albiez dissertation

albiez dissertation
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
Michael Albiez
born in: Freiburg i.Br., Germany
Oral examination: October 19, 2005
Observation of nonlinear tunneling of a
Bose-Einstein condensate in a single
Josephson junction
Referees:
Prof. Dr. Markus K. Oberthaler
Prof. Dr. Jörg Schmiedmayer
Beobachtung von nichtlinearer Tunneldynamik eines Bose-Einstein Kondensates in einem Josephsonkontakt
In der vorliegenden Arbeit wird die erste experimentelle Realisierung eines Josephsonkontaktes für Bose-Einstein Kondensate beschrieben. Das physikalische System zweier
schwach gekoppelter BECs steht in enger Analogie zu den bekannten Josephsonkontakten in Supraleitern und Suprafluiden. Die schwache Kopplung wird in unserem Experiment mit Hilfe eines optischen Doppelmuldenpotentials realisiert, in das das BEC
hinein geladen wird. Es entsteht durch die Superposition eines eindimensionalen optischen Gitters und zweier gekreuzter fokussierten Laserstrahls, die einen dreidimensionalen harmonischen Einschluß erzeugt. Die Tunneldynamik in diesem bosonischen
Josephson Kontakt zeigt zwei deutlich unterscheidbare dynamische Regimes. Für kleine
anfängliche Besetzungzahlunterschiede der beiden Potentialminima werden nahezu sinusförmige Josephson-Oszillationen beobachtet, die sowohl durch eine oszillierende Besetzung als auch durch eine oszillierende relative Phase der beiden BECs charakterisiert
ist. Wird der anfängliche Besetzungsunterschied größer gewählt als ein kritischer Wert,
beobachtet man, daß sich die Besetzung im Laufe der Zeit kaum mehr ändert, wohingegen
die relative Phase der beiden Wellenpakete linear anwächst. Dieses dynamische Verhalten, das als macroscopic self-trapping“bezeichnet wird, wird dadurch verursacht, daß
”
resonantes Tunneln aufgrund stark unterschiedlicher Wechselwirkungsenergien in den
beiden Potentialtöpfen unterdrückt ist und hat keine direkte Analogie in supraleitenden
Josephsonkontakten.
Observation of nonlinear tunneling of a Bose-Einstein condensate in a single
Josephson junction
In this thesis, I present the first experimental implementation of a Josephson junction for
Bose-Einstein condensates. The weak link between two BECs constitutes the nonlinear
generalization of the well known Josephson junction of weakly-coupled superconductors
that are separated by a thin insulating barrier. In our experiment, the required overlap
of two macroscopic wavefunctions is provided by loading a BEC into an optical double
well potential. It is realized by a superposition of a one-dimensional optical lattice
with a focused laser beam optical dipole trap. The tunneling dynamics between the
two potential wells exhibits two distinct dynamical regimes. For small initial population
imbalances of the two wells we observe nearly sinusoidal Josephson tunneling oscillations,
which are characterized by an oscillating population and relative phase. The situation
changes drastically, if the initial population imbalance is chosen above a critical value. In
this case, resonant tunneling between the two wells is prohibited because the difference
between the on-site particle interaction energies in the two wells exceeds the tunneling
energy splitting. As a consequence, the atomic distribution becomes self-locked and the
relative phase evolves unbound in time. This regime of prohibited tunneling, which has
no analogon in superconducting Josephson junctions, is called “macroscopic quantum
self-trapping”.
To Karen, Clara and Luca.
Contents
1
Introduction
2
Josephson effect in superconductors and superfluids
2.1 Fundamental description of Josephson tunneling: the Josephson equations
2.2 RCSJ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Shapiro effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Ultrasmall Josephson junctions: Coulomb blockade . . . . . . . .
2.3 Josephson effect in superfluid helium . . . . . . . . . . . . . . . . . . . .
5
6
10
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14
3
Theory of Josephson junctions for Bose-Einstein condensates
3.1 Meanfield description for BECs: the Gross-Pitaevskii equation
3.2 Non-polynomial nonlinear Schrödinger equation . . . . . . . .
3.3 Two-mode approximation . . . . . . . . . . . . . . . . . . . .
3.3.1 Zero phase modes . . . . . . . . . . . . . . . . . . . . .
3.3.2 Macroscopic quantum self-trapping . . . . . . . . . . .
3.3.3 π-phase modes . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Phase plane portrait . . . . . . . . . . . . . . . . . . .
3.3.5 Limits of the two-mode description . . . . . . . . . . .
3.4 Extended two-mode approximation: variable tunneling model
3.5 External and internal bosonic Josephson junctions . . . . . . .
3.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Split-step Fourier method . . . . . . . . . . . . . . . .
3.6.2 Ground state: imaginary time propagation . . . . . . .
3.6.3 Numerical results: stationary states . . . . . . . . . . .
3.6.4 Numerical results: dynamics . . . . . . . . . . . . . . .
3.7 Feasibility study for bosonic Josephson junction experiments .
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4
Experimental setup and procedure
4.1 Laser system . . . . . . . . . . . . . . . . . . . . . .
4.2 Vacuum, funnel and MOT . . . . . . . . . . . . . . .
4.3 Magnetic trap . . . . . . . . . . . . . . . . . . . . . .
4.4 Optical dipole potentials . . . . . . . . . . . . . . . .
4.4.1 Crossed optical dipole trap . . . . . . . . . . .
4.4.2 Realization of an optical double well potential
4.5 Stabilization of intensity and phase . . . . . . . . . .
4.6 Mechanical stability and beam position control . . . .
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i
Contents
4.7
4.8
4.9
5
6
7
Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Imaging setup and optical resolution . . . . . . . . . . . . . . . . . . . .
New experimental setup for the optical lattice . . . . . . . . . . . . . . .
Experimental results
5.1 Calibration of the double well potential . . . . . . . . .
5.1.1 Harmonic confinement . . . . . . . . . . . . . .
5.1.2 Lattice spacing . . . . . . . . . . . . . . . . . .
5.1.3 Potential depth . . . . . . . . . . . . . . . . . .
5.2 Determination of dynamical variables . . . . . . . . . .
5.2.1 Fractional population imbalance . . . . . . . . .
5.2.2 Relative phase . . . . . . . . . . . . . . . . . . .
5.3 Preparation of the initial population imbalance . . . .
5.4 Observation of Josephson oscillations and self-trapping
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Experiments with BECs in optical lattices
6.1 Dispersion management . . . . . . . . . . . . . . . . . . . . .
6.2 Continuous dispersion management . . . . . . . . . . . . . . .
6.3 Bright atomic gap solitons for atoms with repulsive interaction
6.4 Macroscopic quantum self-trapping in periodic potentials . . .
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81
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. 95
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Conclusion and outlook
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7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
References
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59
60
64
108
1 Introduction
One of the most prominent and striking features of quantum mechanics is the tunneling of massive particles through classically forbidden regions. Although tunneling is a
purely quantum mechanical effect, it can be observed on a macroscopic scale in a system
described by two weakly linked macroscopic wavefunctions with global phase coherence.
This macroscopic tunneling, which is referred to as the “Josephson effect”, was predicted
by Brian D. Josephson in 1962 [1], who was rewarded the nobel price in physics in 1973.
The first experimental evidence of the corresponding current-phase relation has already been observed in 1963 [2] in a superconducting Josephson junction, which is a
device consisting of two coupled superconductors separated by a thin insulating barrier.
The discovery of the Josephson effect opened not only a new important chapter of physics
but also new horizons for a wide variety of applications [3]. Nowadays Josephson junctions are used for example in SQUIDS (Superconducting Quantum Interference Devices)
which allow to measure extremely small magnetic fields with a resolution on the order
of 10−14 T.
The theory of the Josephson effect was developed in the context of superconductors
but can nevertheless be applied to all physical systems described by weakly coupled
macroscopic wavefunctions. Due to the difficulty of creating a weak link between two
neutral superfluids, the Josephson effect in superfluid helium has only been observed in
1997 [4]. In those experiments, the weak link between two macroscopic wavefunctions
was realized by means of a membrane with nanoscopic holes.
In this thesis, the first experimental implementation of a single Josephson junction
in a new experimental system, which can be described by phase-coherent macroscopic
wavefunctions, is presented: macroscopic tunneling in Bose-Einstein condensates.
Nonlinear macroscopic quantum mechanics
Based on the quantum statistical description of indistinguishable bosonic particles by
Satyendra Nath Bose [5], Einstein predicted a quantum phase transition to a fascinating new state of matter [6]. This transition occurs, when the de Broglie wavelength
of the atoms becomes comparable to the distance between the atoms. The successful
achievement of this so-called Bose-Einstein condensation (BEC) in a dilute gas in 1995 [7,
8, 9] was made possible by the development of novel cooling techniques - laser cooling and
evaporative cooling of neutral atoms. The transition to the quantum-degenerate regime
allows for studying matter at the level of a single macroscopically occupied quantum
state. This provides a new source of coherent macroscopic matterwaves, which can
be regarded as an ideal state for studies and applications of quantum mechanics. BoseEinstein condensates of weakly interacting alkali atoms allow the investigation of neutral
superfluids in a controlled and experimentally tunable environment. Nowadays, Bose-
1
Chapter 1 Introduction
Einstein condensates of alkali atoms are routinely generated in many laboratories all
over the world.
A striking feature of Bose-Einstein condensates is their coherent nature on the micrometer scale. This phase coherence has been impressively demonstrated in the first
interference experiments at the MIT [10] where two independent Bose-Einstein condensates were created in a double well potential, realized by focussing a blue detuned laser
beam into the center of the confining magnetic trap. After turning off all traps the two
BECs expand and start to overlap forming clear atomic interference patterns.
Many interesting properties of BECs result from the atom-atom interaction that
can be either attractive or repulsive. Combined with the ability to manipulate BECs
with magnetic and optical potentials, this opened up the way to many outstanding
experiments [11]. The peculiarity of these trapped alkali vapors is that their densities
are sufficiently low to be treated in the framework of the well established theory of weakly
interacting Bose gases. This description provides a simple language to explain most of
the so far performed experiments with BECs in terms of a few physical quantities.
The new field of nonlinear matter wave optics is a highly interdisciplinary branch of
physics which is closely connected to statistical mechanics, atomic physics, photon optics
and solid state physics. The realization of atom lasers [12] for example has pointed out
analogies to photon optics. Nonlinear optics, where the nonlinear Kerr effect strongly
affects the propagation of intense light pulses through optical fibers, found its parallel
in atom-optics by the creation of dark [13] and bright [14, 15, 16] matter wave solitons.
These non-spreading localized wavepackets form, if the linear dispersion of the atoms
is canceled by the nonlinear particle interaction. Additionally, parallels to condensed
matter physics have been demonstrated by the observation of vortices in a stirred BoseEinstein condensate [17].
Bose-Einstein condensates in optical lattices
During the last years, the dynamical behavior of Bose-Einstein condensates in optical
lattices has been intensely investigated theoretically and experimentally. As known from
solid state physics, electrons in periodic potentials exhibit a modified dispersion relation.
The concepts developed in solid state theory can be directly transferred to the description
of Bose-Einstein condensates in optical lattices. This analogy has been confirmed by the
observation of Bloch-oscillations [18], Wannier-Stark ladders [19], Bragg-scattering [20]
[21] and Landau-Zener tunneling [22] for Bose-Einstein condensates. At the beginning
of my PhD period, we have implemented “dispersion management” [23, 24] for atomic
matterwaves, which has allowed for the creation of bright atomic gap solitons of atoms
with repulsive interaction [16]. The highlight of the experiments with Bose-Einstein
condensates in optical lattices was the observation of the quantum phase transition from
a superfluid to a Mott insulating phase of atoms [25], which directly shows the analogies
to solid state physics.
Josephson effect in Bose-Einstein condensates
The first observation of a Josephson atomic current in a one-dimensional Josephson junction array formed by a deep optical lattice has been reported in [26]. However, because
of the small spacing of approximately 400 nm of the individual Josephson junctions, the
2
dynamics of a single junction could not be resolved experimentally. This restricted the
observation to a measurement of the center of mass oscillations inside the superimposed
harmonic trap and to a continuous probing of the phase coherence of the system. Recently, we have observed the nonlinear effect of self-trapping in a Josephson junction
array [27] with a similar experimental setup.
In order to investigate the direct analogon of a superconducting Josephson junction,
one has to implement a single weak link between two Bose-Einstein condensates. This can
be done in two different experimental systems. The most straight forward realization
of the weak coupling is loading a BEC into a double well potential which provides a
small wavefunction overlap (external Josephson effect). This physical system has been
extensively studied theoretically [28, 29, 30, 31, 32, 33, 34].
Additionally, Josephson effects have also been predicted to exist in a system of two
BECs in different hyperfine levels trapped in a single harmonic trap in the presence of a
weak driving field that couples these hyperfine states [35, 36] (internal Josephson effect).
The effect of macroscopic tunneling in a single bosonic Josephson junction manifests
itself in a coherent exchange of atoms between the two wells or correspondingly between
the two hyperfine states. If a BEC is loaded into a double well potential with a small
initial population imbalance, it can oscillate between the wells by quantum mechanical
tunneling. This regime of Josephson oscillation is characterized by an oscillating population and relative phase around a zero mean value. However, due to the nonlinearity
arising from particle interactions, new dynamical tunneling regimes are predicted in a
weak link of two Bose-Einstein condensates. The most striking effect is the inhibition
of tunneling oscillations between the two wells, if the initial population imbalance is
chosen above a critical value. In this case, resonant tunneling between the particles on
each side of the barrier is prohibited because the difference of the interaction energy between the two components exceeds the tunneling energy splitting. Therefore, the atomic
distribution is self-locked inside a single well, and the relative phase of the two wavepackets increases unbound. This new dynamical regime is called “macroscopic quantum
self-trapping”. It has no direct analogon in superconducting Josephson junctions, where
nonlinear self-interaction effects can generally be neglected because of the small fraction
of 10−10 of tunneling Cooper pairs compared to the total number of electrons.
Double well potentials for Bose-Einstein condensates have been created in timeorbiting magnetic traps [37], by a combination of two focused red detuned laser beams
[38] and on atom chips [39]. It has recently been shown that Bose-Einstein condensates
in a double well potential can be prepared with a fixed phase-relation [40] (each realization of the experiment reproduces the same relative phase), which is a prerequisite for
the experimental observation of macroscopic quantum tunneling. However, up to now
nobody has succeeded in observing the Josephson effect between two weakly coupled
Bose-Einstein condensates. The main reason is that in order to realize tunneling times
on an observable timescale on the order of 100 ms, the distance of the two wells has to be
on the scale of a few micrometers [40]. For usual atom numbers of several ten thousand
atoms, working with such small traps implies that the dynamics is always meanfield
dominated which inhibits the occurrence of Josephson tunneling between the two wells.
In principle, Feshbach resonances[41] could be used to tune the strength of interatomic
interaction in order to make Josephson tunneling observable. However, they are difficult
to implement experimentally for the case of 87 Rb [42].
3
Chapter 1 Introduction
In our experiment, we prepare 87 Rb BECs consisting of approximately 1000 atoms
in a double well potential with a well-spacing of 4.4 µm. The stable experimental setup
combined with the optical resolution of 2.7(2) µm of our imaging system makes it possible
for the first time to observe Josephson tunneling in a single bosonic Josephson junction.
This experiment is a generalization of the familiar superconducting Josephson contact
and reveals new insight into the well known tunneling dynamics. Since the relative phase
between two BECs can be directly deduced by observing atomic interference fringes,
the full complex macroscopic wavefunction can be determined experimentally. Both
dynamical regimes - Josephson tunneling and macroscopic quantum self-trapping - are
experimentally accessible. In the presented experiments, the quantum mechanical effect
of tunneling is indeed macroscopic in a sense that the tunneling of massive particles can
be observed directly by means of a simple optical imaging system.
Outline of this thesis
This thesis consists of three sections: an introductory part (chapter 2), a theoretical part
(chapter 3) and a technical and experimental part (chapters 4, 5 and 6).
Chapter 2 gives a fundamental description of the physics of two weakly coupled
macroscopic wavefunctions and a short introduction to Josephson effects in superconductors and superfluids. Chapter 3 treats the theoretical description of Josephson junctions
for Bose-Einstein condensates. After a short introduction to the general theory of BoseEinstein condensates, the two-mode approximation for a BEC in a double well potential
is presented. Subsequently, an extended two-mode model will be discussed and compared
to the results of a numerical integration of the Schrödinger equation. The experimental
procedure and the setup for cooling 87 Rb atoms to quantum degeneracy and preparing
the Bose-Einstein condensate in a double well potential is described in chapter 4. At
this I have focused on the active stabilization of the optical potentials that is crucial
for the realization of a bosonic Josephson junction. In Chapter 5, the experimental results on macroscopic tunneling of Bose-Einstein condensates are presented. Experiments
that have been performed during the first two years of my PhD period are presented in
chapter 6. The results of four publications are discussed: dispersion management and
continuous dispersion management in weak periodic potentials, the observation of bright
atomic gap solitons and the transition from the tunneling to the self-trapping regime in a
one-dimensional optical lattice. Finally, the conclusion and an outlook to feasible future
experiments, which can be performed with the current setup is given in chapter 7.
4
2 Josephson effect in superconductors
and superfluids
The phenomenon of superconductivity was first observed by Heike Kamerlingh Onnes
in 1911. He succeeded in producing liquid Helium at a temperature of 4.2 K in 1908
and was able to subsequently cool other materials down to 1.7 K. When measuring the
electric resistance of mercury, he observed a phenomenon which was incompatible with
the state of knowledge of solid state physics. At that time, the behavior of the electrical resistance close to zero temperature was extensively and controversially discussed.
Lord Kelvin of Largs (1824-1907) predicted an infinite resistance at T = 0 assuming
that electronic motion freezes in at low temperatures. In contrast, Heinrich Matthiessen
(1830-1906) predicted a finite resistance and Heike Kamerlingh Onnes supposed an asymptotic decrease to zero.
When he cooled mercury with liquid Helium, the resistance decreased at first continuously as predicted by solid-state theory. But at a temperature of 4.2 K it changed
abruptly to a non-measurable level as shown in figure 2.1. Kamerlingh Onnes called this
phenomenon superconductivity and the corresponding temperature critical temperature.
Figure 2.1: First observation of superconductivity in mercury cooled by superfluid helium (Leiden
1911, from [43]).
Only twenty years after the observation of superconductivity, theoreticians came up
5
Chapter 2 Josephson effect in superconductors and superfluids
with first theoretical models. The interpretation of superconductivity as a macroscopic
quantum phenomenon was proposed by F. London in 1935 [44]. Ginzburg and Landau
[45] developed a modification of the London theory by introducing a complex parameter
Ψ which is the quantum mechanical wavefunction of superconducting electrons. This
single wavefunction describes a macroscopic number of electrons which are assumed to
“condense” into the same macroscopic quantum state. They succeeded in expanding
the theory of thermodynamic phase transitions which allowed to explain all fundamental
features of superconductivity.
The microscopic basis and a quantum mechanical description of superconductivity
was first developed in 1957 by Bardeen, Cooper and Schrieffer in the framework of the
BCS theory [46]. Within this approach, the effect of superconductivity is attributed to
the existence of paired electrons, so-called Cooper pairs. The electron-electron attraction
leading to this pairing arises from electron-phonon interaction, which is required to
be larger than the inter-electronic Coulomb repulsion. In contrast to single electrons,
Cooper pairs have an integer spin and therefore obey Bose-Einstein statistics instead
of Fermi-Dirac statistics, which explains the existence of a single macroscopic quantum
mechanical wavefunction. Within the framework of the BCS theory, the existence of
supercurrents (currents without resistance) is explained by the appearance of a finite
energy gap in the excitation spectrum of Cooper pairs. This gap corresponds to the
Cooper pair binding energy which is typically on the order of a few meV.
One of the most important features of superconductors is the phase correlation which
means that the quantum mechanical phase is constant over the whole superconductor
allowing for macroscopic tunneling as we will see in the following section.
2.1 Fundamental description of Josephson tunneling: the
Josephson equations
A Josephson junction consists of two superconducting electrodes separated by a thin
insulating barrier. For a barrier thickness of more than typically 30 Å, no electronic
transport between the two superconductors takes place. When reducing the barrier
thickness to 30 Å, single electrons can tunnel and the barrier basically acts as an electrical resistance between the two superconductors. If the thickness of the barrier is further
reduced to typically 10 Å or below, the quantum mechanical macroscopic wavefunctions
of the two superconductors begin to overlap within the classically forbidden region. This
system is often referred to as weak link of two superconductors. In a semi-classical particle model, this overlap corresponds to Cooper pair transport from one superconductor
to the other, a phenomenon called Josephson tunneling. This macroscopic tunneling
effect was theoretically predicted [1] by the graduate student Brian D. Josephson who
investigated weak links of superconductors at the University of Cambridge in 1962.
In these Josephson junctions the phase correlation is transmitted through the dielectric barrier leading to supercurrents without any electrical resistance, i.e. without any
voltage drop across the barrier. The phenomenon of tunneling in a Josephson junction is
therefore often called “weak superconductivity”. For details on geometry and fabrication
of the various junction types I refer to [47].
In deriving the basic Josephson equations I follow the approach of Feynman [48],
which can also be found in superconductor textbooks [3, 43, 49]. Let us consider two
6
2.1 Fundamental description of Josephson tunneling: the Josephson equations
superconductors SL and SR separated by an insulating barrier as shown in figure 2.2.
SL
JR,L
Insulator
SR
JL,R
I
R
Uext
Figure 2.2: Superconducting tunnel junction provided by a thin insulating layer with a typical
thickness of 1 nm between two superconductors SL and SR . JLR and JRL denote the tunneling current
densities in both directions. The weakly overlapping macroscopic wavefunctions are indicated by ΨL
and ΨR . Tunneling Cooper pairs are replaced by the external current source, which therefore suppresses
charge-imbalances across the junction.
If the barrier is thick enough so that the two superconductors are insulated, the
Schrödinger equation for each macroscopic wavefunction reads
i~
∂
ΨL = EL ΨL
∂t
(2.1)
i~
∂
ΨR = ER ΨR .
∂t
In the presence of a weak coupling between the superconductors the two equations couple
via an additional tunneling energy term
i~
∂
ΨL = EL ΨL + KΨR
∂t
(2.2)
i~
∂
ΨR = ER ΨR + KΨL ,
∂t
where K is the coupling constant for the two wavefunctions having a weak overlap in the
barrier region. The two macroscopic wavefunctions can be described using the following
ansatz
ΨL =
√
ρL eiφL ;
ΨR =
√
ρR eiφR ,
(2.3)
where ρL , ρR denote the Cooper pair densities and φL , φR the phases of the Cooper pair
wavefunctions that are assumed to be constant over the individual superconductors.
Substituting this Ansatz into the coupled wave equations 2.2, separating the results
into real and imaginary part and subtracting the equivalent equations yields
7
Chapter 2 Josephson effect in superconductors and superfluids
∂
2K √
(ρL ) =
ρR ρL sin(φR − φL )
∂t
~
(2.4)
∂
1
(φR − φL ) =
(ER − EL ) + K cos(φR − φL )
∂t
~
r
ρL
−
ρR
r
ρR
ρL
,
∂
∂
where we employed the continuity equation ∂t
ρL = − ∂t
ρR for the exchange of Cooper
pairs between left and right. If we define the relative phase between the two superconductors φ = φR − φL and, for the sake of convenience, assume identical superconductors
(i.e. ρL = ρR = ρ), we get
∂
2K
∂
(ρL ) =
ρ sin(φ) = − (ρR )
∂t
~
∂t
(2.5)
∂
1
(φ) =
(ER − EL ).
∂t
~
Note that although ρL and ρR are assumed to be constant, they have a non-vanishing
∂
time derivative ∂t
(ρ). There is no contradiction if we take into account the continuous
replacement of tunneling Cooper pairs by the external current source. Those feeding
currents are not included in our derivation, but they would not change our expression
for macroscopic tunneling1 . We can derive the electrical current density by multiplying the Cooper pair current density with the Cooper pair charge 2e. For the case of
−EL
applied to the
superconductors, the energy difference is given by a voltage U = ER2e
Josephson junction. This leads to the two Josephson equations for the current density
and the evolution of the relative phase
J = J0 sin(φ)
(2.6)
∂
2eU
φ =
∂t
~
where J0 = 4eK
ρ is the maximum allowed Cooper pair tunneling current density.
~
Note that the derivation of the two Josephson equations is only based on the existence of macroscopic wavefunctions and is therefore also valid for other physical systems
like superfluid helium or Bose-Einstein condensates in the non-interacting limit (when
replacing the applied voltage across the Josephson junction by a more general energy
difference between left and right macroscopic wavefunction).
In the absence of an external voltage U = 0 the phase difference is constant, but
not necessarily zero. In this case, equation 2.6 predicts, that a finite current with a
maximum value I0 = J0 A can flow through the barrier without any voltage drop across
the junction, where A is the contact area of the Josephson junction. This phenomenon is
called DC Josephson effect [1] and has already been demonstrated in 1963 by Anderson
and Rowell [2] in their implementation of the first Josephson junction.
1
8
For a complete self-consistent derivation see for example [50].
2.1 Fundamental description of Josephson tunneling: the Josephson equations
Figure 2.3: Typical DC current-voltage characteristics of a Josephson junction at zero temperature.
Currents up to a current density J0 can flow without applied external voltage (DC-Josephson effect).
For junction voltages larger than the quasiparticle-gap 2∆
e the junction has a finite resistance (single
electron tunneling) and the current contains an oscillating part with frequency ω = 2eV
~ (AC-Josephson
effect)
If a constant Voltage U 6= 0 is applied to the junction, it follows from equation 2.6
t is linearly increasing in time which leads to an alternating
that the phase φ = φ0 + 2eU
~
current with a frequency ω = 2eU
= 2π · 483.6 MHz
U . This is the essence of the AC
~
µV
Josephson effect which was experimentally demonstrated by Shapiro in 1963 using the
so-called Shapiro effect (see section 2.2.1).
How does the DC current-voltage characteristics of a Josephson tunnel junction look
like, if we only take Cooper pair tunneling into account? This question is answered by
figure 2.3. If the current flowing across the junction is smaller than the critical current
J0 given above, the system will be in a state with zero voltage drop. For J > J0
the current has to originate from single electron (quasiparticle) tunneling with a finite
at the
resistance R. Therefore the voltage across the junction has to jump to V = 2∆
e
critical current corresponding to a potential difference between the two superconductors
sufficiently large to break up Cooper pairs (2∆ is the energy gap of the superconductor).
For higher currents, the voltage current characteristics is linear according to Ohms law.
In order to compare the maximum Cooper pair current to the normal single electron
, Ambegaokar and Baraktoff [51] derived the following
current at the voltage-threshold 2∆
e
expression at T = 0:
I0 R(T → 0) =
π
∆.
2e
(2.7)
This result implies that the maximum Cooper pair current I0 is approximately 80% of
the single electron threshold current. The critical current density is typically on the order
A
3 A
2
of 102 cm
this yields a typical
2 − 10 cm2 . For a junction with a contact area of 10 µm
critical current of I0 = 500 µA. Using equation 2.7, this implies a resistance R ≈ 10 Ω
9
Chapter 2 Josephson effect in superconductors and superfluids
Figure 2.4: Equivalent circuit of a real current biased Josephson junction in the framework of the
RCSJ-model. R represents the resistance for single-electron current and C is the finite capacitance of
the junction.
for a typical value ∆ = 3 meV.
2.2 RCSJ-model
Josephson tunneling involves quantum effects that are difficult to grasp intuitively. The
current-voltage characteristics of a real Josephson junction including single electron tunneling is well explained by the resistively and capacitively shunted junction (RCSJ)
model which was first used by Stewart [52] and McCumber [53] in 1968. Within this
approach, the weak link is represented by the simple equivalent circuit in figure 2.4.
According to Kirchhoff‘s law the total current in this circuit is the sum of the superconducting Josephson current Is = I0 sin φ, a quasiparticle tunneling current Iqp = UR
∂
U which takes into account the finite capacitance
and a displacement current IC = C ∂t
of the Josephson junction. Therefore, the current balance equation reads
∂
U
U + + I0 sin φ.
(2.8)
∂t
R
Note that the RCSJ model neglects the effect of thermal noise and any spatial dependency of junction parameters within the junction cross section. The most important
simplification is the assumption of a constant resistance R, both below and above the
gap voltage.
Using the second Josephson relation in equation 2.6 we find the differential equation
for the phase difference φ across the junction
I=C
I=
~C ∂ 2
~ ∂
φ+
φ + I0 sin φ.
2
2e ∂t
2eRn ∂t
(2.9)
~ ∂
h ∂t φit yields the dc currentSolving this differential equation using the relation Udc = 2e
voltage characteristics. For small relative phases φ ≈ 0 and for I → 0 this equation can
be linearized leading to small sinusoidal phase-oscillations with the plasma frequency
0
ωp = 2eI
.
~C
Equation 2.9 can be solved analytically only in the case of vanishing capacitance C.
However, one can gain insight into the possible solutions by bearing in mind a simple
mechanical analog, which is described by a comparable differential equation.
10
2.2 RCSJ-model
Figure 2.5: Mechanical analogons for a Josephson junction: rigid pendulum (a) and a particle
moving on a tilted washboard potential in a viscous fluid (b). This analogons can provide intuitive
insight into the tunneling dynamics in the framework of the RCSJ-model.
Let us consider a rigid pendulum of mass m and length l as shown in figure 2.5a. In
the presence of a restoring torque due to gravity and a damping torque due to friction
this pendulum can be described by the following differential equation for the pendulum
angle φ:
∂
∂2
φ
+
D
φ + mgl sin φ,
(2.10)
∂t2
∂t
where τ is an external constant torque, M = ml2 is the moment of inertia, D is the
damping coefficient and g is the acceleration of gravity. This equation obviously has the
same structure as equation 2.9 with corresponding quantities as given in table 2.1.
τ =M
RCSJ-model
applied current
average (dc) voltage term
phase difference
capacitance term
critical current
rigid pendulum
I
↔
τ
hU it 2e
~
↔
ω=
φ ↔
external torque
∂
φ
∂t
average angular velocity
φ
~
C
2e
↔
M
I0
↔
mgl
angular displacement
moment of inertia
critical torque
Table 2.1: Equivalence between physical corresponding quantities describing a Josephson junction
in the framework of the RCSJ-model and the classical rigid pendulum.
