− |2 /ℏ. ese quantities are illustrated in Figure 2.5. For small diﬀerences in the scaering lengths a11 − a22 ≪ a11 , a22 , a12 , as it is the case for 87 Rb, Equation 2.23 can be wrien as c2± = 1n (g11 + g22 ± 2g12 ) 4m (2.24) is simpliﬁcation changes the values of c± by less than 10−3 for our experimental parameters. e equation ℏ2 1 1 mc2− = ngs = (2.25) 2 2m ξ 2s relates the properties of the excitation spectrum to the characteristic length scale of the ground state, the spin healing length, via √ √ kc = 2m|c− |/ℏ = 2/ξ s and kf = kc / 2 = 1/ξ s (2.26) 2.4.3. Bogoliubov spectrum of a dressed system e Bogoliubov theory for binary condensates has been expanded to include a linear coupling between the components [51, 45, 41, 46]. It was found that a linear coupling ﬁeld can stabilize the spin excitation branch, which is unstable in the case of immiscible scaering parameters. In addition the dispersion becomes gapped in the long-wavelength limit k → 0 resulting in massive excitation modes. is section summarizes the results of Tommasini et al. [41] and reformulates them in terms of experimental quantities. 15 2. eory of linearly coupled interacting Bose-Einstein condensates 5000 1000 3000 (Hz2 ) (Hz2 ) 2 3000 2000 1000 0 0 -1000 0.00 -1000 0.00 0.05 0.10 0.15 0.20 wave vector k (1/ m) =0Hz =20Hz =40Hz =60Hz =80Hz 4000 2 4000 2000 5000 =0Hz =10Hz =20Hz =30Hz 0.25 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 Figure 2.6.: Bogoliubov spectra in the miscible (Δ > 1, le panel) and immiscible (Δ < 1, right panel) regime for various amplitudes of the linear coupling Ω. Sum density excitations (ω 2+ , dashed lines) are independent of Ω. For miscible parameters the linear coupling introduces an energy gap in the long wavelength limit k → 0. In the immiscible regime an increasing coupling strength shis the instability of the spin excitations (ω 2− , solid lines) towards smaller wave vectors until the system is stable. A further increase of Ω additionally creates an energy gap. e crossing of the branches is physical only if a11 = a22 ; otherwise the two branches become coupled resulting in an avoided crossing. e two branches of the excitation spectrum in the presence of a linear coupling are given by [41] ℏ2 ω 2+ (k) = c22 ℏ2 k2 + e2k ℏ2 ω 2− (k) = c21 ℏ2 k2 + e2k + g2 (2.27) with ℏ2 k2 2m g2 = G11 nℏΩ + (ℏΩ)2 (2.28) ek = (2.29) the sound velocities c21 = G11 n ℏΩ + 2m m c22 = G22 n 2m c212 = G12 n 2m (2.30) and the dressed interaction constants 1 G11 = (g11 + g22 − 2g12 ) 2 1 G22 = (g11 + g22 + 2g12 ) 2 1 G12 = (g22 − g11 ) (2.31) 2 In the absence of a linear coupling ﬁeld, Ω = 0, Equation 2.27 collapses to Equation 2.22.⁹ However, for Ω ̸= 0 the dispersion for the spin excitation branch ω − qualitatively deviates from the previous expressions as it acquires an energy gap g at k = 0. e excitation spectra for both miscible and immiscible scaering parameters and various coupling strengths are shown in Figure 2.6. In excitation spectra without instabilities the branches for the sum density and the spin excitations ⁹Note that due to the use of the dressed interaction constants Gij , the sound velocities c1 and c2 correspond to c− and c+ used in the previous discussion of the undressed case (rather than c̃1 and c̃2 ). However, the relevant expressions (c21 + c22 ) and ((c21 − c22 )2 + 4c412 ) have the same values for ci and c̃i . 16 2.4. Bogoliubov theory cross because g2 > 0 and c21 < c22 . However, diﬀerent intra-species coupling strengths a11 ̸= a22 couple the density and spin excitations. is leads to a hybridization of the modes and the branches are separated by an avoided crossing [46]. An example of hybridized modes is illustrated in the right panel of Figure 2.7. It is calculated using [41] ℏ2 ω 2± (k) = 1( 2 2ek + (c21 + c22 )ℏ2 k2 + g2 2 ) √ 2 2 4 2 2 2 4 4 2 2 2 2 4 ± [(c1 − c2 ) + 4c12 ]ℏ k + 2[2jα + (c1 − c2 )g ]ℏ k + g (2.32) with j2α = G212 n2 ℏΩ 2m (2.33) In the following we will discuss the properties of the excitation spectra shown in Figure 2.6. For simplicity we will use the unhybridized dispersion relations given by Equation 2.27 for calculations. is can be done as the modiﬁcations due to the hybridization are relevant only close to the avoided crossing and do not aﬀect the results. e sum density excitations ω + are not aﬀected by the linear coupling and our discussion will focus on the spin excitations. In the miscible regime Δ > 1 the presence of a linear coupling causes an energy gap g at k = 0 in the excitation spectrum. In addition the dispersion starts quadratically for small k and thus corresponds to a ’massive’ mode, i.e. a ﬁnite amount of energy is required to excite the system. Another important consequence of the gap is a ﬁnite reaction time of the system to sudden quenches which populate excited modes. e time evolution of each excited mode is given by its energy ℏω − . us, a ﬁnite value of the gap introduces a characteristic time scale for the evolution of the excitations. e implications of this ﬁnite reaction time will be discussed in chapter 5 and chapter 6. For immiscible scaering parameters Δ < 1 the linear coupling shis the region of unstable modes towards longer wavelengths. If it exceeds a critical value Ω > Ωc all eigenenergies are real and the system becomes stable. Similarly to the miscible case the spin excitation branch acquires an energy gap. For values Ωc /2 < Ω < Ωc , the most unstable mode is given by k = 0. e critical coupling strength Ωc marks the transition from an unstable system to a stable one as already seen in the discussion of the ground state properties in subsection 2.3.2. is miscible-immiscible phase transition will be one of the main topics of this thesis. e value of Ωc can be calculated from the vanishing of the energy gap given in Equation 2.29. e condition g(Ωc ) = 0 yields 1 ℏΩc = −nG11 = − n(g11 + g22 − 2g12 ) = −ngs 2 (2.34) in agreement with the previous result in the context of the ground state properties. In terms of the critical coupling strength the gap can be wrien as g2 = G11 nℏΩ + (ℏΩ)2 = ℏΩ(ℏΩ − ℏΩc ) (2.35) In the single spatial mode limit k = 0 the system reduces to the internal Josephson junction discussed in subsection 2.2.2. In this context the gaps in the miscible and immiscible cases corresponds to the frequencies of plasma and π oscillations, respectively.¹⁰ e real and imaginary parts of the ¹⁰e anharmonicity of plasma and π oscillations for large amplitudes is not captured in the Bogoliubov theory as it is valid only in the linear response, i.e. small deviations from the stationary state. 17 2. eory of linearly coupled interacting Bose-Einstein condensates 100 5000 80 40 (Hz2 ) 3000 60 2000 2 energy gap (Hz) =30Hz 4000 1000 avoided crossing 0 20 c 0 -40 -20 0 20 40 60 80 100 120 Rabi frequency (Hz) -1000 0.00 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 Figure 2.7.: (le) Real (solid line) and imaginary (dashed line) parts of the energy gap vs the linear coupling strength. e critical coupling Ωc marks the transition from an immiscible to a miscible system. (right) Example of the dispersion of hybridized modes for the case a11 ̸= a22 . e crossing of the two branches is avoided. gap for immiscible scaering parameters are ploed in the le panel of Figure 2.7. Physical mechanism for stabilization by dressing e physical mechanism for the stabilization of an immiscible system by a linear coupling ﬁeld can be visualized intuitively in analogy to a spin chain interacting with a magnetic ﬁeld. e homogeneous two-component system can be thought of as an array of pseudo-spins in analogy to the Bloch sphere picture of a two level system in a single spatial mode (see section 2.2). We associate an atom of component 1 with spin-down and an atom of component 2 with spin-up. Consequently an equal superposition of the two components corresponds to a horizontal spin vector. e emergence of spin domains during the demixing dynamics of an initial superposition state can be visualized as a local rotation of the spins out of the horizontal plane towards one of the poles. As discussed in subsection 2.2.1 a resonant linear coupling of the two components corresponds to a rotation of the spin around an axis through the equator. e preparation of a dressed state the rotation axis is aligned parallel with the atomic pseudo-spin. is conﬁguration is analogous to an array of spins aligned in an external magnetic ﬁeld. If the eﬀective magnetic ﬁeld associated with the linear coupling is strong enough, i.e. Ω > Ωc , the spins stay aligned with the axis of the linear coupling. A rotation away from the rotation axis and thus demixing of the two components is suppressed. Connection of ground state properties to Bogoliubov spectra Characteristic length scales of the excitation spectra are the wave vectors of the most unstable mode kf and of the unstable mode with the smallest wavelength kc . In the previous section we have related these modes to the spin healing length in the absence of a linear coupling ﬁeld. We will now perform a similar analysis including the linear coupling. e sound velocity c1 can be wrien in terms of the critical coupling as c21 = G11 n ℏΩ 1 + = (ℏΩ − ℏΩc /2) 2m m m (2.36) Using Equation 2.35 and Equation 2.36 the spin branch of the excitation spectrum can be expressed 18 2.4. Bogoliubov theory in terms of Ω and Ωc resulting in ( ℏ2 ω 2− = ℏ2 k2 2m )2 + ℏ2 k2 (2ℏΩ − ℏΩc ) + ℏΩ(ℏΩ − ℏΩc ) 2m (2.37) e values for kf and kc can be calculated using the conditions dωdk− = 0 and ω − = 0, respectively, leading to √ √ √ √ kf = 2m/ℏ Ωc /2 − Ω and kc = 2m/ℏ Ωc − Ω (2.38) Comparing these expressions to the spin healing length derived from the ground state in the √ presence of a linear coupling Equation 2.13, the relation kc = 2/ξ s still holds. is characteristic length scale changes proportional to (Ωc − Ω)−1/2 and this scaling behavior will be discussed in the context of phase transitions in chapter 5. Imbalance eﬀects In order to estimate the sensitivity of the excitation spectrum to experimental imperfections we ana1 lyze the eﬀect of diﬀerent populations in the two atomic clouds, i.e. a non-zero imbalance z = NN21 −N +N2 . is eﬀect is relevant for experiments because a small detuning of the linear coupling or an imperfect preparation pulse directly translate to a population imbalance (see also subsection 2.2.1). In addition this analysis provides a crude estimation for the excitation spectra beyond linear response. When unstable spin excitations grow they create density modulations on the atomic density proﬁles. e Bogoliubov theory discussed in this section is valid only in the linear response regime, i.e. if the modulation depth is small compared to the atomic background density. As the excitations create local imbalances the eﬀect on the dispersion relations can be roughly estimated by considering an imbalanced system. e excitation spectra of imbalanced systems in the immiscible regime are shown in Figure 2.8 for various values of z and both without and with a linear coupling. As an imbalance reduces the eﬀective overlap of the two components it leads to a smaller eﬀective interaction parameter ngs . us the unstable region is shied towards larger wavelengths and the growth rates become slower. In the presence of a linear coupling this eﬀect is ampliﬁed because a smaller eﬀective ngs additionally reduces the critical coupling strength and thus increases the relevant parameter Ω − Ωc for a given value of Ω. e imbalance in our experiments is typically −0.2 < z < 0.2 and changes the excitation spectra by less than 10%. 2.4.4. Eﬀects of a trap and numerical Bogoliubov-de Gennes analysis e presented analytical Bogoliubov theory requires a homogeneous one-dimensional system. In experiments the atomic clouds are conﬁned in an elongated trap and thus inhomogeneous and of ﬁnite size. If the spatial extent of the atomic cloud is large compared to the characteristic length scale of the excitations given by kf , the system can be treated as locally homogeneous and the homogeneous theory is expected to be a reasonable approximation. We will now analyze the deviations of the excitation spectra for harmonically trapped condensates from the homogeneous case by employing a numerical Bogoliubov - de Gennes stability analysis (see subsection B.2.2). As the numerical analysis requires a stationary background state we assume equal values for the intra-species scaering lengths a11 = a22 . For an absolute comparison of the length scales and growth rates we choose a density for the analytical calculations corresponding to about 90% of the maximum density of the inhomogeneous cloud. Figure 2.9 compares the analytical theory to the numerical results for two diﬀerent experimental 19 2. eory of linearly coupled interacting Bose-Einstein condensates (Hz2 ) 2000 2 3000 1000 4000 3000 (Hz2 ) 4000 5000 z=0.0 z=0.2 z=0.4 z=0.6 2000 2 5000 1000 0 0 -1000 0.00 -1000 0.00 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 z=0.0 z=0.2 z=0.4 z=0.6 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 Figure 2.8.: Excitation spectra for diﬀerent values of the imbalance z and for immiscible scaering parameters Δ < 1. In the absence of a linear coupling (le) the eﬀect of an imbalance is less pronounced than in the presence of a coupling ﬁeld with Ω = 50 Hz = 0.7 Ωc (right). Typical experimental imbalances are −0.2 < z < 0.2 which cause only minor changes in the excitation spectrum. e corrections for diﬀerent imbalances are the symmetric for ±z. conﬁgurations in optical dipole traps we refer to as ’waveguide’ and ’charger’ (see section 3.1 for details on the experimental setup). While elementary excitation form a continuous spectrum in the homogeneous case the spectrum becomes discrete for a ﬁnite size system. e density of excitation modes in k-space is directly given by the size of the experimental system. is eﬀect is apparent in the results for the charger (right panel of Figure 2.9), in which the atomic cloud has a longitudinal size of about 40 μm. e ﬁnite number of available modes aﬀects the demixing dynamics, which is dominated by the modes with the largest growth rates. In large atomic clouds many modes have similar growth rates and the emerging spin paern is random depending on small variations of the initial seed of the modes. In contrast, only two or three excitation modes dominate the dynamics in the charger and the shot-to-shot ﬂuctuations in the spin paern will be small. In other words, the boundary conditions given by the ﬁnite size of the system pin the positions of the spin domains if the size of the domains is not much smaller than the extent of the atomic cloud. Besides the discreteness of the spectrum the numerical results for the charger also deviate significantly in the wave vectors and the growth rates of the unstable modes. ey are shied towards smaller wavelengths and the growth rates are increased. is eﬀect signiﬁcantly modiﬁes the demixing dynamics in the charger compared to the analytical predications. In summary the properties of atomic clouds conﬁned in the charger deviate signiﬁcantly from the homogeneous theory while the results for the waveguide agree well with the analytical predictions. is justiﬁes the use of the Bogoliubov theory for a homogeneous system to model the experimental observations in the waveguide presented in this thesis. 2.5. Extension to negative coupling strengths In the previous discussion about the excitation spectra we have assumed a relative phase between the linear coupling and the atoms of φ = 0. In the limit of strong linear coupling this conﬁguration corresponds to the |+⟩ state, i.e. the ground state of the system (see subsection 2.2.1). e stationarity of |+⟩ for Ω ≫ Ωc implies a stable system and the absence of unstable modes in the excitation 20 2.5. Extension to negative coupling strengths 2000 1000 0 (Hz2 ) 3000 2000 2 3000 (Hz2 ) 4000 2 4000 1000 0 -1000 0.00 0.05 0.10 0.15 0.20 0.25 wave vector k (1/ m) -1000 0.00 0.05 0.10 0.15 0.20 0.25 wave vector k (1/ m) Figure 2.9.: Comparison of the analytical Bogoliubov theory for a homogeneous system to the results of a numerical Bogoliubov-de Gennes analysis for typical experimental conﬁgurations in the waveguide (le) and the charger (right). While the excitation spectrum for an atomic cloud in the waveguide agrees well with the predications for the homogeneous system it deviates signiﬁcantly in the charger. In particular the discrete nature of the spectrum for the ﬁnite size system becomes apparent by the separation of the modes in k-space. e mode density is given by the size of the atomic cloud and does not depend on numerical parameters such as the spatial extent of the grid or the number of grid points used for the computation. spectrum. As Ω is reduced and the strong coupling limit breaks down, |+⟩ is not stationary anymore and excitation modes become unstable for Ω < Ωc . However, one can similarly determine the excitation spectrum for a relative phase of φ = π corresponding to the |−⟩ state. As this is an excited state additional instabilities appear in the spectrum of the homogeneous system, which will be discussed in this section. As in subsection 2.2.1 we have associated diﬀerent choices of φ with the preparation of diﬀerent states. However, as φ is the relative phase, i.e. the diﬀerence between the atomic phase and the phase of the linear coupling, it can also be incorporated in the Hamiltonian instead of the prepared state. In this picture the same initial state is prepared but subsequently evolves under diﬀerent Hamiltonians. is equivalence can be understood by comparing the respective actions of the resonant linear coupling Hamiltonian Hcpl Equation 2.2 on the diﬀerent atomic states:¹¹ 1 Hcpl (φ = 0)|−⟩ = − ℏ(+Ω)(ψ ∗1 ψ 2 + ψ ∗2 ψ 1 )|−⟩ 2 1 = − ℏ(−Ω)(ψ ∗1 ψ 2 + ψ ∗2 ψ 1 )|+⟩ 2 = Hcpl (φ = π)|+⟩ (2.39) e ﬁrst expression incorporates φ = π in the phase of the atomic components resulting in the |−⟩ state. e last term includes the value of φ in the Hamiltonian and suggests an alternative interpretation of the conﬁguration: We can assume that the system is always prepared in the |+⟩ state and subsequently evolves under the action of diﬀerent Hamiltonians. e two phases φ = 0 and φ = π of the linear coupling can be interpreted as positive or negative values of Ω. When using ¹¹e other terms of the Hamiltonian H0 and Hint do not contain terms ψ ∗i ψ j with i ̸= j. us, their actions on the |+⟩ and the |−⟩ states are identical. 21 2. eory of linearly coupled interacting Bose-Einstein condensates (Hz2 ) 2000 2 3000 1000 4000 3000 (Hz2 ) 4000 5000 =0Hz =-20Hz =-50Hz =-70Hz 2000 2 5000 1000 0 0 -1000 0.00 -1000 0.00 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 =0Hz =-20Hz =-40Hz 0.05 0.10 0.15 0.20 wave vector k (1/ m) 0.25 Figure 2.10.: Excitation spectra in the miscible (le) and immiscible (right) regime in the presence of a linear coupling ﬁeld with Ω < 0. (le) For miscible parameters long wavelength modes become unstable and the instability is shied towards shorter wavelength modes in the same way as for 0 < Ω < Ωc in the immiscible regime (see Figure 2.6). (right) In the immiscible regime the unstable modes are shied towards short wavelengths while long-wavelength excitations become stable again and the k = 0 mode acquires a gap. the concept of negative Ω it is redundant to distinguish between the preparation of two states. Preparing a |+⟩ state and evolving it with +Ω0 (−Ω0 ) in the Hamiltonian is equivalent to preparing a |−⟩ state and subsequently choosing −Ω0 (+Ω0 ). is picture of an eﬀective negative coupling strength has the advantage that quantities depending on Ω can be ploed in a single graph having one unambiguous axis for Ω. We will use this notation in the remainder of this thesis. However, it is important to remember that the corresponding eigenstate is not the lowest energy state. e excitation spectra for negative values of linear coupling strength can be obtained directly by using values Ω < 0 in the formulae in subsection 2.4.3. Examples of dispersion relations for various negative values of Ω are shown in Figure 2.10 both for miscible and immiscible scaering parameters. In the immiscible case the instability is shied towards shorter wavelength modes while long wavelength modes are stabilized. An energy gap appears at k = 0, which corresponds to plasma oscillations of the Josephson Hamiltonian (see subsection 2.2.2). A system that is miscible in the absence of a linear coupling aains unstable modes for negative values of Ω. Similarly to the immiscible case long wavelength modes become unstable ﬁrst and the instability is shied towards smaller wavelengths as Ω is decreased. e change in the spectrum of unstable modes with the coupling strength is summarized in Figure 2.11. e square-root scaling of the characteristic size of the unstable modes is smoothly continued as Ω becomes negative. us measurements of the mean ﬁeld scaling behavior can be extended to negative coupling strengths in order to increase the dynamic range. e miscible case shows the same characteristic scaling of the instability region, but it is shied towards smaller values of Ω. e system becomes unstable for Ω < 0 (in analogy to Ω < Ωc in the immiscible case) and the k = 0 mode becomes stable again for Ω < Ωc < 0 (in analogy to Ω < 0). e width of the instability region is constant for diﬀerent values of k and given by |Ωc |. e ground state properties of the binary condensate are not changed for negative values of Ω, which in this case it is given by the |−⟩ dressed state. is conﬁguration is energetically identical to the case of positive Ω and a |+⟩ state as we have seen in Equation 2.39. Consequently the extension to negative values of Ω is does not have physical consequences in terms of the ground state of the 22 100 100 50 50 0 0 -50 -50 (Hz) (Hz) 2.5. Extension to negative coupling strengths -100 -100 -150 -150 -200 -200 -2500.0 0.1 0.2 0.3 wave vector k (1/ m) 0.4 -2500.0 0.1 0.2 0.3 wave vector k (1/ m) 0.4 Figure 2.11.: Unstable modes versus linear coupling strength for miscible (le) and immiscible (right) scaering parameters. (le) e miscible system becomes unstable for Ω < 0 and a decreasing Ω shis the region of unstable modes (shaded region) towards larger wave vectors with a square root behavior. e width of the instability region at ﬁxed k is constant and given by |Ωc |. (right) For immiscible parameters, the characteristics of the instability region are the same as for the miscible case but shied by Ωc such that the system becomes stable for Ω > Ωc . system as all of its properties remain the same under the transformation Ω → −Ω and |+⟩ → |−⟩. In other words the ground state of the system always has a relative phase of φ = 0 between atomic states and the linear coupling. If the phase of the linear coupling is changed by π this is ’compensated’ by a phase ﬂip in one of the components. 2.5.1. Phase and stability diagrams and summary e properties of homogeneous two-component Bose-Einstein condensates in the presence of a linear coupling ﬁeld can be summarized in phase diagrams for the ground state and the modulational stability of an equal superposition of the components. ese diagrams are shown in Figure 2.12. e system can be either miscible or immiscible, which manifests itself in the overlap of the atomic clouds in the ground state. e overlap is maximal for the miscible system and minimized in the immiscible case. e notion of negative coupling strengths does not aﬀect the ground states, which are equivalent for positive and negative values of Ω as indicated by the symmetry with respect to Ω = 0 in the phase diagram (le panel of Figure 2.12). e stability of the system is determined from the linear response of an equal superposition of the two components to spin ﬂuctuations. An excitation mode with an imaginary eigenenergy grows exponentially and is considered unstable. Negative coupling strengths can be realized by changing the relative phase of the linear coupling and the atoms by π. As stability depends on the preparation of the experimental state negative coupling strengths lead to new physical eﬀects. In particular a miscible system can become unstable as shown in the right panel of Figure 2.12. e scaling of the relevant length scales of unstable modes (e.g. the smallest unstable wavelength) is continued smoothly across Ω = 0. In this chapter we have introduced the theoretical description of linearly coupled and interacting two-component Bose gases in a one-dimensional geometry. We have seen that a linear coupling ﬁeld can tune a system with immiscible scaering parameters to miscibility, i.e. the ground state wave functions of the two components overlap and their excitation spectrum does not have any unstable 23 100 100 50 50 0 immiscible (Hz) (Hz) 2. eory of linearly coupled interacting Bose-Einstein condensates miscible -50 -100 stable 0 unstable -50 0.6 0.8 1.0 1.2 1.4 -100 0.6 0.8 1.0 1.2 1.4 Figure 2.12.: Phase diagram summarizing the ground state (le) and stability properties of the system (right) as a function of Δ and Ω. (le) e ground state of the system becomes miscible as |Ω| exceeds the critical value Ωc (solid line). It is symmetric with respect to Ω = 0 as the ground state conﬁgurations for positive and negative coupling strengths are equivalent (see text). (right) For positive Ω a system with a miscible ground state is also stable against spin ﬂuctuations. However, the system can be prepared in an excited state corresponding to negative coupling strengths, which is always unstable. modes. e relevant energy scale is the energy contained in a domain wall ngs . is parameter determines all relevant length and time scales of the system, e.g. the critical coupling strength, the spin healing length and the wavelengths and growth rates of the unstable modes. In the following chapter we will introduce the experimental realization of such a system. 24 3. Experimental system and analysis methods Aer having discussed the theoretical description of the ground state and excitation properties of linearly coupled binary Bose-Einstein condensates, we will now focus on the experimental implementation of such a system. We will introduce the experimental apparatus and methods for the generation and detection of ultracold atomic clouds. e methods for the analysis of the experimental data will be presented by discussing the free, i.e. uncoupled, evolution of an equal superposition of two interacting condensates. e dynamics of the density distributions of the two components depends on their relative interaction strengths. By employing a Feshbach resonance, we will discuss both the regimes of potential and phase separation [43]. Modulational instabilities dominate the dynamics in the laer case and lead to regions where only one component is present surrounded by domains of the other component. As these paerns correspond to modulations in the diﬀerence of the density proﬁles we will refer to them as spin domains. A similar experiment employing comparable analysis methods was reported recently by the Spielman group [60]. 3.1. Experimental system In order to experimentally explore the interplay of interactions, linear coupling and spatial degrees of freedom, we employ Bose-Einstein condensates of 87 Rb. ey are well isolated from the environment and can be created with good reproducibility making them an ideal model system for such studies. e experimental apparatus and the details of the generation of the condensates and their detection have been described in previous theses [61, 62, 63, 32]. Here, we will brieﬂy summarize the aspects of the experimental system and sequence that are relevant for the experiments presented in this thesis. Experimental challenges for the creation of one-dimensional condensates in optical dipole traps will be outlined. e numerical values of the relevant experimental parameters are summarized in Appendix A. e 87 Rb atoms are initially conﬁned in a magneto-optical trap and then transferred to a magnetic quadrupole trap. Subsequent evaporative cooling using a time orbiting potential (TOP) creates an atomic cloud of nearly 106 atoms close to degeneracy in the |F = 1, mF = −1⟩ state of the 5S1/2 ground state manifold. Aer a transfer to an optical dipole trap, degeneracy is reached with further evaporative cooling by lowering the intensity of one of the trap beams. e trap lasers with a wavelength of 1064 nm are far red detuned from the atomic resonances at λ D2 = 780 nm and λD1 = 795 nm. e details of the employed optical dipole trap depend on the performed experiments and will be discussed below. Aer condensation, a magnetic bias ﬁeld for tuning of the inter-species scaering length is ramped to its target value and the actual experiment is conducted by a sequence of microwave and radio frequency pulses. Finally, the atomic clouds are imaged destructively by high-intensity absorption imaging [64, 26]. e duration of one experimental cycle is about 40 s. 3.1.1. Optical dipole traps e experiments described in this thesis were performed in two diﬀerent optical dipole traps. e experimental sequence for the preparation of the atomic cloud depends on the details of the em- 25 y ( m) y ( m) 3. Experimental system and analysis methods 0 4 8 12 0 50 100 x ( m) 150 200 0 4 8 12 0 50 100 x ( m) 150 200 Figure 3.1.: Absorption images of 3000 atoms in the charger (top) and 45000 atoms in the waveguide (boom). A high atomic column density is encoded in color, while white regions indicate the absence of atomic absorption. e diﬀerent aspect ratio of the two traps and the increased longitudinal extent of the atoms in the waveguide are clearly visible. ployed trap such as its aspect ratio and trapping frequencies. We will now describe the speciﬁcs of these two traps called ’charger’ and ’waveguide’. Charger e ’charger’ consists of a single laser beam focused down to a waist of about 5 μm [65]. Due to this strong focus, the Rayleigh length is small enough to oﬀer suﬃcient conﬁnement along the longitudinal direction, such that the cold atomic cloud can be transferred directly from the magnetic trap into the charger. A second ’crossed dipole beam’ for additional longitudinal conﬁnement during the preparation of the condensate is not required. e use of a single beam makes the position of the prepared Bose-Einstein condensate very reproducible and spatial excitations such as breathing or sloshing are below the detection limit. e tight conﬁnement leads to fast thermalization rates and the duration of the ﬁnal evaporation ramp is about 100 ms. e aspect ratio of the transverse and longitudinal trapping frequencies is ω ⊥ /ω x = 21 [65]. For the experiments performed in the charger, the longitudinal trap frequency is ω x = 2π × (22 . . . 24) Hz with a transverse conﬁnement of ω ⊥ = 2π × (460 . . . 500) Hz. e exact values depend on the speciﬁc seings for each experiment and will be given in the corresponding section. e condensates in the charger consist of a few thousand atoms with a negligible thermal fraction. A typical image of a condensate in the charger is shown in Figure 3.1. While the preparation of Bose-Einstein condensates in the charger is very reliable and robust, the small aspect ratio leads to a longitudinal size < 30 μm of the atomic cloud. us, the inhomogeneous atomic cloud cannot be described as locally uniform and the deviations from the homogeneous theory are signiﬁcant as discussed in subsection 2.4.4. Furthermore, the typical size of spin paerns created by spatial demixing dynamics is in the range of 5 to 20 μm, which is not much smaller than the extent of the atomic cloud. Boundary eﬀects will dominate the position and size of the spin domains and the low statistics due to the occurrence of only few domains in a single shot limits the accuracy of the determination of their size. ese limitations can be relaxed by employing a diﬀerent trap, the ’waveguide’. 