thesis eike nicklas

thesis eike nicklas
y ( m) y ( m)
A new tool for miscibility control:
Linear coupling
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Eike Nicklas
2013
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Dissertation
submied 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 vielältige 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 Eigenschaen
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 . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . .
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5. A miscible-immiscible phase transition
5.1. Non-adiabatic generation of dressed states . . . . . . . . . . . . . . . . . . . . . . .
5.1.1. Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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 . . . . . . . . .
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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 . . . . . . . .
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7. Conclusion and Outlook
A. Summary of atomic and experimental parameters
A.1. Properties of 87 Rb . . . . . . . . . . . . . . . . . . . . .
A.1.1. Scaering lengths . . . . . . . . . . . . . . . . .
A.1.2. Loss coefficients . . . . . . . . . . . . . . . . . .
A.1.3. Scaering lengths near the Feshbach resonance
A.2. Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3. Optical dipole traps . . . . . . . . . . . . . . . . . . . .
A.3.1. Charger . . . . . . . . . . . . . . . . . . . . . .
A.3.2. Waveguide . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . .
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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 aer 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 laer 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 laices [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 laer 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.
Aer having independently studied the relevant ingredients for the miscible-immiscible transition we characterize the phase transition and measure the linear response of the system aer 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: Aer 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 aer 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 scaering. us, atomic interactions are characterized by a single parameter,
the s-wave scaering length as . A positive scaering length as > 0 denotes repulsive interactions,
while as < 0 implies araction 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 scaering 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
scaering 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
¹e 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 wrien 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 wrien
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
²is 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 wrien
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 oen 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
Aer 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 scaering 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 scaering 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 scaering 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 scaering 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
Aer 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 aractive 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 arac-
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 aractive 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)
⁶e (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 scaering 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.
Aer 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 wrien 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 scaering lengths a11 − a22 ≪ a11 , a22 , a12 , as it is the case for 87 Rb,
Equation 2.23 can be wrien 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 scaering 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 shis 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 scaering 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 scaering parameters Δ < 1 the linear coupling shis 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 wrien 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
¹⁰e 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 scaering parameters are ploed 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 wrien 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 shied 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 scaering 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 scaering
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 paern 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 paern 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 shied 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
¹¹e 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 shied 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 shied 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 ploed 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 scaering
parameters. In the immiscible case the instability is shied 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 aains unstable modes for negative values of Ω. Similarly to the
immiscible case long wavelength modes become unstable first and the instability is shied 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 shied 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)
scaering parameters. (le) e miscible system becomes unstable for Ω < 0 and a
decreasing Ω shis 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 shied 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 scaering 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
Aer 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 laer case and lead to regions where only one component is present surrounded
by domains of the other component. As these paerns 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. Aer 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. Aer condensation, a magnetic bias field for tuning of the inter-species
scaering 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
(boom). 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 seings 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 paerns 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 paerns 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 aer 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 beer
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 spliing of
6.834 GHz [67]. e required radio frequency is determined by the Zeeman spliing, 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 shis the relative spliing 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 shis 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 shis 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 seings used during the initial π/2-pulses light shis 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 scaering 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.: Scaering 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
scaering length a12 of these states by up to 30%. e background scaering 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 paerns 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 paern 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
spliing 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 scaering lengths of 87 Rb. Aerwards 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 scaering 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 scaering 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
scaering 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 scaering 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. Aer
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 scaering 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 beer 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 shis 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 aer t = 32 ms) are shown in false color. e density
paern 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
paern of spin domains, which will be dominated by the unstable modes of the excitation spectrum.
As the resulting spatial paern 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 paerns 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 paerns.