By means of this analogy, we can predict how the Josephson junction will reply with
a dc-voltage when increasing the current I starting from zero. For I < I0 , corresponding
to an applied torque being to small to bring the pendulum to the top position, the
mean angular velocity and correspondingly the mean voltage drop across the junction
will be zero. At I = I0 the pendulum reaches the top position and every further increase
will cause the pendulum to rotate, which implies a non-zero dc voltage drop across the
junction. For very large currents, the mean value of the Josephson term I0 sin φ and
11
Chapter 2 Josephson effect in superconductors and superfluids
Figure 2.6: Current-voltage characteristics obtained by numerically solving the RCSJ-model (eq.
2.9) without (black line) and with (red line) an applied rf-current Irf = 490 µA. The parameters are
I0 = 500 µA, R = 10 Ω, and C = 10−13 F .
the capacitance term in equation 2.9 can be neglected. This leads to a simple linear
dependence I = hURit (Ohm‘s law). A typical current-voltage curve obtained in the
RCSJ-model is shown in figure 2.6 (black line).
Some textbooks (for example Chesca et al. [47]) use another descriptive mechanical
analogon in order to illustrate the RCSJ-model: a particle moving in a tilted washboard
~
(I0 (1 − cos φ) − Iφ) inside a viscous medium (see figure 2.5b). This
potential U = 2e
system is described by a differential equation similar to equation 2.10 and the same
arguments as given for the pendulum analogy also hold in this case.
2.2.1 Shapiro effect
In section 2.1, we found that applying a dc voltage across a Josephson junction causes the
flow of an ac current. In the inverse Josephson effect dc voltages are induced across an
unbiased junction by applying an external microwave field which leads to the appearance
of current steps at constant voltages. The easiest way to understand these steps is to
represent the Josephson junction by the parallel equivalent RCSJ-circuit of figure 2.4,
but now with a total current containing an additional rf-current
It = I + Irf cos(ωrf t).
(2.11)
A numerical solution of this equation provides a staircase current-voltage characteristics
which is often referred to as Shapiro steps. This structure is indicated with the red line
in figure 2.6. For a detailed discussion I refer to [54].
It is mathematically easier to analyze this problem in terms of a circuit, where we
apply a voltage
U (t) = U0 + Urf cos(ωrf t)
(2.12)
to the Josephson junction.
If we insert this voltage into the second Josephson equation 2.6 we get
12
2.2 RCSJ-model
φ(t) = φ0 +
2e
2eUrf
U0 t +
sin(ωrf t),
~
~ωrf
(2.13)
where φ0 is a constant of integration. This yields the following dc Josephson current
2e
2eUrf
I(t) = I0 sin φ0 + U0 t +
sin(ωrf t) .
(2.14)
~
~ωrf
This equation can be expanded in terms of Bessel-functions Jn
I(t) = I0
∞
X
n=1
n
(−1) Jn
2eUrf
~ωrf
sin
2eU0
− nωrf
~
t + φ0 .
(2.15)
0
The time average over the oscillating term is only different from zero if 2eU
= nωrf .
~
This means that if the frequency of the ac-Josephson current corresponds to an integer
multiple of the radio frequency, we get steps of constant voltage in the voltage-current
characteristics of the Josephson junction. These voltages are given by
Un = ±n
~
ωrf .
2e
(2.16)
The height of the n-th Shapiro-step is given by2
In = I0 Jn (
2eUrf
) sin(φ0 ).
~ωrf
(2.17)
2.2.2 Ultrasmall Josephson junctions: Coulomb blockade
So far I have depicted a Josephson junction as an object whose characteristics were
described by the two Josephson equations 2.6. The description in the framework of the
RCSJ-model was exclusively classical and led us to the descriptive mechanical analog of
a rigid pendulum.
Now I want to point out a quantum mechanical feature of Josephson junctions which
is related to the finite capacitance of the junction and based on the quantization of the
electric charge.
We assume a mesoscopic Josephson junction with a capacitance3 C ≈ 10−15 F . As
≈ 3 mV is built
a result of the tunneling of a single Cooper pair a voltage of U = 2e
C
2
up across the junction. This voltage is accompanied by a charging energy EC = 2eC
~
which can be larger than the Josephson coupling energy EJ = 2e
I0 . This leads to
an inhibition of tunneling of further Cooper pairs, provided that thermal fluctuations
are small compared to the charging energy, which implies operation temperatures well
below 1 K. The next Cooper pair cannot tunnel through the barrier until the voltage
dissipates after a typical timescale δt = RC. This inhibition of tunneling due to Coulomb
interaction has been termed Coulomb blockade. It allows a control of the transfer of
a single electron despite of the huge number of typically > 109 free electrons in the
2
As shown in [54], the current steps are all positive because the voltage source has a fixed polarity,
so the sign of φ0 adjusts accordingly.
3
This implies a junction contact area of ≈ 0.01 µm2 , corresponding to an edge length of 100 nm.
13
Chapter 2 Josephson effect in superconductors and superfluids
junction, which makes a realization of a single-electron transistors (SET) possible [55].
For a detailed review about this subject see for example [56].
One effect related to the Coulomb blockade occurs in ordered one or two-dimensional
arrays of Josephson junctions, which can nowadays be manufactured by means of lithographic methods. Here the Mott insulator transition from a macroscopic superconductive
state to an insulating state can be observed when the charging energy of the superconductors becomes larger than the Josephson-tunneling energy. In this case, the number
of Cooper pairs per superconductor gets fixed leading to an undefined relative phase4 .
As we will see in chapter 3.3 the repulsive interparticle interaction of the Bose-Einstein
condensates can lead to an inhibition of macroscopic tunneling. This corresponds to some
extent to the continuous version of the Coulomb blockade described here.
2.3 Josephson effect in superfluid helium
We have seen in section 2.1 that Josephson tunneling of massive particles occurs in systems which can be described by a coherent macroscopic wavefunction. Another example
of a macroscopic quantum system with a global phase coherence is superfluid 3 He and
4
He [57], where the Josephson effects described above have recently been found to have
analogues. In this system, massive atoms can tunnel between weakly linked superfluids.
Here, the viscosity corresponds to the electrical resistance and the Landau critical velocity to the critical current density in superconductors. For a detailed comparison of the
different systems exhibiting Josephson effects see for example [58].
The tunneling experiments I will refer to in this section have been performed in the
group of Packard with superfluid 3 He [59, 4] and in the group of Pearson with 4 He [60].
In the 3 He Josephson experiments, the overlap of two macroscopic wavefunctions is
provided by an array consisting of 4225 small apertures, each of diameter 100 nm and
length 50 nm. In order to realize a weak link between the superfluids, the aperture size
has to be smaller than the superfluid coherence length ξ, which is often referred to as
healing length at the operation temperature
ξ = ξ0 [1 − T /TC ]−1/2 ,
(2.18)
where ξ0 = 65 nm and TC is the transition temperature. In the reported experiments,
the individual apertures in the array act coherently. There is no interference between
adjacent apertures so that the whole array behaves like a single weak link with 4225-fold
increased current. The setup of the 3 He-experiment is shown in figure 2.7.
By applying a pressure gradient ∆P between the two reservoirs, the difference in the
chemical potential becomes ∆µ = 2m3 ∆P/ρ where 2m3 is the mass of a 3 He Cooper
pair and ρ is the liquid density5 . This results in a linear time evolution of the relative
phase φ of the two superfluids according to the second Josephson equation
φ(t) =
4
∆µ
t,
~
(2.19)
The Mott insulator transition has also been observed for a BEC in a three dimensional optical
lattice [25].
5
This pressure gradient is experimentally implemented by changing the potential of the electrode in
figure 2.7 with respect to the metallized soft membrane.
14
2.3 Josephson effect in superfluid helium
Figure 2.7: Experimental setup for the measurement of Josephson oscillations in superfluid 3 He.
The relative phase can be changed by applying a pressure gradient between the two reservoirs on either
side of the microaperture array. This leads to AC-Josephson currents through the weak link. The
particle current can be observed by measuring the displacement of the soft membrane (from [59]).
which leads to ac tunneling-currents with a frequency of 183.7 · 103 Hz/Pa. This small
frequency, which has to be compared to 4.82 · 1014 Hz/V in the superconducting analog,
allows for an observation of the particle tunneling currents.
The instantaneous mass flow of 3 He Cooper pairs through the microaperture array
can be obtained by measuring the displacement of the soft membrane in figure 2.7
I(t) = ρA
∂
x,
∂t
(2.20)
where A is the membrane area. Additionally the membrane’s displacement from the
equilibrium position xe is proportional to the pressure gradient: ∆P = α(x − xe ). The
knowledge of α allows for the determination of the instantaneous phase difference φ(t)
by integrating equation 2.19:
−2m3
φ(t) =
ρ~
Zt
α(x(t) − xe )dt.
(2.21)
0
Equations 2.20 and 2.21 show that it is possible to measure the current-phase relation of
the superfluid weak link by measuring the position of the soft membrane as a function
of time. For temperatures close to TC corresponding to a healing length larger than
the aperture size, the measured I(φ) dependency is nearly sinusoidal as predicted by
the first Josephson equation. For lower temperatures (small healing length compared to
the aperture diameter) the current-phase characteristics I(φ) evolves from sine-like to
almost linear because the weak-link condition is no longer satisfied.
Additionally to the measurement of the current-phase relation, the same group has
also shown the existence of a metastable superfluid state with a phase difference between
15
Chapter 2 Josephson effect in superconductors and superfluids
the two weakly coupled reservoirs which oscillates around a mean value of π [61]. This
state is the analog to the so-called π-junction observable in superconductors in connection
with extrinsic paramagnetic impurities [62] or in connection with the d-wave symmetry
in high Tc superconductors [63]. As we will see in section 3.3 similar π-oscillations can
also be realized in bosonic Josephson junctions.
It is important to note that weakly coupled systems are often discussed in terms
of quantum mechanical tunneling. These tunneling barriers are not necessary for the
occurrence of Josephson effects as shown by the described experiments with liquid helium. The physics governing the effect is only based on the existence of a weak link
or in other words on a small overlap between two macroscopic wavefunctions. It is not
relevant whether this overlap is provided by a tunneling barrier or by small apertures
with dimensions smaller than the healing length of the system.
16
3 Theory of Josephson junctions for BoseEinstein condensates
In our experiment, we use atomic Bose-Einstein condensates as a source of coherent
macroscopic matter waves. The quantum statistics for indistinguishable bosons was introduced by Satyendra Bose [5] in 1924. The phenomenon of Bose-Einstein condensation
of massive bosons has subsequently been predicted by Albert Einstein [6] in 1925. BEC
is the basic process for the occurrence of superconductivity (BEC of cooper pairs) and
superfluidity (BEC of 4 He and pairs of 3 He).
In the following, I want to give a short summary of the theory and characteristics of
Bose-condensed atomic vapors to an extent which is needed for the description of the
performed experiments. For a detailed description of the quantum statistics leading to
BEC see for example [64] and [65]. For further insight into the special case of BECs of
atomic vapors, I recommend the various books and review articles, for example [66, 67,
68].
In the following, a three dimensional trapped gas of bosonic atoms, which for the
moment are assumed to be non-interacting, is described within the framework of the
grand canonical ensemble. The atoms are trapped in an external potential Vext (r) and
occupy the single-particle states of the trap with eigenenergies k . The occupation of
these states is given by the Bose-distribution function
hnk i =
1
(3.1)
e(k −µ)/kB T − 1
where the atom number N is fixed by
N=
X
hnk i.
(3.2)
k
The total number of atoms in the excited states can be written as
Z∞
N − N0 =
6=0
ρ()
− e
0
kB T
d,
(3.3)
−1
where ρ() is the density of states of the atomic gas trapped in the external potential
Vext (r). Here we have replaced the chemical potential by the ground state energy. At
large temperatures the chemical potential is large and negative. As the atomic gas is
cooled down, µ approaches the ground state energy 0 until at a critical temperature
TC it becomes equal to 0 . In this case, the maximum number of atoms in the excited
states becomes limited and the occupation of the ground state N0 becomes a macroscopic
17
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
fraction of the total atom number N . In the limit T → 0 all atoms are in the ground state
of the external confinement. This phenomenon is called Bose-Einstein condensation.
Note, that in all BEC experiments the energy associated with the temperature is much
bigger than the ground state energy1 kB T 0 . One finds that, independent of the trap
geometry, Bose-Einstein condensation occurs if the so-called phase-space density at the
center of the trap reaches the critical value
n0 λ3dB = ζ(3/2) ≈ 2.61.
(3.4)
Here n0 denotes the maximum atomic density,
λdB = √
h
2πmkB T
(3.5)
the thermal de Broglie wavelength, m the atomic mass and ζ Rieman’s zeta-function.
Equation 3.4 reflects the fact that BEC occurs if the de Broglie wavelength is on
the order of the interparticle separation which means that the quantum mechanical
wavefunctions of the atoms start to overlap.
In the case of a harmonic trap, which is encountered in most experimental realizations,
the density of states is given by
1
ρho () = (~ω̄)−3 2 ,
2
where we have defined the mean trapping frequency
(3.6)
1
ω̄ = (ωx ωy ωz ) 3 .
(3.7)
The critical temperature for this case can be obtained by substituting 3.7 into equation
3.3 and integrating over . The phase transition occurs when N0 becomes larger than
zero which leads to
kB TC = ~ω̄
N
ζ(3)
13
1
≈ 0.94 ~ω̄N 3 .
(3.8)
In our experiments with typical trap frequencies ω̄ = 2π · 80 Hz and atom numbers
N = 1000 this equation leads to a critical temperature of TC ≈ 40 nK. The fraction of
atoms in the BEC below the critical temperature is given by
N0
=1−
N
T
Tc
3
.
(3.9)
The presented derivation of the BEC phase-transition has assumed non-interacting bosons
in the thermodynamic limes (N → ∞). This derivation is valid as long as the atomic gas
satisfies the condition for a weakly interacting gas ρ0 a3 1, where a = 5.32 nm is the
s-wave scattering length for 87 Rb. The influence of finite atom numbers and interatomic
interactions slightly decreases the fraction of condensed atoms for a given temperature
[67]. However, for typical experimental parameters this effect is on the scale of less than
one percent.
1
From a classical point of view this is very counterintuitive since according to Boltzmann statistics,
a macroscopic occupation of the ground state is only possible at kB T < 0 .
18
3.1 Meanfield description for BECs: the Gross-Pitaevskii equation
3.1 Meanfield description for BECs: the Gross-Pitaevskii
equation
It is a very difficult task to solve the equations that describe N interacting bosons.
However, Bose-Einstein condensates of atomic vapors are dilute gases of a typical density ρ0 = 1014 /cm3 . In this regime, the interatomic distance is much larger than the
s-wave scattering length a, which describes an effective range of the atom-atom interaction. Therefore elastic two-body low energy collisions are dominant and the scattering
potential can be approximated by a delta-shaped potential
U (r − r0 ) = g3d δ(r − r0 ),
(3.10)
2
a
where g3d = 4π~
denotes the three-dimensional coupling constant. The scattering length
m
a is positive for repulsive interaction and negative for attractive interaction. By assuming
this interaction potential and applying the Bogoliubov approximation2 , E.P. Gross [69,
70] and L.L. Pitaevskii [71] have developed a mean field theory for the approximate
description of BECs at zero temperature. In this framework the condensate is described
by a macroscopic one-body wavefunction Ψ(r, t) and obeys the Gross-Pitaevskii-equation
(GPE)
~2
∂
2
∆ + Vext (r) + g3d N |Ψ(r, t)| Ψ(r, t).
(3.11)
i~ Ψ(r, t) = −
∂t
2m
This description provides a simple language to explain most of the so-far performed
experiments with BECs in terms of a few physical quantities. In this representation,
the condensate wavefunction is normalized to unity and the square of its absolute value
determines the atomic density ρ(r, t) = N ·|Ψ(r, t)|2 . The total energy consists of a kinetic
energy, a potential energy inside the external trap and a meanfield-interaction energy
which is proportional to the square of the wavefunction and therefore often referred to as
nonlinearity. The Gross-Pitaevskii equation can in general not be solved analytically. In
the limiting case of dominant kinetic energy, the GPE reduces to the linear Schrödinger
equation and one recovers the ground state wavefunction in the external potential. In the
opposite case, we can neglect the kinetic energy term, which corresponds to the so-called
Thomas Fermi approximation. The density distribution is then given by
ρ(r) = g −1 (µ − Vext (r)) Θ(µ − Vext (r))
(3.12)
where Θ is the unit step function. In the case of a harmonic external confinement the
shape of the cloud is described by an inverted parabola.
3.2 Non-polynomial nonlinear Schrödinger equation
The numerical integration of the three-dimensional Gross-Pitaevskii equation 3.11 with
an acceptable spatial resolution is a task that requires a sophisticated implementation
2
In the framework of the Bogoliubov theory, the bosonic fieldoperator which annihilates a particle
at position r is replaced by its mean value Ψ(r) = hΨ̂(r)i. This condensate wavefunction is a classical
field which plays the role of the order parameter in the Bose-Einstein phase transition.
19
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
and a huge calculation effort. There have been several attempts to solve the physical
problem by using a one-dimensional equation which to some extent includes the transverse motion of a Bose-Einstein condensate in the corresponding potential [72, 73]. The
necessary approximations are especially justified for cigar shaped BECs where the dynamical evolution is mainly restricted to the longitudinal axis and the transverse states
are close to the harmonic transverse groundstate. We will see in section 3.6 that this
condition is fulfilled for our experimentally realized atom numbers and potentials. As
already mentioned in the PhD thesis of Bernd Eiermann [74], a comparison of the approach of Salasnich et al. [73] and the full numerical solution of the 3d-Gross-Pitaevskii
equation has provided the best agreement. This approach assumes that the transverse
spatial wavefunction of the Bose-Einstein condensate can be described by a Gaussian
distribution
y2 + z2
,
(3.13)
Ψ(y, z, t, σ⊥ (x, t)) ∝ exp −
2σ⊥ (x, t)2
where the transverse Gaussian width σ⊥ (x, t) depends on the longitudinal coordinate x
and on time. The expression for σ⊥ (x, t) is obtained by minimizing the action integral
Z
S=
∂
~2 2
1
2
dt dr Ψ (r) i~ +
∇ − Vext (r) − g3d N |Ψ(r)| Ψ(r)
∂t 2m
2
∗
(3.14)
with respect to σ⊥ (x, t) and yields
p
(3.15)
σ⊥ (x, t)2 = a2⊥ 1 + 2aN |Ψ(x, t)|2 ,
q
√
~
where a⊥ =
is the 1/ e width of the transverse groundstate wavefunction in
mω⊥
a harmonic confinement with a trapping frequency ω⊥ . The effective one-dimensional
Schrödinger equation is then given by [73]
"
i~
2
2
2
#
~ ∂
|Ψ(x, t)|
∂
Ψ(x, t) = −
+ Vext (x) + g1d N p
Ψ(x, t)
2
∂t
2m ∂x
1 + 2aN |Ψ(x, t)|2
"
!#
p
~ω⊥
1
p
+
+ 1 + 2aN |Ψ(x, t)|2
Ψ(x, t),
(3.16)
2
1 + 2aN |Ψ(x, t)|2
where we have introduced the one-dimensional coupling constant g1d = 2~aω⊥ . Equation 3.16 is often referred to as non-polynomial nonlinear Schrödinger equation (NPSE)
and can be numerically implemented with comparatively small effort. This effective
one-dimensional description of the system provides an excellent agreement with our experimental results on Josephson effects between two coupled BECs (see section 3.6.)
3.3 Two-mode approximation
In order to realize a weak link between two 87 Rb Bose-Einstein condensates we load a
BEC into an external effective double well potential, which can be approximated by
1
V (x, y, z) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) + V0 · (cos(πx/q0 ))2 ,
2
20
(3.17)
3.3 Two-mode approximation
where m = 1.44 · 10−25 kg is the mass of a Rubidium atom, ωx = 2π·78 Hz, ωy = 2π·66 Hz
and ωz = 2π·90 Hz are the harmonic trapping frequencies, q0 = 5.2 µm is the spacing
and V0 = h · 413 Hz the potential depth of the optical lattice. The central part of the
density profile in x-direction is sketched in figure 3.1.
Figure 3.1: Double well potential for Bose-Einstein condensates (black line). The localized modes
Φ1 and Φ2 in well 1 and well 2 are represented with red and green lines. E10 and E20 denote the respective
zero point energies. These energies are equal for a symmetric double well potential.
In principal, one has to integrate the GPE or the NPSE given in the last section
by substituting Vext with the effective double well potential in order to describe the
dynamics of the system. Such a numerical solution will be presented in section 3.6.
The condensate tunneling between two adjacent traps which results in an oscillatory
exchange of the atoms between the traps has already been investigated in 1986 [28].
However, the author did not include the many-body interaction between atoms inside a
single well (on-site interaction) which, as we will see in the following chapter is responsible
for a rich variety of different dynamical regimes of the system.
In this chapter an approximation is discussed, which can at least qualitatively describe
the different regimes of nonlinear dynamics of the system of two weakly linked BoseEinstein condensates. The total wavefunction is written as a superposition of two timeindependent spatial wavefunctions localized in the respective well. In such a way, a formal
description of the system in terms of only two dynamical parameters, the fractional
population imbalance and the relative phase can be obtained. These two variables are
canonically conjugate and satisfy two coupled differential equations, which only contain a
few static properties. The mechanical analogon of a nonrigid pendulum will give intuitive
insight into the complicated dynamics of this nonlinear generalization of superconducting
Josephson junctions.
In the following the dynamics of the two coupled systems is constrained to a subspace
spanned by the two-mode wavefunction
Ψ(r, t) = Ψ1 (t)Φ1 (r) + Ψ2 (t)Φ2 (r)
(3.18)
with
Ψ1,2 (t) =
q
N1,2 (t)eiφ1,2 (t)
(3.19)
21
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
where N1,2 is the number of atoms and φ1,2 the phase in the corresponding well. The
normalization ofR the total wavefunction Ψ(r, t) is fixed by the total atom number NT =
2
N
R 1 + N2 and by |Φ1,2 | dr = 1. In order to fulfill the condition for a weak link, we claim
Φ1 Φ2 dr 1, which is often referred to as “tight binding approximation”. The spatial
wavefunctions Φ1 (r), Φ2 (r) can be obtained by a linear combination of the normalized
lowest symmetric Φs (r) and antisymmetric Φas (r) stationary eigenstates of the GrossPitaevskii equation:
Φ1 (r) =
Φs (r) + Φas (r)
√
2
(3.20)
Φ2 (r) =
Φs (r) − Φas (r)
√
.
2
Due to the existence of the nonlinear interaction term the Gross-Pitaevskii equation
does in general not support the superposition principle. Nevertheless, ansatz 3.18 is
justified for two very weakly overlapping clouds, for which the interaction in the overlap
region can be neglected. Additionally, the use of time-independent spatial wavefunctions
Φ1 (r), Φ2 (r) restricts the description to the regime, where atomic interaction produces
only small modifications of the ground state properties in the individual wells.
If we substitute ansatz 3.18 into the Gross-Pitaevskii equation 3.11, multiply by Φ∗1 (r)
and integrate over the spatial dimensions, we find the following equation3
i~ ∂∂t Ψ1 (t) =
R h
i ~2
− 2m
Φ1 ∇2 Φ1 + Φ21 Vext + N1 Φ41 g + Φ31 Φ2 Ψ∗2 Ψ1 g + 2N2 Φ21 Φ22 g dr Ψ1
R h
i ~2
− 2m
Φ1 ∇2 Φ2 + Φ1 Vext Φ2 + N2 Φ1 Φ32 g + Φ21 Φ22 Ψ∗1 Ψ2 g + 2N1 Φ31 Φ2 g dr Ψ2
(3.21)
and a corresponding equation for Ψ2 (t) by replacing all “1”s by “2”s and vice versa.
In the following we only retain on-site interaction, since the spatial overlap of the two
modes is assumed to vanish. Therefore we neglect all mixed terms in Φ1 and Φ2 of order
larger than two4 . If we define the constant parameters
+
~2
2
2
=
|∇Φ1,2 | + |Φ1,2 | Vext dr
2m
Z 2
~
K = −
∇Φ1 ∇Φ2 + Φ1 Vext Φ2 dr
2m
Z
U1,2 = g |Φ1,2 |4 dr,
0
E1,2
3
Z (3.22)
In the following, the localized modes Φ1 and Φ2 are assumed to be real, since they are linear
combinations of the stationary states Φs and Φas .
4
All terms in equation 3.21 can be included in the framework of the extended two mode model, which
will be described in section 3.4
22
3.3 Two-mode approximation
we get the following two coupled Schrödinger equations for the BEC amplitudes in the
two wells
i~
∂
Ψ1 = (E10 + U1 N1 )Ψ1 − KΨ2
∂t
(3.23)
i~
∂
Ψ2 = (E20 + U2 N2 )Ψ2 − KΨ1 .
∂t
These equations are very similar to equations 2.2 which describe the weak coupling of two
superconductors. However, the nonlinearity arising from interatomic interaction gives
rise to an additional term which leads to new effects unobservable in superconducting
Josephson junctions (SJJ).
In analogy to the description of SJJs in section 2.1, we define the relative phase of
the wavefunctions in the two wells
φ(t) = φ2 (t) − φ1 (t)
(3.24)
and the fractional population imbalance
N1 (t) − N2 (t)
.
(3.25)
NT
For the definition of the relative phase, we have assumed a constant quantum phase of a
localized mode. The numerical calculations in chapter 3.6 will show, that this assumption
is justified for our experimental parameters. The fractional population imbalance has
no analog in the case of superconducting Josephson junctions, since the external circuits
suppresses charge imbalances5 (zSJJ = 0).
In close analogy to the derivation of the Josephson equations in section 2.1, equation
3.23 leads to a system of two coupled differential equations for the two dynamic variables
z and φ:
z(t) =
p
ż(t) = − 1 − z 2 (t) sin φ(t)
(3.26)
z(t)
φ̇(t) = ∆E + Λz(t) + p
1 − z 2 (t)
where the time is expressed in units of
meters
∆E =
~
2K
cos φ(t),
and we have defined the dimensionless para-
E10 − E20 U1 − U2
+
NT
2K
4K
(3.27)
Λ =
U1 + U2
NT .
4K
√
Equations 3.26 describe a nonrigid pendulum with a length l = 1 − z 2 if we identify
the relative phase φ with the displacement angle and the fractional population imbalance
As discussed in section 2.2.2, small fractional charge imbalances on the order of z ≈ 10−9 can occur
in mesoscopic superconducting Josephson junctions.
5
23
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
z with the angular momentum of the pendulum. The interwell trap tunneling current is
given by
I=
√
ż(t)NT
= I0 1 − z 2 sin φ,
2
(3.28)
T
corresponds to the maximum allowed Cooper pair current in the
where I0 = − KN
~
superconducting analog. This equation is very similar to the first Josephson equation
describing the dc Josephson effect in superconductor tunneling junctions. However,
it differs from the SJJ-equation in the nonlinearity in z. Equations 3.26 can also be
derived employing the concepts of nonlinear guided wave optics [75]. They resemble
the dynamic equations for the guided power and phase difference of two interacting
orthogonally polarized optical modes in a birefringent fiber.
In contrast to the pendulum analogy in the framework of the RCSJ-model in superconductor Josephson junctions, the atomic counterpart in a symmetric double well
potential has no external torque6 . Nevertheless, we can reach a regime, in which the
nonrigid pendulum rotates with non-vanishing mean angular momentum (z(t) 6= 0 for
all t) by choosing a large initial angular momentum z(0) (see section 3.3.2). This means,
that the population gets locked inside one well which, in the RCSJ pendulum analogy
corresponds to a dc-voltage drop across the Josephson junction.
Another way to deduce the two-mode equations 3.26 is to express the total conserved
energy of the system in terms of the canonically conjugate two-mode dynamical variables
z and φ:
√
Λz 2
+ ∆Ez − 1 − z 2 cos φ.
2
and use the Hamilton equations of motion
H=
ż = −
∂H
∂φ
and
φ̇ =
∂H
.
∂z
(3.29)
(3.30)
In the following, I will restrict the discussion of the Josephson junction dynamics to the
case of a symmetric double well potential with E10 = E20 , U1 = U2 and therefore ∆E = 0.
3.3.1 Zero phase modes
In this section, I investigate the first class of Josephson tunneling dynamics, in which
the time-average of both dynamical variables z and φ is zero.
Non-interacting limit: Rabi-oscillations
For symmetric double well potentials and negligible interatomic interaction (U = Λ = 0)
the differential equations 3.26 reduce to
z̈ = −z.
6
(3.31)
A time-dependent barrier moving adiabatically across the trapping potential has been investigated
in [76]. In this case, the phase dynamics are governed by a driven pendulum dynamics, analog to
current-driven SJJ.
24
3.3 Two-mode approximation
The corresponding solution is a sinusoidal Rabi-like oscillation of the population imbalance z with a frequency (in unscaled units)
2
K.
(3.32)
~
The Rabi-oscillations are shown with dashed red lines in figure 3.2a. This limit of vanishing interaction between the Bose-Einstein condensed atoms can in principle nowadays
be realized using Feshbach resonances [41] in which a homogenous magnetic field is used
to adjust the s-wave scattering length7 .
ωR =
Small amplitude oscillations: Josephson plasma frequency
For a symmetric double well potential, the equilibrium value of the fractional population
imbalance and the relative phase is z = 0 and φ = 0. Linearization of the equation of
motion 3.26 around these values yields
z̈ = −(Λ + 1)z.
(3.33)
This equation describes sinusoidal small amplitude oscillations with an unscaled frequency
√
2K
ωpl = Λ + 1
.
(3.34)
~
In this regime, the oscillation frequency is independent of the initial conditions for z
and φ as long as the small amplitude approximation holds. These oscillations are often
referred to as plasma oscillations in analogy to superconducting Josephson junctions.
Figure 3.2a shows the sinusoidal small amplitude oscillations for Λ = 15 and z(0) = 0.1
(black line). For our experimentally realized potential and atom numbers, the plasma
frequency is typically on the order of 2π · 30 Hz, which has to be compared to typical
SJJ plasma frequencies on the order of several gigahertz.
Large amplitude oscillations
In the general case of interacting atoms with large tunneling amplitudes, equations 3.26
can be solved in terms of Jacobian elliptic functions as shown in [30]. The large amplitude oscillations are the anharmonic nonlinear generalization of the sinusoidal tunneling
dynamics in superconductor Josephson junctions. In this regime, the oscillation frequency depends on the initial population imbalance z(0) and the relative phase φ(0).