26 3.1. Experimental system x = 1.89 Hz 15000 50 atom number center of mass ( m) 100 0 -50 -100 0 200 400 600 time (ms) 800 1000 10000 5000 0 124 126 128 130 132 modulation frequency [Hz] 134 Figure 3.2.: Trap frequency measurements in the waveguide. e longitudinal trap frequency ω x is extracted from a sinusoidal ﬁt to the center of mass motion of an atomic cloud that was initially displaced from the trap center (le). e transverse frequency ω ⊥ is determined by measuring atom loss due to parametric heating when modulating the intensity of the trap laser (right). Waveguide Like the charger, the waveguide consists of a single focused beam. However, its waist is larger¹ resulting in smaller trapping frequencies and a larger aspect ratio ω ⊥ /ω x > 60. In the experiments described in this thesis, the trap frequencies were measured to be ω x = 2π × 1.9 Hz and ω ⊥ = 2π × 128 Hz. e longitudinal trap frequency was deduced from the center of mass motion of the atomic cloud which was initially displaced from the trap center. e transverse conﬁnement was measured by atom loss due to parametric heating by an intensity modulation of the trap laser. e experimental data is shown in Figure 3.2. e shallow longitudinal conﬁnement leads to an atomic cloud with a length > 250 μm, which is signiﬁcantly larger than the typical spin paerns due to demixing. e ﬁeld of view of our camera is about 215 μm wide, such that only a part of the atomic cloud can be detected as shown in the lower panel of Figure 3.1. e downside of the shallow longitudinal conﬁnement is an increased sensitivity to spatial excitations. us, the preparation of the atomic cloud in the waveguide is more complex than for the charger. In particular, the mode overlap with the magnetic trap is strongly reduced and the longitudinal conﬁnement is not strong enough to directly transfer the atoms from the magnetic trap into the waveguide. us, an additional crossed dipole trap called ’Xdt’ has to be used for the transfer from the magnetic trap into the optical traps, as well as for the evaporation where high densities are required for fast thermalization. Only aer the evaporation ramp (where the intensity of the waveguide is linearly reduced), the Xdt can be switched oﬀ. As the longitudinal trap frequency of the waveguide is much smaller than the additional conﬁnement by the Xdt and has a small absolute value, the power of the Xdt has to be reduced slowly in order to minimize spatial excitations. By employing a ramp consisting of three segments with decreasing slopes we can limit the duration of this process to about 1 second. Due to the low intensity of the Xdt beam in the last segment of the ramp small power ﬂuctuations can excite breathing modes of the atomic cloud. Furthermore, external forces can lead to a signiﬁcant displacement of the atomic cloud in the longitudinal direction. Magnetic ﬁeld gradients can cause such forces, because the atoms are in a magnetically sensitive state. For example, the displacement of the trap minimum due to the ﬁrst ¹A waist of 30 μm was measured in [65]. However, the ﬁber outcoupler was changed since. 27 3. Experimental system and analysis methods order Zeeman shi and a magnetic ﬁeld gradient of dB/dx = 50 μG/100 μm is Δx = mF μdB/dx = 11 μm mω 2x with the magnetic moment μ = 700 kHz/G. us, the transfer of atoms from the initial state |F = 1, mF = −1⟩ to the state required for experiments employing the Feshbach resonance, |F = 1, mF = 1⟩, can initiate signiﬁcant spatial oscillation dynamics in the presence of magnetic ﬁeld gradients. In order to minimize such eﬀects, the ﬁeld gradient was canceled by positioning external permanent magnets. In addition, the Feshbach ﬁeld is ramped up slowly on the timescale of 1 second in order to minimize spatial dynamics caused by a changing ﬁeld gradient during the ramp-up. In summary, the waveguide allows to prepare an elongated atomic cloud that is well suited for analyzing the formation of spin domains in one dimension. However, the increased sensitivity to external perturbations such as magnetic ﬁeld gradients or power ﬂuctuations of the trap beams leads to larger shot-to-shot ﬂuctuations on the position of the condensate and requires a longer and more complex preparation sequence. Dimensionality e cigar-shaped atomic clouds trapped in the charger or the waveguide are not strictly one-dimensional, as the chemical potential is not smaller than the transverse trapping frequency. For example, the chemical potential for commonly employed atom numbers in the waveguide is μ ≈ 300 Hz, which is larger than the transverse trapping frequency ω ⊥ = 2π × 128 Hz. e system is beer described as quasi-one-dimensional. subsection B.1.1 contains more information about this dimensionality regime and its theoretical treatment.² However, the spin degree of freedom, i.e. the diﬀerence density, can treated as one-dimensional. e smallest possible structure size in the spin is given by the spin √ healing length ξ s and the smallest unstable wavelength due to modulational instability is λ c = 2πξ s as discussed in chapter 2. e full width at half maximum transverse extent of the atomic cloud in the trap is 1.1 μm in the charger and 2.0 μm in the waveguide. e smallest spin healing length we can produce is ξ s = 1.3 μm close to the Feshbach resonance in the center of the trap, which is on the same order as the transverse size of the cloud. us, transverse spin excitations are strongly suppressed and cannot be observed in our experiments. e spin dynamics is restricted to one spatial dimension. 3.1.2. Linear coupling During the condensation process, the atoms are in the |F = 1, mF = −1⟩ state. Using microwave and radio frequency radiation, we can transfer atoms to any state within the 5S1/2 manifold, which is sketched in Figure 3.3. e frequency of the microwave ﬁeld is given by the hyperﬁne spliing of 6.834 GHz [67]. e required radio frequency is determined by the Zeeman spliing, whose linear contribution is μ = 700 kHz/G leading to a few MHz at magnetic ﬁelds of a few Gauss. Typical single-photon Rabi frequencies for microwave transitions are 10 kHz and for radio frequency transitions 20 kHz, respectively. Two-photon transitions are detuned by 200 kHz from the intermediate level, such that typical two-photon Rabi frequencies are 500 Hz. ese coupling strengths are large enough, such that the non-linear eﬀects of interactions can be ignored during π/2 or π pulses which transfer atoms between diﬀerent sublevels. ²In the language of the one-dimensional systems, our experimental parameters correspond to dimensionless interaction and temperature parameters of γ ≈ 6 × 10−5 and t ≈ 2.6 × 103 . is corresponds to being at the cross-over from thermal to the quantum quasicondensate regime. A phase diagram and can be found in [66]. 28 3.1. Experimental system mF=2 mF=1 F= 2 mF=0 mF=-1 mF=-2 6.834 GHz Ω F= 1 mF=-1 mF=0 mF=1 Figure 3.3.: Level scheme of the 5S1/2 ground state of 87 Rb. e pair of states featuring the Feshbach resonance are marked by ellipses, the ones with a ’magic ﬁeld’ by rectangles. A linear coupling via two-photon microwave - radio frequency radiation is sketched by gray lines for the example of the Feshbach states. As the radio frequency signal is generated using an arbitrary waveform generator³, all its characteristics such as amplitude, phase and frequency can be controlled and changed arbitrarily on time scales < 1 μs. is high level of control makes the linear coupling an ideal control parameter for phase transitions as will be discussed in the following chapters. Resonant linear coupling is critical for the experiments described in this thesis. e correct frequencies for a given magnetic ﬁeld can be calculated using the Breit-Rabi formula. However, as we employ a two-photon transition the resonance condition is modiﬁed by the ac Zeeman shi (also referred to as ’light shi’) due to the coupling radiation itself. Both the microwave and the radio frequency radiation are detuned from the intermediate level by about 10-20 times the coupling strength. e ac Zeeman eﬀect of each ﬁeld shis the relative spliing of the involved atomic states (see [68] for details). e combined eﬀect of the microwave and the radio frequency ﬁeld has to be compensated by adjusting their frequencies. For our parameters this eﬀect shis the resonance by about −80 Hz to −120 Hz. As the exact value of the light shi depends on the amplitude of the coupling ﬁelds it has to be adjusted when changing the coupling strength. e values of the individual light shis stemming from the microwave or radio frequency ﬁeld can be measured by a Ramsey sequence where the corresponding coupling ﬁeld is present during the interrogation time. For the power seings used during the initial π/2-pulses light shis of −120 Hz and +35 Hz were measured for the microwave and the radio frequency ﬁeld, respectively. Another eﬀect that modiﬁes the resonance condition is the mean ﬁeld shi due to diﬀerent interaction parameters of the two atomic components. Diﬀerent intra-species scaering lengths a11 and a22 cause a diﬀerence in the chemical potentials of the two components when an equal superposition of them is prepared. is corresponds to an eﬀective detuning δ MFS ∝ (g11 − g22 ) · n [34, 41]. As this eﬀect depends on the atomic density, it creates a space-dependent detuning when working with harmonically conﬁned and thus inhomogeneous atomic clouds, δ MFS = δ MFS (x). e typical amplitude of the mean ﬁeld shi is δ MFS ≈ 10 Hz in the center of the atomic cloud for our experimental parameters. e average mean ﬁeld shi can be compensated by adjusting the frequency of the radio frequency ﬁeld. However, spatial variations of a few Hertz due to the inhomogeneity of the atomic cloud remain. is eﬀect is negligible for Ω ≫ δ MFS , but can have signiﬁcant implications if this condition is not fulﬁlled. ³Agilent 33522A 29 a12 (aB ) 3. Experimental system and analysis methods 130 1.6 120 1.4 110 1.2 100 1.0 90 0.8 80 0.6 70 9.00 9.05 9.10 9.15 magnetic field (G) miscible immiscible 0.4 9.00 9.05 9.10 9.15 magnetic field (G) Figure 3.4.: Scaering lengths a12 (le) and miscibility parameter Δ (right) around the Feshbach resonance. e experimental data (black circles) was deduced from the frequency of plasma and π oscillations (see subsection 2.2.2) and independently conﬁrmed by the experiments presented in this and the following chapters. e solid lines are taken from [32], the dashed lines correspond to the background value of a12 . 3.1.3. Employed atomic states and Feshbach resonance An interesting pair of states in the ground state manifold of 87 Rb is |1⟩ = |F = 1, mF = −1⟩ and |2⟩ = |2, 1⟩, since their diﬀerential Zeeman shi (which is already the quadratic shi) cancels to ﬁrst order at the ’magic ﬁeld’ B = 3.23 G [69]. In this conﬁguration these states are only weakly sensitive to magnetic ﬁeld ﬂuctuations and very long coherence times have been reported [70, 71]. Similarly, magnetic ﬁeld gradients act as common-mode forces on atoms in these two states and the diﬀerential eﬀects are minimized. Other interesting states are |1⟩ = |F = 1, mF = 1⟩ and |2⟩ = |2, −1⟩, because they feature a narrow Feshbach resonance at B = 9.09 G [72, 73, 74], which allows to change the inter-species scaering length a12 of these states by up to 30%. e background scaering lengths of 87 Rb are all equal within 5% and the system is close to the miscibility threshold Δ = 1 far from resonance (see section 2.3). us we can tune the system both into the miscible and the immiscible regime by employing this Feshbach resonance as shown in Figure 3.4. ese two pairs of states will be used for the experiments described in this thesis. ey are illustrated in Figure 3.3. Magnetic field stabilization Near the Feshbach resonance the diﬀerential shi of the involved atomic states due to the quadratic component of the Zeeman shi is 10 Hz/mG, such that a stable magnetic ﬁeld is required to increase the coherence time and to keep the two-photon coupling resonant. For example, when dressing the atoms with a coupling strength of a few ten Hertz, the detuning has to be kept below a few Hertz, which requires a magnetic ﬁeld stability well below 1 mG. In addition, the Feshbach resonance has a small width of a few milligauss and our measurements require interaction parameters that are constant over the duration of one experiment and shot-toshot reproducible. For example, the relevant interaction parameter gs (equivalent to the nonlinearity χ in the notation of the Josephson junction) increases by 50% when changing the magnetic ﬁeld by 10 mG from 9.07 G to 9.08 G. us a magnetic ﬁeld variation of 1 mG changes the interaction parameter by more than 5% when working close to resonance. 30 3.2. Free evolution experiments and their analysis In order to fulﬁll these stability requirements we synchronize the experimental cycle to the 50 Hz power line and additionally employ an active ﬁeld stabilization. A feedback loop reduces the shotto-shot variations due to low frequency magnetic ﬁeld ﬂuctuations below 50 μG. A feed forward suppresses the amplitude of the remaining 50 Hz power line component from 200 μG [32, 39] below the shot-to-shot ﬂuctuations. e magnetic ﬁeld is generated by large coils with a size of 1 m × 1 m in Helmholtz conﬁguration in order to ensure ﬁeld homogeneity. 3.1.4. Detection of the atomic cloud e observation of spatial paerns requires in-situ imaging of the atomic cloud. e two components are subsequently detected in the trap by high-intensity absorption imaging [64, 26] with a temporal delay between the pictures of 780 μs. e images are followed by a reference picture not containing any atomic signal in order to eliminate fringe noise. e imaging resolution of our experimental setup is 1.1 μm according to the Rayleigh criterion [75]. In previous experiments, the atomic populations of the involved states was ’frozen’ by transferring them to the F = 1 manifold where losses are negligible. e magnetic ﬁeld was subsequently reduced and the image was taken at a low magnetic ﬁeld of a few hundred milligauss for optimized atom number resolution [26]. As the spatial spin paern critically depends on the exact interaction parameters and adjusts to external changes within a few milliseconds, the density distribution cannot be frozen for these experiments. Instead the absorption images are taken at the magnetic ﬁeld close to the Feshbach resonance at 9.09 G where the experiment was conducted. e Zeeman spliing of diﬀerent mF levels is on the same order as the line width of the employed D2 transition, which requires adjustments to the imaging calibration. e necessary steps are summarized in Appendix C. 3.2. Free evolution experiments and their analysis In this section, we will discuss the experimental results on the free evolution of an equal superposition of the two components. By ’free’, we mean the absence of any linear coupling, such that the dynamics is governed by the atomic interactions. We will begin with experiments performed in the charger at the ’magic ﬁeld’ with the background scaering lengths of 87 Rb. Aerwards we will focus on similar experiments in the waveguide but close to the Feshbach resonance, where the dynamics is dominated by modulational instabilities. Along the way, we will introduce the analysis methods for the interpretation of the experimental results that will be used throughout this thesis. 3.2.1. Free evolution far from the Feshbach resonance in the charger We will ﬁrst consider the time evolution of a superposition of atomic clouds far from the Feshbach resonance, i.e. with the background interaction parameters of 87 Rb. For maximum stability, we work at the ’magic’ ﬁeld of B = 3.23 G using the states |1⟩ = |F = 1, mF = −1⟩ and |2⟩ = |2, 1⟩. ese measurements were conducted in the charger with a longitudinal trap frequency of ω x = 2π × 23.4 Hz and a transverse conﬁnement of ω x = 2π × 490 Hz. A condensate of about 5600 atoms in state |1⟩ is prepared, an equal superposition of |1⟩ and |2⟩ is created by a two-photon π/2pulse and the subsequent dynamics is observed. is experiment is similar to the ﬁrst measurements on component separation in 87 Rb [76]. e background interaction parameters of 87 Rb are close to the miscibility threshold, Δ ≈ 1. Commonly used scaering lengths yield Δ = 1.0001 [77], 0.9966 [78] and 0.9980 ± 0.0008 [79], so the literature is not deﬁnite about whether the system is miscible or immiscible (see Appendix A 31 3. Experimental system and analysis methods 0 |1 |2 |1 |2 time (ms) 50 100 150 200 -20 0 x ( m) 20 -20 0 x ( m) 20 -20 0 x ( m) 20 -20 0 x ( m) 20 Figure 3.5.: Time evolution of an equal superposition of two condensates in the charger at the background scaering lengths. e le panels show the experimentally obtained density timetrace in the two components, while the corresponding numerical simulations are shown on the right. Component |1⟩ is pushed to the edges of the trap due to its larger scaering length a11 > a22 . is dynamics is referred to as potential separation (see text). e excellent agreement with numerics demonstrates the experimental stability and reproducibility when working in the charger and at the ’magic ﬁeld’. for the values of the scaering lengths). Independent of the sign of Δ − 1, the dynamics following the π/2-pulse will not be governed by unstable modes. If Δ > 1, there are no unstable modes and considering Δ = 0.99 < 1, the fastest predicted growth rate of 1 Hz is too slow to have an inﬂuence on the timescale of our measurements. Furthermore, the corresponding unstable mode has a wavelength exceeding 50 μm and cannot grow as it is larger than the size of our system. us, the dynamics is purely deterministic and reproducible. e observed time evolution of the atomic density proﬁles is shown in Figure 3.5. Atoms in state |1⟩ are pushed to the edges of the trap while component |2⟩ occupies the trap center. Aer about 80 ms this dynamics is reversed and the density proﬁles oscillate back towards the initial conﬁguration. e agreement with numerical simulations⁴ is excellent. e spatial dynamics is caused by the asymmetry in the intra-species scaering lengths a11 > a22 , which favors lower densities of component |1⟩. In miscible systems such dynamics occurs only if the translational symmetry is broken by the presence of an external trapping potential. No redistribution of the atomic density is expected in a homogeneous system as discussed in subsection 2.3.3. For this reason, the observed time evolution is referred to as potential separation [43]. e measurements can be understood as the evolution of each component in an eﬀective potential given by the external trapping potential and the repulsive mean ﬁeld interactions Veﬀ, i (x) = V(x) + gii |ψ i (x)|2 + gij |ψ j (x)|2 . Before the initial π/2-pulse all atoms are in state |1⟩ and the density distribution is given by the corresponding ground state. e fast coupling pulse creates an equal superposition of the two states, each having the same spatial density proﬁle as the initial state before the pulse. As this conﬁguration is not the ground state of the two-component system, the mean ﬁeld potentials Veﬀ, i (x) will initiate a redistribution of the atomic clouds. is eﬀective potential is not a static one and changes with the spatial density proﬁles of the atoms, but it provides an intuitive approach for understanding the resulting dynamics. e redistribution of the atomic clouds reduces the total potential energy of the system consisting of the external trapping potential and the mean ﬁeld interaction energies. e kinetic energy ⁴A summary of the employed methods can be found in Appendix B, while the atomic and experimental parameters are given in Appendix A. 32 160 140 120 100 80 60 40 20 0 |1 |2 3000 2500 atom number linear density (atoms/ m) 3.2. Free evolution experiments and their analysis 2000 1500 1000 500 -20 -10 0 x ( m) 10 20 00 |1 |2 50 100 time (ms) 150 200 Figure 3.6.: Le panel: Comparison of the temporal mean of the measured density timetrace averaged over 0 ≤ t ≤ 140 ms (solid lines) with the numerically computed ground state (dashed lines). e mean atom number in each component in the chosen time range was used for the simulation. Right panel: Time evolution of the atom number in each component. Loss in state |2⟩ is increased due to spin relaxation loss in agreement with the simulations (solid lines). is increased by the dynamics. Similar to a classical particle, which is initially displaced from the minimum of a harmonic potential, the density conﬁguration will move towards the potential minimum (representing the ground state) and pass through this point due to the accumulated kinetic energy. Assuming symmetric evolution around the energy minimum, we can estimate the ground state conﬁguration of each component by calculating the temporal mean of their density proﬁles. e density proﬁles averaged over the ﬁrst oscillation period of the time evolution are compared to the numerically computed ground state in the le panel of Figure 3.6. One possible dissipation process for the oscillatory dynamics is atom loss. While loss in component |1⟩ is negligible, spin relaxation loss limits the lifetime of component |2⟩. e right panel of Figure 3.6 compares the measured loss with the numerical simulations employing the literature loss coeﬃcients (see Appendix A). 3.2.2. Formation of spin domains near the Feshbach resonance in the waveguide We have seen in the previous section that the dynamics in the charger far from the Feshbach resonance is governed by potential separation, while unstable excitation modes do not play a role. Only close to the Feshbach resonance, instabilities can be observed in the charger, but they are strongly inﬂuenced by the ﬁnite longitudinal size of the cloud and the inhomogeneity due to the external trapping potential. e waveguide is beer suited for the analysis of unstable modes. e time scale for potential separation depends on the chemical potential μ and the longitudinal trapping frequency ω x . In the waveguide μ is smaller by a factor of 2 and ω x by a factor 10 compared the charger, such that potential separation does not play a role on the time scale of our experiments. In addition, the increased size of the atomic cloud shis the infrared cutoﬀ in the excitation spectrum to larger wavelengths. e increased homogeneity leads to comparable experimental conditions over a large spatial range, such that the analytic predictions for homogeneous systems can be applied.⁵ ⁵Validity criteria for the description of excitations in an elongated atomic cloud by a homogeneous theory have been discussed in [33] in the context of sound propagation in Bose-Einstein condensates [80, 81]. 33 3. Experimental system and analysis methods shot number 0 5 10 15 20 0 100 200 300 x ( m) 400 500 Figure 3.7.: Randomness of domain structures generated by phase separation due to unstable modes. e F = 2 density proﬁles of subsequent realizations under the same experimental conditions (B = 9.08 G, image taken aer t = 32 ms) are shown in false color. e density paern does not systematically repeat itself. We can examine the unstable region of the Bogoliubov spectrum in the waveguide by preparing an equal superposition of the two components and subsequently analyzing the emerging spatial paern of spin domains, which will be dominated by the unstable modes of the excitation spectrum. As the resulting spatial paern is based on instability, small density variations in the prepared superposition act as a seed causing the growth of diﬀerent modes. Due to the destructive detection method we cannot observe the dynamics in a single experimental realization and a new independent condensate has to be prepared for each absorption image. us, the position of the domains will not be deterministic but randomly vary from shot to shot. is randomness is illustrated in Figure 3.7. For the experiments in the waveguide near the Feshbach resonance, the connection to the ground state cannot be drawn by the temporal mean of the density proﬁle as it was the case for the potential separation in the charger. us, diﬀerent analysis methods have to be applied, which will be introduced in the following section. 3.2.3. Analysis methods: Counting, Fourier spectra and correlations e characteristic signature of spatial spin paerns is the time evolution of the typical size of the domains and their modulation depth. is section evaluates a few methods for the extraction of this data from the experimental images. One common requirement for all methods is that they must not rely on a reproducible position of the domains due to their random locations. e input signal for all these methods is the experimental spin proﬁle. It is calculated from the longitudinal density proﬁle ni (x) of each component, which is extracted by summing over the rows around the corresponding absorption signal for each pixel column. e spin proﬁle Jz (x) is deﬁned 1 (x) as the normalized diﬀerence proﬁle, Jz (x) = nn21 (x)−n (x)+n2 (x) . As the extent of the atomic cloud is larger than the ﬁeld of view of the camera, no extra care has to be taken to exclude boundary region with increased noise due to a vanishing atom signal. However, some experiments critically depend on a homogeneous density distribution. In this case, only a part of the spin proﬁle around the center of the atomic cloud is used for further analysis. e requirements and the details of this spatial post-selection will be explained in the description of the respective measurements. Counting domains One analysis method proposed and employed in [18] is counting the number of domains in a given spatial region and deducing the ’domain density’ i.e. the number of domains per unit length. For example, domain boundaries can be extracted from the spin proﬁle as zero-crossings. Alternatively, domain centers can be localized by identifying local maxima and minima. ese approaches require 34 3.2. Free evolution experiments and their analysis the use of detection thresholds in order to distinguish the domain signal from detection noise. As atoms are lost during the time evolution, the signal-to-noise ratio is reduced and the threshold criteria change. e required choice of a threshold makes this method unreliable for small modulation depths of the domains. Fourier analysis A more robust analysis method is employing the Fourier spectrum of the spin proﬁle, where the most dominant oscillatory mode, i.e. the typical domain size, is directly given by the highest peak in the spectrum. In order to have suﬃcient spatial resolution, the size of the domains must be signiﬁcantly smaller than the spatial range where the spin proﬁle is evaluated. e amplitude of the detected peak is directly proportional to the modulation depth of the spin paerns. Autocorrelation function Similarly, the autocorrelation function of the spin proﬁle can be calculated. As every point in the correlation function is averaged over the whole proﬁle, this intrinsic averaging suppresses technical noise. e autocorrelation function is connected to the power spectrum of the spin proﬁle via a Fourier transformation as stated by the convolution theorem. us, its information content is equivalent to the previously mentioned Fourier spectra. For periodic structures, a peak in the Fourier spectrum can be found more reliably than the oscillation period of the correlation function. Consequently, the Fourier method is preferred in this case. However, the autocorrelation method additionally allows the analysis of spatial spin correlations even if the system is not prepared in the immiscible regime, where the growth of unstable modes leads to periodic spin structures. In the miscible regime, the autocorrelation function of the ﬂuctuations on top of the spin proﬁle is predicted to decay to zero with a power law or exponentially depending on the system parameters and the details of the measurement [82]. e length scale of the decay depends on the distance to the critical point and can be associated with a correlation length. Measurements in this regime will be discussed in subsection 5.3.1. Both the Fourier spectra and the correlation functions do not depend on shot-to-shot variations in the positions of the domains, i.e. phase shis of the oscillatory structures. us they can be averaged over diﬀerent experimental realizations of the same physical seings (e.g. same magnetic ﬁeld and evolution time). is averaging increases the signal to noise ratio and yields reliable results also for small modulation depths around 5% even in the presence of detection noise such as photon shot noise. Post-processing of the spin profile In many experiments presented in this thesis, we are interested in the linear response of the system to a sudden parameter change. e linear response regime is valid for small perturbations, e.g. periodic structures or correlations on top of the background state. In these measurements, we are analyzing the spin proﬁle, such that inhomogeneity eﬀects of the spatial sum density proﬁle are already canceled. However, both the Fourier and the autocorrelation methods are prone to asymmetries of the spin proﬁle, e.g. to a mean imbalance z ̸= 0 due to diﬀerent densities in the two components. Such problems can occur in the experimental data for multiple reasons: • Imperfect π/2 preparation pulses, e.g. due to temporal dris or spatial gradients in the microwave or radio frequency ﬁelds 35 muon |1 0.1 (Jz (x)) |2 F 0 4 8 12 0 4 8 12 1 0 -1 1 0 -1 0 0.00.0 0.1 1 Jz (x) Jz (x) Jz Jz y ( m) y ( m) 3. Experimental system and analysis methods 100 x ( m) 200 0.2 k (1/ m) 0.3 10 15 20 x-x' ( m) 25 0 -10 5 30 Figure 3.8.: Illustration of the post-processing of the spin proﬁle and its impact on Fourier spectra and correlation functions. From the experimental images (top le), the spin proﬁle is extracted (center le). e eﬀect of spurious muon impacts on the CCD is visible as sharp spikes in the spin proﬁle. e post-processing removes these artefacts and centers the spin proﬁle around zero with the result shown on the boom le. e Fourier spectra and correlation functions corresponding to the raw (black lines) and corrected (gray lines) spin proﬁles are shown on the right. e ’muon’ causes a kink near the origin in the correlation function. e trend removal reduces low-frequency components in the Fourier spectrum and centers the correlation function around zero. • Species dependent, asymmetric loss, such as spin relaxation loss in the F = 2 component • Detuning of a linear coupling ﬁeld • Diﬀerent detectivity of the imaging process to the diﬀerent species, such that a balanced state is detected to be imbalanced (see Appendix C) In addition the spin proﬁle can have an overall trend across the spatial cloud caused by • a local detuning of the linear coupling due to the mean ﬁeld shi, which depends on the spatial density proﬁle • a gradient in the amplitude of the linear coupling ﬁeld (see subsection 4.1.1) leading to an inhomogeneous temporal evolution and a local detuning due to the spatial variations of the light shi. ese eﬀects are small, but can dominate the deduced Fourier spectrum or autocorrelation function as illustrated in Figure 3.8. For this reason, the spin proﬁle extracted from the absorption images is post-processed before performing the analysis. is post-processing consists of the following steps: 1. Compensate for known diﬀerent detectivities of the imaging system (see Appendix C) 2. Remove muons. Impact of particles from cosmic background radiation (which we refer to as ’muons’) on the CCD chip cause bright or dark pixels on the absorption images roughly once every 50 shots. In some cases, the position of these pixels is within the picture of the atomic cloud and they aﬀect the integrated density proﬁles. As these artefacts usually occur only on one of the absorption images of the two components, the muons manifest themselves in the spin proﬁle as single pixels with an unphysical imbalance z ≫ 1. ese pixels are corrected 36 0 4 B=9.08G t=30ms 8 12 0 4 8 12 0 50 |2 |1 100 x ( m) 150 200 |2 |1 100 x ( m) 150 y ( m) y ( m) 0 4 B=9.05G t=164ms 8 12 0 4 8 12 0 50 y ( m) y ( m) y ( m) y ( m) y ( m) y ( m) 3.2. Free evolution experiments and their analysis 200 0 4 B=9.06G t=103ms 8 12 0 4 8 12 0 50 0 4 B=9.12G t=30ms 8 12 0 4 8 12 0 50 |2 |1 100 x ( m) 150 200 |2 |1 100 x ( m) 150 200 Figure 3.9.: Absorption images of the atomic density proﬁles at diﬀerent magnetic ﬁelds close to the Feshbach resonance at 9.09 G. As both the growth rate of the domains and the loss rate are increased when approaching the resonance from the immiscible side, the images were taken aer diﬀerent evolution times as indicated in the graphs. e typical domain size decreases when approaching the resonance. e reduced atom number in state |2⟩ due to spin relaxation loss causes an asymmetry in the domain sizes of the two components. is eﬀect becomes less important when approaching the resonance as symmetric Feshbach loss dominates. No phase separation occurs on the miscible side of the resonance (boom right panel). by replacing them with the average imbalance of their neighboring pixels. Less than 1% of the shots are aﬀected. 