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 shis of the oscillatory structures. us they can be averaged
over different experimental realizations of the same physical seings (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 dris 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 boom 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 aer 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 (boom 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 omied 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 scaering length a12 . When approaching the Feshbach resonance a12 is increased which shis 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. Fiing 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 scaering 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 aer 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 shied 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 paerns 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 paerns. 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 dris 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 paern 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 paerns. 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, oen 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 aer ≈ 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, aer 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, aer
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 laice sites by a standing wave potential. In each well, Rabi oscillations are
measured for 130 ms and fied with a sine. e accumulated phase difference aer 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 aribute 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 laice 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 fied 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 laice 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 aributed 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 shis 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 rewrien 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
²e 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 scaering lengths a++ = a−− = 14 (a11 + a22 + 2a12 ) and a+− =
1
2 (a11 + a22 ) [33, 89], which take over the role of the atomic scaering 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 scaering lengths, this corresponds to a12 > 12 (a11 + a22 ). us, for equal intra-species
scaering 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 shied 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 (boom
row) are inferred. For comparison, the dashed lines show the corresponding data of the
numerical simulations, where the dressed state profiles in the boom 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 fiing 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
aributed to phase separation of the dressed states, whose miscibility criterion is opposite to that of
the atomic states. In particular, the intra-species scaering 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 scaering 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. Aer 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 shied 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 boom 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⟩ aer 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 scaering 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 aer 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 aer 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 (boom 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 aer 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 aer 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 (boom 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 (boom 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 aributed 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 aer 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 aer 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 paerns. 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 aer 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 aer
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 spliing 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 aer the π/2-pulse by dressing the atoms with a
strong linear coupling field of Ω = 2π × 340 Hz aer 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 aer 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 aer 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 aer 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 aer 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
aer 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 aer
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. Aer
the quench we observe a linear growth in ξ with a slope independent of the value of ε Aer 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 ploed for different distances ε from the critical point. Aer
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 aer 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 scaering 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. Seing 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 aer 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 fiing 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 aer 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 aer the kink with hold times where the measured
correlation lengths ξ deviate from the fied slope of the initial dynamics. e evolution of the
correlation lengths in this regime is also fied 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 ploed in the le panel. e overlap of the correlation
functions increases significantly aer 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 aer 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 aer 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.
²e 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
Aer 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 shied 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 paern 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 aer 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 aer 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 aer 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 aer 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
aer the quench are depicted in Figure 5.11. e unstable excitation modes cause the growth of a
periodic spin paern, 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 ploed 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 scaering 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 shied 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
paern. 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
paern with the point spread function of the imaging system. As the domain size decreases, this
effect eventually reduces the amplitude of the observed paern 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 paern aer 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. Aer 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 ploed 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 scaering 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 scaering 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 scaering 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 scaering
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 aer 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 aer 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 aer 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 paern. us the characteristic length scale, the size of the domains, is determined as soon as the modulation depth of the spin paern 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. Aer 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 aer 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 aer 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
aer 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 aer 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 aer 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 aer 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. Aer 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 paern. 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 maer 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 aer 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.
¹is 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 paern. 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.
²is 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 ploed 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
aer 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 aer 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 aer 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 aer 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 aer 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
aer 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
aer the crossing of the critical point remain unclear [114]. It was argued that the size of the defects is actually determined aer 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 aer the initial π/2-pulse with ramp durations of τ Q = 1 . . . 20 ms. e
linear coupling field is switched off aer 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 aer 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 aer fast quenches τ Q < 9 ms, while large structures emerge aer 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 aer 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 aer 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 aer 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 aer 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 aer 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 aer 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 paern 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 paern 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 paern 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
aer the crossing of the critical point on the emerging domain paern 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 aer 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 scaering
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 laices [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. Scaering 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 scaering 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. Scaering 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 scaering 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 spliing 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 beer 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 aer 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 laer 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 aer 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-aer 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 scaering 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-Kua 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-Kua 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-Kua
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 scaering 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. Aerwards 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 aer 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 aer 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 aer 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 aer 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. Aer 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
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111
Danksagung / Acknowledgments
• Zu allererst möchte ich mich bei Markus Oberthaler daür 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 Mahew 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 durchührt 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 Einühren 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 hae 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 Maerwavers zur phantastischen Stimmung und Solidarität in
der Gruppe bei. Ob beim Diskutieren über Physik, beim Miagessen, bei Kaffee oder Bier, im
Schnee, auf Bergen oder im Keller. Ich hae immer einen riesigen Spaß mit euch Nalis, Aas
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 Auaus.
• 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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