A numerical solution of equations 3.26 in the regime of large amplitude oscillations is
shown in figure 3.2 b and 3.2c. An increase of the initial population imbalance z(0) adds
higher harmonics to the sinusoidal oscillations, which leads to more and more anharmonic tunneling dynamics. The oscillation period now increases with z(0) until, at a
certain population imbalance zc , the dynamics undergoes a critical slowdown (dashed red
line in figure 3.2c). This happens, when the initial angular momentum in the nonrigid
pendulum analogon is just enough to reach a maximum phase of π during the dynamical
evolution. The pendulum stops at the top position which corresponds to a zero angular
momentum, i.e. a population imbalance z = 0.
7
For the case of 87 Rb, the width of the Feshbach resonances are on the order of 1 mG [42]. Therefore,
they are very hard to implement experimentally.
25
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.2: Dynamical evolution of the population imbalance z and relative phase φ for Λ = 15. The
initial phase φ(0) is chosen to be zero for all trajectories. The small figures on top show the trajectory
of the nonrigid pendulum analog in the corresponding regimes. a) An initial population imbalance
z(0) = 0.1 results in sinusoidal small amplitude oscillations (black line). The dotted red line shows
the evolution expected for non-interacting atoms (Λ = 0, Rabi oscillations). b) Non-sinusoidal large
amplitude oscillations for z = 0.4. c) Tunneling dynamics slightly below (black line , z = 0.49) and
at (dotted red line) the selftrapping threshold zc = 0.499. d) Macroscopic quantum self-trapping for
z = 0.8 > zc . The population oscillates around a non-zero mean value and the relative phase increases
monotonically (running phase modes).
3.3.2 Macroscopic quantum self-trapping
As shown in the last section, the dynamics of the system changes drastically, when
the initial population imbalance approaches a critical value zc . For larger population
imbalances z(0) > zc the tunneling is strongly suppressed, which leads to self-trapped,
26
3.3 Two-mode approximation
nearly stationary modes localized inside a single well (hzit 6= 0).
The critical value is reached, when the atomic tunneling between the two wells is no
longer fast enough to counteract the evolution of the relative phase due to the difference
of the chemical potentials in the left and the right well. According to equation 3.28, the
tunneling current reverses, when the relative phase exceeds the value π. If this happens
before the population imbalance has been inverted due to tunneling, the population is
locked inside one well and the relative phase will evolve monotonically and unbound
(running phase mode).
This leads to a condition for zc : the initial energy for z(0) = zc and φ(0), given by
equation 3.29 has to be large enough to reach φfinal = π at zfinal = 0 which corresponds
to an energy Hfinal = 1:
Λ 2 p
z − 1 − zc2 cos φ(0) ≡ 1.
(3.35)
2 c
For a zero initial phase, this equality is only solvable for Λ > 2 and a positive K.
For smaller Λ, there is no transition to self-trapping, even at arbitrary large initial
population-imbalances z(0). Using equation 3.35 we find, that the critical population
imbalance for macroscopic quantum self-trapping (MQST) for φ(0) = 0 is given by
√
2 Λ−1
for Λ > 2.
(3.36)
zc =
Λ
From our experiments, we deduce a critical population imbalance zc ≈ 0.5, which means
that the left localized mode Ψ1 contains 75% of the total number of atoms (see chapter
5). If we assume the validity of the two-mode approximation, this value corresponds to
Λ ≈ 15, for which the different dynamical regimes are shown in figure 3.2. However, a
numerical calculation of the symmetric and antisymmetric states yields Λ ≈ 76. The
reason why the two-mode approximation fails to quantitatively explain our experimental
results is discussed in section 3.3.5.
In order to reach the self-trapping threshold, we can increase z(0) above the critical
value zc for a fixed value of Λ, or alternatively increase Λ by increasing the total atom
number NT and keeping z0 fixed. From equation 3.35 we get an expression for the scaled
critical on-site interaction energy
p
1 + 1 − z(0)2 cos(φ(0))
.
(3.37)
Λc = 2
z(0)2
H0 ≡ H(z(0) = zc , φ(0)) =
The dynamical evolution in the regime of MQST is shown in figure 3.2d for z(0)=0.8. The
oscillation period of the population imbalance z increases and the amplitude decreases
with increasing z(0). The phase difference between the two Bose-Einstein condensates
in the left and right well evolves unbound. The effect of MQST can be visualized in
the nonrigid pendulum analogy. There the self-locked population imbalance corresponds
to a steady self-sustained rotation of the pendulum with non-vanishing mean angular
momentum z and phase φ. The effect of macroscopic self-trapping is a nonlinear effect
arising from the scaled on-site interaction energy of the Bose-Einstein condensates in the
individual wells. It is self-maintained in a closed conservative system without an external
drive. There is no analogon for MQST in superconductor Josephson junctions since the
running phase mode in the RCSJ-model is a driven steady state which is independent
of the initial relative phase of the two superconductors. It is also different from the
27
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.3: Scaled critical on-site interaction energies Λc (black line) and Λc,2 (red line) as a
function of the initial population imbalance z(0) for an initial phase φ(0) = π. The green shaded
region corresponds to the regime of π-oscillations. The yellow shaded region indicates the regime of
π-phase MQST. The running-phase MQST-regime (unshaded region) which is the same as the MQST
for φ(0) = 0 can be reached for Λ > Λc,2 .
discrete Coulomb blockade effect discussed in section 2.2.2 which occurs as a result of
the tunneling of a single Cooper pair leading to a charging energy EC larger than the
Josephson coupling energy EJ .
The first experimental observation of Josephson tunneling oscillations and macroscopic quantum self-trapping for Bose-Einstein condensates will be presented in chapter 5.
3.3.3 π-phase modes
Additionally to the discussed zero-phase modes and MQST, Bose-Einstein Josephson
junctions have another rich class of tunneling dynamics in which the system evolves
with a time-averaged value of the relative phase of hφit = π. Those π-phase modes have
not been observed so far in Josephson junctions for Bose-Einstein condensates and I will
therefore restrict the discussion to the basic features of this dynamical regime. For a
detailed description see for example [30].
The momentum dependent length of the pendulum allows the pendulum to perform
small and large amplitude π-oscillations with hzit = 0 around the unstable equilibrium
top position. Similar to the zero phase modes discussed above the dynamics changes to
macroscopically self-trapped modes with non-zero average of the population imbalance if
the scaled on-site interaction energy Λ exceeds the critical value given by equation 3.37.
In contrast to the zero initial phase self-trapping discussed above, there are two
distinct self-trapping regimes for φ(0) = π which are separated by another critical onsite interaction energy [30]: Λc,2 = √ 2 2 . The two critical parameters as a function
1−z(0)
of the initial population imbalance are shown in figure 3.3 for φ(0) = π.
28
3.3 Two-mode approximation
Figure 3.4: Phase-plane portrait of the bosonic Josephson junction in the two mode approximation
for different scaled on-site interaction energies Λ. Trajectories with initial phase φ(0) = 0 are depicted
in red, those with φ(0) = π in blue. The self-trapped modes are drawn with dashed lines. The indicated
numbers represent the different initial population imbalances z(0). The black lines for Λ = 3 and Λ = 15
depict the respective separatrix, which is the phase-plane trajectory for z(0) = zc .
For Λc < Λ < Λc,2 the system evolves in the regime of π-phase macroscopic selftrapping (yellow shaded region in graph 3.3). In this regime the phase is still localized
around π while the population imbalance is locked with hzi =
6 0. In the mechanical
analogon of a nonrigid pendulum, this regime corresponds to a closed-loop trajectory
around the vertical instable equilibrium. For Λ > Λc,2 the phase additionally becomes
unbound (unshaded region). This regime is the same as the MQST regime discussed
above for zero initial phase.
It is important to note, that the critical Λc for φ(0) = π is always between one and
two, which means that for all Λ > 2 the system is always self-trapped. Therefore the
small and large amplitude π-oscillation regimes are not accessible with our current setup,
where a total number of atoms of NT = 1150 corresponds to Λ ≈ 76. This parameter
only allows trajectories in the running phase MQST regime (see figure 3.3). Note that, as
already mentioned for the zero-phase modes, the crossovers between different dynamical
regimes can also be realized by fixing Λ and changing the initial population imbalance.
29
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
3.3.4 Phase plane portrait
The various regimes of the nonrigid pendulum dynamics discussed above can be summarized very intuitively in terms of a phase plane portrait, where constant energy lines
are plotted in a z-φ diagram. Figure 3.4 shows phase space plots for different on-site
interaction energies Λ, obtained by numerically solving the coupled differential equations
3.26. The first graph shows the accessible regimes for Λ = 1.2, where all trajectories
with initial phase φ(0) = 0 are untrapped, even for arbitrary large initial population
imbalances z(0). The situation changes for φ(0) = π where the transition to π-phase
MQST occurs at zc ≈ 0.75. In the second graph the phase-plot is shown for Λ = 3.
In this regime, a transition from zero-phase oscillations to MQST can be observed for
initial zero phase if the initial population imbalance is increased above the critical value
zc = 0.943. The separatrix8 is indicated with a thin black line. For an initial phase
φ(0) = π all trajectories are self-trapped. The transition from running phase-MQST
to π-phase MQST happens at zc,2 ≈ 0.75. Finally, the third graph shows the various
regimes for the scaled on-site interaction energy Λ = 15. The phase plot shows, that
both regimes for zero initial phase, Josephson oscillations and MQST, are accessible.
The transition occurs at zc ≈ 0.5. As we will see in chapter 5, this value is convenient
for our experiments in order to investigate the two regimes. As already shown for Λ = 3,
all trajectories for φ(0) = π are self-trapped. Unfortunately only the running-phase
MQST regime would be accessible for our current experimental parameters, even if we
were able to prepare a π initial phase.
3.3.5 Limits of the two-mode description
The two-mode approximation describes the system in terms of only two time-independent
spatial modes, each of them localized inside a single well. These modes are calculated
by linearly combining the symmetric and the antisymmetric groundstate of the GrossPitaevskii equation. This assumption implies that the many-body interactions produce
only small modifications of the groundstate properties of the individual wells. This
situation is surely encountered, if the on-site interaction energy is much smaller than the
level spacing of the external trap, which leads to a coarse upper bound for the total atom
number. In order to give a rough estimate of the maximum atom number, a harmonic
approximation to the confinement inside a single well
1
V (r) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )
2
(3.38)
is applied. Assuming, that the wavefunctions of the interacting atoms can still be described by the Gaussian groundstate for non-interacting atoms in the external harmonic
confinement inside a single well9 , we obtain
ψ(r) =
8
1
πa20
3/4
− 12
e
2
x2
+ y2
a2
ay
x
2
+ z2
az
(3.39)
The separatrix is the trajectory for z(0) = zc , which separates the different dynamical regimes
This assumption overestimates the interaction energy since the real groundstate of the system is
broadened due to the interatomic repulsion.
9
30
3.3 Two-mode approximation
p
where ax,y,z =
~/mωx,y,z are the respective harmonic oscillator lengths and ā =
(ax ay az )1/3 is their geometric average. This leads to the on-site interaction energy of
N particles inside a single well
Z
Uint = N g
|ψ|4 dr =
Ng
(2π)3/2 r03
(3.40)
which has to be small compared to the level-spacing of the harmonic trap leading to the
estimate
r
N
π r0
.
2 a
(3.41)
For our experimental setup, the mean trapping frequency is on the order of 100 Hz
corresponding to r0 ≈ 1 µm which means
N 200.
(3.42)
A numerical simulation (see section 3.6) of the localized mode yields an on-site interaction
energy of h·105 Hz for N = 700 atoms inside a single well which shows that the expression
given above indeed overestimates the interaction. Nevertheless, the requirements for the
two-mode approximation to apply are not strictly fulfilled, since the minimum condensate
atom number we can generate reproducibly is about NT = 1000. Although the two-state
model can not explain our experimental data quantitatively, it still provides an intuitive
understanding of the various dynamical regimes.
Another approximation we applied to get the coupled two-mode differential equations
3.26 was the neglect of mixed interaction terms in equation 3.21, which led us to a
constant tunneling energy K. An inclusion of the neglected terms still allows a similar
description in terms of a set of two coupled differential equations involving only one
additional physical quantity. This is shown in the next section, where a time-dependent
tunneling energy is included. This small extension of the two-mode model provides
surprisingly good quantitative agreement with the numerical solution and experimental
results.
The two-mode description is based on the Gross-Pitaevskii equation and therefore
does not take into account the effect of quantum fluctuations. These effects have been
included for example in [31, 77, 78].
Additionally, the effect of thermal fluctuations due to the residual thermal cloud
is neglected in the two-mode description presented here. However, as shown in [34] it
can be included by means of a heuristic damping term in equation 3.26. The central
result of this work is that damping produces characteristic decay trajectories to the
final equilibrium φ = z = 0 for all different dynamical regimes discussed above. Finite
temperature effects have also been investigated in [32] where it has been assumed that
the thermal cloud results in a non-coherent dissipative atomic current. Ruostekoski and
Walls [79] have included the effect of thermal fluctuations by coupling the Bose-Einstein
condensate to a thermal reservoir of non-condensed atoms. They also show, that on a
time scale of several tunneling times, the self-trapped state decays to an equal population
in both wells.
31
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
3.4 Extended two-mode approximation: variable tunneling model
As already discussed in section 3.3, the assumption of time-independent spatial wavefunctions and the neglect of mixed interaction terms led to a constant tunneling energy,
which is a severe limitation for the description of real bosonic Josephson junctions.
Stefano Giovanazzi et al. [76] have included the leading mixed interaction term already
in 2000. Recently, David Ananikian and Tom Bergeman [80] have included all terms
in equation 3.21 and derived differential equations for a more exact two-mode model
that accounts for a variable tunneling rate depending on the instantaneous values of the
population imbalance z and the relative phase φ. This model has been termed variable
tunneling model (VTM) in contrast to the normal constant tunneling model (CTM) described above. The VTM assumes the same two-mode approximation (equation 3.18) as
the CTM which implies time-independent local modes Φ1 and Φ2 . These local modes are
again calculated in terms of the symmetric ground state Φs and the first antisymmetric
excited state Φas (see equation 3.20). However, this is the only assumption for the VTM
and no further approximations are applied in this theory. This means in particular, that
the tunneling parameter in front of Ψ2 in equation 3.21 is no longer constant since it
contains terms in Ψ1 and Ψ2 which are time dependent.
Despite the complexity of the additional terms compared to the CTM, the extended
model leads to the following coupled set of differential equations for the symmetric double
well case:
p
C
ż(t) = − 1 − z 2 (t) sin φ(t) + z cos(2φ)
B
(3.43)
φ̇(t) =
z(t)
A
C
z(t) + p
cos φ(t) − z cos(2φ),
2
B
B
1 − z (t)
where the time has been re-scaled as
A =
B
t
~
→ t and we have defined the following constants:
1
· (10γs,as − γs,s − γas,as )
4
B = βas − βs +
C =
1
· (γs,s − γas,as )
2
(3.44)
1
· (γs,s − γas,as − 2γas,as )
4
. Here βs,as represent the chemical potential of the states Φs,as and
Z
γi,j = gNT |Φi |2 |Φj |2 dr, for i, j ∈ {s, as}.
(3.45)
Note, that all constants are expressed in terms of the symmetric and antisymmetric
stationary states of the symmetric double well potential, instead of the localized modes
used in the CTM.
32
3.4 Extended two-mode approximation: variable tunneling model
The differential equations 3.43 can be written in Hamiltonian form using the scaled
Hamiltonian
A z2 √
1C
− 1 − z 2 cos φ +
(1 − z 2 ) cos(2φ).
(3.46)
B 2
2B
Equation 3.43 has a similar form as the analogous equations in the CTM. However, the
additional term in C in the VTM can be significant for large atom numbers10 . A direct
comparison of the relevant constant parameters of the VTM and CTM yields:
H=
1
2K = B − (γas,as − γs,s )
4
(3.47)
NT U = A + 2C.
The different parameters are calculated as a function of the total atom number in [80].
The authors show that the interaction effects are better captured by the VTM than by
the CTM especially for large atom numbers. In particular, K can become negative which
implies imaginary plasma-oscillation frequencies, whereas B is shown to be positive for
all atom numbers.
A very important parameter for the experimental implementation of a bosonic Josephson junction is the critical population imbalance zc , which constitutes the transition between Josephson tunneling and self-trapping regime. The value for zc can be obtained
by equating H(0, π) ≡ H(zc , 0) using the same arguments as in section 3.3:
2
[B(A − B − C)]1/2 .
(3.48)
A−C
In order to compare the two different two-mode approaches and the numerical integration
of the non-polynomial nonlinear Schrödinger equation11 , the group of Tom Bergeman
has numerically calculated the ground state and the first excited antisymmetric state
of the 3d-Gross-Pitaevskii equation for our experimental atom number NT = 1150 and
double well potential. The numerical integration has been implemented by diagonalizing
the DVR Hamiltonian [81] using sparse matrix techniques [82]. The results for the
relevant parameters are A = 2.377, B = 0.073, C = 0.0046, K = 0.0156 and Λ =
76. The critical population imbalance given by equation 3.48 for self-trapping within
the VTM is zc ≈ 0.35 which is in good agreement with the value zc = 0.39 obtained
by integration of the three-dimensional Gross-Pitaevskii equation. The corresponding
phase-plane trajectories for the indicated initial population imbalances are shown with
blue lines in figure 3.5 and are compared to the results from a numerical integration of the
NPSE (red lines). The two predictions are in excellent agreement for small population
imbalances z(0) < 0.4. However, in the self-trapped regime the VTM slightly deviates
from the numerical solution. This can be explained by realizing, that for z(0) = 0.7
the chemical potential of the BEC is already on the order of µ = h · 300 Hz. This is is
approximately half a harmonic oscillator level spacing above the potential barrier, which
zc,V T M =
10
In this context large means that the scaled interaction gN is bigger than unity, where the scattering
length a is given in units of the oscillator length inside the harmonic trap in x-direction and we have
set ~ = m = 1. In our experimental realization: gN ≈ 60.
11
The details of the numerical methods are discussed in section 3.6.
33
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.5: Phase plane trajectories for the variable tunneling model (blue lines) and the numerical solution of the non-polynomial nonlinear Schrödinger equation (red lines) for our experimental
parameters. The indicated numbers denote the initial population imbalances. The black line shows the
separatrix for the VTM at zc = 0.35. The two theoretical predictions show good quantitative agreement.
has a height of Vb = h · 263 Hz. Therefore the BEC can couple to excited states of
the external potential and the assumption of only two modes is no longer strictly valid.
Note that the numerically calculated value Λ = 76 corresponds to a critical population
imbalance of zc,CT M = 0.23 in the framework of the CTM. This shows, that the usual
two-mode approximation fails for the experimentally implemented parameters.
The phase-plane plots do not show the time-scale of the ongoing dynamics. A comparison of the predicted tunneling times as a function of the total atom number in the
Josephson tunneling regime (z(0) = 0.28) is shown in figure 3.6. The CTM, VTM and
a full numerical solution of the 3d-Gross-Pitaevskii equation (TDGPE) with our experimental parameters are taken from [80]. The additional green crosses show our numerical
solution of the nonlinear non-polynomial Schrödinger equation for the same parameters.
This graph shows again the advantage of the VTM compared to the normal CTM especially for large atom numbers. The main feature of the graph is, that for atom numbers
up to 100 all four different theoretical curves predict the same tunneling time. For larger
atom numbers, the CTM begins to fail, since the dynamics becomes self-trapped even
at z(0) = 0.28. The VTM and the numerical integration of the TDGPE and the NPSE
predict the same tunneling time up to 1500 atoms with a deviation below 12%. All three
theories can explain our experimentally measured tunneling time indicated with the red
square and error bars. Note that the deviation of the NPSE compared to the GPE is
below 2% in the complete range of considered atom numbers.
34
3.5 External and internal bosonic Josephson junctions
Figure 3.6: Comparison of the calculated Josephson tunneling times using the CTM, VTM and the
numerical integration of the TDGPE and NPSE. The tunneling frequencies are given in units of the
frequency of the harmonic confinement in x-direction (ωx = 2π · 78 Hz, see the definition of the potential
3.17). All theoretical predictions agree for atom numbers below 100. For larger atom numbers, the
CTM fails, whereas the VTM still predicts the right tunneling frequency. The red square and error
bars indicate the tunneling time deduced from our experiment. This graph is taken from [80] except
the points for the NPSE.
3.5 External and internal bosonic Josephson junctions
In the discussion of the Josephson dynamics given above, the weak link between two
Bose-Einstein condensates was provided by loading a BEC into an external double well
potential (external Josephson effect) with a small spatial wavefunction overlap. The
experimental implementation of this system will be presented in chapter 4. However, it
is also possible to provide the weak link using a two-component Bose-Einstein condensate
consisting of two different hyperfine levels trapped inside a single harmonic trap. The
weak coupling between two wavefunctions, i.e. the two hyperfine states, is provided by a
weak driving field (internal Josephson effect). This coupling, which results in a transfer
from one internal state to the other can be done in a coherent manner, so that the phases
of the two condensates couple.
The first Experiment with a two-component BEC was performed in 1997 [83] with
87
Rb atoms evaporatively cooled in the |F = 1, mF = −1i spin state. Those atoms
cool an additional cloud in the |2, 2i state by sympathetic cooling12 . However, this
experiment can not produce arbitrary relative atom numbers in the two states. In 1998,
the same group has succeeded in using a coherent two-photon transition consisting of a
microwave photon, which is a few MHz red detuned from the hyperfine transition with
ν0 ≈ 6.8 GHz and a radio frequency photon of 1-4 MHz to convert an arbitrary fraction
of a |1, −1i condensate to the |2, 1i state [84, 36]. In this experiment, the coherence
between the two states has been shown explicitly by measuring the relative phase of
the two condensates using a technique based on Ramsey‘s method of separation [85].
12
Sympathetic cooling means that a cloud of atoms is cooled by thermal contact with a colder sample.
This technique has allowed the experimental realization of quantum degenerate Fermi gases.
35
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Note that in the absence of the coupling drive, interconversion between the two species
is negligible. Motivated by the experimental implementation of the coupling of different
hyperfine states, Williams et al. [35] investigated the possibility of creating Josephsontype oscillations in two-component Bose-Einstein condensates. The proposed experiment
starts with a short driving pulse that produces condensates in both states with a defined
relative population imbalance z(0). Due to the different magnetic moments of the two
spin states and the presence of the earth gravitational field, the two clouds see different
harmonic traps, separated along the direction of gravity. After a given time which is
determined by the transient motion of the two clouds to the new equilibrium position,
a weak driving field is turned on, coupling the two condensates in the overlap region,
which can be controlled by the magnetic confinement [36]. The time, at which this drive
is turned on determines the initial relative phase φ(0) of the internal Josephson junction.
As shown in [35] the dynamics of the system in the two-mode approximation
Ψ=
p
p
N1 (t)eiφ1 (t) Φ1 (r) + N2 (t)eiφ2 (t) Φ2 (r)
(3.49)
is governed by the following coupled differential equations
p
ż(t) = − 1 − z 2 (t) sin φ(t)
(3.50)
φ̇(t) = −
[µ1 − µ2 + δ]
z(t)
+p
cos φ(t),
K
1 − z 2 (t)
where δ is the detuning of the two-photon drive with respect to the energy separation of
the two hyperfine states, K is the tunneling energy in units of ~ and µ1,2 are the chemical
potentials of the two modes Φ1 and Φ2 , which are assumed to be time-independent13 .
In order to make equation 3.50 look similar to the differential equations 3.26 of the twomode approximation discussed above, I have re-scaled the time according to t → Kt.
Equation 3.50 is a nonlinear version of the usual Josephson equation and is very similar
to the two-mode differential equation 3.26 for the external bosonic Josephson junction.
Note, that in contrast to the two-mode approximation for two BECs in a double well
trap described above, where we have only included the on-site interaction terms, the
interaction between the two condensates has to be included in the internal Josephson
junction, since the wavefunctions have significant spatial overlap. To my knowledge, the
internal Josephson effect has not been experimentally realized so far.
3.6 Numerical solution
As we have seen in section 3.3 the constant tunneling two-mode approximation qualitatively describes the different dynamical regimes of the bosonic Josephson junction in
terms of only two physical parameters Λ and K. In order to check the validity of the
CTM and VTM, we have to compare it to a numerical simulation. This chapter deals
with the numerical solution of the Gross-Pitaevskii equation and the non-polynomial
nonlinear Schrödinger equation.
13
The assumption of time-independent spatial wavefunctions and chemical potentials is often referred
to as adiabatic approximation.
36
3.6 Numerical solution
3.6.1 Split-step Fourier method
The numerical integration of the GPE (equation 3.11) and the NLSE (equation 3.16) is
performed by applying the infinitesimal time evolution operator Û (dt) to the wavefunction in each infinitesimal time step dt:
Ψ(x, t + dt) = Û (dt)Ψ(x, t)
(3.51)
For the case of a time-independent Hamiltonian Ĥ, Û (dt) is simply given by
i
Û (dt) = e− ~ Ĥdt .
(3.52)
In order to apply this time evolution operator, we split the Hamiltonian into a kinetic
p̂2
and a spatially and time-dependent part S(r̂, t), which is given by
part K(p̂) = 2m
S(r̂, t) = Vext (r̂) + gN |Ψ|2
(3.53)
for the three-dimensional Gross-Pitaevskii equation and by
|Ψ(x, t)|2
S(x̂, t) = Vext (x̂) + g1d N p
+
~ω⊥
2
1 + 2aN |Ψ(x, t)|2
!
p
1
p
+ 1 + 2aN |Ψ(x, t)|2
1 + 2aN |Ψ(x, t)|2
(3.54)
for the NPSE.
The infinitesimal time evolution is symmetrically separated in order to evaluate K(p̂)
in momentum space and S(x̂, t) in coordinate space:
i
dt
i
i
dt
Û (dt) ≈ e− ~ K(p̂) 2 e− ~ S(x,t)dt e− ~ K(p̂) 2 .
(3.55)
This separation introduces an error on the order of dt3 since x̂ and p̂ do not commute14
[86]. However, this error can be neglected by choosing dt sufficiently small. The whole
operator is now applied to the wavefunction in the following way15 :
i p2 dt
i
i p2 dt
Ψ(x, t + dt) = F −1 e− ~ 2m 2 F e− ~ V (x,t)dt F −1 e− ~ 2m 2 F Ψ(x, t),
(3.56)
where the symbol F represents a Fourier transformation. This way to integrate the
Schrödinger equation is called split-step Fourier method and allows a time propagation,
which only uses multiplication and Fourier transformation of the wavefunction. We use a
typical time resolution of 1 µs, which is chosen such, that the maximum phase evolution
per integration step satisfies | ~i Ĥdt|max 2π.
14
If we had split the time evolution operator simply into two parts in momentum and coordinate
space, the error would be on the order of dt2 .
15
This is possible since the momentum operator is diagonal in momentum space and the coordinate
operator is diagonal in coordinate space.
37
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.7: Numerical integration of the NPSE in imaginary time for NT = 1150 atoms. a) Symmetric ground state (blue) and first excited antisymmetric state (red) of the symmetric effective double
well potential (black). b) Localized modes in the left (blue) and right (red) well. c) Two-dimensional
plot of the ground state wavefunction. d) Transverse RMS-width of the BEC scaled in units of the
transverse ground state oscillator length.
3.6.2 Ground state: imaginary time propagation
In order to numerically investigate the Josephson dynamics of the Bose-Einstein condensate in the realized potential, we first have to calculate the initial wavefunction, which is
given by the groundstate in the three-dimensional double well potential. A very effective
way to calculate groundstate wavefunctions is the method of numerical propagation in
imaginary time [87]. The Gross-Pitaevskii equation (or alternatively the non-polynomial
nonlinear Schrödinger equation) is hereby changed into a diffusion equation by applying
a Wick rotation
t → τ = −it.
(3.57)
This transformation results in the necessary energy diffusion but also in a loss of atoms,
which can be compensated by a re-normalization of the wavefunction in each integration
step. As a starting wavefunction for the imaginary time propagation we take an analytical
approximation to the ground state (Thomas Fermi parabola or Gaussian ground state)
and use the split-step Fourier method (with dt → −i dt).
If the chosen initial wavefunction is sufficiently close to the real ground state and if we
choose a sufficiently small imaginary time step i dt ≈ i · 5 µs the wavefunction converges
38
3.6 Numerical solution
Figure 3.8: Ground state population imbalances as a function of the energy difference ∆E of the
two wells. The symmetric double well potential corresponds to ∆E = 0
quickly to the ground state within a few i· ms. A very useful feature of the imaginary time
propagation is, that it converges to the first excited antisymmetric state in the double
well potential, if we start with an appropriate antisymmetric initial wavefunction. We
need both, the symmetric and the first excited antisymmetric state in order to calculate
the localized modes Φ1 and Φ2 used for the two-mode approximation described in section
3.3.
3.6.3 Numerical results: stationary states
The symmetric and antisymmetric stationary states of the 87 Rb BEC in the considered
external double well potential are calculated using imaginary time propagation of the
NPSE. The considered coordinate space has a total size of 50 µm and is divided into
2048 spatial steps of 24.4 nm. The chosen potential has the following form
1
V (x, y, z) = m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) + V0 cos2 (π(x − ∆x)/q0 )2 ,
(3.58)
2
where ωx = 2π · 78 Hz, ωy = 2π · 66 Hz and ωz = 2π · 90 Hz are the harmonic trapping
frequencies, q0 = 5.2 µm is the periodicity and V0 = h · 413 Hz the potential depth of the
optical lattice. For this set of parameters, the distance of the two wells is d = 4.2 µm and
the potential barrier has a height of Vb = h · 263 Hz. A nonzero shift ∆x of the harmonic
confinement in x-direction with respect to the periodic potential results in an asymmetric
double well, for which the two wells have an energy difference ∆E 6= 0. Figure 3.7 shows
the numerical results for 1150 atoms in a symmetric double well potential (∆x = 0).
The symmetric ground state and the first excited antisymmetric state are shown in
graph 3.7a. The numerical integration yields the chemical potentials βs = h · 279.7 Hz
and βas = h · 289.5 Hz. The two localized modes Φ1 and Φ2 , which are the starting point
for the two-mode models are shown in graph b). A two-dimensional plot of the ground
state wavefunction is shown in graph c)16 . The reason why the effective one-dimensional
NPSE is sufficient to describe the physics in the considered regime is shown in figure
16
Note that the transverse shape of the wavefunctions is restricted to be Gaussian in the NPSE.