3. (optional) Remove an overall trend in the spin proﬁle. In this step, the spin proﬁle is smoothed by a Gaussian ﬁlter. e width of the Gaussian has a typical value of 80 μm and is thus much larger than the observed domain sizes of 2 to 10 μm. is smoothed proﬁle is subtracted from the original spin proﬁle, such that the small structures and ﬂuctuations are conserved but centered around zero imbalance. is procedure corresponds to a applying a high-pass to the spin proﬁle and is omied for the analysis of phase separation dynamics near the immiscibility threshold, where the domain size exceeds 10 μm. 4. (optional) Discard the edges of the spin proﬁle in order to neglect regions with a reduced atom density due to the inhomogeneity of the condensate. is ensures comparable experimental parameters over the remaining spin proﬁle. In addition, this measure avoids artiﬁcial boundary eﬀects introduced by the previous step. 3.2.4. Mapping out the Feshbach resonance Using these analysis methods, we can investigate the demixing dynamics of an equal superposition of the two components prepared at various magnetic ﬁelds on the immiscible side of the Feshbach resonance. e phase separation is governed by the unstable modes of the Bogoliubov spectrum, which depends on the value of the inter-species scaering length a12 . When approaching the Feshbach resonance a12 is increased which shis the most unstable mode towards smaller wavelengths and increases its growth rate. On the miscible side of the Feshbach resonance the overlap of the two atomic clouds remains high and no demixing occurs. is behavior is qualitatively illustrated by exemplary experimental images shown in Figure 3.9. 37 3. Experimental system and analysis methods 140 50000 120 30000 20000 B=9.060G B=9.070G B=9.080G B=9.087G 10000 00 20 40 60 time (ms) 80 1 / lifetime (1/s) atom number 40000 100 80 60 40 20 0 100 9.05 9.06 9.07 9.08 magnetic field (G) 9.09 Figure 3.10.: Atom loss at diﬀerent magnetic ﬁelds close to the Feshbach resonance. e lifetime of the atomic cloud strongly depends on the distance to the resonance. Fiing an exponential to the initial evolution of the detected atom numbers (solid lines in le panel) yields the dependence of the lifetime on the magnetic ﬁeld (right panel). While the Feshbach resonance oﬀers control of the interaction parameters it has the disadvantage of increased three-body loss as summarized in Figure 3.10. is limits the lifetime of the condensate and thus the range of accessible evolution times. In addition, the temporal change in the linear atom density n aﬀects both the wavelength and the growth rate of the most unstable mode, such that they dynamically change during the time evolution. is complicates the comparison with theoretical predictions. However, it will turn out that using the mean density yields reasonable agreement with experimental observations. For a quantitative description of the observed dynamics, we recall the results of the Bogoliubov analysis in subsection 2.4.2. As derived in Equation 2.26, the wavelength of the most unstable mode λ f (i.e. twice the domain size) and the smallest unstable wavelength λ c are given by λf = 2πℏ 2π = 2πξ s = √ kf −mngs and λc = λf 2π =√ kc 2 (3.1) e most unstable mode grows exponentially with the time constant τf = 2ℏ ngs (3.2) e inter-species scaering length a12 , which is changed in the vicinity of the Feshbach resonance, enters via the interaction parameters gs (see Equation 2.11). e Bogoliubov analysis predicts the linear response of an initially homogeneous superposition of the two components. e linear response corresponds to the initial dynamics of the spin proﬁle when the modulation depth of the emerging spin structures is small. At longer evolution times when the modulation depth is a signiﬁcant fraction of the atom density in each component, the growth will no longer be exponential but saturate. Beyond the linear response the Bogoliubov spectrum is modiﬁed by the changing density proﬁles of the two components. e growth rates may be changed and even previously stable modes may become unstable. ese ’secondary’ excitation modes have been predicted in the non-equilibrium dynamics aer collisions of heavy nuclei [83]. We did not observe any additional unstable modes in our experiments when studying the time evolution of spin proﬁles beyond the linear response regime. 38 3.2. Free evolution experiments and their analysis 0.05 B=9.060G B=9.070G B=9.080G B=9.087G FFT amplitude 0.04 0.03 0.02 0.01 0.000.00 0.05 0.10 0.15 0.20 0.25 wave vector k (1/ m) Figure 3.11.: Averaged Fourier spectra at diﬀerent magnetic ﬁelds for a modulation depth of about 30%. e peak of the spectrum is shied towards smaller wavelengths when approaching the Feshbach resonance indicating the decreasing size of the domains. e displayed spectra are obtained by averaging over about 20 realizations for each value of the magnetic ﬁeld. We analyze the observed spin paerns via their Fourier spectra. For each experimental run we calculate the Fourier spectrum of the spin proﬁle and average over about 10 . . . 20 experimental realizations taken under the same conditions, i.e. magnetic ﬁeld and evolution time. is averaging process increases the signal-to-noise ratio and allows the detection of structures with a small modulation depth, i.e. in the linear response regime. From the peak of this mean spectrum we extract both the amplitude and the wavelength λf of the most unstable mode. e domain size is given by half of the wavelength corresponding to the position of the peak in the spectrum. Similarly, we associate the smallest unstable wavelength λ c with the position of the small-wavelength edge of the spectrum, i.e. the smallest wavelength where the amplitude of the spectrum exceeds a threshold of 0.01. Exemplary Fourier spectra are shown in Figure 3.11. From these averaged Fourier spectra, we deduce the time evolution of both the modulation depth and the domain size. e amplitude of the spin structures grows exponentially and with an increasing growth rate when approaching the Feshbach resonance. e typical domain size decreases close to resonance and grows in time due to atom loss. ese observations are summarized in Figure 3.12. For a quantitative comparison to the Bogoliubov predictions for a homogeneous system, we average the domain sizes extracted from the mean Fourier spectra over diﬀerent evolution times with a modulation depth between 5% and 15%. We similarly determine the smallest unstable wavelength λ c . A comparison of the experimental results to the Bogoliubov predictions for the homogeneous system is given in Figure 3.13. Magnetic ﬁelds between 9.07 G and 9.08 G are suited best for studying phase separation in our experimental conﬁguration. e lifetime of an initial superposition state is 30 . . . 90 ms and thus larger than typical formation times of the spin excitations of 15 . . . 40 ms. On the other hand, the symmetric Feshbach loss is strong enough to dominate over the asymmetric spin relaxation loss aﬀecting only atoms in state |2⟩. is imbalance between the two components causes diﬀerently sized domains in the two components and thus complicates the extraction of typical domain sizes as the analysis methods rely on the periodicity of the spin paerns. is eﬀect along with the large domains limits the experimental accuracy when working further away from the Feshbach resonance. Closer to the resonance, both the loss rate and the sensitivity of the interaction parameter to ﬂuctuations and dris in the magnetic ﬁeld increase rapidly. In this chapter we have discussed the experimental system along with its capabilities and limi- 39 3. Experimental system and analysis methods 1.0 modulation depth 0.6 0.4 B=9.060G B=9.070G B=9.080G B=9.087G 0.2 0.00 20 40 60 time (ms) 80 FFT peak position ( m) 20 0.8 15 10 B=9.060G B=9.070G B=9.080G B=9.087G 5 00 100 20 40 60 time (ms) 80 100 Figure 3.12.: Time evolution of modulation depth (le panel) and domain size (right) corresponding to the peak in the Fourier spectra at diﬀerent magnetic ﬁelds. e growth rate is increased at magnetic ﬁelds close to the Feshbach resonance. e reduction in modulation depth following the maximum is due to a lower signal-to-noise ratio as atoms are lost. Atom loss also causes the increase of the domain size in time. e initial noise is due to an amplitude of the spin paern below the detection threshold. 12 10 15 FFT edge ( m) FFT peak position ( m) 20 10 5 0 8 6 4 2 9.05 9.06 9.07 9.08 magnetic field (G) 9.09 0 9.05 9.06 9.07 9.08 magnetic field (G) 9.09 Figure 3.13.: Most unstable mode (le) and smallest-wavelength unstable mode (right) at diﬀerent magnetic ﬁelds. e most unstable mode is extracted from the position of the peak of the Fourier spectrum, while the smallest-wavelength unstable mode is associated with the position of the small-wavelength edge of the spectrum, i.e. the largest wave vector whose Fourier amplitude is larger than a threshold of 0.01. e solid lines are Bogoliubov predictions for kf and kc without free parameters based on the values of a12 summarized in Figure 3.4 and the initial linear atom density in the center of the atomic cloud. Our experimental ﬁndings conﬁrm that the assumption of a homogeneous system is a valid description of our experiment close to the Feshbach resonance. e constant oﬀset in the value of the smallest unstable mode is due to the ﬁnite amplitude threshold necessary to exclude detection noise eﬀects. tations. e Feshbach resonance was presented as a tool to change the miscibility of two atomic clouds. e two dynamical regimes of potential and phase separation were studied experimentally and compared to theoretical predictions. Finally, we have discussed analysis methods for quantifying the relevant properties of the emerging spin paerns. ese techniques will be applied throughout the remainder of this thesis. 40 4. Interacting dressed states In the previous chapter, we have introduced the experimental system and a Feshbach resonance as a means to tune inter-atomic interactions. Now we will employ this Feshbach resonance to study the interplay between interactions and a linear coupling of two atomic states. In particular, we will analyze the dynamical amplitude reduction of Rabi oscillations in the presence of interactions and its connection to the density distribution of the atomic clouds. e results can be explained in the picture of interacting dressed states, which spatially separate similar to the atomic states in the experiments discussed in the previous chapter. Our observations are published in [22], which this chapter is based on while providing more detailed explanations. 4.1. Rabi oscillations in the presence of interactions A linear coupling ﬁeld resonantly acting on a two-level system will induce coherent oscillation of the population between the states. ese ’Rabi oscillations’ were ﬁrst demonstrated with BoseEinstein condensates by the group of Eric Cornell in 1998 [84] (see [85] for a review). Figure 4.1 1 shows the evolution of the imbalance z = NN21 −N +N2 for Rabi oscillations in our experimental setup using two-photon microwave - radio-frequency radiation. N1 and N2 denote the populations of the two atomic levels. e amplitude A of the oscillations, oen called visibility, is compatible with 1, the Rabi frequency Ω is about 2π × 520 Hz. Remembering the theoretical description in subsection 2.2.1 these oscillations are modeled by |ψ(t)⟩ = cos(Ω/2 t)|1⟩ + sin(Ω/2 t)|2⟩ ) 1 ( = √ e−iΩ/2 t |+⟩ + e+iΩ/2 t |−⟩ 2 1.0 (4.1) = 519Hz A = 0.98 imbalance 0.5 0.0 -0.5 -1.0 0 1 2 4 3 time (ms) 5 6 Figure 4.1.: Rabi oscillations driven by two-photon microwave - radio-frequency radiation. e solid line is a sinusoidal ﬁt to the evolution of the measured imbalance (black circles) yielding an oscillation amplitude A compatible with 1 and a corresponding frequency Ω of about 2π × 520 Hz. 41 4. Interacting dressed states 1.0 1.0 B=9.03G B=9.17G 0.8 amplitude imbalance 0.5 0.0 -0.5 -1.0 0.6 0.4 0.2 0 100 200 300 time (ms) 400 500 0.00 100 200 300 time (ms) 400 500 Figure 4.2.: Amplitude of Rabi oscillations in the miscible regime at B = 9.17 G (black) and for immiscible parameters (gray, B = 9.03 G). e raw data in the le panel shows a reduction in the oscillation amplitude in the miscible regime aer ≈ 200 ms, which does not occur in the immiscible case. Individual oscillation cycles are not resolved due to the diﬀerent time scale compared to Figure 4.1. For a quantitative analysis, the envelope of the oscillations is found by extracting the maximum imbalance in each cycle and averaging it over ten subsequent periods (right panel). e error bars correspond to two standard deviations of the mean value. Numerical simulations without free parameters reproduce the results in the immiscible regime, but deviate signiﬁcantly for miscible parameters (dashed lines). However, aer including a gradient in the Rabi frequency the simulations reproduce the observed dynamics well (solid lines). with the atomic states |1⟩ and |2⟩ and the dressed states |+⟩ and |−⟩ as deﬁned in Equation 2.5. us, the oscillation corresponds to the interference of an equal superposition of dressed states. e population of each dressed state is constant and the dynamics occurs only in their relative phase. 4.1.1. Amplitude of long Rabi oscillations In order to examine the inﬂuence of interactions on the oscillation dynamics we measure Rabi oscillations in an elongated Bose-Einstein condensate near the Feshbach resonance. e employed atomic states |1⟩ = |F = 1, mF = +1⟩ and |2⟩ = |2, −1⟩ are tuned immiscible at a magnetic ﬁeld of 9.03 G and miscible at 9.17 G. e cloud containing 4400 atoms is conﬁned in the ’charger’ (see subsection 3.1.1) with trap frequencies of ω x = 2π × 22.0 Hz and ω x = 2π × 460 Hz. e system is initially prepared in state |1⟩ before the linear coupling ﬁeld initiates the oscillation dynamics. On short time scales of a few oscillation cycles, the Rabi oscillations are not aﬀected by the interactions and no diﬀerence can be seen between the miscible and the immiscible case. However, aer an evolution time of 200 ms corresponding to roughly 100 cycles, the spatially averaged oscillation amplitude is strongly reduced in the miscible case as shown in Figure 4.2. Here, absorption images of the two atomic clouds reveal a reduced spatial overlap of the two components, i.e. one component occupies the wings of the trap while the other is in the center. Counter-intuitively, this spatial separation occurs only in the miscible regime, while the overlap of the atomic clouds remains high in the immiscible case. We can model our experiments by numerical simulations without free parameters employing the nonpolynomial nonlinear Schrödinger equation (subsection B.1.2). While the agreement in the immiscible regime is good, the simulations do not correctly capture the observed amplitude reduction 42 4.1. Rabi oscillations in the presence of interactions 529 1.0 Rabi frequency (Hz) imbalance 0.5 0.0 -0.5 -1.0 50 well #1 well #8 51 52 54 53 time (ms) 55 56 528 527 526 525 524-20 -15 -10 -5 0 5 x ( m) 10 15 20 Figure 4.3.: Spatial dependence of the Rabi frequency. e elongated atomic cloud is split into eight independent laice sites by a standing wave potential. In each well, Rabi oscillations are measured for 130 ms and ﬁed with a sine. e accumulated phase diﬀerence aer 50 ms (26 cycles) in the outermost wells due to diﬀerent oscillation frequencies is illustrated in the le panel. e extracted frequencies (right panel) reveal the spatial variation of the Rabi frequency. Error bars are the two s.d. uncertainty of the ﬁt. e solid line is a linear ﬁt with a slope of 2π × (94 ± 6) mHz/μm. in the miscible case (dashed lines in Figure 4.2). We aribute this deviation to an inhomogeneity in the linear coupling strength, Ω = Ω(x). e spatial dependence of the Rabi frequency can be characterized independently by slicing the atomic cloud into eight separate wells using an optical standing wave potential with a laice period of 5.5 μm. is technique allows to create independent condensates, each populating a single spatial mode as employed for the experiments in [86, 23, 24, 25]. us, the local resonant Rabi frequency can be measured in each well, which reveals a spatial gradient of κ ≡ ∇Ω = 2π × (94 ± 6) mHz/μm corresponding to about 2π × 3.8 Hz or 0.7% across the entire atomic cloud. Coherent oscillations over 130 ms or 70 cycles were ﬁed in order to obtain this accuracy. ese experimental results are summarized in Figure 4.3 and we conﬁrmed that the gradient is independent of the magnetic ﬁeld. Using Ramsey spectroscopy in each laice site, we independently checked that the spatial variation in the oscillation frequency does not result from a local detuning √ δ(x), e.g. due to a magnetic ﬁeld gradient, that changes the oscillation frequency via Ωeﬀ (x) = Ω2 + δ(x)2 . We determined an upper bound of 0.1% for the contribution of a detuning to the observed gradient in the oscillation frequency.¹ Including this spatial dependence of the linear coupling strength, the numerical simulations (solid lines in Figure 4.2) yield good agreement with the experimental observations. As we are working in the proximity of the Feshbach resonance atom loss is enhanced. Both at 9.03 G and 9.17 G, the 1/e-lifetime is 310 ms. Due to the ongoing interconversion of the two components, which is much faster than the loss rate, the spin-relaxation loss of the F = 2 component does not cause any asymmetries. ese loss eﬀects are included in the numerical simulations as depicted in Figure 4.4. Note that a similar amplitude reduction of Rabi oscillations in cold atoms was observed in [87, 88]. is eﬀect was aributed to diﬀerent trapping potentials for the two atomic states as they had different magnetic moments and the magnetic trapping resulted in diﬀerent gravitational sags. We ¹Later experiments revealed an inhomogeneous power distribution of the radio frequency radiation to be the cause of this gradient. It can be reduced by a diﬀerent spatial conﬁguration of the antennas. 43 4. Interacting dressed states B=9.03G B=9.17G atom number 4000 3000 2000 1000 00 100 200 300 time (ms) 400 500 Figure 4.4.: Simulated and measured atom loss. At both magnetic ﬁelds, the total atom number decays with a 1/e-lifetime of 310 ms. e loss due to the Feshbach resonance is included in the simulations (solid lines) as a three body process. have checked that a similar eﬀect cannot explain our observations. As we employ optical dipole potentials both components experience the same conﬁnement. External forces, e.g. caused by magnetic ﬁeld gradients acting diﬀerently on the two atomic components due to diﬀerent second order Zeeman shis could not explain the observations either. 4.2. Interacting dressed states In order to understand the reduced amplitude of Rabi oscillations on the miscible side of the Feshbach resonance, we describe our system in the basis of dressed states. In contrast to the discussion in subsection 2.2.1, we will include the eﬀects of interactions, which leads to a break down of the single spatial mode approximation. is approach provides an intuitive explanation of the observations as demixing dynamics of the dressed states. e properties of interacting dressed states were derived by Search et al. [33] and Jenkins et al. [89, 90] and these papers serve as the theoretical foundation for this section. e equations of motion Equation 2.3 can be rewrien in the dressed state basis, i.e. in the eigenstates of the linear coupling Hamiltonian when neglecting interactions. In this basis, the linear coupling terms vanish, which simpliﬁes the description of the dynamics because the population in each dressed state is conserved. When including interactions, this advantage does not hold any more as the interactions introduce two classes of additional terms: Some conserve the number of particles in the new basis states while others exchange population between them [33]. However, in the limit of strong coupling, when the energy diﬀerence between the dressed states dominates over the mean ﬁeld interactions, ℏΩ ≫ ngij , two-body interactions can be simpliﬁed and the non-population-conserving terms are suppressed.² is eﬀect can be illustrated by the following energetic consideration [33]: e conversion from one dressed state into the other requires the energy ℏΩ. If the inequality above is fulﬁlled, the mean ﬁeld energy is not suﬃcient to overcome this energy diﬀerence. erefore, Rabi oscillations can be expressed as a superposition of interacting dressed states, each of whose populations are conserved in the limit of strong linear coupling. e equations of motion ²e condition for the strong coupling regime was derived for the general case of arbitrary detuning. In our case of a resonant coupling, it relaxes to ℏΩ ≫ n2 (g11 − g22 ), n2 (g11 + g22 − 2g12 ). is condition is fulﬁlled for our experimental parameters as 520 Hz ≫ 25 Hz. 44 4.2. Interacting dressed states for the dressed states are equivalent to those for atomic states in the experiments of the previous chapter, where we discussed the evolution of an initial superposition of two components in the absence of linear coupling. For resonant coupling, the interactions between the dressed states are parametrized by the eﬀective scaering lengths a++ = a−− = 14 (a11 + a22 + 2a12 ) and a+− = 1 2 (a11 + a22 ) [33, 89], which take over the role of the atomic scaering lengths a11 , a22 , a12 in the equations of motion. In analogy to the (im-)miscibility condition for atomic condensates (Equation 2.9), the condition for stability against demixing reads a2+− < a++ a−− for the dressed states [89]. In terms of the atomic scaering lengths, this corresponds to a12 > 12 (a11 + a22 ). us, for equal intra-species scaering lengths a11 = a22 , which is a good approximation for 87 Rb, the miscibility conditions for atomic and dressed states are mutually exclusive – dressed states are immiscible where atomic states are miscible and vice versa. In this context, the slow dynamics in the envelope of the Rabi oscillations results from a reduced overlap of the dressed states due to spatial separation [90]. e complementary stability condition for atomic and dressed states is the reason for the counter-intuitive behavior of a reduced oscillation amplitude in the miscible regime of the atoms. In the numerical simulations, both the amplitude and the relative phase of the two atomic components are known, such that the density proﬁles of the corresponding dressed states can be calculated. e results are shown in Figure 4.5. e dressed state picture also provides an intuitive explanation for the role of the gradient κ in the linear coupling strength. As a++ = a−− the two dressed states are initially symmetric with respect to spatial separation and in a metastable state. Small perturbations, e.g. in the relative population of the dressed states, lead to symmetric demixing as shown in the le panel of Figure 4.5. However, the symmetry can be broken, for example by external state-dependent forces. As the energy shi of the dressed states is given by ± ℏΩ 2 (see subsection 2.2.1), the gradient leads to an eﬀective state-dependent potential V± (x) = V(x) ± ℏ2 κx. e minima of V± (x) are displaced by ℏκ Δx = ± 2mω 2 = ±11 nm with respect to the state-independent optical dipole potential V(x). is x small perturbation leads to biased antisymmetric demixing and thus a qualitative change in the dynamics, which demonstrates the criticality of the system to symmetry breaking. Furthermore, the persistent spatial overlap in the miscible regime of the dressed states shows that the bare eﬀect of the state dependent eﬀective potential is small. As Rabi oscillations result from interference of the dressed states, the oscillation amplitude decreases as the proﬁles of the dressed states deviate from a balanced superposition. is connection is essential for the reconstruction of spatial dressed state proﬁles from the experimental data and will be discussed in the next section. 4.2.1. Reconstruction of dressed states from spatially resolved Rabi oscillations In the experimentally observed Rabi oscillations only the density proﬁles of the atomic states are measured and their relative phase is not directly accessible. us, the spatial proﬁles of the dressed states cannot be directly calculated. For their reconstruction we need a deeper understanding of the connection between the amplitude and phase of Rabi oscillations and the underlying dressed states. It follows that the density proﬁles of the dressed states can be extracted from an analysis of the local amplitude A(x) and phase φ(x) of the oscillations. Let us expand Equation 4.1 to include a possible imbalance in the dressed states as well as a relative 45 4. Interacting dressed states |+ 0 | 100 100 200 200 time (ms) time (ms) 0 300 |+ | 300 400 400 500 -20 -10 0 10 20 -20 -10 0 10 20 x ( m) x ( m) 500 -20 -10 0 10 20 -20 -10 0 10 20 x ( m) x ( m) Figure 4.5.: Simulated dressed state density timetrace at B = 9.17 G, i.e. in the dressed state immiscible regime, without (le) and with (right) the gradient in the coupling strength. In numerical simulations, both the probability amplitude proﬁle and the relative phase of the atomic states are accessible allowing to directly deduce the density timetrace of the dressed states. Ignoring the gradient in the coupling strength leads to symmetric component separation (le). e symmetry breaking caused by the inhomogeneity in the coupling strength manifests itself in a faster, more pronounced and antisymmetric spatial separation of the dressed states (right). Data shown in the le panel corresponds to the dashed black line in Figure 4.2, the right panel to the solid one. phase φ |ψ(t)⟩ = cos α · e−iΩt/2 e−iφ/2 |+⟩ + sin α · e+iΩt/2 e+iφ/2 |−⟩ 1 = √ (cos α · e−i(Ωt+φ)/2 + sin α · e+i(Ωt+φ)/2 )|1⟩ 2 1 + √ (cos α · e−i(Ωt+φ)/2 − sin α · e+i(Ωt+φ)/2 )|2⟩ 2 (4.2) where the mixing angle α describes the relative population of the dressed states and the use of sin () and cos () ensures normalization. e corresponding imbalance z of the atomic states is given by z= N2 − N1 |⟨2|ψ(t)⟩|2 − |⟨1|ψ(t)⟩|2 = = −2 sin α · cos α · cos (Ωt + φ) N1 + N2 |⟨1|ψ(t)⟩|2 + |⟨2|ψ(t)⟩|2 (4.3) us, the local phase φ(x) of the Rabi oscillations is equivalent to the relative phase of the dressed states, while a spatial change in the amplitude A(x) = |2 · sin α(x) · cos α(x)| = | sin (2α(x))| (4.4) corresponds to a changing imbalance of the dressed states. e amplitude is maximal for an equal superposition of the dressed states, α = π/4, and is reduced as their imbalance increases. us, the decrease of the integrated, i.e. global oscillation amplitude shown in Figure 4.2 can be explained by a reduced overlap of the dressed states due to their spatial separation. Due to the π/2-periodicity of the amplitude A in α, the mixing angle deduced from the measured oscillation amplitude can not be unambiguously mapped onto the amplitudes of each dressed state. For example, the superpositions of dressed states |ψ 1 ⟩ = cos α|+⟩ + sin α|−⟩ and |ψ 2 ⟩ = sin α|+⟩ + 46 |2 x ( m) -20 -10 0 10 20 -20 -10 0 10 20 |1 x ( m) x ( m) x ( m) 4.2. Interacting dressed states 187 189 time (ms) 191 -20 -10 0 10 20 -20 -10 0 10 20 |2 |1 187 189 time (ms) 191 Figure 4.6.: Spatially resolved Rabi oscillations. At B = 9.03 G (le), the oscillations are in phase over the extent of the atomic cloud. In contrast at B = 9.17 G (right) the phase of the oscillations in the wings of the cloud is shied by π with respect to its center and the oscillation amplitude is reduced for intermediate distances from the cloud center. is information about the local amplitude and phase allows to reconstruct the proﬁles of the dressed states. cos α|−⟩ both yield the same amplitude in the corresponding Rabi oscillations. Consequently, only the diﬀerence of the amplitudes at each point in space can be deduced and it is not clear which amplitude is mapped onto which dressed state. Further information is required to reconstruct the proﬁles of the dressed states. For example, a phase jump of π in the oscillations corresponds to a node in the amplitude of one of the dressed states, i.e. it changes its sign with respect to the other state. Furthermore, we can assume the integrated populations of the dressed states to remain equal during the time evolution, as their reduction in amplitude due to atom loss is symmetric. Using these phase and symmetry arguments, the relative probability amplitudes of the dressed states can be reconstructed. eir absolute density proﬁle can be deduced by multiplying the square of the relative amplitudes with the measured atomic sum density proﬁle. As an example, we consider the Rabi oscillations at B = 9.03 G and B = 9.17 G around t ≈ 190 ms shown in Figure 4.6. A sinusoidal ﬁt to the time evolution in each spatial bin yields the local amplitude and phase of the oscillations. Using these ﬁt results, the proﬁles of the dressed states can be reconstructed using the procedure outlined above. Figure 4.7 illustrates the method along with the results. In order to assign the extracted dressed state density proﬁles to the |+⟩ or the |−⟩ state, we remember that the gradient κ causes an eﬀective potential for the dressed states. As the Rabi frequency increases for increasing x, the energy of the ground state |+⟩ is decreased. us, we identify the state on the right with |+⟩ and the le one with |−⟩. e dressed state proﬁles reconstructed from the Rabi oscillations at B = 9.17 G conﬁrm the model of antisymmetric demixing in the dressed basis. eir overlap is minimized, where the |+⟩ state occupies the right half of the trap and the |−⟩ state is on the le. At B = 9.03 G, the overlap of the inferred dressed state proﬁles is only slightly decreased. is demonstrates the miscibility of the dressed states and conﬁrms that the gradient in the linear coupling strength is only a small perturbation. e separation between the maxima of the dressed state densities is about 4 μm. is increase compared to the shi of the eﬀective potentials is due to the remaining repulsive interactions between the miscible dressed states. 47 4. Interacting dressed states 1.0 amplitude 0.8 0.6 0.4 0.2 0.0 2.0 phase ( ) 1.5 1.0 0.5 linear amplitude 0.0 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 |+ | -15 -10 -5 0 5 10 15 x ( m) -15 -10 -5 0 5 10 15 x ( m) linear density |+ | Figure 4.7.: Reconstruction of the dressed state proﬁles from the local amplitude (top row) and phase (second row) of the Rabi oscillations at B = 9.03 G (le column) and B = 9.17 G (right column) shown in Figure 4.6. On the le, neither amplitude nor phase of the oscillations vary in space. On the right, the oscillations in the center of the cloud are out of phase with respect to its edges. is phase jump is accompanied by a reduction in oscillation amplitude. e increased noise of the ﬁt results at the edges of the atomic cloud is due to the reduced atom density. Using Equation 4.4 the relative probability amplitudes of the dressed states are deduced at each pixel (third row), but cannot be assigned to the dressed states (i.e. it is not clear how to connect the points in order to yield the dressed state proﬁles). Using the fact that a phase jump of π (vertical lines) corresponds to a node in the amplitude of one of the dressed states and assuming equal populations of the two components, the normalized density proﬁles of the dressed states (boom row) are inferred. For comparison, the dashed lines show the corresponding data of the numerical simulations, where the dressed state proﬁles in the boom row have been calculated directly from the amplitudes and relative phase of the atomic wave functions. 48 4.2. Interacting dressed states |+ | -15-10 -5 0 5 10 15 x ( m) -15-10 -5 0 5 10 15 x ( m) time (ms) 100 150 200 Figure 4.8.: Timetrace of the dressed state density proﬁle reconstructed from the observed Rabi oscillations. e demixing dynamics of an initially overlapping superposition of dressed states is in good agreement with numerical simulations shown in Figure 4.5. is reconstruction procedure can be repeated at diﬀerent evolution times in order to obtain the full timetrace of the demixing dynamics of an equal superposition of interacting dressed states. e results are illustrated in Figure 4.8 and are in good agreement with the simulations shown in Figure 4.5. Note that the reconstruction of the dressed states relies on ﬁing a sine to the local oscillations for the extraction of amplitude and phase and thus requires coherent oscillations. is requirement is not necessary for analyzing the dynamics in the envelope of the oscillations shown in Figure 4.2. ere, only the ’single shot amplitude’ is relevant, i.e. the maximum observed amplitude per oscillation cycle. is makes the reconstruction of the dressed states unreliable for evolution times > 230 ms. In summary, we have prepared a superposition of interacting dressed states, realized by Rabi oscillations in an interacting Bose-Einstein condensate. e reduced oscillation amplitude could be aributed to phase separation of the dressed states, whose miscibility criterion is opposite to that of the atomic states. In particular, the intra-species scaering lengths of the dressed states are equal resulting in an intrinsically symmetric system ideal for the study of criticality and symmetry breaking. In our experiments, the symmetry is broken by an inhomogeneous Rabi frequency, which results in a qualitative change in the demixing dynamics and illustrates the criticality as revealed by comparison with numerical simulations. e spatial proﬁles of the dressed states were reconstructed by a local analysis of the amplitude and phase of the Rabi oscillations. 49 5. A miscible-immiscible phase transition In the previous chapter we have presented the concept of interacting dressed states, which result from the interplay between atomic interactions and a linear coupling in a two-component BoseEinstein condensate. We have analyzed Rabi oscillations between atomic states as a superposition of interacting dressed states and found that their miscibility condition is opposite to the atomic states. We begin this chapter by reporting on the experimental generation of single dressed states via a novel non-adiabatic preparation scheme. Dressed states are an equal superposition of the atomic states and are stationary if the amplitude of the linear coupling exceeds the energy scale of the atomic interactions. eir stationarity implies that the two atomic states are eﬀectively miscible even if their scaering parameters in the absence of the linear coupling are immiscible. e linear coupling strength acts as a control parameter for this theoretically predicted miscible-immiscible phase transition [17, 18]. We extend the discussion beyond the strong coupling limit of the dressed states and observe the phase transition from miscible to immiscible. We characterize the phase transition by the linear response of the system to sudden quenches to the proximity of the critical point. We observe a power law scaling in the characteristic length scales on both sides of the phase transition in agreement with mean ﬁeld predictions. e dynamic range for the scaling measurements is expanded by employing eﬀectively negative coupling strengths. 5.1. Non-adiabatic generation of dressed states e concept of dressed states has been introduced in subsection 2.2.1 and the previous chapter discussed eﬀective interactions in a superposition of dressed states. Single dressed states have been generated in quantum gases [87, 91] following an adiabatic preparation scheme [31] as follows: All atoms are initially prepared in a single atomic hyperﬁne state, which in the limit of large detuning is an eigenstate of the linear coupling Hamiltonian. e linear coupling ﬁeld is switched on with a low amplitude and far detuned from the atomic transition. e amplitude is subsequently increased and the detuning is decreased slowly such that the atomic state adiabatically follows. When the frequency of the coupling ﬁeld reaches resonance with the atomic transition a dressed state is prepared. For a large amplitude of the coupling ﬁeld dressed states are very stable against external perturbations, e.g. magnetic ﬁeld ﬂuctuations. For example it has been demonstrated that a microwave dressing increases the coherence times in trapped ions by more than two orders of magnitude while still allowing for fast quantum logic with the ions [92]. is experiment employed a similar adiabatic preparation scheme via an incomplete stimulated Raman adiabatic passage and has been proposed as a route to improve coherence times for quantum computing [92]. ese adiabatic methods for the generation of dressed states require a controlled change of the detuning from the atomic transition. e duration of the state preparation is given by multiple Rabi periods. In this section we present a novel non-adiabatic scheme for the generation of dressed states which does not require a sweep in the detuning but only a sudden change in the phase of the linear coupling [22]. For transitions involving radio-frequency radiation this method is straightforward 51 5. A miscible-immiscible phase transition to implement as arbitrary waveforms can be generated for radiation in the Megahertz range. e time scale for the preparation is given by a quarter of a Rabi period and is signiﬁcantly shorter than the adiabatic scheme. is is advantageous in the presence of increased atom loss. In our experiments the duration of the π/2-pulse ranges from tens to hundreds of microseconds depending on the employed transitions. 5.1.1. Experimental sequence e experimental sequence begins with a Bose-Einstein condensate of atoms in a single substate of the F = 1 manifold. A resonant π/2-pulse creates an equal superposition of two hyperﬁne components |1⟩ and |2⟩ corresponding to a coherent spin state. Aer the pulse we switch the phase of the coupling ﬁeld by Δφ = π/2 relative to the ﬁrst pulse within 1 μs. is phase shi aligns the rotation axis of the linear coupling to be parallel with the atomic state on the Bloch sphere. As discussed in subsection 2.2.1 the eﬀective spin is stationary under the action of the linear coupling Hamiltonian and a |+⟩ dressed ground state is prepared. Similarly a phase shi of Δφ = 3π/2 aligns the rotation axis to be antiparallel to the atomic state and corresponds to the generation of an excited |−⟩ dressed state. In the following, we refer to a linear coupling whose rotation axis is aligned with the pseudo-spin of the atomic states as a dressing ﬁeld. e dressed state picture is valid if the energy of the linear coupling exceeds the energy scale of the interactions and can be applied if Ω ≫ Ωc = −ngs as deﬁned in subsection 2.4.1. In the context of the internal Josephson junction discussed in subsection 2.2.2 this scheme for the generation of dressed states is equivalent to zero-amplitude plasma and π-oscillations and the limit of strong coupling corresponds to being deep in the Rabi regime [24]. 5.1.2. Experimental results We have experimentally implemented the experimental sequence sketched above and realize dressed states in a Bose-Einstein condensate of 3500 atoms trapped in the charger (see subsection 3.1.1 for an introduction to the experimental system). In order to probe the eﬀect of atomic interactions in the strong linear coupling regime we prepare dressed states both on the miscible (B = 9.17G) and the immiscible (B = 9.05G) side of the Feshbach resonance. e Rabi frequency for the initial π/2pulse as well as the subsequent dressing is Ω ≈ 2π × 600 Hz, which is much larger than the critical coupling strength of Ωc ≈ 2π × 30 Hz. In the single spatial mode approximation a dressed state is represented by a stationary pseudo-spin vector on the Bloch sphere. Both the population imbalance of the two states and their relative phase are constant. In order to verify the generation of a dressed state we test for these two characteristics. If the eﬀect of interactions can be neglected also the extended atomic cloud in the charger can be described by a single spatial mode and no spatial structure in the imbalance or relative phase of the condensates is expected. e corresponding experimental observations are summarized in Figure 5.1. e upper rows show the density timetraces of the two atomic states revealing equal density in the two components for the experimental time scale of 500 ms. Furthermore no spatial structure can be detected within the cloud, also at the immiscible side of the Feshbach resonance. is demonstrates that a dressing ﬁeld prevents demixing dynamics because the system can be described as a stationary dressed state as predicted in [17]. e observed change in the atomic density proﬁles is due to atom loss. In order to detect the relative phase of the two components we apply an additional π/2-pulse before detection, which translates the relative atomic phase into a population imbalance. In particular the |+⟩ state is mapped onto the atomic state |2⟩ and |−⟩ onto |1⟩. is sequence corresponds to a Ramsey interferometer [93] with a dressing ﬁeld during the interrogation time. e results shown 52 5.2. Dynamics beyond the strong coupling limit |1 0 |2 100 100 200 200 time (ms) time (ms) 0 300 400 |2 | |+ 300 400 | 0 |+ 100 100 200 200 time (ms) time (ms) 0 |1 300 400 300 400 -20 -10 0 10 20 x ( m) -20 -10 0 10 20 x ( m) -20 -10 0 10 20 x ( m) -20 -10 0 10 20 x ( m) Figure 5.1.: Density timetrace of dressed states prepared at 9.05 G (le, immiscible) and 9.17 G (right, miscible). We generate a |+⟩ dressed state employing the non-adiabatic preparation scheme. e overlap of the density proﬁles of the atomic states (top row) remains high both in the miscible and the immiscible regime and conﬁrms that the dressing ﬁeld stabilizes the system. An additional phase shied π/2-pulse before detection allows to measure the relative phase of the two components and maps the dressed state |+⟩ onto the atomic state |2⟩ and |−⟩ onto |1⟩. e small population of the |−⟩ state demonstrates the high ﬁdelity of the preparation scheme of more than 96%. in the boom row of Figure 5.1 demonstrate a constant phase over the time evolution, i.e. phase coherence over hundreds of milliseconds. e population in |1⟩ aer the last π/2-pulse can be used to estimate the ﬁdelity of the preparation scheme yielding > 96%. In conclusion we have experimentally generated dressed states using a non-adiabatic scheme. e preparation time is given by the duration of the initial π/2-pulse. We have shown that in the limit of strong coupling dressed states are stationary and that demixing dynamics of immiscible states is prevented. 5.2. Dynamics beyond the strong coupling limit In order to study the eﬀect of interactions and the resulting break-down of the dressed state picture we measure the time evolution in the presence of weak dressing ﬁelds. is allows us to observe the transition from the stationary dressed states presented in the previous section to interactiondominated evolution for weak linear couplings strengths. e experiments are performed in the charger at the ’magic ﬁeld’ of B = 3.23 G. e employed atomic states are |1⟩ = |1, −1⟩ and 53 5. A miscible-immiscible phase transition |2⟩ = |2, +1⟩ and their the scaering parameters are close to the miscible-immiscible threshold resulting in a critical coupling strength of Ωc ≈ 0. e interaction properties of these states and their dynamics in the absence of a linear coupling ﬁeld was discussed in subsection 3.2.1. We employ the same experimental sequence as for the generation of dressed states, but reduce the amplitude of the microwave radiation aer the initial π/2-pulse. e change in amplitude is performed along with the phase shi within 1 μs. e strength of the coupling ﬁeld is reduced to various values ranging from Ω = 2π × 0 Hz to 2π × 45 Hz in order to study the dependence of the dynamics in the two components on the linear coupling strength. e power reduction in the coupling ﬁelds also causes a change in the light shi and their frequencies have to be adjusted accordingly to maintain resonance as discussed in subsection 3.1.2. Note that the power of the initial π/2-pulse is not reduced in order to minimize nonlinearity eﬀects resulting e.g. in a phase error of the prepared superposition state. We ﬁrst discuss the dynamics for a phase shi of Δφ = π/2 aer the initial coupling pulse. e observed dynamics for various coupling strengths is illustrated in the le panel of Figure 5.2. In the absence of a linear coupling we observe the same potential separation dynamics as presented in subsection 3.2.1. Atoms of component 2 are pushed to the edges of the trap and component 1 gathers in the trap center before the atomic clouds ’oscillate back’ towards the spatial superposition state. As the dressing amplitude Ω is increased the depth of the density modulation in the trap center is reduced and the corresponding oscillation frequency increased. For Ω = 2π × 45 Hz (boom row of the Figure 5.2) the oscillations are barely visible. e density proﬁles of the atomic clouds are stationary and can be described as a |+⟩ dressed state. As discussed in subsection 3.2.1 the temporal average of the density proﬁles can be used as an estimate for the ground state of the system. e right panel of Figure 5.2 compares the mean of the observed density proﬁles of each component to the numerically computed stationary states employing Newton’s method and the nonpolynomial nonlinear Schrödinger equation including the linear coupling (see subsection B.1.2 for details on the numerical methods). e transition to the dressed state for increasing Ω is apparent in the increasing overlap of the two components and the agreement of the experimental observations with the numerical calculations is good. We similarly examine the break-down of the |−⟩ dressed state by changing the phase of the radio frequency radiation by Δφ = 3π/2 aer the initial coupling pulse. e resulting dynamics is summarized in Figure 5.3. e striking diﬀerence to the previous measurements occurs at small coupling strengths of about Ω = 2π × 7 Hz, where the frequency of the oscillation dynamics in the two components is reduced. In addition the role of the two components is reversed as component 1 is predominantly found in the center of the trap. However, as Ω is further in increased the overlap of the components is maximized again and the |−⟩ dressed state is observed. ese features are well visible in the temporal mean of the density proﬁles shown in the right panel of Figure 5.3. As the |−⟩ state is not the ground state the numerically calculated proﬁles correspond to lowest energy stationary state with a relative phase of π between the atomic state and the linear coupling (corresponding to a negative value of Ω as discussed in section 2.5). While stationary, this state is not the ground state of the system. e measured mean density proﬁles agree well with the computed stationary states. 5.3. Linear response to quenches near the critical point While the charger is well suited for experiments estimating stationary states via potential separation dynamics, the small longitudinal extent of the atomic cloud prevents a detailed study of the miscible immiscible transition. A quantitative characterization of the phase transition requires an experimental system much larger than the characteristic length scales to be measured. In addition, 54 5.3. Linear response to quenches near the critical point 0 |1 |2 linear density time (ms) 50 100 150 0 =0.0Hz 100 150 =6.3Hz 100 150 0 =7.7Hz 100 150 0 =12.8Hz 100 150 =45.0Hz -20 0 20 -20 0 20 x ( m) x ( m) 2 6 4 2 6 4 2 6 4 2 0 8 linear density time (ms) 50 4 0 8 linear density time (ms) 50 |1 |2 0 8 linear density time (ms) 50 6 |1 |2 0 8 linear density time (ms) 50 0 8 6 4 2 0 -20-10 0 10 20 x ( m) -20-10 0 10 20 x ( m) Figure 5.2.: Spatial dynamics of the two components for various amplitudes Ω of the linear coupling ﬁeld. e phase is changed by Δφ = π/2 aer the initial π/2-pulse. (le panel) Component separation is suppressed by the dressing ﬁeld while the frequency of the density oscillations in the center of the trap increases. At Ω = 45 Hz (boom row) no component separation is observed and a |+⟩ dressed state is generated. (right panel) e temporal mean of the measured density proﬁles is in good agreement with numerically simulated ground state proﬁles in the presence of the linear coupling. 55 5. A miscible-immiscible phase transition 0 |1 |2 linear density time (ms) 50 100 150 0 =0.0Hz 100 150 =6.3Hz 100 150 0 =7.7Hz 100 150 0 =12.8Hz 100 150 =45.0Hz -20 0 20 -20 0 20 x ( m) x ( m) 2 6 4 2 6 4 2 6 4 2 0 8 linear density time (ms) 50 4 0 8 linear density time (ms) 50 |1 |2 0 8 linear density time (ms) 50 6 |1 |2 0 8 linear density time (ms) 50 0 8 6 4 2 0 -20-10 0 10 20 x ( m) -20-10 0 10 20 x ( m) Figure 5.3.: Spatial dynamics of the two components for various amplitudes of the linear coupling ﬁeld. e experimental sequence is the same as for Figure 5.2 but Δφ = 3π/2. is corresponds to the generation of the |−⟩ dressed state for strong coupling (boom row). (le panel) Compared to the absence of a coupling ﬁeld the roles of the two states are inverted for small Ω as atoms in state |2⟩ are pushed to the edges of the trap while component |1⟩ occupies the trap center. (right) is behavior is apparent in the temporally averaged density proﬁles in agreement with numerically computed stationary states of the system shown on the right. e reduced modulation depth of the experimental proﬁles may be aributed to spin relaxation loss in state |2⟩. 56 5.3. Linear response to quenches near the critical point the value of the critical coupling strength marking the transition point depends on the atom density, which in inhomogeneous systems leads to a spatial dependence of the critical point. In order to reduce this eﬀect an atomic cloud with a close-to-homogeneous density distribution is required. As discussed in subsection 2.4.4 and subsection 3.2.2 the waveguide fulﬁlls these requirements. We concluded in chapter 3 that magnetic ﬁelds of 9.07 G to 9.08 G near the Feshbach resonance are the optimal working points for the study of the phase transition as they oﬀer a good compromise of small characteristic length scales, fast growth rates, a suﬃcient life time and atom loss which is symmetric in the two components. e experiments presented in the remainder of this chapter are conducted at these magnetic ﬁelds. e interaction parameters along with our typical linear atom densities of about 230 atoms/μm result a critical coupling strength of Ωc ≈ 2π × 50 Hz at 9.07 G and Ωc ≈ 2π × 70 Hz at 9.08 G.¹ We characterize the phase transition by measuring the response of the system to sudden quenches of the control parameter Ω. We parametrize the distance to the critical point by the dimensionless quantity ε = (Ω − Ωc )/Ωc which vanishes at the critical point. ε > 0 (ε < 0) denotes the miscible (immiscible) side of the phase transition. e quenches start deep in the miscible regime ε ≫ 1 and end in the proximity of the critical point. e experimental observable in our measurements is the autocorrelation function of the spin proﬁle (or equivalently the corresponding Fourier spectrum) at a hold time t aer the quench (see subsection 3.2.3 for a summary of the analysis methods). On the miscible side of the phase transition, Ω > Ωc , we expect correlations in the ﬂuctuations on top of a ﬂat spin proﬁle, i.e. the autocorrelation function decays to zero on a characteristic length scale ξ c . e value of ξ c depends on the distance to the critical point as ξ c (ε) = ξ 0 /|ε ν | with a mean ﬁeld scaling exponent of ν = 1/2. ξ 0 is proportional to the spin healing length ξ s as discussed in chapter 2. e spin correlations need a ﬁnite time to develop aer the quench. is relaxation time is given by τ = ℏ/g(ε), where g denotes the energy gap in the Bogoliubov spectrum as discussed in subsection 2.4.3. On the immiscible side of the phase transition, Ω < Ωc , the initial equal superposition of the two components is unstable. Small ﬂuctuations of the spin proﬁle act as a seed for excitation modes and are ampliﬁed in a range of wave vectors resulting in the formation of spin paerns. e characteristic length scale in the immiscible regime is given by the unstable mode with the √ smallest wavelength kc . As we have seen in subsection 2.4.3 it diverges near the critical point like Ωc − Ω. Using the same notation as in the miscible regime the characteristic √ length shows the same behavior ξ c (ε) = ξ 0 /|ε ν | with ν = 1/2 and ξ 0 = 2π/kc (Ω = 0) = 2πξ s . e characteristic time scale in the immiscible regime is the maximum growth rate of the unstable modes. Close to the critical point at Ωc /2 < Ω < Ωc it is given by the inverse modulus of the gap, but stays constant for Ω < Ωc /2. us, we expect temporal scaling only close to the critical point. e mean ﬁeld spatial and temporal scaling properties are summarized in Figure 5.4. e experimental sequence for the implementation of the sudden quenches is identical to the one employed in the previous section for the experiments in the charger. We prepare an equal superposition of the two atomic clouds by an initial π/2-pulse and subsequently change the phase of the radio frequency radiation by Δφ = π/2. e amplitude of the coupling ﬁeld is simultaneously reduced to a value near the critical coupling Ωc . As we will now argue, the initial conﬁguration aer the π/2-pulse corresponds to the equilibrium state for Ω ≫ Ωc . As the pulse duration of τ π/2 ≈ 700 μs is shorter than the typical time scale for the dynamics of the atomic cloud and the spin correlations, the density proﬁle of each component aer the pulse is the same as the spatial proﬁle of the |1⟩ component before the pulse (only reduced in amplitude by a factor 2). us, the density ﬂuctuations in the two components are in-phase, and the ¹For simplicity the values for Ω are given in some graphs without the preceding factor of 2π. e unit Hertz always denotes temporal frequencies and not angular frequencies in order to avoid ambiguity. 57 5. A miscible-immiscible phase transition -1 0 1 immiscible 2 miscible gap (Hz) / 0 6-2 4 2 0 100 50 0-50 0 50 (Hz) 100 150 Figure 5.4.: Scaling of the characteristic length and time scales at the miscible-immiscible phase transition in the mean ﬁeld approximation. e length scale on the miscible side of the phase transition is given by the correlation length of the spin ﬂuctuations and on the immiscible side by the wavelength of the unstable modes in the excitation spectrum. Both diverge at the critical point with a scaling exponent of ν = 1/2. e relaxation time scales as the inverse of the energy gap in the excitation spectrum. Close to Ωc the corresponding scaling exponent is νz = 1/2. e arrows illustrate the experimental sequence of quenches from Ω ≫ Ωc to the proximity of the critical point. corresponding spin proﬁle is ﬂat. Only uncorrelated shot noise due to the spliing process is present. As the π/2-pulse creates a coherent state, the shot noise ﬂuctuations have no preferred length scale and all spin excitation modes are equally populated with a small amplitude. is conﬁguration corresponds to a dressed state, i.e. the eigenstate in the system in the limit of large coupling Ω ≫ Ωc . In the eﬀective magnetic ﬁeld picture of the dressing ﬁeld presented in subsection 2.4.3, this conﬁguration corresponds to a strong magnetic ﬁeld. Each spin is aligned to the axis of the ﬁeld and all spin ﬂuctuations are suppressed, which leads to a vanishing of the associated correlation length. We have tested these assumptions about the state aer the π/2-pulse by dressing the atoms with a strong linear coupling ﬁeld of Ω = 2π × 340 Hz aer the pulse. No dynamics in the density proﬁles or in the spin correlations could be detected within the spatial resolution of the imaging system. 5.3.1. Scaling on the miscible side of the transition We begin the discussion of the experimental observations with sudden quenches to the miscible side of the phase transition Ω ≳ Ωc as illustrated by the arrows in Figure 5.4. We quench to several values of ε and measure the spin correlations of the system at diﬀerent hold times t aer the quench. As our detection method is destructive, we cannot detect the dynamics of the correlations in a single experimental realization but have to repeat the experiment under the same conditions and vary the hold time aer the quench. We average over 10 − 20 experimental realizations for each hold time t and distance from the critical point ε. When experimentally probing the dynamics in the spin correlations it is important to keep the distance from the critical point ε constant. Due to the proximity to the Feshbach resonance, atom loss is strongly enhanced and the 1/e-lifetime of the atomic cloud for these measurements is approximately 30 ms. Many parameters of the system depend on the linear atom density, in particular Ωc ∝ n. In order to compensate for the changing atom density, we adjust the amplitude of the linear coupling Ω(t) ∝ n(t), such that the ε remains constant. is procedure compensates for the ﬁrst order eﬀects of the atom loss, but some other eﬀects remain, e.g. a change in the spin healing length √ ξ s ∝ 1/ n. Note that whenever Ω is used as a control parameter in the remainder of this chapter, for example as an axis label in graphs, we refer to its initial value Ω(t = 0). 58 5.3. Linear response to quenches near the critical point 0 1 = 0.31 4 Jz (x) Jz (x) time (ms) 2 6 8 0 1 0 1 10 120 2 4 6 8 10 12 14 x-x' ( m) 0 0 t = 2ms t = 5ms t = 12ms 5 10 x-x' ( m) 15 Figure 5.5.: Dynamics of spin correlations aer a sudden quench to an exemplary value of ε on the miscible side of the phase transition. (right panel) e autocorrelation function of the spin proﬁle (black lines) decays to zero with in a few micrometer. e decay length increases with the hold time t aer the quench. An exponential ﬁt (gray lines) is employed to extract the characteristic length scale. (le) Time evolution of the correlation functions in false color illustrating the growth of the correlation length. e linear atom density not only changes temporally, but also spatially as the atomic cloud is conﬁned in an elongated harmonic trap. e spatial proﬁle n(x) results in a spatial variation of the previously mentioned parameters such as the critical coupling strength, Ωc = Ωc (x). In order to reduce these inhomogeneity eﬀects we restrict the analysis to the central part of the atomic cloud where the gradient of the density proﬁle is minimal. We choose a region with a width of about 150 μm around the trap center. e density at the edge of this analysis region is reduced by about 15% compared to the peak density, such that the value of Ωc is smeared by about this amount in the experiments. An example of the observed spin correlation functions and their change in time is shown in Figure 5.5. e normalized correlation functions decay to zero within a few micrometers. Immediately aer the quench the length scale of the decay is minimal and is determined by the detection limit of our imaging system given by the resolution of imaging optics. e size of one pixel of the CCD chip corresponds to 420 nm in the plane of the atomic cloud and is thus smaller than the resolution of the imaging optics of 1.1 μm in the Rayleigh criterion [75]. In order to reduce the eﬀect of photon shot noise we bin the spin proﬁle over three neighboring pixels before calculating the correlation functions. e decay length of the correlations functions increases during the time evolution aer the quench. In order to quantify the characteristic length scale of the system, we associate the correlation length ξ with the 1/e-decay length of an exponential ﬁt to the autocorrelation functions. Using this method we extract the correlation length from the averaged correlation functions at each value of ε and hold time t. e time evolution of the correlation length for three values of ε is shown in Figure 5.6. Aer the quench we observe a linear growth in ξ with a slope independent of the value of ε Aer a characteristic time depending on ε, this growth rate is reduced and the correlation length saturates. We identify the saturation value of the correlation length with the characteristic length scale of the spin ﬂuctuations ξ c at a given ε. Note that the numerical value of ξ c is not necessarily identical to the equilibrium spin correlation length. However, in the linear response regime it is expected to show the same scaling behavior with ε. e resulting scaling of ξ c with ε is summarized in Figure 5.7. ξ c increases when approaching the critical point Ωc and agrees well with a power law ﬁt. e power law scaling becomes apparent by 59 5. A miscible-immiscible phase transition 2.5 =0.17 =0.31 =0.55 ( m) 2.0 1.5 1.0 0.50 2 4 6 8 time (ms) 10 12 Figure 5.6.: Temporal evolution of the correlation length. e correlation length is extracted using an exponential ﬁt to the autocorrelation function of the spin proﬁle at each value of ε and t. Its time evolution is ploed for diﬀerent distances ε from the critical point. Aer an initial linear growth with a rate independent of ε the growth slows down and the value of the correlation length saturates at ξ c . is saturation value increases when approaching the critical point. 