39
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.9: Quantum phase of the wavepacket after one quarter of the oscillation period. The phases
of the two localized modes and therefore the relative phase are obtained by taking the unweighted mean
value over the shaded regions, where the atomic density (black line in arbitrary units) is at least ten
percent of the maximum inside the respective well.
3.7d, where the scaled transverse RMS-width of the groundstate is plotted as a function
of x. It shows, that the atomic cloud is nearly in the transverse groundstate of the
external potential.
The preparation of the initial population imbalance can be implemented by choosing an asymmetric double well with a non-zero energy difference ∆E of the two wells.
The resulting atom
inside each Rwell is obtained by integrating the atomic denR 0 number
∞
2
sity using N1 = −∞ |Ψ| dx and N2 = 0 |Ψ|2 dx. Figure 3.8 shows the corresponding
groundstate population imbalances z as a function of the energy shift ∆E.
3.6.4 Numerical results: dynamics
The numerical integration of the NPSE using the split-step Fourier method is carried
out for the same potential and atom number and with the same numerical parameters
as given above. As a starting wavefunction, we use the ground state wavefunction in
an asymmetric double well potential. The potential is subsequently shifted to the symmetric case, which initiates the Josephson tunneling dynamics. This shift is applied
non-adiabatically17 in order to prevent atomic tunneling during the shifting process.
As we have seen in the previous chapters, the dynamics of the system can be characterized by the evolution of the two dynamical parameters, the population imbalance z
and the relative phase φ.
The phase of the wavepacket is calculated using the relation
Im(Ψ(x))
.
(3.59)
φ(x) = arctan
Re(Ψ(x))
Figure 3.9 shows a typical spatial distribution of the numerically calculated phase (red
line) and the corresponding atomic density (black line) of the BEC after a quarter of
17
In this context, non-adiabatically means that the shifting process is much faster than the tunneling
time scale, which is typically on the order of 50 ms.
40
3.6 Numerical solution
Figure 3.10: Result of the numerical integration of the NPSE for NT = 1150 atoms in the experimentally realized double well potential. The time evolution of the two dynamical variables z and φ
is shown for different initial population imbalances as indicated in the legend. The dynamics shows
Josephson tunneling for z(0) < zc,N P SE = 0.375 and self-trapping for z(0) > zc .
the oscillation period. The assumption of a constant phase inside a single well, which
is crucial for the two-mode description discussed above, is justified by the numerical
calculation. Since the expression for the phase (equation 3.59) diverges for small real
parts of the wavefunction, the phases of the BECs in the left and right well are calculated
by taking the mean value over a spatial region, where the atomic density is bigger than
ten percent of the respective maximum (gray shaded regions). The relative phase of the
two localized modes in figure 3.9 is φ ≈ π/2.
The results of the time evolution of z and φ in the different dynamical regimes are
shown in figure 3.10. The different lines correspond to different initial asymmetries
and therefore to different initial population imbalances as indicated in the graph. Note
that the initial relative phase of the two wavepackets is zero for all curves, since the
propagation starts with the groundstate of the shifted potential. As predicted by the twomode models, the system shows two distinct dynamical regimes: Josephson tunneling
and macroscopic quantum self-trapping. The numerical integration of the NPSE yields
a critical value zc,N P SE = 0.375 for the initial population imbalance, which is in good
agreement with the value zc,T DGP E = 0.39 given in [80] obtained by integrating the
full three-dimensional Gross-Pitaevskii equation. For z(0) < zc the system evolves in the
Josephson-tunneling regime which is characterized by an oscillating population imbalance
and a relative phase with zero time-average. The self-trapping regime for z(0) > zc
manifests itself in a monotonically increasing phase and a locked population imbalance.
In this regime, the tunneling of atoms can no longer counteract the evolution of the
41
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
Figure 3.11: Time evolution of the transverse RMS-size of the left localized mode in the Josephson
oscillation regime (z(0) = 0.3). The fact, that the spatial extent of the wavepacket does not change significantly during the tunneling evolution justifies the two-mode ansatz, which assumes time-independent
spatial wavefunctions Φ1 and Φ2 .
relative phase due to different on-site interaction energies leading to relative phases of
φ > π. As we will see in chapter 5, this numerical solution is in excellent agreement with
our experimental results.
The two-mode model as well as the extended two-mode model assume time-independent localized modes. In order to check this assumption, I have numerically calculated
the time evolution of the transverse size of the left mode in the Josephson-tunneling
regime for z(0) = 0.3. The result is shown in figure 3.11 and reveals that in fact, the
transverse wavepacket size only shows an evolution on the order of ten percent although
the atom number in this well oscillates between N1 = 750 and N1 = 400. This explains
the good agreement between the extended two-mode model described in section 3.4 and
our experimental findings, since the assumption of a constant spatial wavefunction and
a constant phase inside a single well are the only approximations in this approach.
3.7 Feasibility study for bosonic Josephson junction experiments
The numerical simulations in chapter 3.6 have shown the existence of a set of parameters,
in which both dynamical regimes, Josephson tunneling and macroscopic quantum selftrapping are in principle accessible. However, for a successful experimental implementation of a bosonic Josephson junction it is important to choose a set of parameters, which
is sufficiently uncritical to small deviations. For this purpose numerical simulations based
on the NPSE have been performed for different periodicities q0 of the periodic potential
in equation 3.58 realizing the effective double well structure. The three-dimensional harmonic confinement in 3.58 is fixed to ωx = 2π · 78 Hz, ωy = 2π · 66 Hz and ωz = 2π · 90 Hz
for all calculations. For each choice of q0 , the remaining two free parameters have been
varied: the total atom number NT and the depth V0 of the periodic potential, which is
directly coupled to the barrier height Vb . A combination of NT and V0 is considered to
42
3.7 Feasibility study for bosonic Josephson junction experiments
Figure 3.12: Parameter space for a possible experimental implementation of a bosonic Josephson
junction. Each point corresponds to a combination of the total atom number NT and barrier height Vb ,
where both dynamical regimes are observable. The dark red square and errorbars indicate the parameter
regime, in which our experiments have been performed. The solid lines represent combinations of the
atom number and potential depth, for which the chemical potential of the BEC coincides with the
potential barrier.
be appropriate, if the following conditions are fulfilled:
• First, both regimes - Josephson oscillations and self-trapping - should be experimentally accessible by varying the initial population imbalance z(0). Since the
preparation of very small and the quantitative analysis of very large population
imbalances are very difficult (see chapter 5), the transition between the two regimes
should occur at a critical value 0.3 < zc < 0.6.
• A second condition is, that the tunneling time scales have to be much smaller than
the lifetime of the BEC, which is on the order of a few seconds. Therefore, only
tunneling dynamics with a period of less than 500 ms are taken into account.
The results of this feasibility study for a 87 Rb BEC are shown in figure 3.12 for three
different lattice spacings q0 = 5.2 µm, 10 µm and 15 µm, corresponding to an effective
double well spacing of d = 4.4 µm, 6.4 µm and 8 µm. Each point in the graph represents
an appropriate combination of the two parameters NT and Vb .
The lower bound for the potential barrier heights Vb for given q0 and NT is determined
by the fact, that the chemical potential is too far above the potential barrier between
the two wells, which means that atoms can reach the less occupied well by moving
over the barrier. In this case, the distribution is not self-trapped even for huge initial
population imbalances. The solid lines represent combinations of the atom number and
barrier heights, for which the chemical potential of the BEC coincides with the potential
barrier. A comparison of these lines with the lower bound of the calculated points
43
Chapter 3 Theory of Josephson junctions for Bose-Einstein condensates
shows, that the regime of self-trapping is still accessible, if the chemical potential is up
to typically 100 Hz above the barrier.
The upper bound is determined by the fact, that the potential barrier is too large
to allow for tunneling with z(0) = 0.3, which makes the Josephson tunneling-regime
inaccessible experimentally.
The feasibility study shows, that Josephson junction experiments have to be performed with very small total atom numbers on the order of18 NT ≈ 1000. The appropriate parameter space gets smaller for increasing lattice spacings and is experimentally
inaccessible for d > 8 µm. On the other hand, our imaging system with a resolution of
2.7(2) µm restricts the double well spacing to be at least 4 µm in order to be able to
quantitatively analyze the ongoing tunneling dynamics. The combination of the ability
to reproducibly produce BECs consisting of 1150(150) atoms with our imaging resolution have made it possible to observe the bosonic Josephson junction dynamics for
q0 = 5.2 µm in the parameter regime indicated with the dark red square and error bars
in figure 3.12.
18
The reproducible realization of such small BECs is one of the experimental problems one has to
face in order to realize a bosonic Josephson junction. A realization of a smaller s-wave scattering length
using a Feshbach resonance would allow for higher total atom numbers.
44
4 Experimental setup and procedure
In this chapter, an overview of the apparatus we use for the realization of Bose-Einstein
condensation and for the performed Josephson junction experiments is presented. A
detailed description of our experimental setup can be found in previous diploma and
PhD theses [74, 88].
In order to cool atoms to quantum degeneracy, a combination of different experimental techniques is required. The atomic sample has to be isolated from the environment.
Therefore the experiment has to be performed in ultra-high vacuum (UHV) conditions.
The rubidium atoms are trapped and pre-cooled in a magneto-optical trap (MOT). They
are subsequently loaded into a magnetic time orbiting potential (TOP) trap, where the
first stage of evaporative cooling is performed. The quantum phase transition to BoseEinstein condensation is finally reached after loading the sample into a crossed optical
dipole trap consisting of two focused laser beams and performing a second evaporation
stage. The Bose-Einstein condensate is subsequently loaded into an optical effective
double well trap. This realizes a weak link of two BECs, which is the analog of the
superconductive Josephson junction.
A schematic overview of the experimental setup is given in figure 4.1. It is divided
into two parts: the laser setup at the top and the darkened area at the bottom of the
graph, where the Bose-Einstein condensation of 87 Rb is realized. The different lasers
produce light for cooling, trapping and absorption imaging of the atomic sample. The
frequency as well as the power of the light, which is coupled into individual optical fibres
can be adjusted by acousto-optical modulators (AOMs). Mechanical shutters in front
of each optical fibre and a curtain around the vacuum chamber hold off residual nearresonant light, which could heat up the atomic cloud. The Bose-Einstein condensate
is produced in the glass cell in the lower right part of figure 4.1, where the MOT, the
TOP-trap, the dipole trap and the double well potentials are overlapped. For the sake
of clarity, figure 4.1 does not contain details of the optical, electronic and mechanical
setup.
The relevant parts of the experimental setup and the sequence which is used for
cooling the atoms to quantum degeneracy will be discussed in the following sections.
4.1 Laser system
We use four laser systems for Bose-Einstein condensation and the realization of the optical double well potential:
• The titanium sapphire laser Ti:Sa I (Coherent, Monolithic-Block-Resonator 110)
45
Chapter 4 Experimental setup and procedure
Figure 4.1: Schematic overview of the experimental setup: the upper part contains the laser systems
for cooling, trapping and absorption imaging of the atomic sample. The green part is darkened by a
curtain in order to prevent scattering of near-resonant photons. Bose-Einstein condensation is reached
in an optical crossed dipole trap after a pre-evaporation cooling stage in the magnetic TOP trap.
46
4.1 Laser system
Figure 4.2: Hyperfine-structure of 87 Rb and employed laser wavelengths. The Bose-Einstein condensate is produced in the 5S1/2 F = 2, mF = 2 state.
is pumped by a frequency-doubled diode laser pumped Nd:YAG Laser (Coherent,
Verdi V10) with 10 W optical output power. The Ti:Sa has a typical output power
of 1.5 W. It is locked to the (F = 2 −→ F 0 = (3, 1)) crossover transition of the
D2 line of 87 Rb. The light is divided into three parts for the funnel, the magnetooptical trap (MOT) and the absorption imaging beam. The different parts pass
individual acousto-optic modulators, since the wavelengths for the MOT, funnel
and imaging beam have to be slightly different. The imaging beam is exactly
shifted to the (F = 2 −→ F 0 = 3) transition, whereas the funnel and the MOT
have a detuning of a few MHz.
• An external cavity diode laser (ECDL, 100 mW output power) in Littrow configuration is locked to the (F = 1 −→ F 0 = 2) transition of the D2 -line. It is used to
repump atoms, which have decayed into the 5S1/2 state back to the MOT circuit.
This laser is often referred to as “repumper”.
• The diode laser pumped Nd:YAG III laser (Spectra-Physics, T40-X30-106QW)
with a wavelength of 1064 nm and a maximum output power of 8 W provides the
light for the optical dipole traps. The maximum power in the TEM00-mode, which
can be coupled into the single mode fibre, is 3 W.
• The light for the optical lattice, which is superimposed on the optical dipole trap
and creates the double well potential, is provided by a second titanium sapphire
47
Chapter 4 Experimental setup and procedure
Figure 4.3: Overview of the vacuum setup. It consists of two chambers, which are connected by a
differential pumping stage. The atomic funnel which is operated at a pressure of ≈ 10−9 mbar creates a
slow atomic beam which passes the differential pumping stage and is captured by the MOT, where the
pressure is below 10−11 mbar.
laser Ti:Sa II (Coherent, 899). This laser is also pumped by a Coherent Verdi V10
and is operated at a typical wavelength of approximately 810 nm and an output
power of 1.5 W.
The relevant atomic levels of 87 Rb and the wavelength of the different lasers used for
cooling, trapping and imaging are shown in figure 4.2.
4.2 Vacuum, funnel and MOT
In order to thermally isolate the cooled atomic cloud from the “hot” environment, an
ultrahigh vacuum chamber is indispensable. A pressure below 10−11 mbar is required
in order to be sure, that scattering with background gas atoms does not affect the
condensation process and the lifetime of the Bose-Einstein condensate in the optical
trap. On the other hand, the loading rate of a magneto-optical trap is proportional to
the partial pressure of rubidium, which should therefore not be below 10−9 mbar. We
therefore use a double MOT setup and a vacuum system consisting of two chambers
which are connected by a differential pumping stage. The schematics of the vacuum
setup is shown in figure 4.3.
In our experiment, dispensers are used as source of atomic rubidium. The partial
pressure of rubidium in the funnel chamber can be adjusted by the electric current
through the dispensers. The atomic funnel [89] is loaded from the background gas and
produces a cold continuous atomic beam, which passes the differential pumping stage
and is captured by the three-dimensional MOT [90] in the glass cell. The typical loading
rate of the MOT is 2 · 107 atoms/s at a temperature of ≈ 160 µK. The atom number in
48
4.3 Magnetic trap
the MOT can be monitored using a photodiode which collects the fluorescence of the
trapped atoms. As soon as the MOT contains typically 2 · 108 atoms, the cloud is further
cooled to 40 − 50 µK by means of an optical molasses. At this stage, our atomic sample
has a phase space density of Ω ≈ 3 · 10−7 .
4.3 Magnetic trap
After optically pumping the atoms to the low-field seeking |F = 2, mF = 2i state,
the atoms are loaded into the magnetic trap, which is spatially overlapped with the
MOT. The principle of magnetic trapping of atoms is based on the interaction of the
permanent magnetic moment µ with an external magnetic field B(r). The magnetic
interaction energy is given by
E(r) = −µ · B(r) = mF gF µB B(r),
(4.1)
where gF is the gyromagnetic factor and mF is the magnetic sub-level of the atom. Dependent on the sign of mF · gF , the atoms can be trapped in high (“high field seeker”)
or low (“low field seeker”) magnetic field strength. The Maxwell equations do not allow for a field maximum in regions without current. Therefore only low-field seeking
states are magnetically trappable. For the case of the ground state of 87 Rb, this restricts
the possible states to the magnetic sublevels1 |F = 1, mF = −1i, |F = 2, mF = 1i,
|F = 2, mF = 2i. In our experiment, we choose |F = 2, mF = 2i, since this state has
the highest magnetic moment, producing the steepest magnetic potentials for a given
magnetic field gradient and thus a spatially small atomic cloud. This allows for an efficient evaporative cooling, since the rate of the necessary collisions for re-thermalization
is proportional to the atomic density.
A magnetic field minimum can be experimentally implemented using a pair of coils in
anti-Helmholtz configuration, which produces a three dimensional spherical quadrupole
magnetic field. Atoms, which are moving in this field adjust their magnetic moment
adiabatically to the instantaneous direction of the magnetic field and therefore stay
spin polarized and trapped in the corresponding potential, except for the trap center,
where the magnetic field vanishes. Here, atoms can undergo Majorana-spinflips [91]
to untrapped magnetic sublevels. There are different possible ways to circumvent this
problem. The most common types of magnetic traps with a non-zero magnetic field
minimum are the Ioffe-Pritchard trap [92], the cloverleaf trap [93] and the time-orbiting
potential (TOP) trap [94], which is used in our setup. The TOP trap uses a superimposed
homogenous bias field B0 , which is rotating with a frequency of ωBias = 2π · 9.8kHz in
the radial x-y plane. This rotation frequency is chosen much larger than the frequency of
the center of mass motion of the atoms in the magnetic potential and much smaller than
the larmor frequency. In this case, the superposition of the two magnetic fields yields
a time-averaged three-dimensional harmonic potential with typical trapping frequencies
ωz
= 2π · 100 Hz.
ωx = ωy = √
8
The zero magnetic field is no longer in the center of the trap, but moves on a radial
circle, the so-called “circle of death” with radius rD = B0 /B 0 , where B 0 is the radial field
gradient of the quadrupole magnetic field. The circle of death can be used for forced
1
Due to the quadratic Zeeman-shift also |F = 2, mF = 0i is magnetically trappable.
49
Chapter 4 Experimental setup and procedure
evaporative cooling. By continuously reducing the bias field B0 from 32.5 G to 1.5 G
and therefore the circle of death from 1.6 mm to 75 µm at a fixed magnetic field gradient
B 0 ≈ 200 G/cm, we end up with approximately 3 · 106 atoms at a phase-space density of
Ω ≈ 5 · 10−3 . This cloud can now be loaded into the optical crossed dipole beam trap,
where the second stage of evaporative cooling to quantum degeneracy is performed.
4.4 Optical dipole potentials
The optical potentials, which are used for the last evaporation stage and the production
of the double well potential, are created by focused laser beams with a frequency, which
is far detuned from any resonance in the spectrum of 87 Rb. This allows for a realization
of largely conservative optical traps for neutral atoms.
In the framework of the electric dipole approximation [95], the conservative dipole
potential results from interaction of induced atomic dipole moments d with the oscillating
electric field E of the laser. In the following, I will only consider the case where the
detuning ∆ = ωL − ω0 of the laser frequency with respect to the considered line of
the rubidium spectrum with frequency ω0 is much larger than the Rabi frequency ΩR =
d·E/~ and the natural line-width Γ of the excited state. In this case, the dipole potential
(“ac Stark shift”) in a two-level system can be approximated by
~|Ω(r)|2
.
4∆
The square of the Rabi-frequency can be obtained by
V (r) =
|ΩR (r)|2 =
Γ2 I(r)
,
2Isat
(4.2)
(4.3)
where I(r) is the intensity of the laser beam and Isat is the saturation intensity of
the considered transition. For atoms with hyperfine structure, we additionally have to
take into account the corresponding Clebsch-Gordan coefficients, which are for example
listed in [96]. The spontaneous scattering rate, which is determined by the coupling of
the atoms to the vacuum modes is given by
Γ3 I(r)
.
(4.4)
Γs (r) =
8Isat ∆2
It determines the lifetime of the BEC inside the optical trap, since every spontaneous
scattering event removes the respective atom from the condensate.
It is important to note, that the potential 4.2 is attractive for red detuned (∆ < 0)
and repulsive for blue detuned (∆ > 0) light fields. Therefore, red detuned focused laser
beams can be used to realize optical traps for Bose-Einstein condensates.
4.4.1 Crossed optical dipole trap
Our three-dimensional optical trap is realized by two focused far red detuned laser beams,
which cross under an angle of nearly 90◦ . The overlap of the two beams with the magnetic
trap is optimized by maximizing the atom number transferred into the optical trap. The
laser light for optical trapping is provided by a Nd:YAG laser with a wavelength of
50
4.4 Optical dipole potentials
Figure 4.4: Schematic top view of the optical setup around the glass cell. Bose-Einstein condensation is performed in the crossed optical dipole trap consisting of the “dipole trap” (DT) and the “crossed
dipole trap”(X-DT). The superposition of the X-DT with an optical lattice of periodicity 5.2 µm creates the effective double well potential. For the sake of clarity, magnetic field coils, MOT-optics and
photodiodes for power stabilization of the different beams are not shown.
1064 nm, which is split into two parts (1 W for the dipole trap and 2 W for the crossed
dipole trap). The two beams pass individual acousto-optic modulators, which allow to
control the optical power and serve as a fast switch. The two AOMs are driven with a
frequency difference of 1 MHz in order to prevent interference of the two crossing traps.
The first diffraction order of the AOMs is coupled into single mode optical fibers, which
guide the light to the experiment. Each of the focused gaussian beams realizes a twodimensional confinement perpendicular to its propagation axis. For a definition of the x,
y and z-direction see figure 4.4. In the following, I will denote the beam propagating in
x-direction “dipole trap beam” (DT) and the beam propagating in y-direction “crossed
dipole trap beam” (X-DT).
The intensity distribution of a linearly polarized focused gaussian beam of wavelength
λD propagating in x-direction is
y2
z2
Imax
exp −2(
+
) ,
(4.5)
I(r) =
1 + (x2 /(xRy xRz ))
σy (x)2 σz (x)2
p
where σy,z (x) = σ0 y,z 1 + (x/xR y,z )2 . Here, σ0y,z is the beam waist in y, z-direction,
πσ
2P
xR y,z = λ0Dy/z is the corresponding Rayleigh length and Imax = πσ0y
is the maximum
σ0z
intensity of the beam with power P .
If the laser has a detuning ∆1,2 with respect to the D1 and D2 line of 87 Rb, the
optical potential can be approximated by
51
Chapter 4 Experimental setup and procedure
Vmax
y2
z2
V (r) =
exp −2(
+
) ,
1 + (x2 /(xR y xR z ))
σy (x)2 σz (x)2
1
2
~Γ2 Imax 1
with the trap depth Vmax = 8Isat 3 ∆1 + ∆2 .
(4.6)
For 87 Rb the saturation intensity is Isat = 1.58 mW/cm2 . The potential in equation
4.6 is the sum of the potentials resulting from the D1 and the D2 line. The factor
1/3 in Vmax takes into account the Clebsch-Gordan coefficients of the different hyperfine
transitions and the factor 2 accounts for the fact, that the D2-line is twice as strong as
the D1-line.
If we expand the potential 4.6 quadratically around the focus, we can calculate the
trapping frequencies inside a single focused laser beam
s
s
s
4|Vmax |
4|Vmax |
2|Vmax |
.
(4.7)
ωy =
,
ωz =
,
ωx =
2
2
mσ0 y
mσ0 z
mxR y xR z
Note that in the presence of the earth gravitational field, the minimum of the optical
potential in z-direction is shifted from the focus about the gravitational sag ∆z = g/ωz2
in the case of a parabolic potential. This effectively reduces all trapping frequencies,
since the potential has to be expanded around this new minimum.
The dipole trap beam in our experiment has a radial symmetry and a waist of
60(5) µm. It provides the confinement in y and z-direction. Since the Rayleigh length
is on the order of 1 cm, the confinement of the DT in x-direction is typically below 1 Hz
and therefore negligible. The crossed dipole beam has a waist of 140(5) µm in z-direction
and 70(5) µm in x-direction. It is chosen asymmetrically in order to be able to adjust
the harmonic confinement along the x-direction (this will be the direction of the double well potential) more or less independently of the transverse confinement in the y-z
plane. The dipole potentials add up in z-direction, which is the direction of gravity.
With a maximum power of approximately 500 mW in the dipole trap beam and 800 mW
in the crossed dipole trap2 , we can realize a maximum three-dimensional confinement of
ωx,max ≈ 2π · 120 Hz, ωy,max ≈ 2π · 170 Hz and ωz,max ≈ 2π · 180 Hz and a maximum trap
depth of approximately 5 µK.
All Josephson tunneling experiments have been performed with ωx ≈ 2π · 78 Hz,
ωy ≈ 2π · 90 Hz and ωz ≈ 2π · 66 Hz.
The spontaneous scattering rate per atom
2
Γ3 Imax 1 1
+
.
(4.8)
Γs =
8Isat 3 ∆21 ∆22
at typical optical powers is approximately 0.01 Hz. This means, that we do not expect any
influence of spontaneous photon scattering during our Josephson tunneling experiments,
which are performed within much less than one second.
The main advantage of the crossed optical dipole trap compared to the magnetic
trap is, that we can more or less independently adjust the confinement in the different
directions. Another advantage is, that the trap can be switched off within one millisecond
2
The loss of optical power on the way from the laser to the atoms is mainly dominated by the
diffraction efficiency of the AOMs, the fiber coupling efficiency (both ≈ 70%) and 10% light reflection
of the glass cell.
52
4.4 Optical dipole potentials
without introducing kinetic energy to the BEC. In contrast, a sudden turn-off of our
magnetic trap always leads to a kick of the BEC which differs from shot to shot. This
kick corresponds to a spatial phase gradient on the BEC and would have made the
measurement of the relative phase of the two localized modes in the double well trap
impossible. Furthermore, the position of our magnetic trap has a shot-to-shot uncertainty
on the micrometer scale. As we will see in chapter 5, this uncertainty does not allows
for a defined initial preparation of the population imbalance of the two wells. A stable
mechanical setup for the optical dipole trap (see section 4.6) allows for a beam stability
on the sub-100 nm level, which is required in order to perform the Josephson tunneling
experiments.
4.4.2 Realization of an optical double well potential
There are different possibilities to produce a double well potential for Bose-Einstein
condensates. The first realized double well potential was obtained by focussing a bluedetuned far off-resonant laser beam into the center of a magnetic trap, generating a
repulsive optical dipole potential. This setup with a well-spacing of typically 50 µm was
used for the first interference measurements with BECs [10]. Another implementation
of a double well potential with a typical spacing of 100 µm in a purely magnetic (time
orbiting potential) trap was reported in [37]. Recently, Shin et al. [38, 40] have implemented a purely optical double well potential formed by two focused laser beams with a
waist of 11.3 µm and a minimum distance of 15 µm.
Without using a Feshbach resonance to suppress the effect of nonlinearity, these
implementations are inappropriate for the realization of a Josephson junction for 87 Rb
Bose-Einstein condensates. In order to be able to observe Josephson tunneling dynamics
on a reasonable time scale with at least 1000 condensed atoms, the double well spacing
has to be well below 10 µm (see section 3.7).
Atom chips, on which microscopic wires are used to generate magnetic trapping
potentials, are predestinated to tailor potentials on a small scale. Double well potentials
on atom chips have served as beam splitters for guided atoms already in 2000 [97].
Recently, Estève et al. have realized a stable one-dimensional magnetic double well
potential with a transverse harmonic confinement on an atom chip [39]. The current
status of this experiment is, that a 87 Rb Bose-Einstein condensate of approximately
2·104 atoms is loaded into a double well trap with a well spacing of 1 µm. The population
imbalance can be adjusted by adding an additional bias field on the order of 1 G. However,
this setup is very sensitive to fluctuating external magnetic fields on the mG-level and
to atom-wire interactions, especially to current distortions inside the wires. This leads
to fragmented condensates along the longitudinal potential, which increases the local
interaction energy. The authors predict, that despite the noisy electromagnetic field
environment, coherent Josephson oscillations should be observable after fixing several
technical problems.
To date, coherent Josephson oscillations could not be observed in any of the described
experimental setups.
We use a new approach to realize an optical double well potential with a sufficiently
small well spacing. This potential is the result of a superposition of the three-dimensional
crossed beam dipole trap discussed above with a one-dimensional optical lattice in xdirection. The periodic potential is realized by a pair of laser beams with parallel linear
53
Chapter 4 Experimental setup and procedure
Figure 4.5: Effective double well potential, which is realized as a superposition of the crossed dipole
trap with a one dimensional optical lattice (periodicity 5.2 µm). The left hand side shows the complete
potential in x-direction. The relevant part of the potential is shown on the right. The Bose-Einstein
condensate is loaded into the two central wells, which have a distance of 4.4 µm.
polarization crossing at a relative angle of α = 9 ◦ . The two beams are aligned symmetrically with respect to the crossed dipole trap, such that the resulting potential is
modulated in x-direction:
VSW (x) = V0 cos2
πx d
with
d=
λsw
.
2 sin( α2 )
(4.9)
The potential depth is given by
√
~Γ2 I1 I2 1 1
2
V0 =
+
2Isat 3 ∆1 ∆2
(4.10)
where I1,2 are the intensities of the two beams and ∆1,2 are their detuning from the D1
and D2 line. Since the laser is red detuned (∆1 , ∆2 < 0), the atoms accumulate in the
intensity maxima of the standing light wave.
The laser light for the optical lattice is provided by a Titanium-Sapphire laser, which
is operated at a wavelength λsw = 811 nm. The light is coupled into a single mode
optical fiber. The stability of the laser mode can be monitored by means of a FabryPerot interferometer. The output of the fibre is split into two parts which pass individual
acousto-optic modulators serving to adjust the intensity of the laser beams and the phase
of the periodic potential. The AOMs are driven by the two channels of an arbitrary
waveform generator (Tektronix AWG 420), whose 10 MHz output signal is mixed with the
output of a voltage controlled oscillator operating at 100 MHz. The resulting 110 MHz
component is filtered and amplified. The beam waist at the position of the atoms is
σsw = 350(20) µm, which is much bigger than the typical extent of the BEC.
Figure 4.5 shows the effective double well potential in x-direction, resulting from the
superposition of the X-DT with the optical lattice. Since the spatial extent of the BEC
is much smaller than the waist of the crossed dipole beam, it can be approximated by
54
4.5 Stabilization of intensity and phase
πx 1
Vdw = mωx2 (x − ∆x)2 + V0 cos2
,
(4.11)
2
d
where ∆x is the relative position shift of the two potentials. For ∆x = 0, a symmetric
double well potential as shown in figure 4.5 can be created. For the Josephson junction
experiments, we use a periodicity d = 5.2(2) µm of the optical lattice and a potential
depth V0 = h · 412(20) Hz, where the numbers in parenthesis represent the experimental
uncertainty in the corresponding parameters.
As we will see in chapter 5, the initial distribution is sensitive to a relative shift on
100 nm-scale. This demands a high passive stability of the crossed dipole beam position
and an active stabilization of the phase of the periodic potential.