4 cr=69.2 ±6.8Hz =-0.51 ±0.26 c ( m) 3 3 2 2 1 1 70 80 90 100 (Hz) 110 120 0.1 1.0 Figure 5.7.: Scaling of the correlation length with the distance from the critical point. e correlation length aer an evolution time of t = 12 ms (black circles) increases with a power law when approaching the critical point. A ﬁt (solid line) yields an exponent of ν = 0.51 ± 0.26, which agrees with the mean ﬁeld prediction of ν = 1/2. A critical coupling strength of Ωc = 69.2 ± 6.8 Hz is found in agreement with the prediction of Ωc = 70.5 Hz based on independent measurements of the atom density and the interspecies scaering length (gray shaded area). e right panel shows the same data on a double-logarithmic scale, where linearity indicates a power law scaling. All errors are given as two standard deviations. the linearity on a double logarithmic scale. Both the scaling exponent ν = 0.51 ± 0.26 and the value of the critical coupling Ωc = 69.2±6.8 Hz are extracted from a power law ﬁt to the experimental data and in good agreement with the theoretical mean ﬁeld prediction of ν = 1/2. ese values result ( )−ν c from a ﬁt ξ c (Ω) = ξ 0 Ω−Ω with three free parameters ξ 0 , ν and Ωc . Reducing the number Ωc of ﬁt parameters by ﬁxing them to the theoretically predicted values for our system parameters yields smaller uncertainties. Seing ν = 1/2 results in a ﬁt value of Ωc = 69.5 ± 1.5 Hz, while using the theoretical prediction Ωc = 70.5 Hz yields ν = 0.47 ± 0.05. e ﬁt with three free parameters yields ξ 0 = 0.79 ± 0.10 μm, which is smaller than the prediction for the equilibrium 60 5.3. Linear response to quenches near the critical point 3.0 relaxation time (ms) ( m) 2.5 2.0 1.5 1.0 0.50 2 4 6 8 time (ms) 10 12 20 16 12 8 /gap z=1/2 experimental data 4 0.1 1.0 Figure 5.8.: Scaling of the relaxation time aer a sudden quench. (le panel) We associate the relaxation time of the system with the position of the kink in the time evolution of the correlation length, which separates the fast initial growth from the subsequent saturation. e kink is found by linearly ﬁing the evolution of the correlation length (blue line) where it deviates from the initial growth (gray shaded area) and calculating the intersection point with a ﬁt to the initial dynamics (black line). is procedure is repeated for each ε. e results are summarized in the right panel and agree well with the prediction of ℏ/gap (red line). e error bars correspond to 2 s.d and are obtained by propagation of the uncertainties of the linear ﬁts. e power law scaling prediction of νz = 1/2 (blue dashed line) is only valid close to the critical point. All theory curves are without free parameters. √ value of ξ 0 = ξ s / 2 = 0.90 μm. All errors are given as two standard deviations corresponding to a 95% conﬁdence interval. A characteristic time scale in the temporal evolution of the correlation length is the position of the kink where the growth of ξ slows down and saturates. We associate the position of the kink with the relaxation time τ of the system. It is determined from the intersection point of linear ﬁts to the evolution of the correlation length before and aer the kink as illustrated in the le panel of Figure 5.8. As the initial growth rate of the correlation length is the same for all values of ε, we model it by a linear ﬁt to the evolution of the smallest value of ε, which remains linear within the measurement time. We associate the evolution aer the kink with hold times where the measured correlation lengths ξ deviate from the ﬁed slope of the initial dynamics. e evolution of the correlation lengths in this regime is also ﬁed linearly and we identify the intersection point of the two linear ﬁts with the relaxation time τ(ε). is procedure is repeated for all values of ε where at least three data points can be used for each of the linear ﬁts. e extracted values for τ are shown in the right panel of Figure 5.8. e predicted mean ﬁeld temporal scaling exponent is given by νz = 1/2 with the dynamical exponent z = 1. However, this prediction is valid only close to the critical point as the square root scaling of the energy gap √ g = ℏ Ω(Ω − Ωc ) changes to a linear behavior as the distance from the critical point increases. e experimental data agrees with the prediction based on the energy gap and deviates from the power law scaling with increasing ε as indicated in Figure 5.8. However, the power law scaling remains within the uncertainty of the experimental data. Another method to visualize the spatial and temporal power law scaling of the spin correlations is to directly compare the observed correlation functions at diﬀerent hold times and ε. is method does not require a reduction of the full information contained in the correlation function to a single number ξ c . When rescaling space and time with the predicted power law, the correlation functions 61 5. A miscible-immiscible phase transition Jz (x) Jz (x) 1.0 =0.10 =0.17 =0.23 =0.31 =0.38 =0.55 =0.74 0.5 0.0 0 5 10 x-x' ( m) 15 0 1 2 3 4 5 6 (x-x')/ -1/2 ( m) 7 8 Figure 5.9.: Rescaling of the correlation functions. e measured spin correlation functions at the relaxation time t = τ(ε) are ploed in the le panel. e overlap of the correlation functions increases signiﬁcantly aer rescaling the spatial coordinate with the predicted power law behavior, which indicates a collapse of the correlation functions to a single curve. However, deviations at large displacements remain signiﬁcant. are predicted to collapse to a single function [94]. For each value of ε we plot the correlation function at the relaxation time t = τ(ε) in the le panel of Figure 5.9. In a second step we normalize the spatial coordinate of the correlation functions by the predicted power law scaling ε −1/2 . e resulting correlation functions collapse to a common curve for small displacements, but a deviation remains on larger length scales. In order to understand the experimental observations we² model the spin correlations using the mean ﬁeld Bogoliubov excitation spectra. For the coherent state immediately aer the initial π/2pulse we assume a population of the excited modes given by the equipartition theorem at high temperatures. e spectrum of the spin ﬂuctuations in k space at thermal equilibrium conﬁguration can be calculated by employing the equipartition theorem. e convolution theorem allows to obtain the spin correlation function in real space from the mode populations in momentum space via Fourier transformation. e resulting correlation function is an exponential with a characteristic √ length scale ∝ Ω − Ωc , which conﬁrms the mean ﬁeld power law scaling of the correlation length near the critical point. A quench in ε projects the initial equal population of the spin excitation modes onto the new basis given by the excitation spectrum ω ε (k) for the conﬁguration aer the quench. In the subsequent time evolution each mode k evolves with its characteristic frequency ω ε (k) and the spin excitations de-phase. e resulting correlation functions and the evolution of the correlation length determined from an exponential ﬁt are compared to the experimental observations in Figure 5.10. e initial growth of the correlation length is well modeled by the mean ﬁeld theory. However, the theory predicts oscillations in the correlation length that are not observed in the experiment. is deviation might be caused by the population of transverse excitation modes as the atomic cloud is not strictly one-dimensional, eﬀects of atom loss or beyond-mean ﬁeld corrections. ²e credit for these ideas and calculations goes to Isabelle Bouchoule. 62 5.3. Linear response to quenches near the critical point 2.5 =0.17 =0.31 =0.55 Jz (x) Jz (x) ( m) 2.0 1.5 1.0 0.50 2 4 6 8 time (ms) 10 12 1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 0.0 0 t = 2ms t = 5ms t = 8ms 5 10 15 x-x' ( m) Figure 5.10.: Comparison of the observed correlation functions and time evolution of the correlation length to the mean ﬁeld prediction. (right) e observed correlation functions (blue circles) are compared to the mean ﬁeld prediction (solid lines) for ε = 0.17 at various hold times. (le) e measured and predicted time evolution of the correlation length is compared. e predicted oscillations in the correlation length are not observed experimentally. 5.3.2. Scaling in immiscible regime Aer discussing quenches to the miscible side of the phase transition, we now focus on quenches through the critical point into the immiscible regime at Ω < Ωc ³. As in the miscible case, the mean ﬁeld linear response of the system to a quench is given by the Bogoliubov spectrum of the spin excitations. However, for Ω < Ωc the dynamics is dominated by modulational instabilities of the spatial superposition of the two components. e growth rate of the unstable modes is given by the imaginary part of the excitation spectrum. Before presenting the experimental observations on the emergence of spin domains we recall the discussion of the imaginary part of the excitation spectrum in the context of Figure 2.11. We have seen that the region of unstable modes is shied towards larger wave vectors with a squareroot scaling as Ω is decreased. is square-root corresponds to the value of the critical exponent ν = 1/2. A quantity that is accessible experimentally is the typical size of the domains in the emerging spin paern as it is given by the strongest mode in the Fourier spectrum of the spin proﬁle. e domain size corresponds to the most unstable mode in the Bogoliubov spectrum. e discussion of the instability diagram revealed that the most unstable mode deviates from k = 0 as Ω < Ωc /2. us, the domain size will diverge at Ωc /2 instead of Ωc . e quantity diverging at Ωc is the largest unstable mode in the system. It can be associated to the smallest structure present in the observed spin proﬁles, which is diﬃcult to determine due to detection noise. However, as the width of the instability region in Ω is constant, the domain size is governed by the same scaling exponent as the largest unstable mode. We perform quenches into the immiscible regime by employing the same experimental sequence as in the previous section. However, the amplitude of the linear coupling is reduced below the critical value aer the initial π/2-pulse. In order to increase the dynamical range for the scaling of the characteristic length we also quench to negative values of Ω. Negative values of Ω are realized by ³Note that we consider only sudden quenches through the critical point, such that dynamical eﬀects depending on the quench rate through the critical point can be ignored. e resulting spin dynamics is determined only from the state before the quench and the system parameters aer the quench [95]. 63 5. A miscible-immiscible phase transition = -2.21 0 20 Jz (x) Jz (x) time (ms) 10 30 40 1 0 1 0 50 0 5 15 10 x-x' ( m) 20 25 t = 4ms t = 26ms (Jz (x)) 1 t = 4ms 2 1 0 t = 26ms 2 1 0 t = 48ms 2 1 00.0 0.1 0.2 0.3 wave vector k (1/ m) F 0 t = 48ms 0 5 10 15 20 25 x-x' ( m) Figure 5.11.: Dynamics of spin correlations aer a sudden quench to an exemplary value of ε on the immiscible side of the phase transition. (middle panel) e autocorrelation function of the spin proﬁle shows oscillatory behavior with a decaying amplitude. e oscillations period corresponds to the typical size of the spin domains and marks a characteristic length scale. e right panel shows the corresponding Fourier spectra. (le panel) Time evolution of the correlation functions in false color showing the growth of the oscillation amplitude with time. changing the phase of the radio frequency radiation by π compared to the conﬁguration at positive Ω. is corresponds to a phase shi Δφ = 3π/2 instead of π/2 aer the initial superposition pulse as discussed in section 2.5. ese experiments are performed at B = 9.07 G, where we expect a critical coupling strength of Ωc ≈ 50 Hz. Exemplary correlation functions and Fourier spectra along with their dependence on the hold time aer the quench are depicted in Figure 5.11. e unstable excitation modes cause the growth of a periodic spin paern, which manifests itself in oscillations in the autocorrelation function of the spin proﬁle. e periodicity of the oscillations corresponds to the typical size of the spin domains, while the decay of the envelope of the oscillations is a measure for the spectral width of the excitations. ese quantities can be equivalently determined from the Fourier spectra of the spin proﬁles, which are a direct measure for the contribution of an excitation mode with a given wave vector. We quantify the dynamics in the domain size by averaging the Fourier spectra at each value of ε and hold time t over about 10 to 20 experimental realizations and determining the wavelength corresponding to the mode with the maximum amplitude in the spectrum. e time evolution of the typical domain size is shown in Figure 5.12 for several values of ε. At short evolution times t < 15 ms the modulation depth of the spin domains is below 8% and the domain size cannot be detected reliably due to photon shot noise in the imaging process. At longer times the observed domain sizes are almost constant for each value of ε. e remaining dri towards larger wavelengths can be explained by atom loss. e small dri in the domain size with time is not self-evident as the excitation spectrum can be modiﬁed by the back-action of the unstable modes onto the excitation spectrum (see the discussion of secondaries in subsection 3.2.4). However, we do not observe any evidence for such eﬀects. e typical size of the domains increases when approaching the critical point. For a quantitative study of the domain size scaling with the distance from the critical point we average the Fourier spectra for hold times of 20 ms < t < 30 ms and subsequently determine the domain size from the peak in the spectrum. We choose this time range as it oﬀers a good signalto-noise ratio at a small modulation depth of 10 . . . 25% ensuring the validity of the linear response 64 8 7 6 5 4 3 2 10 =-1.11 =-1.43 =-2.21 =-4.42 10 20 30 40 time (ms) 50 60 0.5 0.4 modulation depth domain size ( m) 5.3. Linear response to quenches near the critical point 0.3 0.2 0.1 0.00 10 20 30 40 time (ms) 50 60 Figure 5.12.: Time evolution of the typical size of the emerging domains at diﬀerent distances ε from the critical point (le) and the modulation depth of the spin domains (right). At short evolution times < 15 ms, the modulation in the spin proﬁle is too small (< 8%) to reliably detect the domain size due to the presence of detection noise. At longer times the increase of the domain size when approaching the critical point is visible. At each ε the typical domain size grows slowly in time, which is compatible with the eﬀects of atom loss (dashed lines). e relative change due to atom loss is independent of ε, such that it does not aﬀect the scaling behavior (see Figure 5.19). approximation⁴. e resulting scaling of the domain size with ε is ploed in Figure 5.13. e linearity of the experimental data points in a double logarithmic plot reveals a power-law scaling in a dynamic range of more than one order of magnitude in ε. A ﬁt yields a scaling exponent of ν = 0.49 ± 0.07, where the error corresponds to two standard deviations. is result is in good agreement with the predicted mean ﬁeld exponent of ν = 1/2. e divergence point for the domain size scaling is Ωc /2, which can be extracted from a power law ﬁt to the experimental data. e resulting value of Ωc /2 = 24.9 ± 4.5 Hz is in agreement with the theoretical prediction for our system parameters of Ωc = 52 ± 3 Hz. e uncertainty in the predictions is given by the accuracy of the determination of the inter-species scaering length a12 . e length scale diverging at Ωc is the largest unstable wave vector in the excitation spectrum, which is diﬃcult to obtain from the Fourier spectra due to detection noise. We determine an approximate value as follows: As a ﬁrst step we determine the full width at half maximum (FWHM) of the peak in the Fourier spectrum. en we associate the largest unstable wave vector with the mode that is shied by one FWHM from the position of the peak. e results are summarized in Figure 5.14 conﬁrm the previously obtained values for the scaling exponent ν and the critical coupling Ωc . e value Ωc = 43.9±15.0 Hz obtained from a power law ﬁt may be oﬀset from the true value due to the diﬃculty of determining an estimate for smallest-wavelength unstable mode, but agrees with both the previously obtained value and the theoretical prediction within the experimental uncertainty. e large range of validity of the power law scaling is remarkable and is due to the fact that we probe the mean ﬁeld scaling behavior. We will now discuss limitations of the dynamic range in our experiments. In the long wavelength limit our measurement range is limited by the size of the atomic cloud. However, the limitation is not directly given by the size of the domains approaching the system size, but rather the inhomogeneity of the atomic density leading to a spatial dependence of the ⁴However, the scaling of the domain size is constant in time as we will discuss towards the end of this chapter (see Figure 5.19). 65 5. A miscible-immiscible phase transition domain size ( m) 8 10 8 cr/2=24.2 ±3.9Hz =-0.49 ±0.07 6 6 4 4 2 010 2 8 6 4 - 2 0 10 - 1 Figure 5.13.: Scaling of the domain size with the distance from the critical point. e data (black circles) is well captured by a power law ﬁt (solid line), also shown by the linearity in a double logarithmic plot (right panel). e extracted scaling exponent agrees with the predicted value of ν = 1/2. e domain size diverges at Ωc /2, whose value extracted from the ﬁt agrees with the theoretical prediction. FFT edge ( m) 10 8 10 cr=43.9 ±15.0Hz 8 =-0.46 ±0.11 6 6 4 4 2 06 5 4 3 - 2 1 0 10 - 1 Figure 5.14.: Scaling of the smallest-wavelength unstable mode with the distance from the critical point. e previous results for the scaling exponent and the critical coupling strength are conﬁrmed. mean ﬁeld shi (see subsection 3.1.2). For positive Ω the local detuning due to the mean ﬁeld shi is such that population is transferred to state |2⟩ in the center of the trap, where the density is highest. e smaller atom density at the wings of the trap causes population transfer to state |1⟩. is conﬁguration acts as a seed for the similarly shaped three-domain ground state conﬁguration (see Figure 2.4). For small values of Ω, where the sensitivity to detuning is large, the amplitude of this seed is suﬃcient such that the three-domain ground state dominates the emerging domain paern. Numerical integration of the equations of motion conﬁrms that this mode grows faster than other instabilities. In the small wavelength limit the dynamic range is limited by the optical resolution of our detection system, which is about 1.1 μm in the Rayleigh criterion [75, 26]. As the domain size approaches the imaging resolution the observed modulation depth is reduced due to the convolution of the spin paern with the point spread function of the imaging system. As the domain size decreases, this eﬀect eventually reduces the amplitude of the observed paern below the detection noise threshold and prevents a reliable determination of the domain size. 66 5.3. Linear response to quenches near the critical point relative FFT amplitude 1.0 0.8 0.6 0.4 = 0.8 m = 1.1 m = 1.3 m 0.2 0.00 5 15 10 wavelength ( m) 20 Figure 5.15.: Inﬂuence of a ﬁnite imaging resolution on the detected modulation depth of sinusoidal structures with diﬀerent wavelengths. e amplitude of the peak in the Fourier spectra of the spin proﬁles at diﬀerent ε decreases for decreasing domain size. e relative change in amplitude (black circles) is compared to the relative modulation depth of a sinusoidal paern aer convolution with a Gaussian of width σ, which models the point spread function the detection optics (solid lines). e good agreement for σ = 1.1 μm conﬁrms the imaging resolution as a limitation for the detectable dynamic range of the domain sizes. is eﬀect can be modeled numerically by convolving a sinusoidal oscillation with a Gaussian proﬁle of width σ as an estimate of the point spread function of the imaging optics. e relative amplitude reduction due to convolution can be compared with the observed relative amplitude of the peaks in the Fourier spectra for diﬀerent domain sizes. e results are summarized in Figure 5.15 and conﬁrm that the decrease in detected modulation depth is compatible with σ = 1.1 μm. Within the dynamic range accessible in our experiment, the scaling of the domain size with ε is well described by a power-law with exponent ν = 1/2. is scaling behavior can also be visualized by comparing directly the autocorrelation functions of the spin proﬁles, where the typical domain size is encoded in the wavelength of the oscillations. Due to the diﬀerent domain sizes at diﬀerent ε, the autocorrelation look very diﬀerent as shown in the le panel of Figure 5.16. Aer rescaling the spatial coordinate by ε −1/2 the correlation functions collapse to a single oscillation frequency demonstrating the power law scaling. e remaining deviation in the oscillation amplitude stems from the previously discussed ﬁnite imaging resolution. e characteristic time scale in the immiscible regime is the maximum growth rate of the unstable modes. As discussed in section 2.5 it is is constant for Ω < Ωc /2. Only for Ωc /2 < Ω < Ωc the maximum growth rate decreases when approaching Ωc . Here, the growth rate of the k = 0 mode is given by the modulus of the energy gap |g(Ω)|, which approaches zero like a square-root corresponding to the temporal critical scaling exponent νz = 1/2. us the temporal scaling is imprinted on the growth rate of the k = 0 mode, which is diﬃcult to access experimentally due to the previously discussed limitations of in the long wavelength regime of the experiments. is problem can be circumvented by using a tight conﬁnement such that the condensates can be described in the single spatial mode approximation and √ the gap in the excitation spectrum corresponds the frequency of π-oscillations given by ω π = Ω(Ω − Ωc ) (see subsection 2.2.2). However, the square root scaling is only valid close to Ωc and its observation requires the measurement of oscillation frequencies ≪ Ωc . For our experimental parameters with Ωc ≈ 50 . . . 70 Hz this corresponds to oscillation frequencies < 10 . . . 20 Hz, which are in the order of the lifetime of atomic cloud of 90 . . . 30 ms. As Ωc and thus the π-oscillation frequency are further reduced by atom loss it is diﬃcult to conduct a 67 5. A miscible-immiscible phase transition 1.0 =-1.01 =-1.22 =-1.31 =-1.43 =-1.61 =-1.86 =-2.22 =-2.72 =-3.43 =-4.43 =-5.84 =-7.84 Jz (x) Jz (x) 0.5 0.0 -0.5 0 5 10 15 20 x-x' ( m) 25 30 0 5 10 15 20 25 30 35 40 (x-x')/ -1/2 ( m) Figure 5.16.: Rescaled correlation functions. e autocorrelation functions of the spin density proﬁle are ploed for diﬀerent distances from the critical point ε and equal hold times of t = 35 ms (le panel). e power law scaling of the domain size becomes apparent by the collapse of the correlation function to a single curve when rescaling the spatial coordinate with ε −1/2 (right panel). e reduced oscillation amplitude for large ε is caused by the ﬁnite resolution of our imaging optics (see Figure 5.15). quantitative analysis of the temporal power law scaling in our experimental system. On the miscible side of the Feshbach resonance the two components are naturally miscible and there is no phase transition in the ground state of the system. However, the system can be destabilized by a negative linear coupling as discussed in section 2.5. e system becomes unstable for Ω < 0, but the most unstable mode diﬀers from k = 0 only for Ω < Ωc /2. Here, Ωc is deﬁned as previously but takes on negative values for miscible scaering parameters. us, the detected domain size will have a divergence point at Ωc /2 as in the previously discussed immiscible case. We perform quench experiments at B = 9.11 G and observe the instabilities induced by the linear coupling ﬁeld. e scaling of the resulting domain sizes is summarized in Figure 5.17. A power-law ﬁt to the experimental data yields Ωc /2 = −13 ± 17 Hz, which is compatible with the expected value of Ωc = −45 ± 3 Hz. e absolute size of the domains is expected to depend only on the distance to the divergence point (Ω − Ωc /2)−1/2 and the atomic density n−1/2 and should be independent of the scaering parameters of the system and the value of Ωc . We compare the domain sizes to the previous measurements in the immiscible regime at B = 9.07 G in the right panel of Figure 5.17. e good agreement of the measurements performed on the two sides of the Feshbach resonance demonstrates the independence of the domain size from the scaering parameters. For this comparison the 9.11 G data was corrected for a total atom density being about 10% smaller than for the 9.07 G measurements. e experimental observations on scaling of the unstable excitation modes can be summarized in an instability diagram similar to the theory prediction given in Figure 2.11. e measured Fourier spectra at diﬀerent values of the coupling strength and both for immiscible and miscible scaering parameters are shown in Figure 5.18. 5.4. Summary, outlook and applications In conclusion we have realized a miscible-immiscible transition by applying a linear dressing ﬁeld to interacting binary Bose-Einstein condensates. We have experimentally conﬁrmed the concept of negative linear coupling strengths, which in the strong coupling limit corresponds to the generation of the excited |−⟩ dressed state. In this regime a miscible system can be destabilized and the two 68 5.4. Summary, outlook and applications domain size ( m) 8 10 8 =0.49 ±0.07 =0.50 ±0.30 6 6 4 4 2 2 0 -250 -200 -150 -100 -50 (Hz) 0 50 100 -( - c ) (Hz) 10 100 100 50 50 0 0 -50 -50 (Hz) (Hz) Figure 5.17.: Scaling of the domain size with the distance from the critical point. Both on the immiscible (B = 9.07 G, black circles) and the miscible (B = 9.11 G, gray squares) side of the Feshbach resonance, the scaling of the domain size is well modeled by a power law ﬁt. e power law scaling is also shown by the linearity in a double logarithmic plot (right panel). e extracted scaling exponents agree with the predicted value of ν = 1/2. e domain size diverges at Ωc /2, whose values extracted from the ﬁt agree with the theoretical predictions. e absolute value of the domain size for a given distance (Ω − Ωc /2) from the critical point is independent of the details of the atomic interactions as demonstrated by the small diﬀerence in domain sizes at B = 9.07 G and 9.11 G (right panel). e 9.11 G data was corrected for a 10% smaller total atom density. -100 -100 -150 -150 -200 -200 -2500.0 0.1 0.2 0.3 wave vector k (1/ m) 0.4 -2500.0 0.1 0.2 0.3 wave vector k (1/ m) 0.4 Figure 5.18.: Experimentally observed spectrum of unstable modes versus the linear coupling strength Ω for immiscible (B = 9.07 G, le) and miscible (B = 9.11 G, right) scattering parameters. e normalized Fourier spectra of the spin proﬁle aer quenches to diﬀerent values of Ω are shown in false color. e square root scaling of the instability spectrum with Ω is clearly visible. Dashed lines correspond to the power law ﬁts to the experimental data and solid lines denote the Bogoliubov prediction. components phase separate. Similarly, a linear coupling of positive value (resembling the preparation of the |+⟩ dressed ground state) tunes an immiscible system to miscible when Ω exceeds a critical value Ωc . We have investigated the scaling of the characteristic length scales on both sides of the phase transition and found power law scaling in good agreement with theoretical mean ﬁeld predictions. We 69 5. A miscible-immiscible phase transition 4 3 2 6 t=7ms =0.34 ±0.21 4 1 2 ( m) 6 t=19ms =0.53 ±0.11 4 4 3 2 t=6ms =0.32 ±0.12 c domain size ( m) t=1ms =-0.04 ±0.06 1 2 4 3 2 6 t=27ms =0.49 ±0.03 4 1 2 4 3 2 6 t=39ms =0.51 ±0.04 4 t=12ms =0.51 ±0.06 1 2 10 t=10ms =0.42 ±0.14 - 1 0.1 1.0 Figure 5.19.: Emergence of the scaling law in the immiscible (le) and miscible regime (right). In the immiscible regime spin domains with a characteristic wavelength grow. is length scale of this spin modulation is immediately imprinted on the spin proﬁle, but can be measured only when the modulation amplitude exceeds detection noise. In the miscible regime the correlation length of the initially uncorrelated system grows with a ﬁnite speed. us the emerging scaling law is visible ﬁrst far from the critical point and then propagates towards ε = 0. In contrast to the domain size, the scaling law of the correlation length cannot be detected immediately but requires a growth of the correlations whose duration depends on the value of ε. Solid lines are power law ﬁts to the experimental data (black circles). Dashed lines are the ﬁts taken from Figure 5.13 and Figure 5.7. will now contrast the emergence of the scaling law on the two sides of the transition as summarized in Figure 5.19. On the miscible side the spin correlations grow from an initially uncorrelated state aer the quench. As the spin correlations propagate with a ﬁnite speed, the duration of the growth of the correlation length needs a ﬁnite time depending on its saturation value. us, the ﬁnal correlation length ξ c is proportional to the relaxation time τ as indicated by the dynamical scaling exponent z = 1 and the scaling in the correlation length can only be observed aer waiting suﬃciently long for the correlations to develop. 70 5.4. Summary, outlook and applications In contrast the emerging spin domains on the immiscible side are the result of a modulational instability. Initial ﬂuctuations of the spin proﬁle, for example due to shot noise of the prepared coherent state, grow exponentially in a range of wavelengths and the fastest growing mode will dominate the emerging domain paern. us the characteristic length scale, the size of the domains, is determined as soon as the modulation depth of the spin paern exceeds the initial spin ﬂuctuations. e scaling law can be observed experimentally as soon as the amplitude of the domains exceeds detection noise. e scaling law does not change in time subsequently. As previously explained the temporal scaling can be observed only in the single spatial mode approximation and close to the critical point where Ωc /2 < Ω < Ωc , which is not accessible in our experiments. e herewith characterized phase transition oﬀers prospects for further studies. e control parameter Ω can be changed on short timescales faster than microseconds and almost arbitrarily when generating the RF radiation with an arbitrary waveform generator. For example, this system has been proposed [18, 56] as a realization for testing the predictions of the Kibble-Zurek mechanism [96, 97] by ramping the amplitude of the linear coupling through the critical point with various speeds. e feasibility of such schemes with our experimental system and the current status of experiments will be discussed in the next chapter. 71 6. Dynamics of phase transitions and the Kibble-Zurek mechanism When studying phase transitions, one usually focuses on the equilibrium scalings of physical properties near the critical point of a homogeneous system [55, 1]. In the previous chapter, we have seen that the corresponding spatial and dynamical scaling exponents ν and z can also be deduced from the linear response of the system to sudden quenches of the control parameter. e Kibble-Zurek mechanism describes the dynamics of phase transitions, for example of a system that is initially in equilibrium and subsequently ramped through a critical point with a ﬁnite speed. is chapter explains the general ideas behind the Kibble-Zurek mechanism and a possible implementation of its quantum version employing the miscible-immiscible phase transition discussed in the previous chapter. We will summarize the status of the experiments and the feasibility and challenges for the observation of the Kibble-Zurek mechanism in our experimental setup. e dynamics of phase transitions was ﬁrst discussed in the context of the early universe. Aer the successful uniﬁcation of weak and electromagnetic interactions involving a spontaneously broken gauge symmetry, Tom Kibble suggested in 1976 that the universe has undergone a series of phase transitions in its early phase aer the big bang as the temperature decreased [98, 96]. He argued that space-like separated points independently choose their symmetry broken states. us causality, i.e. the speed of light, limits the size of the domains over which the choice of the symmetry broken state is propagated. Kibble discussed the resulting topological structures in the early universe that may have survived and triggered the formation of structures still visible today (see [99] for a review from Kibble’s perspective). Kibble’s ideas have been generalized and applied to laboratory-scale systems by Wojciech Zurek in 1985. He associated cosmological strings with vortex lines in superﬂuids that spontaneously form for example aer a pressure quench in 4 He [97]. e spontaneous formation of vortices was predicted to cause a measurable macroscopic rotation of superﬂuid Helium in an annular geometry aer the transition to superﬂuidity. Zurek calculated the dependence of the mean velocity on the rate at which the temperature is quenched through the normal to superﬂuid transition. A general model was developed that predicts the number of topological defects aer ﬁnite-time quenches through the critical point of a second order thermodynamic phase transition. e scaling of the number of defects, or equivalently their size, with the speed of the quench is the main prediction of the Kibble-Zurek mechanism. is concept has been applied to systems ranging from low temperature Bose-Einstein condensates to the energy scales of grand uniﬁcation in cosmology. A review by Zurek is found in [100]. e Kibble-Zurek mechanism has been tested both numerically and experimentally for second order classical phase transitions in various experimental system ranging from liquid crystals [101, 102] to annular Josephson junctions [103, 104] and nonlinear optical systems [105]. e 4 He experiment originally proposed by Zurek was initially reported to be conducted successfully [106], but the results had to be retracted [107] concluding that the vortex production in the experiment is at least two orders of magnitude lower than predicted by the Kibble-Zurek mechanism. However, similar experiments were successfully performed in 3 He [108, 109]. In the context of cold atomic gases the formation of defects aer reaching degeneracy has been predicted. e production of vortices 73 6. Dynamics of phase transitions and the Kibble-Zurek mechanism was observed experimentally [7, 110] in three-dimensional atomic clouds. e formation of solitons in one-dimensional systems has been predicted [111, 112] and is currently under experimental investigation. e ideas of the Kibble-Zurek mechanism have been expanded from second order classical phase transitions to quantum phase transitions and were found to be compatible with the Landau-Zener model [113]. e prediction of the Kibble-Zurek mechanism agrees with analytical solutions of the quantum Ising model [114, 115]. Symmetry breaking at quantum phase transitions has been observed in antiferromagnetic spinor gases [8, 9], at the Dicke phase transition [11] and in immiscible binary condensates (see [60] and this thesis). Another well characterized quantum phase transition is the Mo-to-superﬂuid transition [4]. However, observations of the dynamical scaling of the number of defects with the quench rate through the critical point are rare. Experiments quenching from the superﬂuid to the Mo regime [5] have studied the microscopic atom number statistics across the transition. e inverse quench has been analyzed and found indications of Kibble-Zurek scaling in the ”amount of excitation produced during the quench” [116]. Very recently the formation of defects in linear ion chains has been studied [117] and their dependence on the quench rate in the longitudinal trap frequency was observed to follow the predicted power law behavior [118]. 6.1. Proposed implementation in binary Bose-Einstein condensates An experimental scheme recently proposed by Sabbatini et al. allows for the direct observation of scaling in the number of topological defects in binary Bose-Einstein condensates [18, 56]. e employed phase transition is the miscible-immiscible transition discussed in the previous chapter. e proposal is supported by numerical simulations, also including ﬁnite-size eﬀects and inhomogeneities. In this section, we discuss this proposal in the context of our experimental parameters and present the basic ideas behind the Kibble-Zurek mechanism along the way. e proposed experimental system consists of a one-dimensional two-component Bose-Einstein condensate with immiscible interaction parameters. As discussed in the previous chapter the system can be stabilized and become miscible in the presence of a linear coupling ﬁeld whose strength exceeds a critical value Ωc . When preparing the system in the miscible regime Ω > Ωc and subsequently ramping Ω below the critical value, the system becomes unstable and spin domains form. In a homogeneous system translational symmetry is spontaneously broken and the position of the domains depends on spin ﬂuctuations acting as a seed for domain formation. us, the absolute position of the domains is random. e dependence of the number of defects, i.e. spin domains, on the speed of a quench from Ωi > Ωc to Ωf < Ωc can be derived from the following adiabaticity argument. 6.1.1. A criterion for adiabatic quenches A general property of second order phase transitions is a divergence of both the relaxation time τ and the equilibrium value of the correlation length ξ at the critical point. We derived the mean ﬁeld scaling of the correlation length in chapter 2√ and found a power law scaling with a critical exponent ν = 1/2, i.e. ξ(ε) = ξ 0 /|ε ν |. Here ξ 0 = ξ s / 2 is given by the spin healing length ξ s of the system and ε = (Ω − Ωc )/Ωc denotes the distance from the critical point. e relaxation time τ is a measure for the duration the system needs to adjust to external changes of the control parameter, in our case Ω. It is given by the inverse of the energy gap in the excitation spectrum τ = ℏ/g. Consistently with a diverging relaxation time the gap must vanish at the critical point. 74 6.1. Proposed implementation in binary Bose-Einstein condensates √ In our experimental system the gap is given by g = ℏ Ω(Ω − Ωc ) (see subsection 2.4.3), which shows a square root behavior in the vicinity of Ωc . us, the associated mean ﬁeld critical exponent is νz = 1/2 and because ν = 1/2 one can conclude that z = 1. e relaxation time close to the critical point scales as τ = τ 0 /|ε|νz with τ 0 = 1/Ωc . It is important to note that this scaling only holds close to the critical point and for Ω ≫ Ωc the gap scales linearly. is change in the scaling exponent was indicated in the experimental observations shown in Figure 5.8. For the following review of Zurek’s adiabaticity criterion we will assume a power law scaling in the relaxation time and ignore the deviations far from the critical point. However, this eﬀect will be included whenever numerical predictions about the scaling behavior in our experimental system are made. e experimental sequence initially discussed by Zurek is the following. e system is prepared in equilibrium on the symmetric side of the phase transition at a given distance from the critical point ε i . Subsequently the control parameter ε is linearly changed to a ﬁnal value ε f on the symmetrybroken side of the transition. We denote the duration of the quench with τ Q . When the relaxation time τ(ε) is smaller than the transition time associated with the quench, the system can adjust adiabatically to the external change and remains in the equilibrium conﬁguration corresponding to the instantaneous value of ε. If the time evolution was always adiabatic through the critical point, the system would smoothly change from the miscible ground state to the one of the equivalent immiscible ground state. However, as τ diverges at the critical point the time evolution eventually becomes non-adiabatic and excitations enter the system. e time when the evolution becomes non-adiabatic is called the freezing time t̂. Mathematically, the freezing time can be determined by equating the relaxation time τ with the transition time τ t given by the relative change of the energy gap due to the quench [28] τ(t̂) = τ t (t̂) ⇔ ℏ g(t̂) = g(t̂) ġ(t̂) ⇔ g(t̂)2 = ℏġ(t̂) (6.1) where the dot indicates the derivative with respect to time. Note that this argument is equivalent to the condition that the equilibrium correlation length ξ grows as fast as the speed of sound given ξ0 by ξ 0 and τ 0 , i.e. dξ dt (ε̂) = 2πτ 0 . In analogy to the freezing time t̂ we deﬁne the correlation length ξ̂ = ξ(t̂) and the distance from the critical point ε̂ = ε(t̂) at the instant when the evolution of the system becomes non-adiabatic. is criterion for adiabaticity is illustrated in Figure 6.1. Equation 6.1 can be solved for the freezing time t̂, which allows to calculate the relevant experimental parameters ε̂ and ξ̂. Assuming a linear ramp from Ωi = 2Ωc to Ωf = 0 with the duration τ Q we obtain 1 νz t̂ = τ 01+νz τ Q1+νz 1 νz ε̂ = ε(t̂) = −1 + 2t̂/τ Q = −1 + 2τ 01+νz τ Q1+νz ν ξ̂ = ξ(t̂) = ξ 0 (τ Q /τ 0 ) 1+νz −1 (6.2) e central assumption behind the Kibble-Zurek mechanism is that the correlations are ’frozen’ as the evolution becomes non-adiabatic. In particular, the characteristic length scale of the system remains ξ̂ throughout the evolution in the non-adiabatic regime. As the system crosses the critical point symmetry is broken and it becomes unstable. e frozen conﬁguration of the correlations acts as a seed for the growth of defects. Causality requires that topological defects are chosen independently in space-like separated areas, i.e. information only propagates with the ﬁnite speed of sound of the system. As ξ̂ is the only characteristic length scale in the system, it determines the typical size of the emerging defects. us, Equation 6.2 predicts a power law scaling of the domain 75 14 12 10 8 6 4 2 0-1 t,1 ( Q =50ms) t,2 ( Q =5ms) time (ms) time (ms) 6. Dynamics of phase transitions and the Kibble-Zurek mechanism ( 1, 1) ( 2, 2) 0 1 14 12 10 8 6 4 2 0-1 t,1 ( Q =50ms) t,2 ( Q =5ms) ( 1, 1) ( 2, 2) 0 1 Figure 6.1.: Illustration of the adiabaticity criterion given by Equation 6.1 for quenches through the critical point. e le panel depicts the situation for the power law scaling in the relaxation time τ ∝ |ε|−νz , which is valid only close to the critical point. e data shown in the right panel is for τ ∝ g−1 . Adiabatic evolution is possible if the relaxation time τ (solid black line) is smaller than the transition time, which depends on the duration of the quench (blue and green solid lines). e freezing points ε̂ are depicted. ν size with the quench duration. e associated scaling exponent is 1+νz . For our experimental system with ν = 1/2 and z = 1 an exponent of 1/3 is predicted for the scaling of the domain size with the duration of a linear quench. e Kibble-Zurek mechanism only predicts a proportionality between ξ̂ and the resulting defect size. Numerical simulations show that the resulting domain structures are larger than ξ̂ by a factor on the order of 10, which depends on the details of the system [114, 111]. By comparing the numerically observed domain sizes [18] with the predictions for ξ̂ in the speciﬁc case of two linearly coupled atomic clouds we ﬁnd that the resulting domain size is larger than ξ̂ by a factor of 18.2. However, the details for an a priori determination of the value of this proportionality factor are not known. One aspect of this question is the range of unstable modes in the symmetry-broken regime. It determines which of the seeded excitation modes are unstable and lead to the formation of topological defects. In our experimental system only modes with a wave vector k < kc are unstable and the value of kc depends on ε (see subsection 2.4.3). When the evolution of the system becomes adiabatic again aer the crossing of√the critical point at ε(t) = −ε̂ [114], the wavelength corresponding to kc is given by λ c (−ε̂) = 2πξ s |ε̂|−1/2 = 2π ξ̂, which is larger than the frozen correlation length ξ̂. us, modes with a wavelength given by the frozen correlation length ξ̂ will not grow when adiabatic evolution is regained. Aer the quench has ended at ε f = −1 (Ωf = 0), the cut-oﬀ is given by kc = 1/ξ s and smaller structures can not be observed in the emerging domain paern. us, the validity of the Kibble-Zurek scaling argument can only hold if the proportionality factor between ξ̂ and the corresponding defect size is large enough. In summary, the Kibble-Zurek mechanism predicts the scaling of number of defects (or equivalently their size) with the quench rate through a symmetry breaking phase transition. e argument is based on the transition from adiabatic to non-adiabatic evolution and the assumption that the system’s correlations are frozen as soon as the evolution becomes non-adiabatic. us, the model requires a Hamiltonian with an energy gap in its excitation spectrum which vanishes at the critical point. If the excitation spectrum of the Hamiltonian does not have an energy gap, long wavelength excitations can enter the system no maer how slow the change in the control parameters is and 76 6.1. Proposed implementation in binary Bose-Einstein condensates adiabatic evolution is not possible.¹ 6.1.2. Numerical simulations and inhomogeneity eﬀects Finite time quenches through the miscible-immiscible transition of linearly coupled Bose-Einstein condensates have been simulated numerically by Jacopo Sabbatini et al. [18]. In a homogeneous system conﬁned in a ring trap they observed the predicted power-law scaling over two orders of magnitude with an exponent of 0.341±0.006 close to the predicted value of 1/3. Similar simulations were performed for the experimentally relevant case of an inhomogeneous one-dimensional atomic cloud conﬁned in a harmonic trapping potential. We will now summarize the insights gained from the numerical simulations [18, 56] on the eﬀects caused by inhomogeneities. e linear density of an atomic cloud conﬁned in a trapping potential is not constant spatially. Several system parameters depend on the atomic density, in particular the critical coupling Ωc ∝ n, √ but also the correlation length ξ 0 ∝ 1/ n and the formation time of spin domains τ f ∝ 1/n. us, the reduced density in the outer region of an inhomogeneous atomic cloud leads to a slower growth of larger domains compared to the trap center. According to simulations the largest eﬀect is the spatial dependence of Ωc (x) [18]. e higher density in the center of the trap leads to a larger value of Ωc . As the coupling strength is reduced during the ramp, the critical point is ﬁrst crossed in the trap center while the edges of the trap are still on the miscible side of the transition. e front of the phase transition subsequently propagates from the trap center to the outside regions with a lower atom density. is moving front leads to a faster decrease of the number of domains with longer quench times due to: 1. Suppression of domain formation. If the velocity of the moving front is smaller than the local speed of sound in the atomic cloud, information about the choice of the broken symmetry propagates along the front of the phase transition. Domains in diﬀerent regions are not formed independently but causally connected, which favors the same choice of broken symmetry in the newly unstable regions and thus suppresses domain formation. is eﬀect only aﬀects long quenches and becomes more dominant as τ Q increases leading to an increased scaling exponent. If the front moves faster that the speed of sound, the defect density is expected to the same as for a homogeneous transition [56]. As estimated in [119] this eﬀect does not aﬀect our experiments. It becomes relevant for ramp times τ Q > 200 ms, which is longer than the life time of the atomic cloud. A detailed discussion of this eﬀect in the context of soliton formation in one-dimensional condensates is found in [111]. 2. Increased annihilation of domains aer their formation. As the front of the transition moves from the trap center to the edges translational symmetry is broken and the domains have a preferred direction of movement. is causes a larger annihilation rate of the domains compared to the homogeneous phase transition. [18] ese eﬀects lead to an increased scaling exponent of ≈ 0.47 in the simulation of the inhomogeneous system. e two aforementioned eﬀects have been studied independently by further ’numerical experiments’ [56]. e suppression of domain formation due to causality can be simulated in a homogeneous atomic cloud with a position-dependent coupling strength Ω(x), which resembles the spatial dependence Ωc (x) of the inhomogeneous system. A similarly increased scaling exponent of 0.497 ± 0.015 was found. e increased domain annihilation was conﬁrmed by simulating the time evolution of seeded domains in an inhomogeneous system. ¹is is the case when implementing the miscible-immiscible phase transition in Bose-Einstein condensates by changing the atomic interactions, for example with a Feshbach resonance. As discussed in subsection 2.4.3 the excitation spectrum is gapless in the absence of a linear coupling ﬁeld. 77 6. Dynamics of phase transitions and the Kibble-Zurek mechanism Another consequence of the inhomogeneous density distribution and the resulting variation in Ωc is a position-dependent quench time. is eﬀective quench time τ Q (x) is decreased in the wings of the trap [56], which partly compensates the previously discussed eﬀects and decreases the domain size in the outer regions of the atomic cloud. e inhomogeneity in the atomic density proﬁle not only causes a spatial dependence of the value of the critical coupling, but also introduces a local detuning of the coupling ﬁeld caused by the mean ﬁeld shi (see subsection 3.1.2). In our experiments we tune the frequency of the linear coupling ﬁeld such that the spatial average of the mean ﬁeld shi is compensated. us atoms in the center of the trap have a detuning of a diﬀerent sign than atoms in the outer regions of the trap. is detuning creates a corresponding local imbalance during the quench. As we will see in subsection 6.2.2 this eﬀect has important consequences for slow ramps through the critical point. e conﬁguration in our experiments is such that the density of |1⟩ atoms is increased in the wings of the trap while the center contains more |2⟩ atoms. is acts as a seed for the three-domain ground state of the atomic cloud presented in subsection 2.3.3. For slow ramps, this seed is large enough and the ground state conﬁguration dominates the resulting domain paern. e formation of smaller domains is suppressed. Numerical time integration of the equations of motion for our experimental parameters revealed that the three-domain conﬁguration grows faster than any other unstable mode. A numerical Bogoliubov - de Gennes analysis is diﬃcult as the asymmetry a11 ̸= a22 breaks the symmetry of the superposition state prepared by the initial π/2 pulse. Consequently, this ’background’ state is not stationary and the Bogoliubov - de Gennes analysis can not be applied (see subsection B.2.2). 6.1.3. Experimental feasibility In order to estimate the experimental feasibility of the Kibble-Zurek scheme in our system we calculate the requirements for adiabatic evolution and compare them to the lifetime of the atomic cloud. We assume the same parameters as for the phase transition experiments discussed in section 5.3, i.e. a critical coupling of Ωc ≈ 2π × 70 Hz and a 1/e lifetime of about 30 ms. Using the adiabaticity criterion Equation 6.1 we calculate the maximum rate of change of the control parameter that allows for adiabatic evolution of the system. Due to the divergence of the relaxation time this quench rate depends on ε (and equivalently Ω) as shown in the le panel of Figure 6.2. From this quench rate one can calculate the minimum duration of a ramp starting at Ω that permits adiabatic behavior. e evolution of the system for shorter ramps is never adiabatic. A further requirement on the ramps is that ε̂ < 1, because the system returns to adiabatic behavior at −ε̂ and the ramps stop at Ω = 0 corresponding to ε = −1. is restricts possible experiments to Ω̂ < 2Ωc = 2π × 140 Hz.² Along with the eﬀects of atom loss these requirements restrict the parameter range for experiments to ramps starting at Ωi = 2π × (110 . . . 140) Hz if the duration of the ramp must not exceed the 1/e-lifetime of the cloud. e largest possible range of ramp durations is τ Q = 15 . . . 30 ms at Ωi = 2π × 140 Hz. is small dynamic range does not allow for an experimental observation of a power law scaling in the domain size with the quench rate. Note that these requirements are a lower bound as the eﬀects of atom loss on adiabaticity criterion were neglected. A reduction in the atomic density decreases the speed of sound in the system and thus requires smaller quench rates and longer ramps. ²is restriction may be circumvented by extending the ramps to negative values of the coupling strengths as deﬁned in section 2.5. 78 6.2. Experimental results 100 minimal ramp duration (ms) ramp speed (Hz/ms) 25 20 15 10 5 0 60 80 100 120 140 160 180 200 (Hz) 80 60 40 20 0 60 80 100 120 140 160 180 200 (Hz) Figure 6.2.: Requirements for adiabatic ramps in our experimental system. (le) e maximum ramp speed allowing for adiabatic evolution is ploed versus Ω. e corresponding minimal duration of ramps starting at Ωi = Ω is shown in the right panel. Faster ramps are never adiabatic. 6.2. Experimental results As previously discussed the limited tunability of the critical coupling strength (and the corresponding time scales for ramps and the subsequent growth of the domains) in combination with the ﬁnite lifetime of the atomic cloud due to atom loss hamper the implementation of the Kibble-Zurek scheme in our experimental setup. However, one can study some aspects of the Kibble-Zurek mechanism using slow ramps towards the critical point or also fast non-adiabatic ramps through the transition. In this section we discuss experimental observations in the context of these modiﬁed experimental schemes. 6.2.1. Adiabatic and non-adiabatic ramps towards the critical point A central element of Zurek’s argument is the transition from adiabatic to non-adiabatic behavior when approaching the critical point with a ﬁnite speed. We probe this transition by ramping towards the critical point with various ramp speeds and observe the spin correlations during the ramp. e experimental system and the parameters are identical to the sudden quenches performed on the miscible side of the phase transition discussed in subsection 5.3.1. e experimental sequence is similar to the aforementioned experiments: We create an equal superposition of the two components employing a fast π/2-pulse, suddenly change the phase of the radio frequency ﬁeld by Δφ = π/2 and simultaneously reduce the amplitude of the ﬁeld to Ωi . In contrast to the previous quench experiments we now change the amplitude of the linear coupling during the subsequent evolution and linearly ramp towards the critical point with a ﬁxed slope Ω(t) = Ωi − dΩ dt t. Atom loss aﬀects relevant system parameters such as the value of Ωc and we compensate for these eﬀects by adjusting Ω(t) such that the slope in ε is constant (see subsection 5.3.1) for the details. As previously the values for Ω and dΩ dt given in the following text and ﬁgures refer to their initial values. In a ﬁrst experiment we quench the system to Ωi = 2π × 120 Hz and subsequently ramp towards the critical point with three diﬀerent slopes ranging from dΩ dt = 2π × 2.0 Hz/ms to 2π × 13 Hz/ms. According to Zurek’s adiabaticity criterion the slowest ramp is expected to be adiabatic during the time scale of our experiments while the faster ramps are or become non-adiabatic. Note that it is important to include the eﬀect of the changing atom density due to atom loss in this calculation, which is the cause for the deviation from the numbers given in Figure 6.2. As in subsection 5.3.1 we 79 6. Dynamics of phase transitions and the Kibble-Zurek mechanism 2.0Hz/ms 2.7Hz/ms 13.0Hz/ms 1.8 ( m) 1.6 1.4 2.5 2.0 1.2 1.5 1.0 0.8 0.1 1.0Hz/ms 2.0Hz/ms 3.0Hz/ms 3.0 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 Figure 6.3.: Correlation length versus ε for ramps of diﬀerent slopes towards the critical point. e slowest ramps are adiabatic, while faster ramps are predicted to become non-adiabatic at ε̂ indicated by the vertical lines. e correlation length at a given value of ε is larger for slow ramps than for fast ramps, which indicates the (non-)adiabaticity. e ramps shown in the le panel started at Ωi = 2π ×120 Hz and the ones on the right at Ωi = 2π ×95 Hz aer an initial equilibration time of 6 ms. e dashed lines indicate the predicted mean ﬁeld equilibrium values of the correlation length. calculate the autocorrelation function of the spin proﬁle and determine the correlation length from an exponential ﬁt. e time evolution of the correlation length during the ramps is shown in the le panel of Figure 6.3 e correlation length of the system is expected to follow the equilibrium conﬁguration for adiabatic ramps. For non-adiabatic ramps the equilibrium correlations grow faster than the system can adjust, which results in correlation lengths smaller than the equilibrium value at the instantaneous value of ε. When adiabatically ramping to a given value of ε we measure a correlation length that is larger than for a non-adiabatic sequence, which indicates the observation of adiabatic and non-adiabatic evolution. We perform a similar experiment by starting the ramps closer to the critical point at Ωi = 2π × 95 Hz and reducing the slopes to dΩ dt = 2π × 1.0 Hz/ms to 2π × 3.0 Hz/ms such that the slowest ramp remains adiabatic. In order to let the system equilibrate before the ramp we hold it at Ωi for t = 6 ms aer the initial quench. e results are summarized in the right panel of Figure 6.3 and show a behavior similar to the previous observations: Slower ramps result in a larger correlation length at a given value of ε. In order to test the prediction that the system follows the equilibrium conﬁguration during adiabatic ramps we compare the results of the slowest ramps to the saturation values of the correlation lengths aer the sudden quenches presented in subsection 5.3.1. As shown in Figure 6.4 the observed absolute value of the correlation lengths and their scaling with ε agree nicely and are close to the theoretical prediction. Note that the saturation value aer the quench is not necessarily the equilibrium value and the absence of oscillations in the correlation length remains to be understood (see discussion in subsection 5.3.1). However, the agreement of the diﬀerent experiments and the consistence with the theoretical prediction provide further indications for observation of adiabatic behavior. 80 6.2. Experimental results 3.0 ( m) 2.5 2.0 1.5 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Figure 6.4.: Growth of the correlation length during the adiabatic ramps (blue circles) compared to the saturation values aer sudden quenches to a ﬁxed ε (black squares; same data as in Figure 5.7). e good agreement of the absolute length scales is a further indication for adiabatic behavior. e dashed line shows the theoretical prediction for the equilibrium correlation length. 6.2.2. Non-adiabatic ramps through the critical point A central ingredient for Zurek’s scaling argument is the transition from adiabatic to non-adiabatic behavior at ε̂ as we discussed in the previous section. e size of the resulting topological defects aer the quench is proportional to the correlation length ξ̂ at this point. However, the value of the proportionality factor and the inﬂuence of the system properties in the symmetry-broken regime aer the crossing of the critical point remain unclear [114]. It was argued that the size of the defects is actually determined aer the critical point was passed and numerical evidence supports this view [120, 56]. While the lifetime of the atomic cloud does not allow for the experimental realization of a wide range of ramps through the critical point with ε̂ < 1, we can perform faster ramps and study the spin structures emerging due to eﬀects in the symmetry-broken regime. Many observations in the context of these experiments are summarized in [119]. We implement the ramps through the critical point by changing the amplitude of the dressing ﬁeld from 1.4Ωc to 0.1Ωc aer the initial π/2-pulse with ramp durations of τ Q = 1 . . . 20 ms. e linear coupling ﬁeld is switched oﬀ aer the ramp, i.e. Ω = 0. In order to have similar mean atomic densities during the diﬀerent ramps, shorter ramps with a duration τ Q < 20 ms are delayed by 12 (max(τ Q ) − τ Q ). During the delay the system is exposed to a strong dressing ﬁeld with Ω = 2π × 340 Hz such that the overlap of the two clouds remains unchanged and the spin correlation length is small. All hold times are given relative to the initial superposition pulse such that all ramps are centered around t = max(τ Q )/2 = 10 ms. In this notation the ramps begin and end at diﬀerent times t depending on τ Q . is experimental scheme is sketched in Figure 6.5. We study the spin structures emerging aer the crossing of the critical point by analyzing the autocorrelation function and the Fourier spectrum of the spin proﬁle. eir dependence on the ramp duration τ Q at diﬀerent hold times t is shown in Figure 6.6. Small spin domains dominate the spectrum aer fast quenches τ Q < 9 ms, while large structures emerge aer long ramps τ Q > 14 ms. e size of the small domains is given by the most unstable mode in the Bogoliubov spectrum as discussed in the context of the free demixing experiments in subsection 3.2.2.³ e large structures correspond to the three-domain ground state conﬁguration we also observed when dressing the ³In the free evolution experiments the linear coupling is switched oﬀ aer the π/2-pulse, which corresponds to a sudden quench with τ Q = 0. 81 6. Dynamics of phase transitions and the Kibble-Zurek mechanism 350 300 =2 200 150 100 /2 pulse (Hz) 250 50 0 0 5 10 time (ms) 15 20 Figure 6.5.: Sketch of the experimental sequence for the ramps through the critical point (dashed line) for three exemplary ramps with durations of 4 ms, 8 ms and 20 ms. e ramps are symmetric around t = 10 ms. system with a weak coupling ﬁeld with an amplitude near Ω ≈ Ωc /2 (see subsection 5.3.2). For intermediate durations of the ramp 9 ms < τ Q < 14 ms both of these length scales are visible in the Fourier spectrum which has bi-modal characteristics. However, at τ Q = 9 ms there is a sharp reduction in the contribution of the long-wavelength mode and the large structures do not occur for faster ramps. We model the spin dynamics during and aer the ramps by integrating over the imaginary part of the mean ﬁeld excitation spectra at each hold time. For Ω < Ωc the excitation spectrum has unstable modes which grow exponentially. e range of unstable modes and their growth rates depend on the value of Ω, which is changed during the ramp. us diﬀerent modes∫grow during the course of the t ramp and their relative amplitude can be calculated by integrating 0 dt′ exp{Im[ω(k, Ω(t′ ))] · t′ }, where Ω(t′ ) models the experimental sequence and depends on τ Q . e results of the numerical integration including the eﬀects of atom loss are shown in the right panel of Figure 6.6. At early hold times it shows a similar crossover of the dominating length scales around τ Q ≈ 10 ms. e size of the small modes agrees well with our observations while the size of the large structures is given by the ﬁnite size of the atomic cloud and thus not contained in the homogeneous theory. When integrating the unstable part of the spectrum for longer times, the short wavelength mode always dominates as it is the fastest growing mode in the absence of a linear coupling aer the ramp. However, it is important to remember that the theoretical model is valid only for small modulation depths of the spin structures as the excitation spectrum may change when the density proﬁles deviates from the equal superposition state. us, the validity of this approach breaks down at the hold time t when the modulation depth exceeds a threshold. e sharp change of the spin structures at τ Q can be understood in this context: If the amplitude of the long-wavelength mode aer the ramp is large enough, the subsequent growth of the small domains is suppressed and the long-wavelength mode prevails also for longer times. If this condition is not fulﬁlled, the growth of the small domains supersedes the large structures. is threshold is passed at τ Q = 9 ms according to the experimental observations. We estimate this threshold to be a modulation depth of approximately 30%, as this the typical modulation amplitude for τ Q = 9 ms and t = 19 ms. In summary, we have discussed the classic Kibble-Zurek scenario. It states that the size of topological defects aer ﬁnite-time quenches through a critical point is proportional to the correlation length at the freezing time. At that instant the relaxation time of the system is larger than the transition time of the quench and the system fails to follow adiabatically. We have observed indications of a transition from adiabatic to non-adiabatic behavior by analyzing the growth of spin correlations 82 ramp time (ms) 6.2. Experimental results t=19ms t=19ms t=19ms t=49ms t=49ms t=49ms 15 10 5 ramp time (ms) 0 15 10 5 00 50 100 x-x' ( m) 150 0.0 0.1 0.2 0.0 0.1 0.2 wave vector k (1/ m) wave vector k (1/ m) Figure 6.6.: Observed correlation functions (le column) and Fourier spectra (center column) in false color for diﬀerent ramp durations and hold times aer the initial π/2-pulse. Red color indicates correlations, blue color anti-correlations in the correlation functions. For the Fourier spectra red color means a strong amplitude of the corresponding wave vector and blue color denotes weakly populated modes. e spin paern is dominated by small structures for τ Q < 9 ms, while large structures prevail for τ Q > 14 ms. e small structures correspond to the most unstable mode for Ω = 0 Hz and the large paern is given by the three-domain ground state in the trap. Both length scales contribute in the intermediate regime 9 ms < τ Q < 14 ms. A sharp change in the amplitude of the longwavelength mode occurs at τ Q = 9 ms. e Bogoliubov prediction for the spectrum of the spin paern is shown in the right column. e imaginary part of the Bogoliubov spectrum is integrated over the experimental sequence employed in the measurements. It reproduces well the experimentally observed cross-over of length scales at τ Q ≈ 10 ms. is model is valid only for small times as it requires a small modulation depth of the spin domains. for diﬀerent ramp speeds towards the critical point. In addition, the inﬂuence of unstable modes aer the crossing of the critical point on the emerging domain paern was demonstrated. 