The combination of the crossed dipole beam trap with a frequency ωx = 2π · 78 Hz
and the standing wave results in a well-spacing of dwell = 4.4 µm and a barrier height
Vb = h · 263(20) Hz. The main advantage of our double well potential is, that in principle
any well spacing can be adjusted by changing the relative angle of the two beams realizing
the periodic potential. Additionally this setup allows to adjust the potential barrier
independently of the harmonic trapping frequencies, which makes it possible to choose
the optimal set of parameters for an experimental observation of nonlinear Josephson
junction dynamics at a given total atom number.
4.5 Stabilization of intensity and phase
The intensity of the dipole trap and of the crossed dipole trap are very critical experimental parameters, since intensity fluctuations on the percent level can already parametrically heat the cloud and inhibit Bose-Einstein condensation of small atom numbers.
Furthermore, intensity fluctuations of the laser beams realizing the one-dimensional optical lattice result in a fluctuating barrier height during a single experimental sequence
and from shot to shot. Therefore, it is important to stabilize all laser beam intensities
as good as possible. For this purpose, each beam is locked by an individual control
circuit. This consists of a monitor photodiode, which picks up the respective reflection
from the glass cell, an AOM and a proportional-integral (PI) loop. The electronic control
allows to stabilize the intensity on the photodiode and therefore the intensity inside the
glass cell to a nominal value given by the computer by adjusting the input power of the
AOM. The resulting relative stability of the beam intensities are better than 10−4 . This
minimizes shot-to-shot fluctuations of the atom number and allows for a reproducible
formation of a BEC consisting of 1000(150) atoms. The condensate lifetime inside the
double well potential is typically on the order of 10 s and does not limit the Josephson
oscillation measurements, which are performed within approximately 100 ms.
The intensity loop has a time scale of approximately 100 µs and therefore allows
for a fast (< 1 ms) turnoff of the optical potentials. This is especially important for
time-of-flight measurements.
Besides the intensity of the laser beams, the relative position of the crossed dipole
beam and the standing wave is crucial for the implementation of a bosonic Josephson
junction. The relative shift determines whether the resulting effective double well is
symmetric or asymmetric. A shift of 350 nm already induces a population imbalance z =
zc = 0.39 corresponding to the self-trapping threshold (see chapter 5). This means, that
55
Chapter 4 Experimental setup and procedure
Figure 4.6: Experimentally measured (red line) and theoretically expected (black line) phase lock
photodiode signal as a function of the relative phase shift of the two beams. This signal is used to
actively lock the relative phase of the one-dimensional optical lattice.
in order to be able to clearly experimentally distinguish the Josephson tunneling regime
from the self-trapping regime, a relative position stability below 100 nm is required.
In order to stabilize the position of the periodic potential, the relative phase of the two
lattice laser beams has to be controlled. For this purpose, two anti-reflection coated glass
plates reflect approximately 2% of the intensity of the two beams under a relative angle
of 6◦ . Those two reflected beams interfere outside the vacuum chamber and produce
an interference pattern with a periodicity of 8 µm. A precision air slit with a width
of 5(1) µm and a length of 3mm (Melles Griot) is placed in the overlap region and is
oriented parallel to the interference fringes. The light intensity transmitted through the
slit, which is dependent on the relative phase of the two beams, is monitored with a
photodiode. The experimental setup for the relevant parts of the phase-lock is shown
in figure 4.7. The phase of one beam can be controlled by adjusting the phase of the rf
signal driving the corresponding AOM. This is achieved by using four rf phase shifters
(Mini-Circuits Jsphs120) connected in series, each with a maximum phase shift of π. The
scaled signal of the photodiode as a function of the rf phase is shown in figure 4.6 (red
line). The maximum of the signal is on the order of 500 mV for typical experimentally
used intensities during the phase lock. The modulation of the signal is nearly 50%. The
black line shows the theoretically expected modulation for a lattice of periodicity 8 µm
and a slit of 5 µm width. The deviation from the measured signal is most likely due to
imperfect alignment of the slit with respect to the direction of the optical lattice. The
signal on the photodiode, and therefore the position of the optical lattice can be locked
at a given value using a proportional-integral loop which utilizes the phase-shifters to
adjust the relative phase of the two beams. We measure the position stability of the
periodic potential by fitting the position of the two Bose-Einstein condensates inside
the double well potential. From this measurement we obtain a standard deviation of
56
4.6 Mechanical stability and beam position control
Figure 4.7: Relevant optical parts for the position stabilization of the periodic potential and the
crossed dipole trap beam. The position of the optical lattice is controlled by adjusting the relative phase
of the two laser beams. The x-position of the X-DT is monitored using a four quadrant photodiode
with a spatial resolution of approximately 30 nm
approximately 100 nm, which corresponds to a phase stability of less than π/50. Since
one pixel of our CCD-chip corresponds to approximately 680 nm in real space, the fit
error of the position is on the order of the measured position fluctuations. Therefore we
believe, that the real position stability of the optical lattice is even better than 100 nm.
One drawback of this implementation of the phase-lock is, that it only works for a
constant intensity. However, for loading the Bose-Einstein condensate in the double well
potential, the periodic potential has to be adiabatically ramped up to a given value. We
therefore lock the phase for two seconds at a very small light intensity, before we ramp
to the final potential depth. This intensity is chosen such, that the resulting potential
barrier is far below the chemical potential. Therefore the initial adjustment of the phase
to the nominal value does not induce collective excitations of the BEC in the crossed
dipole trap. A sample and hold integrated circuit is used to fix the input signal of the
phase shifters as soon as we start ramping the periodic potential depth to its final value.
The standard deviation of the population imbalance directly after the ramp and after
two seconds inside the double well potential is comparably small. Therefore we do not
expect that the unlocked phase during the Josephson dynamics affects the tunneling
dynamics.
57
Chapter 4 Experimental setup and procedure
4.6 Mechanical stability and beam position control
The last section described the active position stabilization of the periodic potential. Since
the relative position of the standing wave with respect to the crossed dipole trap beam
is the relevant experimental parameter, we have to make sure that also the x-position of
the X-DT is stable on the same level. This implies a very stable mechanical setup, since
the distance of the atoms from the optical table is as large as 22 cm.
In the original setup for the crossed dipole trap individual conventional posts and
postholders (from Thorlabs Inc.) for mounting the optical parts have been used. The
optical fibre used for the crossed dipole trap beam was assembled by ourselves. In order to
control the x-position of the X-DT, a piezo stack was built into the mirror mount carrying
the fibre outcoupler. The standard deviation of the x-position of the X-DT using this
setup was approximately 800 nm for 30 successive shots. This value is unacceptable for
the reproducible adjustment of the initial population imbalance. Therefore we decided
to build a breadboard 19 cm above the optical table. This breadboard is fixed on six
stable aluminum mounts. A very stable piezo-actuated mirror mount (Thorlabs KC1PZ), which is directly screwed and glued onto the breadboard is now used to control
the position of the X-DT beam. The laser light is focused by a single lens, which
is directly glued onto the outcoupler of the new optical fibre (Schäfter + Kirchhoff
PMC-1060-P). The optical path has been reduced from 50 cm to 20 cm and now only
contains an additional cylindrical lens and an anti-reflection coated glass plate (see figure
4.7). This new setup allows for a standard deviation in the X-DT position below 80 nm.
Combined with the position stability of the periodic potential, this is sufficient for the
reproducible preparation of a defined initial population imbalance of the two BECs inside
the double well potential.
The anti reflection coated glass plate in the optical path of the crossed dipole trap
beam reflects a part of the light onto a four-quadrant photodiode (see figure 4.7). This
photodiode allows to monitor the position of the beam with a spatial resolution of
approximately 30 nm. We have actively stabilized the beam position to the photodiode signal using another PI-loop which acts on the piezo of the piezo actuated mirror
mount. However, it turned out that the mechanical setup of the four quadrant photodiode and the glass plate was not more stable than the unlocked position of the X-DT
itself. Therefore the active position control did not improve the stability.
In order to initially prepare a given population imbalance, we use a relative shift
∆x 6= 0 of the X-DT with respect to the optical lattice. This shift is experimentally
implemented with the piezo actuated mirror mount. After the initial preparation of the
BEC, the X-DT beam is shifted back to ∆x = 0, which initiates the Josephson tunneling
dynamics. The piezo setup has a response time of 7 ms, which allows to shift the beam
non-adiabatically with respect to the Josephson tunneling time scales which are on the
order of 50 ms. However, the finite response time has to be taken into account in the
numerical integration of the NPSE in order to quantitatively explain our experimental
results on Josephson tunneling. The shift could in principle also be done by shifting the
phase of the periodic potential. However, this would move the position of the minima
of the double well potential, which are basically given by the optical lattice. Therefore,
unintentional collective oscillations of the BEC inside the double well potential would
be excited.
58
4.7 Experimental sequence
Figure 4.8: Experimental sequence for the observation of Josephson junction dynamics. The BEC
is formed inside a crossed optical dipole trap and subsequently loaded into an asymmetric double well
potential. After this preparation stage, the potential is shifted to the symmetric double well case, which
initiates the Josephson junction dynamics. The sequence ends with the imaging of the wavepacket after
a variable propagation time. The black dotted line indicates the activation of the position lock for the
periodic potential and the dashed line shows the moment, when the sample-and-hold circuit is activated.
4.7 Experimental sequence
The experimental sequence used for Bose-Einstein condensation of the atomic cloud and
preparation in the double well potential is shown in figure 4.8. The figure does not show
the MOT-loading process (≈ 15 s), optical molasses, transfer into the magnetic trap and
circle-of-death cooling (≈ 30 s). At time zero, the transfer from the magnetic trap into
the crossed optical dipole trap is initiated by ramping up the intensities of the two laser
beams (dark blue and green line). At this stage, the atomic cloud has a phase space
density of Ω ≈ 5 · 10−3 . As soon as the laser beam intensities have reached the maximum
value, we ramp down the magnetic quadrupole field indicated by the red line. The whole
transfer process takes 2.5 s, which is chosen in order to prevent collective excitations of
the atomic cloud. We subsequently perform forced evaporative cooling by lowering the
laser beam intensities within 3.5 s. The minimum beam intensities determine the final
atom number in the Bose-Einstein condensate. We can reproducibly generate BECs
consisting of 5 · 104 down to 1 · 103 atoms in the crossed optical dipole trap. As soon as
the desired atom number is reached, the dipole trap beam intensities are again ramped up
and at the same time the periodic potential (light blue line) is increased to a given value.
This creates two weakly linked BECs inside an asymmetric double well potential with
given harmonic trapping frequencies and potential barrier. After the initial preparation,
the potential is shifted to the symmetric case by shifting the X-DT position (black line),
whereby the Josephson tunneling dynamics are initiated. After a given propagation time,
the potential barrier is suddenly ramped up and the X-DT is switched off. This results
in dipole oscillations of the atomic clouds around two neighboring minima of the periodic
59
Chapter 4 Experimental setup and procedure
potential. The atomic distribution is imaged at the time of maximum separation using
absorption imaging techniques.
The whole sequence from the beginning of the MOT loading to the destructive absorption image of the cloud takes approximately one minute. In order to measure the
Josephson dynamics with a sensible statistics and time resolution, a BEC in the double well potential has to be produced several hundred times with reproducible initial
population imbalances. This demands for a exceptional long run stability of the system
(especially atom numbers, laser beam intensities and positions) and endurance of the
experimenters.
Computer control
The experimental sequence is controlled with a time resolution of 1 ms using a programm
written in “LabView”. We use 16 analog output channels (-10V to 10V, 1mV RMS
noise), 16 digital output channels and 8 analog inputs to control all relevant experimental
parameters. A second computer controls the CCD camera taking the absorption images.
The first evaluation of the taken images is performed directly after each experimental
sequence.
4.8 Imaging setup and optical resolution
The imaging system is crucial for experiments with cold atoms, since all experimental
results are extracted from the taken images. We use destructive absorption imaging,
which is appropriate for the small atom number N < 104 used in our experiments. For
an overview over non-destructive imaging methods see for example [66].
A collimated gaussian σ + polarized laser beam with a waist of 1.9mm, which is
resonant with the F = 2 → F 0 = 3 transition of the D2-line illuminates the atomic
cloud. In order to keep the atoms spin-polarized inside the optical dipole trap, we
apply a homogenous magnetic field in the direction of the imaging beam. A part of the
laser light is absorbed by the BEC. The shadow of the atoms is approximately 10-fold
magnified and imaged onto a CCD camera (Theta-System SiS s285M) by a commercial
aspheric lens system (Zeiss Plan-Apochromat S, focal length f = 10 cm). The CCDchip consists of 1040 × 1392 pixels with a pixel size of 6.45 µm ×6.45 µm and has a
quantum efficiency of approximately 30%. The BEC is heated up and destroyed during
the imaging pulse. However, the broadening of the cloud during the exposure time of
4 µs is below one micrometer and therefore negligible. For each realization, we take three
images of the transmitted intensity with a time lag of 500 ms: one image Ipic of the light
in the presence of the atoms, one reference image Iref without atoms and one background
image Iback without imaging light. The spatial dependent relative intensity is calculated
using
T (x0 , z 0 ) =
Ipic (x0 , z 0 ) − Iback (x0 , z 0 )
.
Iref (x0 , z 0 ) − Iback (x0 , z 0 )
(4.12)
Here, y 0 denotes the propagation direction of the imaging beam, which encloses an angle
of 12.3◦ with the crossed dipole trap beam (see figure 4.4) and the x0 − z 0 -plane is
perpendicular to y 0 . All taken images are projections of the atomic cloud onto the imaging
60
4.8 Imaging setup and optical resolution
Figure 4.9: Measurement of the magnification of the imaging system. The BEC is released from
the optical trap and accelerated in the earth gravitational field. The magnification deduced from a
parabolic fit is M = 9.5(4).
plane. Expression 4.12 eliminates the influence of background light and of stationary
interference fringes caused by imaging optics. However, time dependent interference
fringes, which are caused by oscillations of the glass cell and the imaging lens or timedependent density fluctuations in the air, limit the quality of the data.
The relative intensity loss inside the atomic cloud is given by
dI(x0 , y 0 , z 0 )
= −n(x0 , y 0 z 0 ) σ(x0 , y 0 , z 0 )dy 0 ,
I(x0 , y 0 , z 0 )
(4.13)
where n(x0 , y 0 z 0 ) is the atomic density. The scattering cross section σ(x0 , y 0 , z 0 ) is dependent on the local light intensity I(x0 , y 0 , z 0 ) [96]. For imaging light which is resonant with
the transition of frequency ω0 , one gets
σ(x0 , y 0 z 0 ) =
1
Γ~ω0
.
0
2Isat 1 + I(x , y 0 , z 0 )/Isat
(4.14)
Using this expression, the atomic density and the total atom number can be obtained
by integrating equation 4.13.
Under the condition s(x0 , y 0 , z 0 ) 1, equation 4.13 can be solved analytically. In this
case, the atom number which is imaged to a single pixel is given by [74]
Npixel (x0 , z 0 ) =
A
(1 − T (x0 , z 0 )),
σM 2
(4.15)
where A is the pixel area, M is the magnification of the imaging system and σ =
σ(x0 , y 0 , z 0 ) is assumed to be constant. In our experiments, the typical saturation is
I/Isat ≈ 10, such that equation 4.15 underestimates the real atom number by typically
less than 10%.
61
Chapter 4 Experimental setup and procedure
Figure 4.10: Determination of the optical resolution of the imaging system by loading a BEC into a
deep optical lattice of period d = 4.46(5) µm. a) Observed density distribution. b) theoretically expected
density distribution without including the finite resolution of the imaging system. c) theoretically
expected density distribution after convolution with an Airy function of width δRayleigh = 4.1(2) µm.
d) Corresponding density profiles in x-direction (red line: experimental data, black line: theoretically
expected density profile, green line: theoretically expected density profile after convolution).
The magnification M of the imaging system is calibrated using a time-of-flight method.
For this purpose, a Bose-Einstein condensate is produced inside the crossed optical dipole
trap and subsequently the trapping laser beams are switched off. The BEC is accelerated
in z-direction due to the earth gravitational field. Figure 4.9 shows absorption images
after different time of flights up to 10 ms. By fitting the center of mass trajectory with
a parabola, we obtain the magnification M = 9.5(4). Therefore, one pixel on the CCD
chip corresponds to 678(30) nm in real space.
Optical resolution of the imaging system
The optical resolution of the imaging system can in principle be calculated using the
Abbe formula [98]
δRayleigh = 0.61
λ
NA
(4.16)
which is based on the Rayleigh criterion of resolution3 . In equation 4.16, NA is the
numeric aperture of the imaging lens, which is NA = 0.218 in our setup. Therefore,
the theoretically expected resolution is δRayleigh = 2.18 µm. Since equation 4.16 does not
take into account any abberations, this is a lower bound for the real optical resolution of
the imaging setup, which has to be determined experimentally. A detailed investigation
of this subject will be presented in the diploma thesis of Jonas Fölling. Therefore, I will
only give a summary of the obtained results.
A BEC consisting of 1.8 · 104 atoms is loaded into a deep one-dimensional optical
lattice with a potential depth V0 = h · 1740(50) Hz and a lattice spacing d = 4.46(5) µm
superimposed on a harmonic confinement with trapping frequencies ωx = 2π· 13.5(5) Hz
3
The optical resolution according to the Rayleigh criterion corresponds to the position of the first
zero of the Airy-function.
62
4.8 Imaging setup and optical resolution
and ωy ≈ ωz = 2π· 116(5) Hz. Figure 4.10a shows an absorption image of the experimentally observed atomic density distribution. The asymmetry of the density distribution
in x-direction is due to a tilt α = 0.63◦ of the dipole trap with respect to the plane perpendicular to the direction of gravity, which results in a weak distortion of the gaussian
potential profile. The density distribution can be compared to the distribution expected
from the numerical solution of the NPSE for the corresponding trapping parameters,
which has to take into account the enclosed angle of 12.3◦ between the imaging plane
and the plane perpendicular to the optical lattice. The theoretically expected atomic
distribution is shown in Figure 4.10b. It is much more modulated than the experimental
data, since it does not include the effect of a finite resolution of the imaging system. The
optical resolution is obtained by convolving this theoretically obtained distribution with
an Airy function of variable size and comparing the result to the measured atomic distribution. The best agreement is obtained using an optical resolution of δRayleigh = 4.1(2) µm.
The corresponding convolved theoretical atomic distribution is shown in figure 4.10c.
Figure 4.10d shows density profiles in x-direction, which are obtained by averaging
over the three central rows of the two-dimensional distributions. The experimental data is
represented with the red line and shows good agreement with the numerically calculated
profile (green line), which has been convolved with an airy function of width δRayleigh =
4.1(2) µm. The black line represents the density profile corresponding to the numerical
solution without taking into account the finite resolution.
The Rayleigh criterion of resolution is rather pessimistic, since the sum of the Airy
functions of two point sources located at a distance δRayleigh still shows a clear minimum with a modulation of approximately 26%. If the two point sources are further
approached, this minimum vanishes at a certain distance δSparrow . The resolution criterion based on this condition is known as the Sparrow-criterion. It represents the far
more appropriate resolution criterion for modern detection devices [99]. The Sparrow
resolution is given by
λ
.
(4.17)
NA
From our experimental data, we obtain δSparrow = 3.2(2). The resolution expected from
equation 4.17 is approximately 45% smaller than the experimentally obtained value.
This deviation has several reasons:
δSparrow = 0.47
• The optical resolution of the experimental imaging setup is not only limited by
diffraction, but also by abberations of the imaging lens.
• The atomic clouds have a waist of approximately 5 µm in z-direction, which has to
λ
be compared to the Rayleigh depth of sharpness ∆zR = NA
2 ≈ 8 µm. Therefore,
the parts of the cloud, which are not in the central focal plane are effectively
broadened by the imaging system.
• The resonant imaging light heats up the atomic clouds and leads to an expansion particularly in imaging direction [74]. This expansion is given by ∆zh =
1/4vR Γt2 ≈ 800 nm, where vR = 5.7 mm
is the recoil velocity of a 87 Rb atom
s
emitting a photon on the D2 transition and t = q
4 µs is the imaging time. The
broadening in the x’-y’ plane is given by ∆z⊥ = Γ3 vR t3/2 ≈160 nm. This effect
additionally broadens the imaged atomic distribution.
63
Chapter 4 Experimental setup and procedure
Figure 4.11:
Improved experimental setup for the standing light wave. All optical parts are
mounted onto a massive aluminum body. The relative phase of the laser beams is adjusted using a
single electro-optic modulator.
Therefore the value for the optical resolution above is only an upper bound for the
real resolution of the imaging setup.
Another way to determine the resolution is to produce a BEC consisting of ≈ 6000
atoms in the crossed dipole trap with frequencies ωx = 2π · 25(2) Hz and ωy ≈ ωz =
116(5) Hz. The experimentally measured 1/e2 width of the wavepacket is σz = 2.9(2) µm.
From a numerical calculation of the corresponding ground state, one obtains a 1/e2
width of 1.7 µm. A deconvolution of the measured profile yields an optical resolution
δSparrow = 2.7(2). This will be referred to as optical resolution of the imaging setup in
the following sections.
4.9 New experimental setup for the optical lattice
As already discussed above, the depth and the position of the potential barrier are crucial
experimental parameters. The lock of the relative phase of the two beams allows for a
sufficient short term and long run stability of the lattice position. In the setup described
above, we use two independent optical paths of approximately one meter and two AOMs
to control the intensity and phase of the individual beams. It turned out, that this setup
shows a long term drift of the position of the individual beams and therefore of the
barrier height. This made a daily calibration of the potential depth necessary. Therefore
we have decided to build a new stable setup for the optical lattice after we have finished
the Josephson junction measurements presented in chapter 5. This new setup is shown
in figure 4.11. All optical parts are mounted onto a single massive aluminum body. The
beam intensities are now controlled using a single AOM in front of the optical fibre.
After the fibre outcoupler, the laser beam is split into two parts using a non-polarizing
beam splitter, which supersedes the use of additional waveplates. The relative phase of
the two beams and therefore the position of the lattice is controlled by a single electro-
64
4.9 New experimental setup for the optical lattice
optic modulator which is placed into one of the two beams. The optical paths could be
reduced to 30 cm. We use the fibre outcoupler to adjust the beam waist inside the glass
cell to approximately 500 µm without using any additional focussing lenses.
This new setup allows us to run the experiment for several days without re-calibration
of the optical potentials.
65
Chapter 4 Experimental setup and procedure
66
5 Experimental results
In this chapter the experimental results on Josephson junction dynamics of a BoseEinstein condensate in a double well potential are presented. These results have been
published in Physical Review Letters [100].
5.1 Calibration of the double well potential
The characterization of the double well potential is crucial for the investigation of Josephson dynamics of two coupled Bose-Einstein condensates. In order to adjust the atom
number such, that we are able to observe both dynamical regimes - Josephson oscillations
and macroscopic quantum self-trapping - all optical potentials have to be calibrated with
a precision of a few percent. The calibration of the double well potential is done in three
steps. We independently measure the trapping frequencies of the three-dimensional harmonic confinement, the lattice spacing and the potential depth of the one-dimensional
optical lattice.
5.1.1 Harmonic confinement
The calibration of the harmonic trapping frequencies in x- and z-direction is performed
by exciting collective dipolar oscillations and deducing the center of mass position as a
function of the evolution time from the taken absorption images. The dipolar oscillation
in z-direction (direction of the earth gravitational field) is excited by reducing the power
of one of the dipole trap beams for one millisecond. After readjusting the initial power,
the BEC starts to oscillate with an amplitude of a few micrometers. The oscillation
in x-direction is simultaneously excited by shifting the crossed dipole trap beam using
the piezo actuated mirror. Figure 5.1 shows the experimental data (red crosses) and a
damped sinusoidal fit (black line) for the parameters we use in the Josephson junction
experiments. The measurement yields harmonic trapping frequencies ωx = 2π · 79(2) Hz
and ωz = 2π · 66.0(2) Hz.
Since we can not directly observe oscillations in y-direction with our imaging setup,
we have to calibrate the third frequency indirectly. The contribution of the X-DT to
this frequency is negligible, since the longitudinal trapping frequency is below 1 Hz for
all used beam intensities. We measure the trapping frequency in z-direction and the
corresponding equilibrium position for four different intensities of the dipole trap beam.
During this measurement, the crossed dipole trap beam is operated at a very small
power and does not contribute to the measured frequencies. The potential in z-direction
is therefore solely given by the dipole trap beam and the earth gravitational potential
67
Chapter 5 Experimental results
Figure 5.1: Calibration of the harmonic confinement in x- and z-direction. A dipolar oscillation
in both directions is simultaneously excited and the center of mass motion of the BEC around the
equilibrium position of the optical dipole trap is deduced from absorption images (red crosses). The
damped sinusoidal fit to the data (black line) yields the corresponding trapping frequencies ωx = 2π ·
78.8(2) Hz and ωz = 2π · 66.0(2) Hz.
V (z) = Vmax,DT e−2(z−z0 )
2 /σ 2
DT
− mgz
(5.1)
By harmonically expanding equation 5.1 around the respective minimum and comparing
the results to the measured frequencies and equilibrium positions, we can deduce the
waist σDT of the DT, the position z0 of the dipole trap beam and the absolute values of
Vmax,DT for the four different adjusted relative beam intensities.
The knowledge of these parameters allows for a calculation of the trapping frequency
in y-direction as follows. The z-position zcloud of the BEC for the trapping parameters we
are interested in can be deduced from the measurement shown in figure 5.1. Using this
value, the gravitational sag ∆z = zcloud −z0 , which is the displacement from the intensity
maximum z0 of the beam can be calculated. The potential in y-direction has a gaussian
shape even in the presence of the earth gravitational field acting in z-direction. Under
the assumption of a radially symmetric dipole trap beam, the corresponding trapping
frequency is given by (see equation 4.7)
s
2
2
4|Vmax,DT |e−2(zcloud −z0 ) /σDT
= 2π · 91(2) Hz
(5.2)
ωy =
2
mσDT
where Vmax,DT is the potential at the intensity of the dipole trap beam used for the
measurement in figure 5.1.
5.1.2 Lattice spacing
The lattice spacing of the periodic potential can in principle be obtained by measuring the
relative angle of the two lattice laser beams. However, due to the restricted access to the
68
5.1 Calibration of the double well potential
Figure 5.2: Calibration of the lattice spacing of the periodic potential. A Bose-Einstein condensate
consisting of 2.8 · 104 atoms is loaded into a deep optical lattice. a) Absorption image of the atomic
cloud b) Density profile in x-direction (red line). The distance of the density maxima corresponds to
the lattice spacing d = 5.18(9) µm, which is obtained by fitting the density profile with a sinusoidally
modulated gaussian (black line).
beams passing below the breadboard, the angle α = 9(1)◦ can only be determined with
an accuracy of approximately 10%. This limits the knowledge of the lattice spacing to
d = 5.17(46) µm, which is not accurate enough for the quantitative comparison between
numerical simulations and the Josephson junction experiment.
Therefore a direct measurement of the lattice spacing has to be performed. For
this purpose, a BEC consisting of 2 · 104 atoms is loaded into a deep optical lattice
(V0 ≈ h · 2000 Hz) superimposed on a harmonic trap of trapping frequencies ωx,y,z =
2π · (5, 120, 140) Hz. For those parameters, the influence of the shallow harmonic potential in x-direction on the spacing of the potential minima is negligible. Therefore, the
spacing of the maxima of the atomic density directly corresponds to the lattice spacing d.
The resulting atomic distribution is shown in figure 5.2, where several potential wells are
occupied. The density profile in x-direction (red line in figure 5.2) is fitted with a sinusoidally modulated gaussian (back line) and the spacing of the atomic density maxima is
extracted. By taking the mean value over five realizations, the projection of the spacing
onto the imaging plane can be determined to dimag = 5.06(9) µm. Since the direction of
the optical lattice encloses an angle of 12.3◦ with the imaging plane, the obtained value
has to be corrected to d = dimag / cos (12.3◦ ) = 5.18(9) µm, which is in agreement with
the estimation given above.
5.1.3 Potential depth
The potential depth of the periodic potential is calibrated by measuring the relative
motion of the two BECs in the double well potential. A dipolar oscillation inside the respective wells is excited by non-adiabatically increasing the well spacing. This is achieved
by loading the BEC into the effective double well potential (black line in figure 5.3a) and
subsequently switching off the crossed dipole beam and increasing the potential depth
by a factor of 4.9 within 2 ms. At the beginning, the double well spacing dwell = 4.2 µm
is determined by both, the lattice spacing d = 5.18 µm of the optical lattice and the
strength of the harmonic confinement in x-direction. By switching off the crossed dipole
beam, we change the double well potential to a purely sinusoidal optical lattice with pe-
69
Chapter 5 Experimental results
Figure 5.3: Calibration of the periodic potential depth. a) The BEC is first loaded into the effective
double well potential (black line). We subsequently switch off the harmonic confinement in x-direction
and increase the depth of the optical lattice. This increases the distance between the two wells and
causes an oscillation of the two BECs in the individual wells. b) Relative position of the the two BECs
in the optical lattice as a function of time after exciting the dipolar oscillation inside the wells. The
experimental data (red crosses) is compared to the numerical simulation (black line), which has only
the lattice depth as a free parameter.
riodicity d = 5.18 µm (red line in figure 5.3a). Therefore the two BECs start oscillating
around the new potential minima. The final potential depth of the periodic potential
is chosen such, that no tunneling can occur during the oscillation. It turned out that
an instantaneous change from the double well to the purely periodic potential also excites collective excitations in the transverse direction, which couples to the x-direction
via the nonlinear interaction term in the Gross-Pitaevskii equation. Therefore we have
chosen a ramp time of 2 ms, which strongly reduces these excitations, but also reduces
the expected oscillation amplitude inside the single wells to approximately 400 nm.
Figure 5.3b shows the distance between the two BECs as a function of time after
exciting the dipolar oscillation in the individual wells (red crosses). The black line is the
result of a numerical simulation of this measurement using the nonlinear non-polynomial
Schrödinger equation. The only free parameter in the simulation is the initial potential
depth V0,initial of the periodic potential, since the trapping frequencies of the harmonic
confinement have already been calibrated. By calculating the numerical solution for
different V0,initial and comparing the results to the experimentally obtained oscillation,
we get a potential depth V0,initial = h · 412(20) Hz. This potential depth corresponds to
a barrier height Vb = h · 263(20) Hz for the experimentally used harmonic confinement
ωx = 2π · 78 Hz.
5.2 Determination of dynamical variables
The macroscopic tunneling dynamics of a Bose-Einstein condensate in a double well
potential is governed by the time evolution of the two dynamical variables, the fractional
population imbalance and the relative phase between the left and right component. Their
experimental determination is discussed in this section.
70
5.2 Determination of dynamical variables
Figure 5.4: Determination of the fractional population imbalance. a) Absorption image of the BEC
in the double well potential after increasing the well spacing (see text). The fringes around the atomic
cloud result from the fact, that the size of the individual BECs in x-direction is approximately a factor
of two smaller than the optical resolution. b) The density profile in x-direction (red line) can be well
approximated by a sum of to gaussian functions (black lines). b) The profile in z-direction is fitted using
a gaussian function.
5.2.1 Fractional population imbalance
The population imbalance is obtained from the absorption images taken after a given
propagation time of the BEC inside the double well potential. In order to be able to
clearly resolve the two localized modes, we increase the distance between the two BECs
before imaging the atomic distribution. This is done by exciting a dipolar oscillation of
the BECs in the individual wells as described in section 5.1.3. Since the lattice depth
is increased by a factor of 4.9, no further Josephson tunneling takes place, so that the
population imbalance is “frozen” during this process. The absorption image of the atomic
cloud is taken at the time of maximum separation ∆t = 1.5 ms. At this time, the two
wave-packets have a distance of approximately 5.5 µm and can therefore be resolved by
our imaging system. A typical absorption image of a BEC consisting of 1200(100) atoms
in the double well potential is shown in figure 5.4a. By fitting the profile of the optical
density in x-direction (red line in figure 5.4b) with a sum of two gaussian functions and
in z-direction (figure 5.4c) with a gaussian function, we obtain the atom numbers in the
individual wells and the fractional population imbalance
z=
N1 − N2
= 0.23(3).
NT
(5.3)
An alternative way to determine the atom number in each well without applying a fit
routine is to sum over the pixels in the respective regions (red rectangles in figure 5.4a).
From this approach we get zcount = 0.25(4), which is consistent with the value obtained
from the fit.
71
Chapter 5 Experimental results
Figure 5.5: Determination of the relative phase of the two localized modes. The figures on the left
hand side show absorption images of the atomic interference pattern after a time of flight of 8 ms after
switching off the trapping potentials. From the fit (black line) to the density profile in x-direction (red
line) we obtain a) φ = 0.1(1)π and b) φ = −0.9(1)π.
5.2.2 Relative phase
In order to measure the relative phase of the localized modes, the wavepackets are
released from the double well potential by suddenly switching off the confining laser
beams. The two BECs are accelerated in the earth gravitational field and expand due to
their dispersion. After a certain time-of-flight, the two clouds overlap and form a double
slit interference pattern [10]. The phase of the atomic interference pattern is directly
connected to the relative phase of the two BECs. In order to extract the relative phase,
we assume, that the density distribution in x-direction of the BEC inside the double well
potential can be approximated by the sum of two gaussians with independent amplitudes1
Ψ(x) = A1 e−
(x−x1 )2
σ2
+ A2 e−
(x−x2 )2
σ2
.
(5.4)
The corresponding interference pattern, which is formed in the far field after releasing
the cloud from the trapping potentials can be obtained by Fourier transformation of
equation 5.4, which is given by
2
−2((k−k0 )/σk )2
|Ψ(k)| = e
2π
2
2
A1 + A2 + 2A1 A2 cos
(k − k0 ) − φ ,
dT OF
(5.5)
where k0 is the center and σk the waist of the gaussian envelope in momentum space, φ
the relative phase of the two localized modes and dT OF the periodicity of the interference
fringes.
1
The width of the two gaussians is assumed to be equal since the difference of the two widths is
below 10%, even at a population imbalance z = 0.6.
72
5.3 Preparation of the initial population imbalance
Figure 5.6: The initial population imbalance z(0) 6= 0 is prepared by adiabatically loading the BEC
into the ground state of an asymmetric double well potential (∆x 6= 0) shown in the left figure. The
potential is subsequently shifted to the symmetric case within 7 ms in order to initiate the macroscopic
tunneling dynamics.
Figure 5.5 shows two typical absorption images of atomic interference patterns formed
after 8 ms time-of-flight. The density profiles along the x-direction are fitted using equation 5.5. We extract a relative phase of φ = 0.1(1) π for the image in figure 5.5a and
φ = −0.9(1) π for figure 5.5b.
The evaluation of the experimentally observed interference pattern using equation
5.5 requires a long time of flight ensuring that the measurement is performed in the far
field. However, the optical density of the atomic cloud drops quickly after releasing a
BEC containing approximately 1000 atoms from the trap, which makes an imaging with
a reasonable signal to noise ratio impossible for time of flights longer than typically 8 ms.
Numerical simulations have shown, that the far field is reached after a time of flight of
5 ms in the Josephson oscillation regime and 8 ms in the self-trapping regime2 .
5.3 Preparation of the initial population imbalance
In order to observe Josephson dynamics of a Bose-Einstein condensate inside a symmetric
double well potential, we prepare an initial population imbalance z(0) 6= 0. This is
experimentally implemented by adiabatically loading the BEC into the ground state3
of an asymmetric double well potential as shown in figure 5.6a. The asymmetry of the
potential and therefore the population imbalance of the corresponding ground state can
be controlled by adjusting the relative position shift ∆x of the crossed dipole trap beam
realizing the harmonic confinement in x-direction with respect to the optical lattice. The
potential is subsequently shifted to the symmetric double well configuration (figure 5.6b
within 7 ms using the piezo actuated mirror which has been calibrated by measuring
2
For those time of flights, the systematic errors of the phases obtained by applying equation 5.5 are
less than 0.1π.
3
It is important to note, that the preparation in the ground state of the shifted potential implies a
zero initial relative phase of the two localized modes independently of the asymmetry of the double well
potential.
73
Chapter 5 Experimental results
Figure 5.7: Theoretically expected ground state population imbalance (black line) as a function of
the relative shift ∆x of the harmonic confinement with respect to the optical lattice. The filled red
circles show the measured population imbalances. The transition is expected to occur at a relative shift
∆x = 350 nm.
the x-position of the BEC in the harmonic trap as a function of the piezo voltage. This
initiates the macroscopic tunneling dynamics which will be presented in the next section.
Figure 5.7 shows the calculated ground state population imbalance as a function of the
relative shift ∆x (black line) obtained by an imaginary time propagation of the nonlinear
nonpolynomial Schrödinger equation using the calibrated potential parameters. The
red points are the measured population imbalances. The error bars represent statistic
and systematic errors4 . Josephson oscillations are expected in the gray shaded region.
The theoretically predicted transition to the self-trapping regime occurs at zc = 0.38
corresponding to a relative position shift of ∆x = 350 nm. The mechanical stability
of our setup allows for a reproducible adjustment of any initial population imbalance
with a standard deviation of ∆z(0) = 0.06. This enables a preparation of the initial
population imbalance corresponding to the Josephson oscillation regime as well as to the
self-trapping regime.
5.4 Observation of Josephson oscillations and self-trapping
In this chapter, the first experimental observation of a Josephson tunneling between two
Bose-Einstein condensates is presented.
After the preparation of a given initial population imbalance and the subsequent shift
to the symmetric double well case, the tunneling massive particles are directly imaged
using absorption imaging. Figure 5.8 shows the taken images (17.5 µm × 9.8 µm) of
the Bose-Einstein condensate after a variable evolution time in the symmetric double
well potential. Before taking the images, the potential barrier is suddenly ramped up
and the harmonic confinement in x-direction is switched off. This results in dipole
4
74
The errors are dominated by fit errors of the lattice position and the population imbalance.
5.4 Observation of Josephson oscillations and self-trapping
Figure 5.8: Observation of macroscopic tunneling dynamics between two weakly linked BECs in
a symmetric double well potential. Shown are absorption images of the atomic cloud after a variable
evolution time. a) Josephson oscillations are observed, if the initial population imbalance z(0) = 0.28(6)
is below the critical value zc = 0.38. b) In the self-trapping regime with z(0) = 0.62(6), the population
inside the individual wells is stationary within the experimental errors.
oscillations of the atomic clouds in the individual wells. The images are taken at the
time (1.5 ms) of maximum separation (5.5 µm) of the two BECs. Each image corresponds
to a new experimental realization. The experimental data are post-selected retaining
only realizations of atom numbers between 1000 and 1300 atoms5 . This is necessary in
order to obtain comparable overlaps of the two localized modes and therefore comparable
tunneling times in different realizations.
Figure 5.8a shows macroscopic tunneling dynamics in the Josephson oscillation regime.
The initial population imbalance is adjusted to z(0) = 0.28(6), which is implemented using a relative position shift ∆x = 240(80) nm of the harmonic confinement in x-direction
with respect to the optical lattice. The time evolution of the population of the left and
right well is directly visible in the absorption images. The atoms tunnel to the right
and left over time. The population imbalance is inverted after approximately 20 ms and
5
approximately 70% of all experimental realizations fulfill this condition.
75
Chapter 5 Experimental results
Figure 5.9: Quantitative analysis of the time evolution of the dynamical variables. a) In the
Josephson oscillation regime (z(0) = 0.28(6)), the population imbalance as well as the relative phase
oscillate around a zero mean value with a timescale of 40(2) ms. The numerical solution of the NPSE
(red line) and the prediction of the VTM (green line) without free parameter show excellent agreement
with the experimental data (filled circles). The gray shaded region in the upper graph indicates the
theoretically expected scattering of the data due to the uncertainty in the initial population imbalance.
b) In the self-trapping regime (z(0) = 0.62(6)) the population imbalance is nearly stationary within the
depicted timescale. The relative phase evolves unbound due to the big difference between the chemical
potentials in the two wells. The deviation of the VTM is due to an overestimation of the difference
between the chemical potentials in the two wells (see text).
reaches the initial value after 45 ms.
The transition from the Josephson tunneling regime to the self-trapping regime is
implemented experimentally by increasing the initial population imbalance z(0) above
the critical value zc = 0.38. Figure 5.8b shows the time evolution of the BEC after
switching to the symmetric double well case for z(0) = 0.62(6) which corresponds to
∆x = 500(80) nm. The population of the two wells is stationary within the experimental
errors. This effect of macroscopic quantum self-trapping is due to the nonlinear interatomic interaction and is absent in any former realization of weak links in superfluids or
superconductors.
The quantitative analysis of the Josephson dynamics is shown in figure 5.9 in both
dynamical regimes. The upper graph in figure 5.9a shows the experimental results on the
time evolution of the population imbalance (filled circles) in the Josephson oscillation
regime with z(0)=0.28(6) for two tunneling periods. The experimentally determined
tunneling time is 40(2) ms which is significantly shorter than the linear tunneling time
76
5.4 Observation of Josephson oscillations and self-trapping
of 500(50) ms expected for non-interacting atoms in the realized potential. This reveals
the important role of the nonlinear atom-atom interaction in bosonic Josephson junction
experiments. The population imbalance oscillates around the mean value hz(t)it =
0, which shows that we have indeed realized a symmetric double well potential. The
results of the numerical integration of the NPSE using the calibrated parameters are
represented by the red line. This simulation includes the finite response time ∆t = 7 ms
of the piezo actuated mirror mount. The theoretical prediction without free parameters
shows excellent agreement with the experimental data. The gray shaded region is the
theoretically expected scattering of the data due to uncertainties of the initial parameters.
In order to obtain this region, the time evolution of the population imbalance is calculated
for different atom numbers 1000 < NT < 1300 and initial relative shifts 160 nm <
∆x < 320 nm. The lower and upper bound of the region is subsequently determined
by taking the minimum and the maximum of all calculated population imbalances for
each evolution time. It broadens for larger evolution times due to different tunneling
frequencies for different initial population imbalances and atom numbers. The results
of the extended two mode approximation (variable tunneling model) is represented with
green lines6 . Since the piezo response time of 7 ms can not be included in this theory,
the results are shifted by 3.5 ms in time. The VTM is in excellent agreement with the
experimental data and the solution of the NPSE.
The lower graph shows the relative phase of the two localized modes (filled circles)
in the individual wells for one tunneling period. This phase is obtained from the double
slit interference patterns formed after releasing the BEC from the double well trap for
a time-of-flight of 5 ms before taking an absorption image. The insets show typical
interference patterns obtained by integrating the absorption images along the y-and zdirection. The indicated times represent the evolution times before switching off the
double well trap. The relative phase oscillates with an amplitude of 0.6(1)π around its
mean value hφit = 0 and the same period of 40(5) ms as the population imbalance. The
indicated error bars denote statistical errors arising from the uncertainty of the initial
population imbalance. Each point in the phase measurement is the result of the average
over at least 5 realizations. The solutions of the NPSE (red line) and of the VTM (green
line) show quantitative agreement with our experimental findings.
The dynamics of the system changes drastically if the population imbalance is increased above the critical value zc . Figure 5.9b shows the time evolution of the dynamic
variables for z(0) = 0.62(6). The experimentally obtained population imbalance, which
is represented with open circles in the upper graph, shows no time evolution within the
experimental errors for the depicted time scale. The theoretical predictions show small
amplitude oscillations of the population imbalance, which can not be resolved experimentally. The relative phase, which is measured after 8 ms time of-flight is depicted in
the lower graph for one oscillation period. It is now unbound and winds up quickly over
time reaching the value φ = π already after 9(1) ms. This quick evolution of the relative
phase is due to the large difference between the on-site interactions of the left and right
well. The resulting phase evolution can no longer be compensated by the atomic tunneling leading to a linearly increasing relative phase. The initial deviation from a linear
time dependence is due to the finite response time of the piezo actuated mirror. In the
6
The symmetric ground state and the first excited antisymmetric state are calculated by numerically
integrating the three-dimensional GPE for the experimentally implemented atom number and double
well potential without free parameters.
77
Chapter 5 Experimental results
Figure 5.10: Time evolution of the population imbalance for long propagation times. a) The
experimentally observed tunneling oscillations (filled red circles with statistic and systematic errors)
are damped out after two oscillations periods, which is in agreement with the numerical solution of the
NPSE for different initial populations (gray shaded region). b) Decay of the self-trapped state for long
evolution times. The black line shows an exponential fit with a 1/e timescale of 280(80) ms. The decay
is due to a finite temperature or an instability of the experimental system.
self-trapping regime, the solution of the VTM shows deviations from the experimental
data. This can be understood by realizing, that the VTM is based on the assumption
of localized spatial wavefunctions, which are independent on the atom number in the
respective mode. These wavefunctions are calculated by superpositions of the symmetric and antisymmetric states of the double well potential and are therefore correct for
N1 = N2 = NT /2. In the self-trapping regime, the large population imbalance z(0) leads
to a broadening of the localized mode containing the larger number of atoms. This is
not captured by the VTM, which therefore overestimates the difference in the chemical
potential leading to a faster increase of the relative phase and a faster evolution of the
population imbalance.
It is important to note, that the constant tunneling two-mode approximation fails to
explain our experimental findings. A calculation of the normalized on-site interaction for
the potential and atom numbers used in the experiments yields Λ = 76 corresponding to
zc,CT M = 0.23. Therefore, the CTM predicts a macroscopically self-trapped state for an
initial population imbalance z(0) = 0.28, which is in contrast to the experimental results
revealing Josephson oscillations.
The evolution of the population imbalance for long propagation times is shown in
figure 5.10. Graph a) shows the experimental data (filled red circles) in the Josephson
oscillation regime for evolution times up to 200 ms and the values, which are theoretically
expected from the solution of the NPSE (gray shaded region). The error bars represent
systematic errors in the determination of the population imbalance as well as statistic
errors. The experimentally observed oscillation of the population imbalance is clearly
damped out after two periods. The scattering of the experimental data especially for
large oscillation times is due to the uncertainty in the initial population imbalance and
can be explained by the numerical solution of the NPSE. However, the experimental
errors are smaller than the expected scattering. This additional damping is most likely
78
5.4 Observation of Josephson oscillations and self-trapping
Figure 5.11: Comparison of the experimentally obtained phase plane trajectories to the predictions
of the extended two-mode model (black lines). The Josephson oscillation regime (gray shaded region) is
characterized by closed trajectories (filled circles). The separatrix, which is represented by the dashed
line, constitutes the transition to the self-trapped regime (open circles).
due to the residual thermal cloud, which is not included in the NPSE. Graph b) shows the
experimentally measured population imbalance in the self-trapping regime for evolution
times up to 1 s. Clearly, the self-trapped state decays to the equally populated state
(zfinal = 0). The black line shows an exponential fit with a 1/e timescale of 280(80) ms.
This timescale is significantly smaller than the lifetime of the BEC in the double well
potential, which is on the order of 10 s. The decay can either be due to an instability
of the experimental setup or to finite temperature effects, which are not included in
the Gross-Pitaevskii equation. Those damping effects have already been investigated
in the framework of the two-mode approximation [34, 79]. However, as discussed in
chapter 3, the two-mode approximation does not apply for the experimentally accessible
atom numbers and well spacings. Therefore, a determination of an upper bound for the
temperature of the BEC from the timescale of the decay of the self-trapped state is only
possible by including finite temperature effects in the numerical integration of the NPSE
or at least in the extended two mode model. We are currently investigating the effect
of thermal and quantum fluctuations by measuring the fluctuations of the relative phase
of the two localized modes as a function of the barrier potential depth. As discussed
in [101], this measurement should allow for a determination of the temperature of the
Bose-Einstein condensate inside the double well potential.
Phase plane portrait
The distinction between the two dynamical regimes becomes very apparent in the phase
plane portrait of the two dynamical variables z and φ. This is shown in figure 5.11, where
the experimental data set is compared to the predictions of the extended two-mode
model. The experimentally obtained closed phase plane trajectory in the Josephson-
79
Chapter 5 Experimental results
regime z(0)=0.28(6) is depicted with filled red circles. The error bars represent statistical errors and mainly result from the high sensitivity of the relative phase on the initial
population imbalance especially for long evolution times. The prediction of the VTM
(black line) is in quantitative agreement with the experimental findings. The dotted line
shows the separatrix for z(0) = zc,V T M = 0.35. It constitutes the crossing between the
Josephson oscillation regime (gray shaded region) and the self-trapped regime, which is
characterized by open phase plane trajectories (open circles).
The presented experiments cover only the phase plane trajectories with zero initial
phase. The dynamical regimes of π−phase oscillations and self-trapping are so far inaccessible experimentally. There is still plenty of future experiments to be done with
Josephson junctions for Bose-Einstein condensates.
80
6 Experiments with BECs in optical lattices
In this chapter, experiments are presented which have been carried out during the first
two years of my PhD period. The experiments investigated nonlinear wavepacket propagation in a one-dimensional optical lattice oriented along a one-dimensional waveguide.
In contrast to the Josephson junction experiment, the optical lattice is realized by two
counter-propagating far red detuned laser beams resulting in a lattice spacing of d ≈
400 nm. By introducing a frequency difference between the two beams, a moving optical
lattice can be realized. The one-dimensional waveguide, or rather the two-dimensional
harmonic confinement, is realized by a focused far off-resonant laser beam, which is
referred to as dipole trap in the previous chapters. After condensation in the crossed
optical dipole trap, the BEC is adiabatically loaded into the optical lattice. The size of
the atomic cloud in x-direction and therefore the number of occupied lattice sites1 can
be adjusted by controlling the intensity of the crossed dipole trap. After this preparation
stage, the BEC is released into the one-dimensional situation by switching off the crossed
dipole trap beam.
The dispersion relation of free particles can be modified by exposing them to a periodic potential. As known from solid state physics [102], stationary solutions of the
Schrödinger equation in the presence of a periodic potential are given by the Bloch
functions
Ψ(x) = φn,k (x) = un,k (x) exp (ikx) ,
(6.1)
where un,k (x) = un,k (x + d) are periodic functions with the same periodicity as the
external potential and k is the quasi-momentum of the atomic cloud, which can be
controlled by adjusting the velocity of the moving optical lattice. The energy eigenvalues
En (k) = En (k + 2π/d) form a band structure, where the different bands are labeled by
the band index n. This modified dispersion relation allows to adjust the dispersion of
the atoms by adiabatically preparing the wavepacket in the first band (n = 0) at a
given quasi-momentum. Combined with the atom-atom interaction, this makes different
regimes of nonlinear wavepacket propagation accessible which will be investigated in the
following sections.
1
The BEC typically occupies more than 20 lattice sites.
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Chapter 6 Experiments with BECs in optical lattices
6.1 Dispersion management
The term “dispersion management” originates from nonlinear photon optics where the
dispersion of light pulses can be controlled using periodic refractive index structures [86,
103]. It is nowadays also used for the altered dispersion of atoms in periodic potentials.
The experiment presented here deals with the wavepacket dynamics in the regime of
weak periodic potentials and weak nonlinearity2 . The experimental results have been
published in Physical Review Letters [23]. This publication is attached on the following
pages. For a detailed description of the experimental system and the underlying theory
see the PhD thesis of Bernd Eiermann [74]. The results can be summarized as follows:
• The first experiment shows in a descriptive way, that the dynamical evolution
of the wavepacket in the first band at the Brillouin-zone edge kb = π/d can be
2 E (k)
0
)−1 . A BEC is first
characterized by an effective negative mass meff = ~2 ( ∂ ∂k
2
prepared at zero quasi-momentum and the crossed dipole trap is subsequently
switched off. After an expansion time of t1 = 17ms, during which the BEC broadens
by approximately 80% due to normal dispersion, the wavepacket is prepared to the
Brillouin-zone edge within dt < 1 ms. The BEC is imaged after a variable evolution
time t2 . The negative sign of the effective mass in the phase evolution factor
k2
t) which is often referred to as anomalous dispersion, can be interpreted
exp(−~ 2m
as a time inversion and leads to a re-compression of the atomic cloud until it reaches
its initial size after t2 ≈ 8 ms
• In a second experiment we investigate the applicability of the approximation of
a constant effective mass. The evolution of the BEC at the edge of the Brillouin
zone is measured for different potential depths of the optical lattice. For small
potential depths V0 < 2EG , the small range of constant effective mass makes the
inclusion of dispersion terms of higher order necessary. This leads to non-Gaussian
wavepacket shapes. For higher potential depths, the wavepacket stays Gaussian
during the expansion, which can now be described using the approximation of
constant effective mass.
• The third experiment is finally dedicated to a systematic investigation of the ex0 (k)
tremal group velocities vg = ~1 ∂E∂k
in the first band. These maximal and minimal
group velocities correspond to the quasi-momenta with infinite positive and negative effective masses.
2
For the atomic densities used in this experiment, the nonlinear atom-atom interaction only results
in small corrections to the wavepacket dynamics expected for non-interacting atoms.
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6.1 Dispersion management
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Dispersion Management for Atomic Matter Waves
B. Eiermann,1 P. Treutlein,1,2 Th. Anker,1 M. Albiez,1 M. Taglieber,1 K.-P. Marzlin,1 and M. K. Oberthaler1
1
2
Fachbereich Physik, Universität Konstanz, Fach M696, 78457 Konstanz, Germany
Sektion Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4, 80799 München, Germany
(Received 14 March 2003; published 8 August 2003)
We demonstrate the control of the dispersion of matter wave packets utilizing periodic potentials.
This is analogous to the technique of dispersion management known in photon optics. Matter wave
packets are realized by Bose-Einstein condensates of 87 Rb in an optical dipole potential acting as a onedimensional waveguide. A weak optical lattice is used to control the dispersion relation of the matter
waves during the propagation of the wave packets. The dynamics are observed in position space and
interpreted using the concept of effective mass. By switching from positive to negative effective mass,
the dynamics can be reversed. The breakdown of the approximation of constant, as well as experimental
signatures of an infinite effective mass are studied.
DOI: 10.1103/PhysRevLett.91.060402
The broadening of particle wave packets due to the free
space dispersion is one of the most prominent quantum
phenomena cited in almost every textbook of quantum
mechanics. The realization of Bose-Einstein condensates
of dilute gases allows for the direct observation of wave
packet dynamics in real space on a macroscopic scale [1].
Using periodic potentials it becomes feasible to experimentally study to what extent the matter wave dispersion
relation can be engineered. This approach is similar to
dispersion management for light pulses in spatially periodic refractive index structures [2].
First experiments in this direction have already been
undertaken in the context of Bloch oscillations of thermal
atoms [3] and condensates [4]. The modification of the
dipole mode oscillation frequency of a condensate due to
the changed dispersion relation in the presence of a periodic potential has been studied in detail [5,6]. In contrast
to these experiments where the center of mass motion
was studied, we are investigating the evolution of the
spatial distribution of the atomic cloud in a quasi-onedimensional situation. Our experiments show that the
dispersion and thus the wave packet dynamics can be
experimentally controlled. This is a new tool which also
allows one to study the interplay between dispersion and
atom-atom interaction and to realize predicted nonspreading wave packets such as gap solitons [7] and
self-trapped states [8].
For atomic matter waves inside a one-dimensional
optical waveguide, we have achieved dispersion management by applying a weak periodic potential with
adjustable velocity. Figure 1 shows the results of an experiment in which the propagation of an atomic wave
packet is studied in the normal [Fig. 1(b)] and anomalous
[Fig. 1(c)] dispersion regime corresponding to positive
and negative effective mass, respectively. A broadening of
the wave packet is observed in both cases. The faster
spreading in the case of anomalous dispersion is a consequence of the smaller absolute value of the negative
effective mass. However, if we switch from one regime to
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0031-9007=03=91(6)=060402(4)$20.00
PACS numbers: 03.75.Be, 03.75.Lm, 32.80.Pj
the other during the propagation by changing the velocity
of the periodic potential, the effects of normal and
anomalous dispersion cancel. The wave packet, which
has initially broadened under the influence of normal
dispersion, reverses its expansion and compresses until
it regains its initial shape [Fig. 1(d)]. This is a direct proof
of the realization of negative effective mass.
The wave packets have been realized with a 87 Rb
Bose-Einstein condensate. The atoms are collected in
a magneto-optical trap and subsequently loaded into a
FIG. 1. Controlling the dispersion of an atomic wave packet
in a waveguide using a periodic potential. Shown are absorption
images of the wave packet averaged over four realizations (left)
and the corresponding density distributions nx; t along the
waveguide (right). (a) Initial wave packet. (b),(c) Images taken
after an overall propagation time of t 26 ms for different
dispersion regimes with different effective masses as indicated.
(d) Wave packet subjected to dispersion management: an initial
stage of expansion for t 17 ms with normal dispersion is
followed by propagation with anomalous dispersion for t 9 ms. The broadening in the normal dispersion regime has been
reversed by anomalous dispersion.
 2003 The American Physical Society
060402-1
83
Chapter 6 Experiments with BECs in optical lattices
84
5 a)
4
E1
3
2
1
E0
0
n (x, t=60 ms) [a.u.]
En / EG
vg / vG
magnetic time-orbiting potential trap. By evaporative
cooling we produce a cold atomic cloud which is then
transferred into an optical dipole trap realized by two
focused Nd:YAG laser beams with 60 m waist crossing
at the center of the magnetic trap. Further evaporative
cooling is achieved by lowering the optical potential
leading to pure Bose-Einstein condensates with up to 3 104 atoms in the jF 2; mF 2i state. By switching
off one dipole trap beam the atomic matter wave is
released into a trap acting as a one-dimensional waveguide with radial trapping frequency !? 2
80 Hz
and longitudinal trapping frequency !k 2
1:5 Hz.
The periodic potential is realized by a far off-resonant
standing light wave with a single beam peak intensity of
up to 5 W=cm2 . The chosen detuning of 2 nm to the blue
off the D2 line leads to a spontaneous emission rate below
1 Hz. The frequency and phase of the individual laser
beams are controlled by acousto-optic modulators driven
by a two channel arbitrary waveform generator allowing
for full control of the velocity and amplitude of the periodic potential. The absolute value of the potential depth
was calibrated independently by analyzing results on
Bragg scattering [9] and Landau Zener tunneling [4,10].
The wave packet evolution inside the combined potential of the waveguide and the lattice is studied by taking
absorption images of the atomic density distribution after
a variable time delay. The density profiles along the
waveguide, nx; t, are obtained by integrating the absorption images over the transverse dimension z.
The concept of effective mass meff [11] allows one to
describe the dynamics of matter wave packets inside a
periodic potential in a simple way via a modified dispersion relation. The periodic potential in our experiments is well described by
V
Vx 0 cosGx
2
with a modulation depth V0 on the order of the grating
recoil energy EG h 2 G2 =8m, with G 2
=d where d 417 nm represents the spatial period. The energy spectrum of atoms inside the periodic potential exhibits a
band structure En q which is a periodic function of
quasimomentum q with periodicity G corresponding to
the width of the first Brillouin zone [Fig. 2(a)]. In our
experiment, we prepare condensates in the lowest energy
band (n 0) with a quasimomentum distribution wq
centered at q qc with an rms width q G [12].
It has been shown by Steel et al. [13] that in this case
the condensate wave function in a quasi-one-dimensional
situation can be described by x; t Ax; t qc x expiE0 qc t=h,
where qc represents the Bloch function in the lowest energy band corresponding to the
central quasimomentum. The evolution of the envelope
function Ax; t, normalized to the total number of
atoms N0 , is described by
@
@
h 2 @2
vg
ih
A g~jAj2 A: (1)
A
@t
@x
2meff @x2
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VOLUME 91, N UMBER 6
0.5 b)
0
-0.5
2 c)
1
0
-1
-2
-1
(q)
w(q)
d)
1.2 EG
e)
2.0 EG
f)
2.7 EG
1
0
1
0
1
0
0
q [G/2]
q∞ 1 q∞
-200
0 -200
x [µm]
FIG. 2. (a) Band structure in the first Brillouin zone for atoms
in an optical lattice with V0 1:2 EG (solid), parabolic approximation to the lowest energy band at q G=2 (dashed),
corresponding group velocity (b), and effective mass (c) in the
lowest energy band. The vertical dashed lines at q q
1 indicate where jmeff j 1. (d) –(f) Spatial densities of the wave
packet after t 60 ms of propagation with qc G=2 for different V0 . The position x along the waveguide is measured in
the moving frame of the optical lattice. The solid lines represent the theoretical predictions using linear propagation with
the exact band structure and the quasimomentum distribution
given in graph (c). The dashed lines in graphs (d) –(f) represent
the prediction of the constant effective mass approximation.
The strength of the atom-atom interaction is given by g~ nl 2h!