83 7. Conclusion and Outlook In this thesis we have experimentally studied the rich interplay of non-linear interactions and a linear coupling ﬁeld in a two-component elongated Bose-Einstein condensate, which results in a miscible-immiscible phase transition. We have discussed the properties of the system far away from the transition point, which are dominated by interaction among the atoms in the limit of weak linear coupling. In the opposite limit of a strong coupling ﬁeld the analog of optical dressed states has been realized in condensates using a non-adiabatic preparation scheme. We have examined the eﬀective interactions between the dressed states by studying the long time dynamics of Rabi oscillations and found their miscibility to be opposite to that of atomic states. e critical point of the phase transition occurs when the energy scales of the interactions and the linear coupling are equal. We have characterized the phase transition by examining the linear response of the spin correlations aer a sudden quench to the proximity of the critical point. A power law divergence has been observed in the characteristic length scale on both sides of the phase transition. e extracted values for the critical exponent ν of 0.49±0.07 and 0.51±0.26 are in good agreement with the mean ﬁeld prediction of ν = 1/2. e relaxation time on the miscible side of the transition agrees with a prediction based on the energy gap in the excitation spectrum of the binary condensate. Furthermore, we have discussed a scheme for the measurement of dynamical scaling exponents in the context of the Kibble-Zurek mechanism and sketched its experimental feasibility. Indications for the transition from adiabatic to non-adiabatic behavior were observed. e experimental system presented in this thesis oﬀers a high level of control for the study of quantum phase transitions. As the control parameter is realized by the amplitude of a radiofrequency ﬁeld, well deﬁned sudden and ﬁnite-time quenches can be realized with sub-microsecond resolution. Further experiments can be made possible by the following improvements to the experimental implementation employed in this thesis. e experimentally accessible time scale for the presented experiments was limited by increased atom loss due to the proximity to the Feshbach resonance. In addition, atom loss modiﬁes relevant system parameters that depend on the atom density such as the critical coupling strength or the spin healing length. While the eﬀect on the critical coupling strength could be compensated by adjusting the amplitude of the linear coupling ﬁeld, unwanted imperfections remain. is problem can be circumvented by utilizing a diﬀerent atomic species with naturally immiscible scaering parameters such that the use of a Feshbach resonance can be avoided. A further requirement for the employed states is a small diﬀerential sensitivity to external perturbations such as magnetic ﬁelds in order to ensure resonance of the linear coupling. Another limitation of the current experimental setup is the ﬁnite longitudinal size of the atomic cloud and the resulting inhomogeneity. is leads to a spatial dependence of the critical coupling strength and restricts the analysis region to the central part of the atomic cloud, which can be treated as locally homogeneous. In addition our atomic cloud is not truly one-dimensional and the ﬁrst few transverse excited states are populated. e role of transverse excitations in the condensates and their consequences for our observations require further investigation. True one-dimensional conﬁnement has been realized both with atom chips [66, 15] and optical dipole potentials [121]. Such a geometry eliminates possible eﬀects of transverse excitations and reduces eﬀects of cloud inhomogeneities. 85 7. Conclusion and Outlook We have detected the atomic clouds by in-situ absorption imaging, which is a destructive process and provides only a snapshot of the spin dynamics in each single experimental realization. In many of the performed experiments the dynamics results from spontaneous symmetry breaking and is not shot-to-shot reproducible such that only realization-independent quantities such as average domain sizes could be analyzed. e evolution of a single experimental realization can be followed by employing non-destructive imaging techniques such as phase-contrast imaging [8, 122], which permits further insights on the growth of spin domains and their subsequent dynamics. Ultracold atoms continue to oﬀer great prospects for the study of quantum phase transitions. Novel schemes for the determination of dynamical scaling exponents have been proposed [94, 16] as alternatives to the classical Kibble-Zurek scenario involving quenches through the critical point. Very recent experiments have explored the dynamics following quenches in one [15] and twocomponent [60] elongated systems as well as one-dimensional laices [123]. e divergence and scaling of ﬂuctuations has been studied at a driven-dissipative Dicke phase transition [124] and universal spin dynamics was observed in two-dimensional Fermi gases [125]. is variety of experiments in the recent months provides only a ﬁrst glimpse of the vast opportunities quantum gases have to oﬀer in this ﬁeld. Further experimental and theoretical insights about universality, phase transitions and quench dynamics near critical points will continue to push the frontiers of this exciting ﬁeld of physics. 86 A. Summary of atomic and experimental parameters A.1. Properties of 87 Rb For an overview of the atomic properties of 87 Rb including optical, magnetic and electronic data we refer to [67]. e interaction and loss parameters that are relevant for our experiments employing states |1⟩ = |F = 1, mF = ±1⟩ and |2⟩ = |F = 2, mF = ±1⟩ are summarized below along with the corresponding references. A.1.1. Scaering lengths a11 a22 a12 100.40 aB 100.44 aB 95.00 aB 95.47 aB 95.44(7) aB 97.70 aB 97.66 aB 98.09 aB 98.006(16) aB [73, 77] [78, 79] [77] [78] [79] [126] [77] [78] [79] For scaering properties within the F = 1 or F = 2 manifold, see [127]. A.1.2. Loss coeﬀicients For the loss coeﬃcients we use the notation KNijk , where N denotes the number of particles involved in the inelastic collision process and i, j, k ∈ {1, 2} are the states of the involved atoms. K1 K211 K222 K212 K3111 K3222 typically 0.05 . . . 1 /s 0 cm3 /s −16 < 1.6 × 10 cm3 /s −14 11.94(19) × 10 cm3 /s 10.4 × 10−14 cm3 /s 8.1(3) × 10−14 cm3 /s 7.80(19) × 10−14 cm3 /s 5 × 10−14 cm3 /s 1.51(18) × 10−14 cm3 /s 5.8(1.9) × 10−30 cm6 /s 18(5) × 10−30 cm6 /s [77, 74] [128] [77] [74] [79] [77] [74] [79] [128] [129] Also see [130] for a discussion of inelastic collisions in the F = 2 manifold. 87 A. Summary of atomic and experimental parameters A.1.3. Scaering lengths near the Feshbach resonance B [G] a12 [aB ] 9.03 100.3 9.05 102.6 9.06 105.6 9.07 112.9 9.08 120 Also see Figure 3.4 for a graphical summary. A.2. Imaging ampliﬁcation CCD pixel size pixel size in object space image width image height 30.96 13 μm 420 nm 512 px, 215 μm 140 px, 59 μm A.3. Optical dipole traps A.3.1. Charger For measurements near Feshbach resonance transverse trap frequency ω ⊥ longitudinal trap frequency ω x 2π × 460 Hz 2π × 22.0 Hz For measurements at the ’magic field’ of 3.23 G transverse trap frequency ω ⊥ longitudinal trap frequency ω x 2π × 490 Hz 2π × 23.4 Hz A.3.2. Waveguide transverse trap frequency ω ⊥ longitudinal trap frequency ω x 2π × 128 Hz 2π × 1.9 Hz Parameters of the atomic cloud typical linear atom density minimum spin healing length at 9.08 G 88 230 atoms/μm 1.3 μm 9.11 84 9.17 94.0 B. Numerical methods for simulating Bose-Einstein condensates is appendix summarizes the theoretical background for numerical simulations of one-dimensional trapped two-component Bose-Einstein condensates in the presence of a linear coupling between them. First, we will outline the underlying equation of motion, the Gross-Pitaevskii equation, and its variants for quasi one-dimensional geometries. e second part covers numerical methods for calculating the ground state, Bogoliubov excitations and time dynamics of Bose condensates. For simplicity, all theoretical and numerical methods will be discussed in the context of a onedimensional single species condensate. e generalization to higher dimensions with more components (possibly with a linear coupling between them) will be outlined only where qualitatively diﬀerent problems arise. B.1. Gross-Pitaevskii equation In the mean ﬁeld approximation, Bose-Einstein condensates at zero temperature, i.e. neglecting thermal excitations, can be described by the Gross-Pitaevskii equation (GPE) [29, 30] [ ] ∂ψ ℏ2 2 2 iℏ = − ∇ + V + g|ψ| ψ (B.1) ∂t 2m where 2πℏ is Planck’s constant, m the atomic mass and V = V(x, y, z) the external trapping potential. 2 g = 4πℏm as quantiﬁes interatomic interactions, parametrized by the s-wave scaering length as . is equation models the temporal and spatial dynamics of the three-dimensional wave ∫function, ψ = ψ(x, y, z, t), which is normalized to the number of atoms N in the condensate, N = |ψ|2 dV. For many trapping geometries, such as a spherical or cylindrical conﬁnement, Equation B.1 can be reduced to lower dimensions in order to simplify analytic calculations and decrease computational complexity. B.1.1. One-dimensional Gross-Pitaevskii equation For cylindrical trapping geometries, where the transverse conﬁnement is much stronger than the longitudinal one, the condensate can be described by a one-dimensional wave function. is is achieved by spliing the three-dimensional wave function into a longitudinal and a transversal part, ψ = ψ 1D (x, t) · ψ ⊥ (y, z). Assuming a transversal proﬁle of Gaussian shape with width σ = a⊥ and integrating out the transverse dimensions, one yields a one-dimensional Gross-Pitaevskii equation (1D GPE) with an eﬀective interaction parameter g′ = g/(2πa2⊥ ) [131] [ ] ∂ψ 1D ℏ2 ∂ 2 ′ 2 iℏ = − + V + g |ψ 1D | ψ 1D ∂t 2m ∂x2 (B.2) √ Here, a⊥ = ℏ/mω ⊥ denotes the transverse harmonic oscillator length and ω ⊥ the corresponding trap frequency. 89 B. Numerical methods for simulating Bose-Einstein condensates 3d GPE 1d GPE NPSE 150 linear density (atoms/ m) linear density (atoms/ m) 200 100 50 0-30 -20 -10 0 x ( m) 10 20 30 160 140 120 100 80 60 40 20 0-30 3d GPE 1d GPE NPSE -20 -10 0 x ( m) 10 20 30 Figure B.1.: Comparison of diﬀerent approximations to the full 3D Gross-Pitaevskii equation. e ground states of single (le) and two-component (right) condensates containing 4000 atoms in the charger are calculated using the 1D GPE (blue) and the NPSE (green). Comparison to the full three-dimensional solution (red) reveals, that the NPSE results agree well with the exact solution, while the 1D GPE deviates signiﬁcantly. e validity of this 1D approximation requires waveguides without longitudinal conﬁnement or a large trapping aspect ratio ωω⊥x ≫ 1. However, it also depends on the atom density, since the assumption of a transverse Gaussian proﬁle, i.e. being in the transverse ground state, requires the chemical potential to be smaller than the transverse trapping frequency. e general condition for the validity of the 1D GPE is given by the dimensionality parameter [132] Nas a⊥ ≪1 a2x with the longitudinal harmonic oscillator length ax . For our experiments performed in the charger (N = 3000, as = 100 abohr , ω x = 23.4 Hz, ω ⊥ = 490 Hz), this dimensionality parameter is 1.6 and for the measurements in the waveguide (N = 50000, as = 100 abohr , ω x = 1.9 Hz, ω ⊥ = 128 Hz) even 4.1. us, the one-dimensional Gross-Pitaevskii equation is not an adequate description of our system. B.1.2. Nonpolynomial nonlinear Schrödinger equation Our cigar-shaped Bose-Einstein condensate is beer described by the nonpolynomial nonlinear Schrödinger Equation (NPSE), which assumes a Gaussian transverse proﬁle as well, but allows its width to vary along the longitudinal trap axis, σ = σ(x). Integration over the transverse coordinates yields [133] [ ( )] ∂ψ 1D ℏ2 ∂ 2 g|ψ 1D |2 1 ℏω ⊥ 1 2 2 iℏ = − +V+ + + σ /a⊥ ψ 1D ∂t 2m ∂x2 2 2πa2⊥ σ 2 /a2⊥ σ 2 /a2⊥ √ where σ 2 = a2⊥ 1 + 2as |ψ 1D |2 . e last term is not a constant oﬀset term (which could be ignored), but has an implicit spatial dependence through σ. Note that in the limit of weak interactions, as |ψ 1D |2 ≪ 1, the NPSE reduces to the 1D GPE, σ ≈ a⊥ . Our experimental parameters mentioned above yield as |ψ 1D |2 ≈ 0.8. us the NPSE is a more appropriate description of our system as demonstrated in Figure B.1, which compares the 90 { B.2. Numerical methods Figure B.2.: Schematic representation of the discretization of the wave function on a spatial grid. e continuous function is mapped onto a vector with Ng elements. ground states of the condensate calculated by the 1D GPE, NPSE and the full 3D GPE. e extension of the NPSE to binary condensates is non-trivial and we will not give the results here but refer to [134]. Note that for simulations of the time evolution including atom loss, the loss coeﬃcients have to be modiﬁed in a similar manner as the interaction parameters when employing the NPSE instead of the 3D GPE. B.2. Numerical methods e Gross-Pitaevskii equation and its variants can be solved analytically only for a few special cases, e.g. in the limit of no interactions or in the limit of strong interactions (omas-Fermi regime). In general, solutions have to be found numerically. is section describes numerical methods for computing stationary states as well as Bogoliubov excitations and time dynamics of the Gross-Pitaevskii equation. It is based on personal communication with Panayotis Kevrekidis¹ during October 2010. All numerical methods presented here require a discretization of continuous functions (e.g. the wave function or the potential) on a spatial grid as outlined in Figure B.2. For a spatial grid consisting of Ng grid points, the wave function ψ is represented by a vector of size Ng , whereas the Hamiltonian takes the form of an Ng × Ng matrix. e distance Δx between two grid points has to be smaller than the smallest features one expects to observe in the simulated wave function. Similarly, the state of two components can be represented by a vector of size 2Ng , where the ﬁrst Ng elements correspond to the wave function of the ﬁrst component and the other entries describe the second component. B.2.1. Computing the ground state e starting point of numerical simulations is calculating the ground state (or more generally, a stationary state) of the system, as this usually is the initial experimental state aer Bose-Einstein condensation. Factoring out the time evolution in Equation B.1, ψ(x, t) = e−iμt ϕ(x), one yields the equation modeling stationary states [ ] ℏ2 ∂ 2 2 μϕ = − + V + g|ϕ| ϕ (B.3) 2m ∂x2 with the chemical potential μ. is section outlines two methods for computing the ground state of the system: Imaginary time propagation (ITP) and Newton’s method for ﬁnding the roots of a function. While the former is restricted to ﬁnding the state of minimal energy, the laer method can converge to any stationary state. ¹Homepage: http://www.math.umass.edu/~kevrekid/ 91 B. Numerical methods for simulating Bose-Einstein condensates Both methods are of iterative nature, i.e. they asymptotically approach the desired stationary state in discrete steps. is requires both a sensible initial guess of the wave function, for example a omas-Fermi or a Gaussian proﬁle, and a break condition for terminating the iteration, such as a lower bound for the relative change of the wave function in one iteration step. Imaginary time propagation e time evolution of the wave function ψ(x, t) under the Hamiltonian Ĥ is calculated using the propagator e−iĤΔt/ℏ , ψ(x, t + Δt) = e−iĤΔt/ℏ ψ(x, t) (B.4) Any wave function can be expressed as a superposition of energy eigenstates ϕ m with time∑ dependent amplitudes, ψ(x, t) = m am (t)ϕ m (x). When rewriting Equation B.4 in this basis and replacing Δt → −iΔt , the amplitudes of the basis states decay exponentially with a decay constant given by the corresponding energy eigenvalue, ψ(x, t + iΔt) = ∑ am (t)e−Em Δt/ℏ ϕ m (x) m Since the eigenstate with the lowest energy decays slowest, iteratively performing propagation steps in imaginary time yields the ground state of the system. Note that this operation is not unitary and thus requires renormalization of the wave function aer each iteration step. On a spatial grid, the determination of the propagator corresponds to the calculation of the matrix ˆ exponential e−iĤΔt/ℏ = e−i(Hp +Ĥx )Δt/ℏ , where Ĥp is the kinetic energy Hamiltonian. e potential and interaction term Ĥx is diagonal and its matrix exponential can be easily computed. e kinetic energy contribution is calculated more conveniently in momentum space, where Ĥp is diagonal, but which requires a Fourier transformation of the wave function. us, one can split one propagation step into three successive steps in position and momentum space as proposed in [135], ψ(x, t + Δt) = e−i(Hp +Ĥx )Δt/ℏ ψ(x, t) ≈ e−iHp Δt/2ℏ e−iĤx Δt/ℏ e−iHp Δt/2ℏ ψ(x, t) ˆ ˆ ˆ is so-called ’split-step fast Fourier transform method’ makes use of the Baker-Campbell-Hausdorﬀ formula, which is only exact for commuting operators Ĥp and Ĥx . However, the operators for kinetic and potential energy do not commute, which leads to an error O[(Δt)3 ]. us, this methods requires small steps in (imaginary) time in order to avoid the accumulation of numerical errors [135, 136]. In summary, the method of imaginary time propagation has the advantage of reliably converging towards the ground state without the requirement of a good initial guess of the ground state wave function. However, due to the diﬀusive nature of the equation, its convergence towards the steady state is slow and requires many iterations as shown in Figure B.3. Newton’s method Newton’s method is used to approximate the roots of a function f(x). Given an initial guess x0 , k) ′ it approaches a root, limn→∞ f(xn ) = 0, by the iteration xk+1 = xk − ff(x ′ (xk ) , where f denotes the derivative of f [137]. is method can be applied to the problem of ﬁnding the ground state of a Bose-Einstein condensate by deﬁning F(ϕ) = Ĥϕ − μϕ = − 92 ℏ2 ∂ 2 ϕ + Vϕ + g|ϕ|2 ϕ − μϕ 2m ∂x2 B.2. Numerical methods 10-1 Newton ITP number of steps duration per step (s) 100 10-2 10-3 10-4 1 10 102 103 number of grid points 104 107 Newton ITP 106 5 10 104 103 102 101 100 -14 -13 -12 -11 -10 -9 -8 -7 -6 10 10 10 10 10 10 10 10 10 relative error tolerance Figure B.3.: Comparison of the scaling characteristics of the Newton method and imaginary time propagation for computing a ground state. e le panel shows the scaling of the computational requirements when improving the number of grid points, i.e. the spatial resolution. e scaling for improved accuracy of the simulation is shown on the right. While single iteration steps take longer for Newton’s method, a fewer iterations are required for high accuracy results. in analogy to the stationary Gross Pitaevskii equation Equation B.3. Each stationary state ϕ 0 is a root of this functional, F(ϕ 0 ) = 0, and can be found by the iteration ϕ k+1 = ϕ k − J−1 (ϕ k ) · F(ϕ k ) with the Jacobian Jij (ϕ k ) = ∂Fi . ∂ϕ kj e computationally expensive part is the calculation of the inverse of the Jacobian. When discretizing space on a grid, F(ϕ) = 0 has to hold for each grid point, i.e. F(ϕ n ) = 0 for each n. With a grid spacing of Δx, the second derivative can be approximated by a second diﬀerence 2 ϕ +ϕ −2ϕ n as ddxϕ2 ≈ n+1 Δxn−1 resulting in a tri-diagonal Jacobian J. us, J can be handled as a sparse 2 matrix in order to speed up the calculation of its inverse. Since the chemical potential μ directly enters F(ϕ), knowledge about this parameters is required. e corresponding atom number N follows from the norm of the resulting stationary state. However, the atom number is easier to determine experimentally, such that it is favorable to provide the atom number instead of the chemical potential as ∫ an 2input parameter for the simulations. e additional constraint of a ﬁxed atom number N = |ϕ| dV can be included via a Lagrangian multiplier κ resulting in an augmented F′ (ϕ) having Ng + 1 dimensions instead of Ng , ( ) F(ϕ) + 2κΔxϕ ′ ∑ F (ϕ) = =0 2 n |ϕ n | Δx − N Newton’s method has the advantage of quadratic convergence, i.e. fewer iteration steps are required to yield a stationary state with high accuracy as depicted in Figure B.3. Furthermore, this algorithm converges not only to the ground state, but also to any stationary state of the system. However, ﬁnding the desired solution requires a good initial guess of the wave function and thus knowledge about the sought-aer state. e robustness of imaginary time propagation can be combined with the fast convergence of Newton’s method by ﬁrst iterating in imaginary time to obtain an estimate for the ground state, which can be used as a sensible starting point for Newton’s method. 93 B. Numerical methods for simulating Bose-Einstein condensates B.2.2. Bogoliubov - de Gennes stability analysis is subsection summarizes the Bogoliubov - de Gennes analysis, a method for numerically computing the excitation spectrum and the corresponding spatial modes on top of a background state ψ 0 (x). We assume ψ 0 (x) to be a known stationary solution of Equation B.1 with a time evolution of ψ 0 (x, t) = e−iμt ψ 0 (x, t = 0) and δψ(x, t) to be a perturbation on top of ψ 0 (x) ψ(x, t) = e−iμt (ψ 0 (x) + εδψ(x, t)) with ε ≪ 1. Inserting this state in Equation B.1 and ignoring terms of second or higher order in ε, we obtain an equation of motion for the perturbation iℏ ∂δψ ℏ2 + μδψ = − ∇2 δψ + Vδψ + 2g|ψ 0 |2 δψ + g|ψ 0 |2 δψ ∗ ∂t 2m (B.5) where ()∗ denotes the complex conjugate. We now assume the perturbation to be of the form δψ(x, t) = a(x)eiωt + b∗ (x)e−iω ∗t with the complex amplitudes a(x) and b(x) and the energy ℏω. We insert this ansatz into Equation B.5 and sort the equation by the linear independent terms e−iωt and eiωt resulting in two coupled equations for the excitation modes ℏ2 2 ∇ a + Va + 2g|ψ 0 |2 a + g|ψ 0 |2 b − μa 2m ℏ2 ℏωb = − ∇2 b + Vb + 2g|ψ 0 |2 b + g|ψ 0 |2 a − μb 2m −ℏωa = − us, the excitation spectrum ω and the corresponding modes a and b are obtained as the eigenvalues and -vectors of the matrix equation ( ) ( )( ) a L1 L2 a ℏω = (B.6) b L3 L4 b with ) ( ℏ2 L1 = − − ∇2 + V + 2g|ψ 0 |2 − μ 2m L2 = −g|ψ 0 |2 L3 = −L∗2 = g|ψ 0 |2 L4 = −L∗1 = − ℏ2 2 ∇ + V + 2g|ψ 0 |2 − μ 2m For the homogeneous system, V(x) = 0, μ = ng, the famous Bogoliubov dispersion law [27] is recovered √ ( ) √ ℏ2 k2 ℏ2 k2 ℏω = L21 − L22 = + 2ng 2m 2m 94 B.2. Numerical methods e generalization of Equation B.6 to the case of two linearly coupled interacting condensates can be calculated using the same method as above and we only give the resulting 4Ng × 4Ng matrix: a1 L11 L12 L13 L14 a1 b1 L21 L22 L23 L24 b1 ℏω a2 = L31 L32 L33 L34 a2 b2 L41 L42 L43 L44 b2 with ℏ2 2 ∇ + V + 2g11 |ψ 1,0 |2 + g12 |ψ 2,0 |2 − μ 1 2m = g11 |ψ 1,0 |2 L11 = − L12 L13 = g12 ψ ∗1,0 ψ 2,0 + ℏΩ/2 L14 = g12 ψ ∗1,0 ψ 2,0 L21 = −L∗12 L22 = −L∗11 L23 = −L∗14 L24 = −L∗13 L31 = g12 ψ ∗1,0 ψ 2,0 + ℏΩ/2 L32 = g12 ψ ∗1,0 ψ 2,0 ℏ2 2 ∇ + V + 2g22 |ψ 2,0 |2 + g12 |ψ 1,0 |2 − μ 2 2m = g22 |ψ 2,0 |2 L33 = − L34 L41 = −L∗32 L42 = −L∗31 L43 = −L∗34 L44 = −L∗33 Here, the strength of the linear coupling is parametrized by the Rabi frequency Ω and gij are the intra- and inter-species interaction parameters. Practical tips We conclude this section with a few tips regarding the numerical solution and the interpretation of the Bogoliubov - de Gennes equations. • e Bogoliubov - de Gennes equations require knowledge of the chemical potential μ, which can be calculated numerically using the known background state ψ 0 and Equation B.3. • e second derivative can be calculated on a grid using ﬁnite diﬀerences as mentioned in the discussion of Newton’s method. is results in the matrices Lii not being diagonal any more but also having entries in the ﬁrst oﬀ-diagonals directly above and below the main diagonal. • e numerically obtained excitation modes in general are not plane waves. us it is not straight-forward to assign a wave-vector k to each mode in order to obtain dispersion re- 95 B. Numerical methods for simulating Bose-Einstein condensates lations as shown in subsection 2.4.4. We estimate k by the determining the position of the largest amplitude in the Fourier spectrum of the diﬀerence proﬁle of the excitation modes of the two components. In addition, each excited mode has to be assigned to one of the two branches of the dispersion. As the gapped branch corresponds to out-of-phase modulation and the other branch to in-phase modulation of the two components, we assign modes whose maximum amplitude in the Fourier spectrum is larger than a threshold to the gapped branch. e threshold value is chosen such that each branch contains the same number of modes. • e derivation of Equation B.6 required the background state ψ 0 (x, t) to be stationary, i.e. ψ 0 (x, t) = e−iμt ψ 0 (x, t = 0). If this condition is not fulﬁlled the Bogoliubov - de Gennes analysis does not yield reliable results. is is the case for a superposition of two condensates in a trap, where the symmetry is broken by diﬀerent intra-species scaering lengths a11 ̸= a22 . us we have to assume a11 = a22 when numerically calculating the excitation spectra for inhomogeneous atomic clouds. While this is not exactly fulﬁlled in 87 Rb it is a good approximation of the experimental system. B.2.3. Time integration In the previous sections we discussed algorithms to numerically compute stationary states of the GPE and their excitation spectra. Now, we will introduce two methods for calculating the time evolution of a wave function by integrating the Gross-Pitaevskii equation: real time propagation, which is straight forward to implement if imaginary time propagation is already available, and a Runge-Kua method, which is a generic, and thus more ﬂexible method for integrating diﬀerential equations. e time dynamics of a state is computed in discrete time steps Δt. Similar to the conditions for the spatial grid, this temporal grid has to be chosen signiﬁcantly smaller than the characteristic timescale of the dynamics. Additionally, small time steps are necessary to avoid the accumulation of numerical errors in each time step. Real time propagation As explained in the context of imaginary time propagation, the time evolution of a wave function is given by the propagator deﬁned in Equation B.4. us, the same split-step fast Fourier transform algorithm as for imaginary time propagation can be applied to calculate the time dynamics of a condensate. e only change is that real time is used instead of imaginary time. Again, this method works well if the real space part of the Hamiltonian Ĥx is diagonal, such that the matrix exponential can be computed eﬃciently. For two-component condensates with a linear coupling between the species, the matrix representation of the Hamiltonian is no longer diagonal, but includes two populated oﬀ-diagonals. In this special case, the Hamiltonian can be diagonalized analytically, but the extension to more complex scenarios is diﬃcult. Runge-Kua method A more universal approach to integrating diﬀerential equations is employing general explicit methods. As a simple example, we will brieﬂy outline the Euler method: Given a diﬀerential equation y′ (t) = f(t, y(t)) (in our case Equation B.1) with an initial value y(t0 ) = y0 (corresponding to the ground state), we can use a ﬁrst order Taylor expansion to approximate y(t0 + Δt) ≈ y(t0 ) + Δt · f(t0 , y0 ). Iteratively repeating this procedure yields an approximate solution of the differential equation. Higher order terms can be included such that the error in each iteration is smaller 96 B.2. Numerical methods at the cost of higher computational complexity. A good compromise is a fourth-order Runge-Kua method. A more detailed introduction to the numerical solution of diﬀerential equations can be found in [138]. e expansion of the one-dimensional example above to the Ng dimensional Gross-Pitaevskii equation on a grid is straight-forward. e only hitch is the kinetic energy term, which can be approximated using second order ﬁnite diﬀerences as explained in the context of Newton’s method. For simple scenarios in which the Hamiltonian can be easily diagonalized, this method is slower than real time propagation, but it can be easily generalized to more complex situations. 97 C. Calibration of in-situ imaging near the Feshbach resonance For the detection of the atomic clouds we employ high-intensity absorption imaging [64]. A detailed description of our imaging setup and the atom number calibration was given in the context of spin squeezing measurements, where knowledge of the absolute atom number is critical [86, 32]. e limits or our imaging setup are discussed in [26]. e observable for the experiments on the miscible-immiscible phase transition discussed in this thesis is the diﬀerence of the density proﬁles of the atomic clouds. Knowledge of the absolute atom density is important as it enters the relevant parameter n · gs , where gs is the interaction parameter and n the linear atom density (see subsection 2.3.1). e accuracy in this parameter is limited by shotto-shot ﬂuctuations in the atom density of about 10% and the systematic uncertainty of the interspecies scaering length a12 close to the Feshbach resonance. us, a calibration of the absolute atom number within these uncertainties is desired. However, the experiments under discussion require in-situ detection of the spatial density distributions of the two components. An imaging pulse duration of 10 μs was chosen in order to minimize blurring due to defocussing and heating while keeping a good signal-to-noise ratio [26]. Ramping down the magnetic ﬁeld from close to the Feshbach resonance at 9.09 G to ﬁelds below 0.5 G for detection takes about 300 ms, which is a lot longer than the time scale of spatial dynamics and atom loss in the condensates. us, we modify the imaging sequence to detect the atomic clouds directly at a magnetic ﬁeld close to the Feshbach resonance where the experiments are performed. e necessary steps are described in this appendix. C.1. Adjusting the imaging frequency for maximum detectivity e linear Zeeman shi at magnetic ﬁelds close to the Feshbach resonance is about 6.3 MHz/(ΔmF ), which is on the same order as the natural linewidth of the employed 87 Rb D2 line of 6.07 MHz [67]. us the frequency of the imaging laser beam has to be adjusted for a maximum atomic absorption. We experimentally determine the best frequency by using the zero-ﬁeld calibration as a starting point and optimizing the signal to noise ratio, i.e. the deduced atom number as shown in Figure C.1. e frequency of the imaging beam is optimized for resonance to the (F = 2) ↔ (F′ = 3) transition of the D2 line. e two atomic clouds are detected consecutively. Atoms in state |2⟩ are imaged ﬁrst and lost due to photon recoil in the absorption process. Aerwards atoms in state |1⟩ are transferred to state |2⟩ by a repumping laser and subsequently detected. As shown in Figure C.1 the sensitivities for atoms in state |1⟩ and |2⟩ are both maximal at the same frequency, but their relative sensitivities are signiﬁcantly diﬀerent. is eﬀect can be caused by an ineﬃcient transfer of atoms from state |1⟩ to |2⟩ by the repumping laser, since its frequency is not adjusted for high ﬁeld imaging. is leads to an underestimation of the atom number in the F = 1 manifold. In addition diﬀerent magnetic sublevels mF are populated when directly imaging atoms in state |2⟩ compared to the situation aer the atoms in state |1⟩ have been repumped to the F = 2 manifold. us, diﬀerent Clebsch-Gordan coeﬃcients inﬂuence the relative detectivity of the two components as well. 99 C. Calibration of in-situ imaging near the Feshbach resonance detected atom number 25000 20000 15000 10000 5000 00 |1 |2 2 4 6 frequency shift (MHz) 8 Figure C.1.: Detected number of atoms in the two hyperﬁne states aer a π/2-pulse when varying the frequency of the imaging laser beam relative to the resonance at low magnetic ﬁeld. e sensitivity is maximal at a shi of 4.7 MHz, but very diﬀerent for the two species. e real number of atoms in the condensate is kept constant. e dashed line indicates the chosen working point. C.2. Absolute atom number calibration In order to estimate the detection error in the absolute atom number we compare the measurement at high magnetic ﬁeld with the well characterized [26] imaging method at low ﬁeld for diﬀerent atom numbers (realized by variation of the ﬁnal value of the evaporation ramp in the optical dipole traps). is comparison is only possible for a single species atomic cloud prepared in the F=1 hyperﬁne manifold, since neither spin-relaxation loss nor Feshbach loss reduce the atom number while ramping down the magnetic ﬁeld. Atom loss due to heating or movement of the atomic cloud (e.g. caused by magnetic ﬁeld gradients during the magnetic ﬁeld ramp) are assumed to be small since neither the shape nor the position of the condensate changes during the ramp. e le panel of Figure C.2 summarizes these calibration measurements along with a linear ﬁt that is used to correct for the detection error when imaging at high ﬁeld. e number of atoms in the F = 1 manifold is underestimated by a factor 1.5. e detection error for atoms in the F = 2 manifold is estimated by comparing the detected atom number with the corrected atom number in F = 1 aer a π/2-pulse, which reliably creates an equal superposition of the two states.