? a, with the s-wave scattering
R length a, and a
renormalization factor nl 1=d d0 dxjqc j4 . Besides
the modification of the nonlinear term the periodic
potential leads to a group velocity of the envelope
Ax; t determined by the energy band via vg qc h 1 @E0 [email protected] [Fig. 2(b)]. In addition, the kinetic
energy term describing the dispersion of the wave packet
is modified by the effective mass [Fig. 2(c)]
2
1
@ E0 q meff qc h 2
:
2
@q qc
Since this approximation of constant effective mass corresponds to a parabolic approximation of the energy band,
it is valid only for sufficiently small q.
The general solution of Eq. (1) is a difficult task, but
simple analytic expressions can be found in the special
cases of negligible and dominating atom-atom interaction. Omitting the last term in Eq. (1) it is straightforward to see that jmeff j controls the magnitude of the
dispersion term and thus the time scale of the wave packet
broadening. A change in sign of meff corresponds to time
reversal of the dynamics in a frame moving with velocity
vg . In the regime where the atom-atom interaction is
dominating, e.g., during the initial expansion of a condensate, the evolution of the envelope function can be
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found in standard nonlinear optics textbooks [2] and
in the form of scaling solutions in the context of
Bose-Einstein condensates [14]. Note that in this regime the kinetic energy term is still relevant and thus a
change of the sign of the effective mass will reverse the
dynamics.
In the following we will analyze the obtained experimental results in more detail. The initial wave packet
shown in Fig. 1(a) contains 2 104 atoms and is characterized by x0 14:86 m (x is the rms width of
a Gaussian fit). Before releasing the atomic cloud into
the one-dimensional waveguide, a weak periodic potential along the waveguide is adiabatically ramped up to
V0 2:82 EG within 6 ms. This turns the initial
Gaussian momentum distribution of the atoms into a
Gaussian distribution of quasimomenta wq centered at
qc 0 with a corresponding effective mass meff 1:25 m. The density distribution shown in Fig. 1(b) is a
result of propagation within the stationary periodic potential
for t 26 ms and exhibits a spread of x :
q
2
xt x20 18:412 m in contrast to xf 20:214 m for expansion without periodic potential.
The resulting ratio xf = x 1:1015 indicates that the
evolution is dominated p
by
the nonlinearity, in which case
one expects xf = x meff =m 1:11 in the short-time
limit [15]. In the case of linear propagation one expects
xf = x meff =m 1:25.
The dynamics in the anomalous dispersion regime
[Fig. 1(c)] are investigated by initially accelerating the
periodic potential within 3 ms to a velocity v vG :
hG=2m,
thus preparing the atomic wave packet at the
edge of the Brillouin zone (qc G=2), where meff 0:5 m. The velocity is kept constant during the subsequent expansion. In the regime of negative mass a condensate exhibits collapse dynamics. Two-dimensional
calculations for our experimental situation reveal that
this collapse happens within the initial 3–6 ms of propagation. Subsequently this leads to an excitation of transverse states and thus to a fast reduction of the density and
the nonlinearity. An indication of the population of transverse states is the observed increase of the transverse
spatial extension of the wave packets by almost a factor of 2. The optical resolution of our setup does not
allow for a quantitative analysis of the transverse broadening. Because of the fast reduction of the nonlinearity
the subsequent expansion should be well described by
linear theory predicting a ratio xf = x 0:5 which is
close to the observed value 0:465 [ x 38:515 m
after 23 ms].
In the case of dispersion management Fig. 1(d) the
wave packet was first subjected to normal dispersion for
17 ms at qc 0. The time of subsequent propagation
with anomalous dispersion at qc G=2 was adjusted
to achieve the minimal wave packet size of x 15:42 m. The minimum was achieved for times ranging from 7 to 9 ms which is in rough agreement with the
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expected
time resulting from effective mass considerap
tions 0:5=1:25 17 ms 10:7 ms.
Since the assumption of a constant effective mass used
so far is only an approximation, it is important to check
its applicability in experiments. Therefore we investigate
the dynamics of wave packets prepared at the Brillouin
zone edge for different potential depths. The observed
density profiles after 60 ms of propagation are shown in
Figs. 2(d) –2(f). While both the initial wave packet shape
nx; 0 and the quasimomentum distribution wq are
measured to be approximately Gaussian, the wave packet
changes its shape during evolution. We attribute this distortion to the invalidity of the constant effective mass
approximation, which assumes that the populated quasimomenta experience the same negative curvature of
E0 q. Since the range of quasimomenta fulfilling this
criterion becomes smaller with decreasing modulation
depth, a more pronounced distortion of the wave packet
shape for weak potentials is expected [see Fig. 2(d)].
This explanation is confirmed more quantitatively by
comparing the observed wave packets with the results of a
linear theory. Since the initial collapse of the condensate
cannot be described by a linear theory, we take a
Gaussian function fitted to the density distribution measured at 20 ms as the initial wave packet for the numerical
propagation. Because of the fact that this is not a minimum uncertainty wave packet we add a quadratic phase in
real space such that the Fourier transform of the wave
packet is consistent with the measured momentum distribution. In first approximation this takes into account
the initial expansion including the repulsive atom-atom
interaction. For the subsequent propagation of 40 ms in
quasimomentum space we use the full expression for
E0 q which is obtained numerically. In Fig. 2(d) –2(f)
we compare the data with the linear theory described
above (solid line) and with the constant effective mass
approximation (dashed line). Clearly the constant effective mass approximation cannot explain the observed
distortion and it strongly overestimates the expansion
velocity for weak potentials. Additionally, for small potential modulation depths new features appear in the
central part of the wave packet which cannot be explained
using the linear theory. We are currently investigating
these features in more detail.
The observed distortion is mainly a consequence of
another very interesting feature of the band structure:
the existence of jmeff j 1 for certain quasimomenta
q q
1 [see Fig. 2(c)]. A diverging mass implies that
the group velocity is extremal and the dispersion vanishes as can be seen from Eq. (1). As a consequence an
atomic ensemble whose quasimomentum distribution is
overlapping q q
1 will develop steep edges as can be
seen in Fig. 2(d) and in Fig. 3. These edges propagate with
the maximum group velocity of the lowest band.
The systematic investigation of the velocities of the
edges is shown in Fig. 3 for different values of V0 . In
order to get a significant overlap of wq with q q
1 , we
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Chapter 6 Experiments with BECs in optical lattices
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1
0.4
0
-0.5
-1
0
6
2
4
modulation depth V0 [EG]
8
FIG. 3. Group velocities of steep edges emerging from an
initial wave packet with significant overlap with q
1 in quasimomentum space. The measured velocities of the indicated
positions (arrows in insets) agree very well with the expected
maximum and minimum velocity in the lowest band (solid
line) corresponding to the infinite masses. The dashed line
represents the prediction of the weak potential approximation
[11]. For potentials smaller than 2 EG (diamonds) data are obtained by preparing the initial wave packet at q G=2 leading
to two steep edges (see inset). For higher potentials (circles) the
wave packet is prepared at q G=4 to ensure population of the
quasimomentum corresponding to infinite mass.
prepare atomic ensembles with q 0:17 G=2 at qc G=2 realized by Bose-Einstein condensates of 2 104
atoms with a spatial extension of x 15 m. The velocities of the edges are derived from two images taken at
20 and 60 ms, respectively. In each image the position of
the edge is evaluated at the levels indicated by the arrows
in the insets of Fig. 3 (50% and 25% of the maximum
density). Since the momentum spread is too small to
populate the infinite mass points for potentials deeper
than 2 EG the atomic ensemble was then prepared at qc G=4. The resulting wave packet shapes are asymmetric
exhibiting a steep edge on one side which becomes less
pronounced for potentials deeper than 5 EG . The obtained
experimental results in Fig. 3 are in excellent agreement
with the numerically calculated band structure predictions. In contrast to the good agreement of the maximum
velocity for all potential depths we find that for V0 >
5 EG the group velocity of the center of mass is only 10%
of the expected velocity. This could be an indication of
entering the tight binding regime where the nonlinear
effect of self-trapping, i.e., stopping and nonspreading
wave packets, has been predicted [8]. We are currently
investigating the transport properties in this regime in
more detail.
In conclusion, we have demonstrated experimentally
that the dispersion of atomic matter waves in a waveguide
can be controlled using a weak periodic potential. Matter
wave packets with positive, negative, and infinite effec060402-4
86
week ending
8 AUGUST 2003
tive masses are studied in the regime of weak and intermediate potential heights. The preparation of matter
waves with engineered dispersion (meff < 0) is an important prerequisite for the experimental investigation of
atomic gap solitons and other effects arising from the
coherent interplay of nonlinearity and dispersion in periodic potentials.
We wish to thank J. Mlynek for his generous support, A. Sizmann and B. Brezger for many stimulating
discussions, and J. Bellanca and K. Forberich for their
help in building up the experiment. This work was supported by Deutsche Forschungsgemeinschaft, Emmy
Noether Program, and by the European Union, Contract
No. HPRN-CT-2000-00125.
Note added.—Only recently, we became aware of an
experimental work [16] which is closely related to the
work presented in this Letter.
[1] Bose-Einstein Condensation in Atomic Gases, edited by
M. Inguscio, S. Stringari, and C. Wieman (IOS Press,
Amsterdam, 1999).
[2] G. P. Agrawal, Applications of Nonlinear Fiber Optics
(Academic Press, San Diego, 2001); Nonlinear Fiber
Optics (Academic Press, San Diego, 1995).
[3] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and
C. Salomon, Phys. Rev. Lett. 76, 4508 (1996).
[4] B. P. Anderson and M. A. Kasevich, Science 282, 1686
(1998); O. Morsch, J. Müller, M. Cristiani, D. Ciampini,
and E. Arimondo, Phys. Rev. Lett. 87, 140402 (2001).
[5] S. Burger, F. S. Cataliotti, C. Fort, F. Minardi,
M. Inguscio, M. L. Chiofalo, and M. P. Tosi, Phys. Rev.
Lett. 86, 4447 (2001).
[6] M. Krämer, L. Pitaevskii, and S. Stringari, Phys. Rev.
Lett. 88, 180404 (2002).
[7] P. Meystre, Atom Optics (Springer-Verlag, New York,
2001), p. 205, and references therein.
[8] A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353
(2001).
[9] M. Kozuma, L. Deng, E.W. Hagley, J. Wen, R. Lutwak,
K. Helmerson, S. L. Rolston, and W. D. Phillips, Phys.
Rev. Lett. 82, 871 (1999).
[10] C. F. Bharucha, K.W. Madison, P. R. Morrow, S. R.
Wilkinson, Bala Sundaram, and M. G. Raizen, Phys.
Rev. A 55, R857 (1997).
[11] N. Ashcroft and N. Mermin, Solid State Physics
(Saunders, Philadelphia, 1976).
[12] J. Hecker-Denschlag, J. E. Simsarian, H. Häffner,
C. McKenzie, A. Browaeys, D. Cho, K. Helmerson,
S. L. Rolston, and W. D. Phillips, J. Phys. B 35, 3095
(2002), and references therein.
[13] M. Steel and W. Zhang, cond-mat/9810284.
[14] Y. Castin, and R. Dum, Phys. Rev. Lett. 77, 5315 (1996);
Yu. Kagan, E. L. Surkov, and G.V. Shlyapnikov, Phys.
Rev. A 54, R1753 (1996).
[15] M. J. Potasek, G. P. Agrawal, and S. C. Pinault, J. Opt.
Soc. Am. B 3, 205 (1986).
[16] L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, cond-mat/0303626.
060402-4
6.2 Continuous dispersion management
6.2 Continuous dispersion management
In the experiment described above, the wavepacket has been prepared to a fixed quasimomentum. The concept of dispersion management can be expanded to a continuously
varying quasi-momentum (“continuous dispersion management”). The corresponding
experiments have been carried out with a slightly modified setup, in which the onedimensional waveguide and the optical lattice are not collinear, but enclose an angle of
α = 21◦ . If the acceleration time of the periodic confinement to a certain velocity corresponding to a given quasi-momentum is not adiabatic with respect to the transverse
harmonic confinement, transverse collective excitations inside the waveguide are excited.
The resulting oscillation of the wavepacket has a component along the optical lattice.
This leads to a continuous oscillation of the quasi-momentum. Combined with the nonlinear nature of atom-atom interaction, a rich variety of different regimes of wavepacket
dynamics in real and momentum space can be realized. The experimental results as well
as a simple theoretical model explaining the data have been published in Optics Express
[24] (attached in the following pages). Two different acceleration schemes to the edge of
the Brillouin zone have been investigated:
• In the first experiment, the BEC is prepared to k = 1.5 kb within 3 ms at a potential
depth V0 = 6Erec . The lattice is subsequently reduced to V0 = 0.52Erec and
decelerated within 3 ms in order to finally prepare the wavepacket at the band edge.
The dynamical evolution of the wavepacket is observed for expansion times up to
50ms. After an initial stage of nonlinear pulse compression [86], the dynamics is
dominated by the continuous change of the quasi-momentum due to the transverse
oscillation in the waveguide. This leads to a periodic change from positive to
negative group velocities, which appears as an oscillation in real space. During the
evolution, the quasi-momentum crosses the points of infinite effective mass. Since
the dispersion vanishes at these points and additionally the dynamics at positive
and negative masses partially compensate, the central wavepacket is only slowly
spreading.
• In the second implemented preparation scheme, the periodic potential is ramped
up to V0 = 0.37 Erec and the wavepacket is subsequently directly prepared to
k = 1.05 kb within 3ms. After an initial stage of compression, the dynamics reveals
a very complex time evolution. The wave packet repeatedly splits up into two parts
in real space and shows a reduced dispersion. This behavior can only be explained
by including the nonlinear atom-atom interaction.
A detailed description of the experiments and the numerical simulations will be presented in the PhD thesis by Thomas Anker.
87
Chapter 6 Experiments with BECs in optical lattices
Linear and nonlinear dynamics of
matter wave packets in periodic
potentials
Th. Anker, M. Albiez, B. Eiermann, M. Taglieber and M. K.
Oberthaler
Kirchhoff Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120
Heidelberg
[email protected]
http://www.kip.uni-heidelberg.de/matterwaveoptics
Abstract: We investigate experimentally and theoretically the nonlinear
propagation of 87 Rb Bose Einstein condensates in a trap with cylindrical
symmetry. An additional weak periodic potential which encloses an
angle with the symmetry axis of the waveguide is applied. The observed
complex wave packet dynamics results from the coupling of transverse and
longitudinal motion. We show that the experimental observations can be
understood applying the concept of effective mass, which also allows to
model numerically the three dimensional problem with a one dimensional
equation. Within this framework the observed slowly spreading wave packets are a consequence of the continuous change of dispersion. The observed
splitting of wave packets is very well described by the developed model and
results from the nonlinear effect of transient solitonic propagation.
© 2004 Optical Society of America
OCIS codes: (270.5530) Pulse propagation and solitons; (020.0020) Atomic and molecular
physics; (350.4990) Particles
References and links
1. “Bose-Einstein condensation in atomic gases,” ed. by M. Inguscio, S. Stringari, and C. Wieman, (IOS Press,
Amsterdam 1999)
2. F.S. Cataliotti, S. Burger, S. C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Ingusio,
“Josephson Junction Arrays with Bose-Einstein Condensates”, Science 293 843 (2001).
3. A. Trombettoni and A. Smerzi, “Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates,” Phys.
Rev. Lett. 86 2353 (2001).
4. M. Steel and W. Zhang, “Bloch function description of a Bose-Einstein condensate in a finite optical lattice,”
cond-mat/9810284 (1998).
5. P. Meystre, Atom Optics (Springer Verlag, New York, 2001) p 205, and references therein.
6. The experimental realization in our group will be published elsewhere.
7. V.V. Konotop, M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev.
A 65 021602 (2002).
8. N. Ashcroft and N. Mermin, Solid State Physics (Saunders, Philadelphia, 1976).
9. A.A. Sukhorukov, D. Neshev, W. Krolikowski, and Y.S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg
scattering in optically-induced lattices,” nlin.PS/0309075.
10. B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion
Management for Atomic Matter Waves,” Phys. Rev. Lett. 91 060402 (2003).
11. M. Kozuma, L. Deng, E.W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S.L. Rolston, and W.D. Phillips, “Coherent
Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg Diffraction,” Phys. Rev. Lett. 82 871
(1999).
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Received 11 November 2003; revised 21 December 2003; accepted 21 December 2003
12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 11
6.2 Continuous dispersion management
12. B.P. Anderson, and M.A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science
282 1686 (1998);
13. O. Morsch, J. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, “Bloch Oscillations and Mean-Field Effects
of Bose-Einstein Condensates in 1D Optical Lattices,” Phys. Rev. Lett. 87 140402 (2001).
14. C.F. Bharucha, K.W. Madison, P.R. Morrow, S.R. Wilkinson, Bala Sundaram, and M.G. Raizen, “Observation of
atomic tunneling from an accelerating optical potential,” Phys. Rev. A 55 R857 (1997)
15. L. Salasnich, A. Parola, and L. Reatto, “Effective wave equations for the dynamics of cigar-shaped and diskshaped Bose condensates,” Phys. Rev. A 65 043614 (2002).
16. G.P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
17. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).
18. R.G. Scott, A.M. Martin, T.M. Fromholz,S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, “Creation of Solitons
and Vortices by Bragg Reflection of Bose-Einstein Condensates in an Optical Lattice,” Phys. Rev. Lett. 90 110404
(2003).
1.
Introduction
The experimental investigation of nonlinear matter wave dynamics is feasible since the realization of Bose-Einstein-condensation of dilute gases [1]. The combination of this new matter
wave source with periodic potentials allows for the realization of many nonlinear propagation
phenomena. The dynamics depends critically on the modulation depth of the potential. For deep
periodic potentials the physics is described locally taking into account mean field effects and
tunneling between adjacent potential wells. In this context wave packet dynamics in Josephson
junction arrays have been demonstrated experimentally [2] and nonlinear self trapping has been
predicted theoretically [3]. In the limit of weak periodic potentials and moderate nonlinearity
rich wave packet dynamics result due to the modification of dispersion which can be described
applying band structure theory [4]. Especially matter wave packets subjected to anomalous dispersion (negative effective mass) or vanishing dispersion (diverging mass) are of great interest.
In the negative mass regime gap solitons have been predicted theoretically [5] and have been
observed recently [6]. Also modulation instabilities can occur [7].
The experiments described in this work reveal wave dynamics in the linear and nonlinear
regime for weak periodic potentials. The observed behavior is a consequence of the special
preparation of the wave packet leading to a continuous change of the effective mass and thus the
dispersion during the propagation. The initial propagation is dominated by the atom-atom interaction leading to complex wave dynamics. After a certain time of propagation slowly spreading
atomic wave packets are formed which are well described by linear theory. In this work we
focus on the mechanisms governing the initial stage of propagation.
The paper is organized as follows: in section 2 we describe the effective mass and dispersion
concept. In section 3 we present our experimental setup and in section 4 the employed wave
packet preparation schemes are discussed in detail. In section 5 the experimental results are
compared with numerical simulations. We show that some features of the complex dynamics
can be identified with well known nonlinear mechanisms. We conclude in section 6.
2.
Effective mass and dispersion concept
In our experiments we employ a weak periodic potential which leads to a dispersion relation
En (q) shown in Fig. 1(a). This relation is well known in the context of electrons in crystals
[8] and exhibits a band structure. It shows the eigenenergies of the Bloch states as a function
of the quasi-momentum q. The modified dispersion relation leads to a change of wavepacket
dynamics due to the change in group velocity vg (q) = 1/h̄ ∂ E/∂ q (see Fig. 1(b)), and the
group velocity dispersion described by the effective mass me f f = h̄2 (∂ 2 E/∂ q2 )−1 (see Fig.
1(c)), which is equivalent to the effective diffraction introduced in the context of light beam
propagation in optically-induced photonic lattices [9]. In our experiment only the lowest band is
populated, which is characterized by two dispersion regimes, normal and anomalous dispersion,
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89
Chapter 6 Experiments with BECs in optical lattices
-1
q 1 q
0
q [G/2]
0.5
b)
meff / m
5 a)
4
3
E1
2
1
E0
0
vg / vG
En / EG
corresponding to positive and negative effective mass. A pathological situation arises at the
quasimomentum q±
∞ , where the group velocity vg (q) is extremal, |me f f | diverges and thus the
dispersion vanishes.
0
-0.5
-1
q 1 q
0
q [G/2]
c)
2
1
0
-1
-2
-1
q 1 q
0
q [G/2]
Fig. 1. (a) Band structure for atoms in an optical lattice with V0 = 1.2 Erec (solid), parabolic
approximation to the lowest energy band at q = π /d = G/2 (dashed), corresponding group
velocity (b) and effective mass (c) in the lowest energy band. The vertical dashed lines at
q = q±
∞ indicate where |meff | = ∞. The shaded region shows the range of quasimomenta
where the effective mass is negative.
In the following we will show that the two preparation schemes employed in the experiment lead to a continuous change of the quasimomentum distribution, and thus to a continuous
change of dispersion. One of the preparation schemes allows to switch periodically from positive to negative mass values and thus a slowly spreading wave packet is formed. This is an
extension of the experiment reporting on dispersion management [10]. The second preparation
gives further insight into the ongoing nonlinear dynamics for the initial propagation.
3.
Experimental setup
The wave packets in our experiments have been realized with a 87 Rb Bose-Einstein condensate
(BEC). The atoms are collected in a magneto-optical trap and subsequently loaded into a magnetic time-orbiting potential trap. By evaporative cooling we produce a cold atomic cloud which
is then transferred into an optical dipole trap realized by two focused Nd:YAG laser beams with
60 µ m waist crossing at the center of the magnetic trap (see Fig.2(a)). Further evaporative cooling is achieved by lowering the optical potential leading to pure Bose-Einstein condensates
with 1 · 104 atoms in the |F = 2, mF = +2 state. By switching off one dipole trap beam the
atomic matter wave is released into a trap acting as a one-dimensional waveguide with radial
trapping frequency ω⊥ = 2π · 100 Hz and longitudinal trapping frequency ω = 2π · 1.5 Hz. It is
important to note that the dipole trap allows to release the BEC in a very controlled way leading
to an initial mean velocity uncertainty smaller than 1/10 of the photon recoil velocity.
The periodic potential is realized by a far off-resonant standing light wave with a single
beam peak intensity of up to 1W /cm2 . The chosen detuning of 2 nm to the blue off the D2 line
leads to a spontaneous emission rate below 1 Hz. The standing light wave and the waveguide
enclose an angle of θ = 21◦ (see Fig. 2(b)). The frequency and phase of the individual laser
beams are controlled by acousto-optic modulators driven by a two channel arbitrary waveform
generator allowing for full control of the velocity and amplitude of the periodic potential. The
light intensity and thus the absolute value of the potential depth was calibrated independently
by analyzing results on Bragg scattering [11] and Landau Zener tunneling [12, 13, 14].
The wave packet evolution inside the combined potential of the waveguide and the lattice is
studied by taking absorption images of the atomic density distribution after a variable time delay. The density profiles along the waveguide, n(x,t), are obtained by integrating the absorption
images over the transverse dimension.
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6.2 Continuous dispersion management
Fig. 2. Scheme for wave packet preparation (a-d). (a) initial wave packet is obtained by
condensation in a crossed dipole trap. (b) A stationary periodic potential is ramped up adiabatically preparing the atoms at quasimomentum qc = 0 in the lowest band. (c),(d) The
periodic potential is accelerated to a constant velocity. (e) shows the numerically deduced
quasimomentum shift for the preparation method I described in the text. (f) The motion
of the center quasimomentum for the preparation method II described in the text. The additional shift to higher quasimomenta for long times results from the residual trap in the
direction of the waveguide. The shaded area represents the quasimomenta corresponding
to negative effective mass.
4.
Dynamics in reciprocal space
In our experimental situation an acceleration of the periodic potential to a constant velocity
leads to a collective transverse excitation as indicated in Fig. 2(d). Since the transverse motion
in the waveguide has a non vanishing component in the direction of the periodic potential due to
the angle θ , a change of the transverse velocity leads to a shift of the central quasimomentum of
the wave packet. The coupling between the transverse motion in the waveguide and the motion
along the standing light wave gives rise to a nontrivial motion in reciprocal (see Fig. 2(e,f)) and
real space.
The appropriate theoretical description of the presented experimental situation requires the
solution of the three dimensional nonlinear Schrödinger equation (NLSE) and thus requires
long computation times. In order to understand the basic physics we follow a simple approach
which solves the problem approximately and explains all the features observed in the experiment. For that purpose we first solve the semiclassical equations of motion of a particle which
obeys the equation F = M ∗x¨ where M ∗ is a mass tensor describing the directionality of the effective mass. We deduce the time dependent quasimomentum qc (t) in the direction of the periodic
potential by identifying h̄q˙c = Fx̂ and x̂˙ = vg (qc ) (definition of x̂ see Fig. 2(b)). Subsequently
we can solve the one dimensional NPSE (non-polynomial nonlinear Schrödinger equation)[15]
where the momentum distribution is shifted in each integration step according to the calculated
qc (t). Thus the transverse motion is taken into account properly for narrow momentum distributions. We use a split step Fourier method to integrate the NPSE where the kinetic energy
contribution is described by the numerically obtained energy dispersion relation of the lowest
band E0 (q). It is important to note, that this description includes all higher derivatives of E0 (q),
and thus goes beyond the effective mass approximation.
In the following we discuss in detail the employed preparation schemes:
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91
Chapter 6 Experiments with BECs in optical lattices
Acceleration scheme I: After the periodic potential is adiabatically ramped up to V0 = 6Erec
it is accelerated within 3 ms to a velocity v pot = cos2 (θ )1.5vrec . Then the potential depth is
lowered to V0 = 0.52Erec within 1.5 ms and the periodic potential is decelerated within 3 ms to
v pot = cos2 (θ )vrec subsequently. V0 and v pot are kept constant during the following propagation.
The calculated motion in reciprocal space qc (t) is shown in Fig. 2(e).
Acceleration scheme II: The periodic potential is ramped up adiabatically to V0 = 0.37 Erec
and is subsequently accelerated within 3 ms to a final velocity v pot = cos2 (θ ) × 1.05 vrec . The
potential depth is kept constant throughout the whole experiment. Fig. 2(f) reveals that in contrast to the former acceleration scheme the quasimomentum for the initial propagation is mainly
in the negative effective mass regime.
5.
Experimental and numerical results
In this section we compare the experimental results with the predictions of our simple theoretical model discussed above. The numerical simulation reveal all the experimentally observed
features of the dynamics such as linear slowly spreading oscillating wave packets, nonlinear
wave packet compression and splitting of wave packets. The observed nonlinear phenomena
can be understood by realizing that in the negative effective mass regime the repulsive atomatom interaction leads to compression of the wave packet in real space and to a broadening
of the momentum distribution. An equivalent picture borrowed from nonlinear photon optics
[16, 17] is the transient formation of higher order solitons, which show periodic compression
in real space with an increase in momentum width and vice versa.
5.1.
Preparation I
The experimental results for the first acceleration scheme discussed in section 4 are shown in
Fig. 3. Clearly we observe that a wave packet with reduced density is formed which spreads out
slowly and reveals oscillations in real space. This wave packet results from the initial dynamics
characterized by two stages of compression which lead to radiation of atoms [18]. The observed
behavior is well described by our numerical simulation which allows further insight into the
ongoing physics.
In Fig. 3(c,d) we show the calculated momentum and real space distribution for the first 14ms
of propagation. As can be seen the acceleration of the standing light wave leads to a oscillatory
behavior in momentum space. For the chosen parameters the wave packet is initially dragged
with a tight binding potential (V0 = 6Erec ) over the critical negative mass regime. While the real
space distribution does not change during this process, the momentum distribution broadens
due to self phase modulation [16, 17]. The subsequent propagation in the positive mass regime
leads to a further broadening in momentum space and real space (t=4-9ms).
The dynamics changes drastically as soon as a significant part of the momentum distribution
populates quasimomenta in the negative mass regime (t=10ms). There the real space distribution reveals nonlinear compression as known from the initial dynamics of higher order solitons.
This compression leads to a significant further broadening in momentum space and thus to
population of quasimomenta corresponding to positive mass. This results in a spreading in real
space due to the different group velocities involved and leads to the observed background. The
change of the quasimomentum due to the transverse motion prohibits a further significant increase in momentum width, since the whole momentum distribution is shifted out of the critical
negative mass regime at t=14ms.
The long time dynamics of the slowly spreading wave packet is mainly given by the momentum distribution marked with the shaded area for t=14ms in Fig. 3(c). The subsequent motion
is dominated by the change of the quasimomentum due to the transverse motion. This leads to
a periodic change from normal to anomalous dispersion and thus the linear spreading is sup#3346 - $15.00 US
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6.2 Continuous dispersion management
Theory
(b)
Experiment
(a)
10
5
100µm
(d)
density [a.u.]
(c)
10ms
100µm
phase [ ]
propagation time [ms]
10ms
1
0
-1
0
10
position [µm]
0
-2
-1
1 0
quasimomentum q [ /d]
20 40 60
position [µm]
80
Fig. 3. Wave packet dynamics for preparation I. (a) Experimental observation of wave
packet propagation. (b) Result of the numerical simulation as discussed in the text. The
data is convoluted with the optical resolution of the experiment. The obtained results are in
good agreement with the experimental observations. The theoretically obtained (c) quasimomentum distribution and (d) real space distribution are given for the initial 14ms of
propagation. The inset reveals the phase of the observed slowly spreading wave packet.
pressed. This is an extension of our previous work on dispersion management for matter waves
- continuous dispersion management.
5.2.
Preparation II
This preparation scheme reveals in more detail the transient solitonic propagation leading to the
significant spreading in momentum space. This results in a splitting of the wave packet which
cannot be understood within a linear theory. The results are shown in Fig. 4 and the observed
splitting is confirmed by our numerical simulations.
In contrast to the former preparation scheme the momentum distribution is prepared as a
whole in the critical negative mass regime. Our numerical simulations reveal that the wave
packet compresses quickly in real space after t=4ms which is accompanied by an expansion
in momentum space. The momentum distribution which stays localized in the negative mass
regime reveals further solitonic propagation characterized by an expansion in real space and
narrowing of the momentum distribution (t=5-10ms). The transverse motion shifts this momentum distribution into the normal dispersion regime after 11ms of propagation resulting in a
wave packet moving with positive group velocity (i.e. moving to the right in fig. 4(b)). The initial compression at t=4ms even produces a significant population of atoms in the normal mass
regime which subsequently move with negative group velocity showing up as a wave packet
moving to the left in Fig. 4(b). Thus the splitting in real space is a consequence of the significant
nonlinear broadening in momentum space.