¹ e right panel of Figure C.2 shows the resulting imbalance versus total atom number. As the imbalance is close to zero the imaging calibration for the F = 2 atoms does not need further adjustment. e remaining dri of the imbalance originates from non-linearities in the calibration of the F = 1 atom number. C.3. Imaging in the presence of a linear coupling field In many experiments the amplitude of the linear coupling ﬁeld is varied during the experimental cycle, for example when performing ramps or to compensate for atom loss. In this case the programming of the arbitrary waveform generator takes up to 20 s depending on the details of the sequence. In order to minimize variations in the duration of the experimental cycle the generator is programmed only once for a series of measurements where the time evolution of the atomic clouds ¹Several oscillation cycles are evaluated for the measurement of the Rabi frequency. A sinusoidal ﬁt to the oscillations provides a value for the Rabi frequency that is nearly independent of the details of the atom number calibration. 100 C.3. Imaging in the presence of a linear coupling ﬁeld 50000 0.2 40000 0.1 imbalance atom number reference 60000 30000 20000 -0.1 10000 00 0.0 20000 40000 60000 detected atom number at high field raw corrected 0 20000 40000 total atom number Figure C.2.: (le) Detected number of atoms in the F=1 manifold against the results of a reference measurement at low magnetic ﬁeld. e detected atom number is underestimated by a factor 1.5. A linear ﬁt (solid line) is used as a calibration curve for deducing the real atom number. (right) e imbalance aer a π/2-pulse both for raw (black circles) and corrected (gray squares) F = 1 atom numbers is used to approximate errors in the detection of F = 2 atoms. Aer correction of the F = 1 atom numbers, the imbalance is close to zero meaning that the detection error on the F=2 atom number is small. e remaining dri of the imbalance with the total atom number is caused by a non-linearity in the atom number detection, i.e. a quadratic contribution which is not compensated when using the linear ﬁt as a calibration. is measured for a given sequence of the linear coupling. us, images at early evolution times (e.g. during the ramp) are taken in the presence of the coupling ﬁeld. By comparing measurements with and without a linear coupling we found the detected number of atoms in the F = 1 (F = 2) manifold to change by a factor 1.02 ± 0.06 (1.04 ± 0.04) compatible with 1. 101 Bibliography [1] S. Sachdev. antum Phase Transitions. Cambridge University Press, 2 edition, 2011. [2] T Donner, S Rier, T Bourdel, A Ol, M Köhl, and T Esslinger. Critical behavior of a trapped interacting Bose gas. Science, 315(5818):1556–8, 2007. [3] Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Baelier, and Jean Dalibard. Berezinskii–Kosterlitz–ouless crossover in a trapped atomic gas. Nature, 441(7097):1118– 1121, 2006. [4] Markus Greiner, Olaf Mandel, Tilman Esslinger, eodor W. Hansch, and Immanuel Bloch. antum phase transition from a superﬂuid to a Mo insulator in a gas of ultracold atoms. Nature, 415(6867):39–44, 2002. [5] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner. Probing the Superﬂuid-to-Mo Insulator Transition at the Single-Atom Level. Science, 329:547, 2010. [6] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger. Dicke quantum phase transition with a superﬂuid gas in an optical cavity. Nature, 464:1301–1306, 2010. [7] C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson. Spontaneous vortices in the formation of Bose-Einstein condensates. Nature, 455:948–951, 2008. [8] L E Sadler, J M Higbie, S R Leslie, M Vengalaore, and D M Stamper-Kurn. Spontaneous symmetry breaking in a quenched ferromagnetic spinor Bose-Einstein condensate. Nature, 443(7109):312–5, 2006. [9] Jochen Kronjäger, Christoph Becker, Parvis Soltan-Panahi, Kai Bongs, and Klaus Sengstock. Spontaneous Paern Formation in an Antiferromagnetic antum Gas. Phys. Rev. Le., 105(9):090402, 2010. [10] Manuel Endres, Takeshi Fukuhara, David Pekker, Marc Cheneau, Peter Schauβ, Christian Gross, Eugene Demler, Stefan Kuhr, and Immanuel Bloch. e/Higgs/’amplitude mode at the two-dimensional superﬂuid/Mo insulator transition. Nature, 487(7408):454–458, 2012. [11] K. Baumann, R. Mol, F. Brennecke, and T. Esslinger. Exploring Symmetry Breaking at the Dicke antum Phase Transition. Phys. Rev. Le., 107(14):140402, 2011. [12] Chen-Lung Hung, Xibo Zhang, Nathan Gemelke, and Cheng Chin. Observation of scale invariance and universality in two-dimensional Bose gases. Nature, 470(7333):236–9, 2011. [13] Xibo Zhang, Chen-Lung Hung, Shih-Kuang Tung, and Cheng Chin. Observation of quantum criticality with ultracold atoms in optical laices. Science, 335(6072):1070–2, 2012. [14] Marc Cheneau, Peter Barmeler, Dario Polei, Manuel Endres, Peter Schauss, Takeshi Fukuhara, Christian Gross, Immanuel Bloch, Corinna Kollath, and Stefan Kuhr. Light-conelike spreading of correlations in a quantum many-body system. Nature, 481(7382):484–7, 2012. 103 Bibliography [15] M Gring, M Kuhnert, T Langen, T Kitagawa, B Rauer, M Schreitl, I Mazets, D Adu Smith, E Demler, and J Schmiedmayer. Relaxation and prethermalization in an isolated quantum system. Science, 337(6100):1318–22, 2012. [16] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalaore. Colloquium : Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83(3):863–883, 2011. [17] Ilya M. Merhasin, Boris A. Malomed, and Rodislav Driben. Transition to miscibility in a binary Bose–Einstein condensate induced by linear coupling. Journal of Physics B: Atomic, Molecular and Optical Physics, 38(7):877, 2005. [18] Jacopo Sabbatini, Wojciech H. Zurek, and Mahew J. Davis. Phase Separation and Paern Formation in a Binary Bose-Einstein Condensate. Phys. Rev. Le., 107(23):230402, 2011. [19] B. D. Esry and Chris H. Greene. Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates. Phys. Rev. A, 59(2):1457–1460, Feb 1999. [20] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases. Rev. Mod. Phys., 82(2):1225–1286, 2010. [21] Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Grynberg, and Pierre Meystre. AtomPhoton Interactions: Basic Processes and Applications. New York: Wiley, 1992. [22] E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, P. G. Kevrekidis, and M. K. Oberthaler. Rabi Flopping Induces Spatial Demixing Dynamics. Phys. Rev. Le., 107(19):193001, 2011. [23] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler. Nonlinear atom interferometer surpasses classical precision limit. Nature, 464(7292):1165–1169, 2010. [24] Tilman Zibold, Eike Nicklas, Christian Gross, and Markus K. Oberthaler. Classical Bifurcation at the Transition from Rabi to Josephson Dynamics. Phys. Rev. Le., 105(20):204101, 2010. [25] C Gross, H Strobel, E Nicklas, T Zibold, N Bar-Gill, G Kurizki, and M K Oberthaler. Atomic homodyne detection of continuous-variable entangled twin-atom states. Nature, 480(7376):219– 23, 2011. [26] W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu, J. Tomkovic, D. Hume, and M. Oberthaler. Optimized absorption imaging of mesoscopic atomic clouds. to be published. [27] Lev Pitaevskii and Sandro Stringari. Bose-Einstein Condensation. Oxford University Press, New York, 2003. [28] Chaohong Lee. Universality and Anomalous Mean-Field Breakdown of Symmetry-Breaking Transitions in a Coupled Two-Component Bose-Einstein Condensate. Phys. Rev. Le., 102(7):070401, 2009. [29] E. Gross. Structure of a quantized vortex in boson systems. Il Nuovo Cimento (1955-1965), 20(3):454–477, 1961. [30] L. P. Pitaevskii. Vortex Lines in an imperfect Bose Gas. Soviet Physics JETP-USSR, 13(2), 1961. [31] Harold J. Metcalf and Peter Van der Straten. Laser cooling and trapping. Springer Verlag, 1999. 104 Bibliography [32] Christian Gross. Spin squeezing and non-linear atom interferometry with Bose-Einstein condensates. PhD thesis, University of Heidelberg, 2010. [33] C. P. Search and P. R. Berman. Manipulating the speed of sound in a two-component BoseEinstein condensate. Phys. Rev. A, 63(4):043612, 2001. [34] M. J. Steel and M. J. Colle. antum state of two trapped Bose-Einstein condensates with a Josephson coupling. Phys. Rev. A, 57(4):2920–2930, 1998. [35] Masahiro Kitagawa and Masahito Ueda. Squeezed spin states. Phys. Rev. A, 47(6):5138–5143, 1993. [36] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein. Atom-chip-based generation of entanglement for quantum metrology. Nature, 464(7292):1170–1173, 2010. [37] B. D. Josephson. Possible new eﬀects in superconductive tunnelling. Physics Leers, 1(7):251– 253, 1962. [38] Anthony J. Legge. Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Rev. Mod. Phys., 73(2):307–356, 2001. [39] Tilman Zibold. Classical Bifurcation and Entanglement Generation in an Internal Bosonic Josephson Junction. PhD thesis, University of Heidelberg, 2012. [40] R. Gati and M. K. Oberthaler. A bosonic Josephson junction. Journal of Physics B: Atomic, Molecular and Optical Physics, 40(10):–61, 2007. [41] Paolo Tommasini, E. J. V. de Passos, A. F. R. de Toledo Piza, M. S. Hussein, and E. Timmermans. Bogoliubov theory for mutually coherent condensates. Phys. Rev. A, 67(2):023606, 2003. [42] P. Ao and S. T. Chui. Binary Bose-Einstein condensate mixtures in weakly and strongly segregated phases. Phys. Rev. A, 58(6):4836–4840, 1998. [43] E. Timmermans. Phase Separation of Bose-Einstein Condensates. Phys. Rev. Le., 81(26):5718– 5721, 1998. [44] P. B. Blakie, R. J. Ballagh, and C. W. Gardiner. Dressed states of a two component Bose-Einstein condensate. Journal of Optics B: antum and Semiclassical Optics, 1(4):378, 1999. [45] C. P. Search, A. G. Rojo, and P. R. Berman. Ground state and quasiparticle spectrum of a two-component Bose-Einstein condensate. Phys. Rev. A, 64(1):013615, 2001. [46] M. Abad and A. Recati. A study of coherently coupled two-component Bose-Einstein Condensates. ArXiv e-prints, 2013. [47] Tin-Lun Ho and V. B. Shenoy. Binary Mixtures of Bose Condensates of Alkali Atoms. Phys. Rev. Le., 77(16):3276–3279, 1996. [48] H. Pu and N. P. Bigelow. Properties of Two-Species Bose Condensates. Phys. Rev. Le., 80(6):1130–1133, 1998. [49] Marek Trippenbach, Krzysztof Góral, Kazimierz Rzazewski, Boris Malomed, and Y. B. Band. Structure of binary Bose-Einstein condensates. Journal of Physics B: Atomic, Molecular and Optical Physics, 33(19):4017, 2000. 105 Bibliography [50] Kevin E Strecker, Guthrie B Partridge, Andrew G Trusco, and Randall G Hulet. Formation and propagation of maer-wave soliton trains. Nature, 417(6885):150–3, 2002. [51] Elena V. Goldstein and Pierre Meystre. asiparticle instabilities in multicomponent atomic condensates . Phys. Rev. A, 55(4):2935–2940, 1997. [52] C. K. Law, H. Pu, N. P. Bigelow, and J. H. Eberly. “Stability Signature” in Two-Species Dilute Bose-Einstein Condensates. Phys. Rev. Le., 79(17):3105–3108, 1997. [53] V. P. Mineev. eory of Solution of two almost perfect Bose Gases. Sov. Phys. JETP, 40:132, 1974. [54] W. B. Colson and Alexander L. Feer. Mixtures of Bose liquids at ﬁnite temperature. Journal of Low Temperature Physics, 33(3):231–242, 1978. [55] Bogdan Damski and Wojciech H. Zurek. antum phase transition in space in a ferromagnetic spin-1 Bose–Einstein condensate. New Journal of Physics, 11(6):063014, 2009. [56] Jacopo Sabbatini, Wojciech H. Zurek, and Mahew J. Davis. Causality and defect formation in the dynamics of an engineered quantum phase transition in a coupled binary Bose–Einstein condensate. New Journal of Physics, 14(9):095030, 2012. [57] P. Ao and S. T. Chui. Two stages in the evolution of binary alkali Bose-Einstein condensate mixtures towards phase segregation. Journal of Physics B: Atomic, Molecular and Optical Physics, 33(3):535, 2000. [58] Kenichi Kasamatsu and Makoto Tsubota. Multiple Domain Formation Induced by Modulation Instability in Two-Component Bose-Einstein Condensates. Phys. Rev. Le., 93(10):100402, 2004. [59] H. Pu and N. P. Bigelow. Collective Excitations, Metastability, and Nonlinear Response of a Trapped Two-Species Bose-Einstein Condensate. Phys. Rev. Le., 80(6):1134–1137, 1998. [60] S. De, D. L. Campbell, R. M. Price, A. Putra, B. M. Anderson, and I. B. Spielman. enched binary Bose-Einstein condensates: spin domain formation and coarsening. arXiv preprint 1205.1888, 2013. [61] Bernd Eiermann. Kohärente nichtlineare Materiewellendynamik - Helle atomare Solitonen -. PhD thesis, University of Heidelberg, 2004. [62] Michael Albiez. Observation of nonlinear tunneling of a Bose-Einstein condensate in a single Josephson junction. PhD thesis, University of Heidelberg, 2005. [63] Andreas Weller. Dynamics and Interaction of Dark Solitons in Bose-Einstein Condensates. PhD thesis, University of Heidelberg, 2009. [64] G. Reinaudi, T. Lahaye, Z. Wang, and D. Guéry-Odelin. Strong saturation absorption imaging of dense clouds of ultracold atoms. Opt. Le., 32(21):3143–3145, 2007. [65] Jens Appmeier. Bose-Einstein condensates in a double well potential: A route to quantum interferometry. Master’s thesis, University of Heidelberg, 2007. [66] ibaut Jacqmin, Julien Armijo, Tarik Berrada, Karen V. Kheruntsyan, and Isabelle Bouchoule. Sub-Poissonian Fluctuations in a 1D Bose Gas: From the antum asicondensate to the Strongly Interacting Regime. Phys. Rev. Le., 106(23):230405, Jun 2011. 106 Bibliography [67] D. A. Steck. Rubidium 87 D Line Data, 2010. [68] Helmut Strobel. antum Spin Dynamics in Mesoscopic Bose-Einstein Condensates. Master’s thesis, University of Heidelberg, 2011. [69] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell. Eﬀect of cold collisions on spin coherence and resonance shis in a magnetically trapped ultracold gas. Phys. Rev. A, 66(5):053616, 2002. [70] C. Deutsch, F. Ramirez-Martinez, C. Lacroûte, F. Reinhard, T. Schneider, J. N. Fuchs, F. Piéchon, F. Laloë, J. Reichel, and P. Rosenbusch. Spin Self-Rephasing and Very Long Coherence Times in a Trapped Atomic Ensemble. Phys. Rev. Le., 105(2):020401, 2010. [71] M. Egorov, R. P. Anderson, V. Ivannikov, B. Opanchuk, P. Drummond, B. V. Hall, and A. I. Sidorov. Long-lived periodic revivals of coherence in an interacting Bose-Einstein condensate. Phys. Rev. A, 84(2):021605, 2011. [72] M. Erhard, H. Schmaljohann, J. Kronjäger, K. Bongs, and K. Sengstock. Measurement of a mixed-spin-channel Feshbach resonance in 87 Rb . Phys. Rev. A, 69(3):032705, 2004. [73] Artur Widera, Olaf Mandel, Markus Greiner, Susanne Kreim, eodor W. Hänsch, and Immanuel Bloch. Entanglement Interferometry for Precision Measurement of Atomic Scaering Properties. Phys. Rev. Le., 92(16):160406, 2004. [74] Satoshi Tojo, Yoshihisa Taguchi, Yuta Masuyama, Taro Hayashi, Hiroki Saito, and Takuya Hirano. Controlling phase separation of binary Bose-Einstein condensates via mixed-spinchannel Feshbach resonance. Phys. Rev. A, 82(3):033609, 2010. [75] Timo Oenstein. A New Objective for High Resolution Imaging of Bose-Einstein Condensates. Master’s thesis, University of Heidelberg, 2006. [76] D. S. Hall, M. R. Mahews, J. R. Ensher, C. E. Wieman, and E. A. Cornell. Dynamics of Component Separation in a Binary Mixture of Bose-Einstein Condensates. Phys. Rev. Le., 81(8):1539–1542, 1998. [77] K. M. Mertes, J. W. Merrill, R. Carretero-González, D. J. Frantzeskakis, P. G. Kevrekidis, and D. S. Hall. Nonequilibrium Dynamics and Superﬂuid Ring Excitations in Binary Bose-Einstein Condensates. Phys. Rev. Le., 99(19):190402, 2007. [78] Yun Li, P. Treutlein, J. Reichel, and A. Sinatra. Spin squeezing in a bimodal condensate: spatial dynamics and particle losses. Eur. Phys. J. B, 68(3):365–381, 2009. [79] M. Egorov, B. Opanchuk, P. Drummond, B. V. Hall, P. Hannaford, and A. I. Sidorov. Precision measurements of s-wave scaering lengths in a two-component Bose-Einstein condensate. ArXiv e-prints, 2012. [80] M. R. Andrews, D. M. Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle. Propagation of Sound in a Bose-Einstein Condensate. Phys. Rev. Le., 79(4):553–556, Jul 1997. [81] M. R. Andrews, D. M. Stamper-Kurn, H. J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Keerle. Erratum: Propagation of Sound in a Bose-Einstein Condensate [Phys. Rev. Le. 79, 553 (1997)]. Phys. Rev. Le., 80(13):2967–2967, Mar 1998. 107 Bibliography [82] J. Dziarmaga. Dynamics of a quantum phase transition and relaxation to a steady state. Advances in Physics, 59:1063–1189, 2010. [83] Jürgen Berges, Sebastian Scheﬄer, and Dénes Sexty. Boom-up isotropization in classicalstatistical laice gauge theory. Phys. Rev. D, 77(3):034504, 2008. [84] MR Mahews, DS Hall, DS Jin, JR Ensher, CE Wieman, EA Cornell, F Dalfovo, C Minniti, and S Stringari. Dynamical response of a Bose-Einstein condensate to a discontinuous change in internal state. Physical Review Leers, 81(2):243–247, 1998. [85] E. A. Cornell, D. S. Hall, M. R. Mahews, and C. E. Wieman. Having it both ways: Distinguishable yet phase-coherent mixtures of Bose-Einstein condensates. J. Low Temp. Phys., 113, 1998. JILA Pub. 6133. [86] J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler. Squeezing and entanglement in a Bose–Einstein condensate. Nature, 455(7217):1216, 2008. [87] M. R. Mahews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J. Holland, J. E. Williams, C. E. Wieman, and E. A. Cornell. Watching a Superﬂuid Untwist Itself: Recurrence of Rabi Oscillations in a Bose-Einstein Condensate. Phys. Rev. Le., 83(17):3358–3361, 1999. [88] J. Williams, R. Walser, J. Cooper, E. A. Cornell, and M. Holland. Excitation of a dipole topological state in a strongly coupled two-component Bose-Einstein condensate. Phys. Rev. A, 61(3):033612, 2000. [89] Stewart D. Jenkins and T. A. Brian Kennedy. Spin squeezing in a driven Bose-Einstein condensate. Phys. Rev. A, 66(4):043621, 2002. [90] Stewart D. Jenkins and T. A. B. Kennedy. Dynamic stability of dressed condensate mixtures. Phys. Rev. A, 68(5):053607, 2003. [91] Y-J Lin, K Jiménez-García, and I B Spielman. Spin-orbit-coupled Bose-Einstein condensates. Nature, 471(7336):83–6, 2011. [92] N Timoney, I Baumgart, M Johanning, A F Varón, M B Plenio, A Retzker, and Ch Wunderlich. antum gates and memory using microwave-dressed states. Nature, 476(7359):185–8, 2011. [93] Norman F. Ramsey. A Molecular Beam Resonance Method with Separated Oscillating Fields. Phys. Rev., 78(6):695–699, 1950. [94] A. Chandran, A. Erez, S. S. Gubser, and S. L. Sondhi. Kibble-Zurek problem: Universality and the scaling limit. Physical Review B, 86(6):064304, 2012. [95] C. De Grandi, V. Gritsev, and A. Polkovnikov. ench dynamics near a quantum critical point. Phys. Rev. B, 81(1):012303, 2010. [96] T. W. B. Kibble. Some implications of a cosmological phase transition. Physics Reports, 67(1):183–199, 1980. [97] W. H. Zurek. Cosmological experiments in superﬂuid helium? Nature, 317:505–508, 1985. [98] T. W. B. Kibble. Topology of cosmic domains and strings. Journal of Physics A: Mathematical and General, 9(8):1387, 1976. 108 Bibliography [99] T.W.B. Kibble. Phase-Transition Dynamics in the Lab and the Universe. Physics Today, 60:47, 2007. [100] W. H. Zurek. Cosmological experiments in condensed maer systems. Physics Reports, 276:177–221, 1996. [101] I. Chuang, B. Yurke, R. Durrer, and N. Turok. Cosmology in the laboratory - Defect dynamics in liquid crystals. Science, 251:1336–1342, 1991. [102] M. J. Bowick, L. Chandar, E. A. Schiﬀ, and A. M. Srivastava. e Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals. Science, 263:943–945, 1994. [103] R. Monaco, J. Mygind, and R. J. Rivers. Zurek-Kibble Domain Structures: e Dynamics of Spontaneous Vortex Formation in Annular Josephson Tunnel Junctions. Phys. Rev. Le., 89(8):080603, Aug 2002. [104] R. Monaco, J. Mygind, and R. J. Rivers. Spontaneous ﬂuxon formation in annular Josephson tunnel junctions. Phys. Rev. B, 67(10):104506, Mar 2003. [105] S. Ducci, P. L. Ramazza, W. González-Viñas, and F. T. Arecchi. Order Parameter Fragmentation aer a Symmetry-Breaking Transition. Phys. Rev. Le., 83(25):5210–5213, Dec 1999. [106] P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock, and C. D. H. Williams. Generation of defects in superﬂuid 4 He as an analogue of the formation of cosmic strings. Nature, 368:315– 317, 1994. [107] M. E. Dodd, P. C. Hendry, N. S. Lawson, P. V. E. McClintock, and C. D. H. Williams. Nonappearance of Vortices in Fast Mechanical Expansions of Liquid 4 He through the Lambda Transition. Physical Review Leers, 81:3703–3706, 1998. [108] V. M. H. Ruutu, V. B. Eltsov, A. J. Gill, T. W. B. Kibble, M. Krusius, Y. G. Makhlin, B. Pla̧cais, G. E. Volovik, and W. Xu. Vortex formation in neutron-irradiated superﬂuid 3 He as an analogue of cosmological defect formation. Nature, 382:334–336, 1996. [109] C. Bäuerle, Y. M. Bunkov, S. N. Fisher, H. Godfrin, and G. R. Picke. Laboratory simulation of cosmic string formation in the early Universe using superﬂuid 3 He. Nature, 382:332–334, jul 1996. [110] D V Freilich, D M Bianchi, A M Kaufman, T K Langin, and D S Hall. Real-time dynamics of single vortex lines and vortex dipoles in a Bose-Einstein condensate. Science, 329(5996):1182– 5, 2010. [111] Wojciech H. Zurek. Causality in Condensates: Gray Solitons as Relics of BEC Formation. Phys. Rev. Le., 102(10):105702, 2009. [112] E. Witkowska, P. Deuar, M. Gajda, and K. Rz̧ażewski. Solitons as the Early Stage of asicondensate Formation during Evaporative Cooling. Physical Review Leers, 106(13):135301, apr 2011. [113] Bogdan Damski. e Simplest antum Model Supporting the Kibble-Zurek Mechanism of Topological Defect Production: Landau-Zener Transitions from a New Perspective. Phys. Rev. Le., 95(3):035701, 2005. 109 Bibliography [114] Wojciech H. Zurek, Uwe Dorner, and Peter Zoller. Dynamics of a antum Phase Transition. Phys. Rev. Le., 95(10):105701, 2005. [115] Jacek Dziarmaga. Dynamics of a antum Phase Transition: Exact Solution of the antum Ising Model. Phys. Rev. Le., 95(24):245701, 2005. [116] David Chen, Mahew White, Cecilia Borries, and Brian DeMarco. antum ench of an Atomic Mo Insulator. Phys. Rev. Le., 106(23):235304, 2011. [117] M. Mielenz, J. Brox, S. Kahra, G. Leschhorn, M. Albert, T. Schaetz, H. Landa, and B. Reznik. Trapping of Topological-Structural Defects in Coulomb Crystals. Physical Review Leers, 110(13):133004, 2013. [118] S. Ulm, J. Roßnagel, G. Jacob, C. Degünther, S. T. Dawkins, U. G. Poschinger, R. Nigmatullin, A. Retzker, M. B. Plenio, F. Schmidt-Kaler, and K. Singer. Observation of the Kibble-Zurek scaling law for defect formation in ion crystals. ArXiv e-prints, 2013. [119] Moritz Höfer. Observation of Paern Formation in a enched Binary Bose-Einstein Condensate. Master’s thesis, University of Konstanz, 2012. [120] Nuno D. Antunes, Pedro Gandra, and Ray J. Rivers. Is domain formation decided before or aer the transition? Phys. Rev. D, 73(12):125003, 2006. [121] Toshiya Kinoshita, Trevor Wenger, and David S Weiss. Observation of a one-dimensional Tonks-Girardeau gas. Science, 305(5687):1125–1128, 2004. [122] Maxime Joos. Phase contrast imaging of mesoscopic Bose-Einstein condensates. Master’s thesis, University of Heidelberg, 2013. [123] Florian Meinert, Manfred J. Mark, Emil Kirilov, Katharina Lauber, Philipp Weinmann, Andrew J. Daley, and Hanns-Christoph Nägerl. Many-body quantum quench in an atomic onedimensional Ising chain. arXiv preprint 1304.2628, 2013. [124] Ferdinand Brennecke, Rafael Mol, Kristian Baumann, Renate Landig, Tobias Donner, and Tilman Esslinger. Real-time observation of ﬂuctuations at the driven-dissipative Dicke phase transition. arXiv preprint 1304.4939, 2013. [125] Marco Koschorreck, Daniel Pertot, Enrico Vogt, and Michael Köhl. Universal spin dynamics in two-dimensional Fermi gases. arXiv preprint 1304.4980, 2013. [126] A. M. Kaufman, R. P. Anderson, omas M. Hanna, E. Tiesinga, P. S. Julienne, and D. S. Hall. Radio-frequency dressing of multiple Feshbach resonances. Phys. Rev. A, 80(5):050701, 2009. [127] Artur Widera, Fabrice Gerbier, Simon Fölling, Tatjana Gericke, Olaf Mandel, and Immanuel Bloch. Precision measurement of spin-dependent interaction strengths for spin-1 and spin-2 87 Rb atoms. New Journal of Physics, 8(8):152, 2006. [128] E. A. Burt, R. W. Ghrist, C. J. Mya, M. J. Holland, E. A. Cornell, and C. E. Wieman. Coherence, Correlations, and Collisions: What One Learns about Bose-Einstein Condensates from eir Decay. Phys. Rev. Le., 79(3):337–340, 1997. [129] J. Söding, D. Guéry-Odelin, P. Desbiolles, F. Chevy, H. Inamori, and J. Dalibard. ree-body decay of a rubidium Bose–Einstein condensate. Applied Physics B: Lasers and Optics, 69(4):257– 261, 1999. 110 Bibliography [130] Satoshi Tojo, Taro Hayashi, Tatsuyoshi Tanabe, Takuya Hirano, Yuki Kawaguchi, Hiroki Saito, and Masahito Ueda. Spin-dependent inelastic collisions in spin-2 Bose-Einstein condensates. Phys. Rev. A, 80(4):042704, 2009. [131] M. Olshanii. Atomic Scaering in the Presence of an External Conﬁnement and a Gas of Impenetrable Bosons. Phys. Rev. Le., 81(5):938–941, 1998. [132] Chiara Menoi and Sandro Stringari. Collective oscillations of a one-dimensional trapped Bose-Einstein gas. Phys. Rev. A, 66(4):043610, 2002. [133] L. Salasnich, A. Parola, and L. Reao. Eﬀective wave equations for the dynamics of cigarshaped and disk-shaped Bose condensates. Phys. Rev. A, 65(4):043614, 2002. [134] Luca Salasnich and Boris A. Malomed. Vector solitons in nearly one-dimensional BoseEinstein condensates. Phys. Rev. A, 74(5):053610, 2006. [135] M. D. Feit, J. A. Fleck Jr., and A. Steiger. Solution of the Schrödinger equation by a spectral method. Journal of Computational Physics, 47(3):412–433, 1982. [136] Juha Javanainen and Janne Ruostekoski. Symbolic calculation in development of algorithms: split-step methods for the Gross–Pitaevskii equation. Journal of Physics A: Mathematical and General, 39(12):–179, 2006. [137] Rolf Rannacher. Einührung in die Numerische Mathematik, 2006. Vorlesungsscriptum. [138] Rolf Rannacher. Numerische Mathematik 1, 2009. Vorlesungsscriptum. 111 Danksagung / Acknowledgments • Zu allererst möchte ich mich bei Markus Oberthaler daür bedanken, dass er mir die Möglichkeit gegeben hat in seiner Gruppe zu arbeiten. Mit seiner unermüdlichen Begeisterung, der oﬀenen Art und dem fairen Umgang sorgt er ür eine phantastische Atmosphäre in der Gruppe. Vielen Dank ür das Vertrauen, die Unterstützung, die erhellenden Diskussionen, die vielen Fragen und noch wesentlich mehr Antworten. • Bedanken möchte ich mich auch bei omas Gasenzer ür die Bereitscha meine Arbeit zu begutachten. Die Diskussionen der letzten Jahre haben mir dabei geholfen viele Aspekte unserer Experimente aus anderen Blickwinkeln zu betrachten. • Panayotis Kevrekidis taught me almost everything I know about numerical methods for the simulation of Bose-Einstein condenstates. anks a lot for the patient and thorough explanations, the enlightening discussions and the great hospitality in Amherst. • A big thank you goes to Mahew Davis and Jacopo Sabbatini, who introduced me to the many fascinating aspects of the Kibble-Zurek mechanism and clariﬁed many of my questions in long discussions by email, on the phone or in person. • Isabelle Bouchoule has contributed fundamental ideas and calculations to the understanding of the experiments in the ﬁnal stages of this work. ank you for the support. • Ein sehr großer Dank gilt Moritz Höfer der mit mir zusammen viele der vorgestellten Messungen durchührt hat. Danke ür die vielen unterhaltsamen und produktiven Tage und Abende im Labor und den unermüdlichen Einsatz bei der Arbeit, aber auch auf dem Rad oder in den Bergen. • Die Unterstützung von Aisling Johnson war großartig. Vielen Dank ür die Erklärungen, das gemeinsame Grübeln und die vielen guten Kommentare und Vorschläge beim Lesen dieser Arbeit. • Wolfgang Müssel und Helmut Strobel danke ich ür die unglaubliche Unterstützung in jeglicher Hinsicht: Für die richtigen Fragen, die klare Sicht auf die Physik, die gründlichen Gedanken und die vielen Stunden in denen ihr diese Arbeit Korrektur gelesen habt. • Bei Tilman Zibold bedanke ich mich ür die vielen gemeinsamen und immer unterhaltsamen Stunden und die immerwährende Diskussionsbereitscha. • Christian Gross danke ich ür das Einühren in das Experiment, die vielen Erklärungen, die tolle Zusammenarbeit, die vielen Stunden im Labor und die Bloch Kugel. • Ich möchte mich bei Jirka Tomkovic ür die schöne gemeinsame Zeit in Büro und Labor bedanken, ür die selbstlose Hilfsbereitscha und die vielen interessanten Diskussionen. • Bei Ion Streoscu und Daniel Linnemann bedanke ich mich ür die vielen guten Vorschläge und aufmunternden Worte beim Schreiben dieser Arbeit und die gute Zusammenarbeit. 113 • I would like to thank David Hume for many helpful discussions and the great collaboration. • Ich hae das große Glück mit den besten Kollegen zusammenarbeiten zu dürfen, die man sich nur vorstellen kann, dem BEC Team: Jerome, Stefano, Andreas, Jens-Philipp, Naida, Elisabeth, Simon, Jonas, Philipp, Maxime, Mike, … • Ebenso tragen alle anderen Maerwavers zur phantastischen Stimmung und Solidarität in der Gruppe bei. Ob beim Diskutieren über Physik, beim Miagessen, bei Kaﬀee oder Bier, im Schnee, auf Bergen oder im Keller. Ich hae immer einen riesigen Spaß mit euch Nalis, Aas und Aegislern! • Unseren team assistants Dagmar und Christiane gebührt ein großer Dank ür die unkomplizierte Hilfe bei der Bewältigung der großen und kleinen bürokratischen Hürden. • Der Verwaltung des KIP möchte ich danken, dass eben diese Hürden auf Minimalhöhe gestutzt wurden. • Der mechanischen Werksta gilt mein Dank ür die präzise Erstellung wesentlicher Komponenten unseres experimentellen Auaus. • Für ihre Geduld und ihr Verständnis möchte ich all denjenigen Freunden danken, mit denen ich in den letzten Monaten viel zu wenig Zeit verbracht habe. • Ich danke meinen Eltern ür die permanente Unterstützung in allen Fragen des Lebens, das Verständnis, die Freude, die langen Spaziergänge und den immer guten Rat. • Meiner Frau Konstanze danke ich, dass sie immer ür mich da ist, mir die Augen öﬀnet, mit mir lacht, mich versteht, mich mitnimmt und zurückholt. Hannes danke ich ür die immer gute Laune, das Lächeln, das Grinsen, das Krächzen, das Lachen und das Schmunzeln. Ebenso gilt mein Dank Emma ür das Klopfen, das nasse Gesicht und den warmen Rücken. 114

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