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Chapter 6 Experiments with BECs in optical lattices
Theory
(b)
Experiment
(a)
(c)
10
5
100µm
phase [ ] density [a.u.]
propagation time [ms]
10ms
10ms
10ms
100µm
(d)
1
0
-1
0
10
position [µm]
0
-2
-1
1 0
quasimomentum q [ /d]
20 40 60
position [µm]
80
Fig. 4. Wave packet dynamics for preparation II. (a) Experimental results on wave packet
propagation. (b) Result of the numerical simulation as discussed in the text. The simulation
reproduces the observed wave packet splitting. The theoretically obtained (c) quasimomentum distribution and (d) real space distribution are given for the initial 14ms of propagation.
The inset reveals that the transient formed wave packet has a flat phase indicating solitonic
propagation.
6.
Conclusion
In this paper we report on experimental observations of nonlinear wave packet dynamics in
the regime of positive and negative effective mass. Our experimental setup realizing a BEC
in a quasi-one dimensional situation allows the observation of wave dynamics for short times,
where the nonlinearity due to the atom-atom interaction dominates and also for long times,
where linear wave propagation is revealed.
We have shown that a slowly spreading wave packet can be realized by changing the quasimomentum periodically from the normal to anomalous dispersion regime. This can be viewed
as an implementation of continuous dispersion management. We further investigate in detail
the formation process of these packets, which are a result of the initial spreading in momentum space due to nonlinear compression. A second experiment investigates in more detail the
nonlinear dynamics in the negative mass regime where the solitonic propagation leads to a significant broadening in momentum space. This shows up in the experiment as splitting of the
condensate into two wave packets which propagate in opposite directions.
The developed theoretical description utilizing the effective mass tensor models the experimental system in one dimension and can explain all main features observed in the experiment.
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6.3 Bright atomic gap solitons for atoms with repulsive interaction
6.3 Bright atomic gap solitons for atoms with repulsive
interaction
Bright solitons, which are non-spreading localized wave packets, are encountered whenever the spreading due to linear dispersion is compensated by nonlinear dynamics. For
atomic matter waves, bright solitons can form if the mass m and the coupling constant g
characterizing the atom-atom interaction have opposite sign mg < 0. They have already
been demonstrated for atoms with attractive interaction [15, 14].
In the experiment presented here, the anomalous dispersion (negative effective mass)
in the first band at the edge of the Brillouin zone is used to implement a bright atomic
gap soliton for atoms with repulsive interaction. The experimental results have been published in Physical Review Letters [16]. The necessary pre-requisites for the observation
of a gap soliton can be summarized as follows:
• The spatial size of the wavepacket has to be much larger than the periodicity d
of the periodic potential. In this case, the BEC can be described by an envelope
f (x, t), which is modulated with the corresponding periodic Bloch functions.
• The momentum spread of the wavepacket has to be much smaller than the size
of the Brillouin zone in order to prepare a constant negative effective mass. This
condition implies a lower bound for the spatial size of the wavepacket.
• The experiments have to be performed in a quasi one-dimensional situation [104],
in which the chemical potential of the BEC is smaller than the level spacing of the
transverse harmonic confinement. This limits the maximum allowed linear atomic
. It is important to note, that the existence of
density to approximately 100 atoms
µm
non-spreading wavepackets has been also predicted in two- and three-dimensional
periodic potentials in the tight-binding regime [105].
• The atom number of the initially prepared wavepacket has to be on the order of the
expected final atom number in the soliton, which is N ≈ 350 for our experimentally
implemented transverse confinement and periodic potential depth.
• The shift in momentum space due to the residual longitudinal confinement of the
waveguide has to be much smaller than the range of negative effective mass. In
our experiment, the longitudinal frequency ωx ≈ 2π · 0.5 Hz limits the observation
time of the soliton to tmax = 100 ms.
A detailed investigation of the experimental setup and the theoretical description of
bright atomic gap solitons can be found in [74].
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Chapter 6 Experiments with BECs in optical lattices
VOLUME 92, N UMBER 23
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Bright Bose-Einstein Gap Solitons of Atoms with Repulsive Interaction
B. Eiermann,1 Th. Anker,1 M. Albiez,1 M. Taglieber,2 P. Treutlein,2 K.-P. Marzlin,3 and M. K. Oberthaler1
1
Kirchhoff Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
2
Max-Planck-Institut für Quantenoptik und Sektion Physik der Ludwig-Maximilians-Universität,
Schellingstrasse 4, 80799 München, Germany
3
Department of Physics and Astronomy, Quantum Information Science Group, University of Calgary,
2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
(Received 16 December 2003; published 8 June 2004)
We report on the first experimental observation of bright matter wave solitons for 87 Rb atoms with
repulsive atom-atom interaction. This counterintuitive situation arises inside a weak periodic potential,
where anomalous dispersion can be realized at the Brillouin zone boundary. If the coherent atomic wave
packet is prepared at the corresponding band edge, a bright soliton is formed inside the gap. The
strength of our system is the precise control of preparation and real time manipulation, allowing the
systematic investigation of gap solitons.
DOI: 10.1103/PhysRevLett.92.230401
Nonspreading localized wave packets [1] —bright solitons— are a paradigm of nonlinear wave dynamics and
are encountered in many different fields, such as physics,
biology, oceanography, and telecommunication. Solitons
form if the nonlinear dynamics compensates the spreading due to linear dispersion. For atomic matter waves,
bright solitons have been demonstrated for which the
linear spreading due to vacuum dispersion is compensated
by the attractive interaction between atoms [2]. For repulsive atom-atom interaction, dark solitons have also
been observed experimentally [3].
In this Letter, we report on the experimental observation of a different type of solitons, which exist only in
periodic potentials —bright gap solitons. For weak periodic potentials, the formation of gap solitons has been
predicted [4], while discrete solitons [5] should be observable in the case of deep periodic potentials. These
phenomena are well known in the field of nonlinear
photon optics where the nonlinear propagation properties
in periodic refractive index structures have been studied
[6]. In our experiments with interacting atoms, a new
level of experimental control can be achieved, allowing
for the realization of gap solitons for repulsive atom-atom
interaction corresponding to a self-defocusing medium. It
also opens up the way to study solitons in two- and threedimensional atomic systems [7].
In our experiment, we investigate the evolution of a
Bose-Einstein condensate in a quasi-one-dimensional
waveguide with a weak periodic potential superimposed
in the direction of the waveguide. In the limit of weak
atom-atom interaction, the presence of the periodic potential leads to a modification of the linear propagation;
i.e., dispersion [8]. It has been demonstrated that with this
system anomalous dispersion can be realized [9], which is
the prerequisite for the realization of gap solitons for
repulsive atom-atom interaction.
Our experimental observations are shown in Fig. 1 and
clearly reveal that after a propagation time of 25 ms a
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96
0031-9007=04=92(23)=230401(4)$22.50
PACS numbers: 03.75.Be, 03.75.Lm, 05.45.Yv, 05.45.–a
nonspreading wave packet is formed. The observed behavior exhibits the qualitative features of gap soliton
formation such as (i) during soliton formation excessive
atoms are radiated and spread out over time, (ii) solitons
do not change their shape and atom number during propagation, and (iii) gap solitons do not move.
The coherent matter wave packets are generated
with 87 Rb Bose-Einstein condensates [Fig. 2(a)]. The
atoms are initially precooled in a magnetic time-orbiting
FIG. 1 (color). Observation of bright gap solitons. The atomic
density in the negative mass regime deduced from absorption
images (430 m 125 m) averaged over four realizations is
shown for different propagation times. After approximately
25 ms, a small peak is formed which does neither change in
shape nor in amplitude. Excessive atoms are radiated and
disperse over time. After 45 ms only the soliton with 250
atoms has sufficient density to be clearly observable. The
second peak at 15 ms shows the atoms which have been
removed by Bragg scattering to generate an initial coherent
wave packet consisting of 900 atoms. For longer observation
times, those atoms move out of the imaged region.
© 2004 The American Physical Society
230401-1
6.3 Bright atomic gap solitons for atoms with repulsive interaction
VOLUME 92, N UMBER 23
PHYSICA L R EVIEW LET T ERS
FIG. 2 (color online). Realization of coherent atomic wave
packets with negative effective mass utilizing periodic potentials. (a) Top view of the crossed dipole trap geometry used for
Bose-Einstein condensation. (b) A periodic potential is ramped
up while the atoms are still trapped in the crossed dipole trap
realizing the atomic ensemble at qc 0. (c),(d) The atoms are
released into the one-dimensional waveguide and, subsequently, the periodic potential is accelerated to the recoil
velocity vr h=m. This prepares the atomic wave packet at
the band edge of the lowest band. (e) Normal and anomalous
(shaded area) dispersion regime in a periodic potential. The
single preparation steps are indicated. The shown band structure is calculated for a modulation depth of V0 1Er .
potential trap using the standard technique of forced
evaporation leading to a phase space density of 0:03.
The atomic ensemble is subsequently adiabatically transferred into a crossed light beam dipole trap ( 1064 nm, 1=e2 waist 60 m, 500 mW per beam), where
further forced evaporation is achieved by lowering the
light intensity in the trapping light beams. With this
approach, we can generate pure condensates with typically 3 104 atoms. By further lowering the light intensity, we can reliably produce coherent wave packets of
3000 atoms. For this atom number no gap solitons have
been observed. Therefore, we remove atoms by Bragg
scattering [10]. This method splits the condensate coherently leaving an initial wave packet with 900(300) atoms
at rest. The periodic potential V V0 sin2 2
x of periodicity d =2 is realized by a far off-resonant standing
light wave of wavelength 783 nm. The absolute value
of the potential depth was calibrated independently by
analyzing results on Bragg scattering and Landau-Zener
tunneling [11].
After the creation of the coherent wave packet, we
ramp-up the periodic potential adiabatically, which prepares the atomic ensemble in the normal dispersion regime at quasimomentum q 0 as indicated in Fig. 2. The
dispersion relation for an atom moving in a weak periodic potential exhibits a band structure as a function of
quasimomentum q known from the dispersion relation of
electrons in crystals [12] [see Fig. 2(e)]. Anomalous
dispersion, characterized by a negative effective mass
230401-2
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meff < 0, can be achieved if the mean quasimomentum
of the atomic ensemble is shifted to the Brillouin zone
boundary corresponding to q =d. This is accomplished by switching off one dipole trap beam, releasing
the atomic cloud into the one-dimensional horizontal
waveguide [Fig. 2(c)] with transverse and longitudinal
trapping frequencies !? 2 85 Hz, and wjj 2 0:5 Hz. Subsequently, the atomic ensemble is prepared at
quasimomentum q =d by accelerating the periodic
potential to the recoil velocity vr h=m. This is done
by introducing an increasing frequency difference between the two laser beams, creating the optical lattice.
The acceleration within 1.3 ms is adiabatic; hence, excitations to the upper bands by Landau-Zener transitions
are negligible [11]. It is important to note that the strength
of the dispersion is under full experimental control. The
absolute value of meff q =d V0 =V0 8Er m
(weak potential approximation [12]) scales with the
modulation depth of the periodic potential, where Er h 2 =2m2 =d2 is the recoil energy.
For weak periodic potentials, the full wave function
of the condensate is well described by x; t Ax; tuq0 c x expiqc x, where uq0 c x expiqc x represents
the Bloch state in the lowest band n 0 at the corresponding central quasimomentum qc . Within the approximation of constant effective mass, the dynamics of the
envelope Ax; t is governed by a one-dimensional nonlinear Schrödinger equation [13]:
@
h 2 @2
2 Ax; t;
ih Ax; t g
jAx;
tj
1d
@t
2meff @x2
with g1d 2ha!
? nl , where nl is a renormalization
factor due to the presence of the periodic potential (nl 1:5 for q =d in the limit of weak periodic potentials
[13]), and a is the scattering length. The stationary solution for qc =d is given by
p
2
eff x0 ;
Ax; t N=2x0 sechx=x0 eiht=2m
where x0 is the soliton width and meff is the effective mass
at the band edge. The total number of atoms constituting
the soliton is given by
h
N
:
(1)
nl !? meff x0 a
This quantitative feature of bright solitons can also be
deduced by equating the characteristic energies for dispersion ED h 2 =meff x20 and atom-atom interaction Enl g1d jAx 0; tj2 .
A characteristic time scale of solitonic propagation
due to the phase evolution can also be identified. In
analogy to light optics, the soliton period is given by
TS meff x20 =2h.
Solitonic propagation can be confirmed experimentally if the wave packet does not
broaden over time periods much longer than TS .
Our experimental results in Fig. 1 show the evolution of
a gap soliton in the negative mass regime for different
230401-2
97
Chapter 6 Experiments with BECs in optical lattices
98
rms width [μm]
a
soliton period
15
meff / m = - 0.1
meff / m = 1
5
200 b
position [μm]
propagation times. The reproducible formation of a single
soliton is observed if the initial wave packet is close to the
soliton condition, i.e., a well-defined atom number for a
given spatial width. The preparation scheme utilizing the
Bragg pulse leads to a wave packet containing 900 atoms
with a spatial size of 2:5 m (rms). The periodic potential depth was adjusted to V0 0:705Er leading to
meff =m ’ 0:1 at the band edge. The soliton can clearly
be distinguished from the background after 25 ms, corresponding to three soliton periods. This is consistent
with the typical formation time scale of few soliton
periods given in nonlinear optics textbooks [14]. After
45 ms of propagation, the density of the radiated atoms
drops below the level of detection and thus a pure soliton
remains, which has been observed for up to 65 ms. It has
been shown that for gap solitons a finite lifetime is expected due to resonant coupling to transversally excited
states [15]. In order to understand the background, we
numerically integrated the nonpolynomial nonlinear
Schrödinger equation [16]. The calculation reveals
that the nonquadratic dispersion relation in a periodic
potential leads to an initial radiation of atoms. However,
the absolute number of atoms in the observed background
(600 atoms) is higher than the prediction of the employed effective one-dimensional model (250 atoms).
Therefore we conclude that transverse excitations have to
be taken into account to get quantitative agreement. This
fact still has to be investigated in more detail.
In the following, we will discuss the experimental facts
confirming the successful realization of gap solitons.
In Fig. 3(a), we compare the spreading of wave packets
in the normal and anomalous dispersion regime which
reveals the expected dramatic difference in wave packet
dynamics. The solid circles represent the width of the gap
soliton for meff =m 0:1, which does not change significantly over time. We deduce a soliton width of x0 6:09 m (xrms 4:5 m) from the absorption images
where the measured rms width shown in Fig. 3(a) is
deconvolved with the optical resolution of 3:8 m
(rms). In this regime, the wave packet does not spread
for more than eight soliton periods [TS 7:723 ms].
Since our experimental setup allows one to switch from
solitonic to dispersive behavior by turning the periodic
potential on and off, we can directly compare the solitonic evolution to the expected spreading in the normal
dispersion regime. The open circles represent the expansion of a coherent matter wave packet with 300(100)
atoms in the normal mass regime meff =m 1.
The preparation at the band edge implies that the group
velocity of the soliton vanishes. This is confirmed in
Fig. 3(b), where the relative position of the soliton with
respect to the standing light wave is shown. The maximum group velocity of the lowest band is indicated by
the dotted lines. In the experiment, care has to be taken
to align the optical dipole trap perpendicular to the
gravitational acceleration within 200 rad. Otherwise
230401-3
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PHYSICA L R EVIEW LET T ERS
0
-200
1500 c
number of atoms
VOLUME 92, N UMBER 23
1000
500
0
20
40
propagation time [ms]
60
FIG. 3. Characteristic features of the observed gap soliton.
(a) Comparison of expansion in the positive and the negative
effective mass regime for 300 atoms. While the soliton does not
disperse at all over a time of 65 ms, corresponding to more than
eight soliton periods (solid circles), a wave packet in the normal
mass regime expands significantly (open circles). Each point
represents the result of a single realization. The solid line marks
the average measured rms width of Gaussian fits to the solitons.
Panel (b) shows the position of the soliton in the frame of the
periodic potential and reveals that a standing gap soliton has
been realized. The dotted lines indicate the positions that
correspond to the maximum and the minimum group velocity
in the lowest band. (c) Number of atoms in the central peak.
The initial atom numbers exhibit large shot to shot fluctuations,
which are reduced during the soliton formation. The predicted
relation between the number of atoms and the soliton width
[Eq. (1)] is indicated by the horizontal bar using the width
deduced as shown in (a). Note that this comparison has been
done without a free parameter since all contributing parameters are measured independently.
the solitons are accelerated in the direction opposite
to the gravitational force revealing their negative mass
characteristic.
The calculated number of atoms [Eq. (1)] is indicated
by the horizontal bar in Fig. 3(c). The width of the bar
represents the expectation within our measurement uncertainties. The observed relation between atom number
and width, characteristic for a bright soliton, is in excellent agreement with the simple theoretical prediction
without any free parameter.
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6.3 Bright atomic gap solitons for atoms with repulsive interaction
PHYSICA L R EVIEW LET T ERS
VOLUME 92, N UMBER 23
-3
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B. Brezger for many stimulating discussions. We thank
O. Vogelsang and D. Weise for their donation of
Ti:sapphire light. This work was supported by the
Deutsche Forschungsgemeinschaft, the Emmy Noether
Program, and by the European Union, Contract
No. HPRN-CT-2000-00125.
x10
4
N x0
3
2
1
6
10
14
18
22
-m/meff
FIG. 4. Scaling properties of a gap soliton. The effective mass
was varied experimentally by changing the periodic potential
depth. The scaling predicted by Eq. (1) is represented by the
solid line and is in good agreement with our experimental
observations. The error bars represent the variation of the
scaling parameter for different realizations.
As an additional check for soliton formation, we determine the product of atom number and soliton width as
a function of the effective mass which is varied by adjusting the modulation depth of the periodic potential. Figure 4 shows the range of effective masses, for which
solitons have been observed. For smaller values of jmeff j,
corresponding to smaller potential depths, Landau-Zener
tunneling does not allow a clean preparation in the negative mass regime, while for larger values the initial number of atoms differs too much from the soliton condition.
The observed product of atom number and wave packet
width after 40 ms of propagation are shown in Fig. 4 and
confirm the behavior expected from Eq. (1). Additionally,
our experimental findings reveal that the change of the
scaling parameter Nx0 in Fig. 4 is dominated by the
change in the atom number, while the soliton width exhibits only a weak dependence on the effective mass.
The demonstration of gap solitons confirms that Bose
condensed atoms combined with a periodic potential
allow the precise control of dispersion and nonlinearity.
Thus, our setup serves as a versatile new model system for
nonlinear wave dynamics. Our experiments show that gap
solitons can be created in a reproducible manner. This is
an essential prerequisite for the study of soliton collisions.
The experiment can be realized by preparing two spatially separated wave packets at the band edge and applying an expulsive potential. Ultimately, atom number
squeezed states can be engineered with atomic solitons
by implementing schemes analog to those developed
for photon number squeezing in light optics [17]. This
is interesting from a fundamental point of view and
may also have impact on precision atom interferometry
experiments.
We wish to thank J. Mlynek for his generous support,
and Y. Kivshar, E. Ostrovskaya, A. Sizmann, and
230401-4
[1] J. S. Russell, Report of the 14th Meeting of the British
Association for the Advancement of Science, Plates
XLVII–LVII (1845), pp. 311–390.
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[4] P. Meystre, Atom Optics (Springer-Verlag, New York,
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Ostrovskaya and Yu. S. Kivshar, Phys. Rev. Lett. 90,
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L. Santos, and M. Lewenstein, cond-mat/0310042.
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Lett. 86, 4447 (2001).
[9] B. Eiermann, P. Treutlein, Th. Anker, M. Albiez,
M. Taglieber, K.-P. Marzlin, and M. K. Oberthaler,
Phys. Rev. Lett. 91, 060402 (2003).
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[14] G. P. Agrawal, Applications of Nonlinear Fiber Optics
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Fiber Optics (Academic, San Diego, 1995).
[15] K. M. Hilligsøe, M. K. Oberthaler, and K.-P. Marzlin,
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[16] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65,
043614 (2002).
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230401-4
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Chapter 6 Experiments with BECs in optical lattices
6.4 Macroscopic quantum self-trapping in periodic potentials
An experiment which is very closely related to the Josephson junction experiment is the
observation of macroscopic quantum self-trapping in a one-dimensional optical lattice
superimposed on a one-dimensional waveguide. This effect has been predicted by A.
Trombettoni et al. [106] and is the analog of MQST in bosonic Josephson junctions.
A BEC consisting of a variable number of atoms NT is formed in the crossed optical
dipole trap and subsequently adiabatically loaded into the first energy band of a deep onedimensional optical lattice (V0 ≈ 11Erec ). After this preparation at zero quasimomentum,
the crossed dipole trap beam is switched off, which releases the wavepacket into the onedimensional waveguide. The subsequent expansion along the waveguide is monitored for
observation times up to tmax = 100 ms.
The tunneling dynamics of a Bose-Einstein condensate in a deep optical lattice can
be theoretically modeled by a tight binding approximation, in which the total wavefunction is written as a sum of individual modes localized in the minima of the periodic
potential. The neighboring modes are coupled via atomic tunneling. This description is
very similar to the two-mode approximation described in chapter 3 and constitutes the
generalization to an array of coupled bosonic Josephson junctions. For small atom numbers, which implies small on-site interaction energy differences between adjacent wells,
Josephson tunneling can occur between the neighboring sites. This leads to a continuously expanding cloud keeping its gaussian shape. The situation changes drastically, if
the total atom number is increased above a critical value Ncrit ≈ 3000. In this case, the
wavepacket develops steep edges due to the population of quasi-momenta corresponding
to infinite effective masses3 . Those steep edges lead to a population imbalance between
two adjacent wells, which is larger than the critical population imbalance z(0) for macroscopic quantum self-trapping. The local effect of the inhibition of tunneling between two
adjacent wells results in a stopping of the expansion of the wavepacket, which is observed
in the experiment.
The transition to the macroscopically self-trapped state is characterized by a single
parameter, the scaled critical on-site interaction energy Λc . The experimental investigation confirms, that all data points for different system parameters (i.e. atom numbers,
initial width and periodic potential depth) collapse onto a single curve, which is exclusively determined by Λc .
A detailed investigation of self-trapping in periodic potentials will be presented in
the PhD thesis of Thomas Anker.
3
The broadening in momentum space during the evolution is dependent on the atom number of the
BEC.
100
6.4 Macroscopic quantum self-trapping in periodic potentials
PRL 94, 020403 (2005)
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21 JANUARY 2005
PHYSICAL REVIEW LETTERS
Nonlinear Self-Trapping of Matter Waves in Periodic Potentials
Th. Anker,1 M. Albiez,1 R. Gati,1 S. Hunsmann,1 B. Eiermann,1 A. Trombettoni,2 and M. K. Oberthaler1
1
2
Kirchhoff Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
I.N.F.M. and Dipartimento di Fisica, Universitá di Parma, parco Area delle Scienze 7A, I-43100 Parma, Italy
(Received 1 October 2004; published 19 January 2005)
We report the first experimental observation of nonlinear self-trapping of Bose-condensed 87 Rb atoms
in a one-dimensional waveguide with a superimposed deep periodic potential . The trapping effect is
confirmed directly by imaging the atomic spatial distribution. Increasing the nonlinearity we move the
system from the diffusive regime, characterized by an expansion of the condensate, to the nonlinearity
dominated self-trapping regime, where the initial expansion stops and the width remains finite. The data
are in quantitative agreement with the solutions of the corresponding discrete nonlinear equation. Our
results reveal that the effect of nonlinear self-trapping is of local nature, and is closely related to the
macroscopic self-trapping phenomenon already predicted for double-well systems.
DOI: 10.1103/PhysRevLett.94.020403
PACS numbers: 03.75.Lm
0031-9007=05=94(2)=020403(4)$23.00
only the nonlinear energy by adjusting the number of
atoms in the wave packet close to (2000 200 atoms)
and above (5000 600 atoms) the critical value. Clearly
both wave packets expand initially. At t 35 ms the wave
packet with higher initial atomic density has developed
steep edges and stops expanding (see inset in Fig. 1). In
contrast, the wave packet with the lower initial atomic
density continues to expand keeping its Gaussian shape.
The coherent matter-wave packets are generated with
87
Rb Bose-Einstein condensates realized in a crossed light
beam dipole trap ( 1064 nm, 1=e2 waist 55 m,
600 mW per beam). Subsequently a periodic dipole poten-
width
16
rms-width [µm]
The understanding of coherent transport of waves is
essential for many different fields in physics. In contrast
to the dynamics of noninteracting waves, which is conceptually simple, the situation can become extremely complex
as soon as interaction between the waves is of relevance.
Very intriguing and counterintuitive transport phenomena
arise in the presence of a periodic potential. This is mainly
due to the existence of spatially localized stationary
solutions.
In the following we will investigate the dynamics of
Bose-condensed 87 Rb atoms in a deep one-dimensional
periodic potential; i.e., the matter waves are spatially localized in each potential minimum (tight binding) and are
coupled via tunneling to their next neighbors. This system
is described as an array of coupled boson Josephson junctions [1]. The presence of nonlinearity drastically changes
the tunneling dynamics [2] leading to new localization
phenomena on a macroscopic scale such as discrete solitons, i.e., coherent nonspreading wave packets, and nonlinear self-trapping [3]. These phenomena have also been
studied in the field of nonlinear photon optics where a
periodic refractive index structure leads to an array of
wave guides, which are coupled via evanescent waves [4].
In this Letter we report on the first experimental confirmation of the theoretically predicted effect of nonlinear
self-trapping of matter waves in a periodic potential [3].
This effect describes the drastic change of the dynamics of
an expanding wave packet when the nonlinearity, i.e.,
repulsive interaction energy, is increased above a critical
value. Here the counterintuitive situation arises that
although the spreading is expected to become faster due
to the higher nonlinear pressure, the wave packet stops to
expand after a short initial diffusive expansion. Since we
observe the dynamics in real space, we can directly measure the wave packet width for different propagation times.
In Fig. 1 we show the experimental signature of the transition from the diffusive to the self-trapping regime. We
prepare wave packets in a periodic potential and change
14
12
10
8
0
20
40
60
expansion time [ms]
80
FIG. 1. Observation of nonlinear self-trapping of Bosecondensed 87 Rb atoms. The dynamics of the wave packet width
along the periodic potential is shown for two different initial
atom numbers. By increasing the number of atoms from 2000 200 (squares) to 5000 600 (circles), the repulsive atom-atom
interaction leads to the stopping of the global expansion of the
wave packet. The insets show that the wave packet remains
almost Gaussian in the diffusive regime but develops steep edges
in the self-trapping regime. These edges act as boundaries for the
complex dynamics inside.
020403-1
 2005 The American Physical Society
101
Chapter 6 Experiments with BECs in optical lattices
n(x)
numerical
experimental
tial Vp sEr sin2 kx, realized with a far off-resonant
standing light wave ( 783 nm) collinear with one of
the dipole trap beams is adiabatically ramped up. The
depth of the potential is proportional to the intensity of
2 2
the light wave and is given in recoil energies Er h2mk with
the wave vector k 2=. By switching off the dipole
trap beam perpendicular to the periodic potential, the
atomic matter wave is released into a trap acting as a
one-dimensional waveguide Vdip m2 !2? r2 !2k x2 with radial trapping frequency !? 2 230 Hz and
longitudinal trapping frequency !k 2 1 Hz. The
wave packet evolution inside the combined potential of
the waveguide and the lattice is studied by taking absorption images of the atomic density distribution after a
variable time delay. The density profiles nx; t along the
waveguide are obtained by integrating the absorption images over the radial dimensions and allow the detailed
investigation of the wave packet shape dynamics with a
spatial resolution of 3 m.
In Fig. 2 the measured temporal evolution of the wave
packet prepared in the self-trapping regime (s 10; 7:65 m initial rms width, 5000 600 atoms) is
shown. The evolution of the shape is divided into two
characteristic time intervals. Initially (t < 20 ms) the
wave packet expands and develops steep edges. This dynamics can be understood in a simple way by considering
that the repulsive interaction leads to a broadening of the
momentum distribution and thus to a spreading in real
space. Since the matter waves propagate in a periodic
potential, the evolution is governed by the modified dispersion (i.e., band structure) Eq 2K cosdq where
d =2 is the lattice spacing, hq
is the quasimomentum,
t=10 ms
-20
0
20 -20
t=20 ms
0
t=30 ms
20 -20 0 20 -20
x [µm]
t=40 ms
0
20 -20
t=50 ms
0
20
FIG. 2. Comparison between theory and experiment for s 10; 7:65 m initial rms width and 5000 600 atoms. The
upper graphs show the measured density distribution for different propagation times. During the initial expansion in the selftrapping regime the wave packet develops steep edges which act
as stationary boundaries for the subsequent internal dynamics.
The results of the numerical integration of Eq. (2) (depicted in
the lower graphs) are in very good agreement. For t 50 ms a
1.5 mrad deviation of the waveguides’ horizontal orientation
(consistent with the experimental uncertainty) is taken into
account and reproduces the experimentally observed asymmetry
(gray line).
and K is the characteristic energy associated with the
tunneling. The formation of steep edges is a consequence
of the population of higher quasimomenta around q =2d where the dispersion is strongly reduced and the
group velocity is extremal. In order to populate quasimomenta jqj > =2d the initial interaction energy has to be
higher than the characteristic tunneling energy K and thus
the critical parameter depends on the ratio between the onsite interaction energy and the tunneling energy as we will
discuss in detail. While in the linear evolution the steep
edges move with the extremal group velocity [5], in the
experiment reported here they stop after their formation.
As we will show, this is a consequence of the high atomic
density gradient at the edge which suppresses tunneling
between neighboring wells. The further evolution is characterized by stationary edges acting as boundaries for the
complex internal behavior of the wave packet shape. The
formation of the side peaks is an indication that atoms
moving outwards are piled up because they cannot pass
the steep edge. Finally the pronounced features of the wave
packet shape disappear and a square shaped density distribution is formed.
In order to understand in detail the ongoing complex
self-trapping dynamics we compare quantitatively our experimental findings with numerically obtained solutions
(see Fig. 2). For our typical experimental parameters of
s 11 and 100 atoms per well, we are in the regime
where the dynamics can be described by a macroscopic
wave function r~; t and thus by the Gross-Pitaevski
equation [6]. Since we use deep optical lattices, the description can be reduced to a one-dimensional discrete
nonlinear equation, which includes the fundamental processes, namely, tunneling between the wells and nonlinear
phase evolution due to the interaction of the atoms [3,7]. In
our experiment the trapping frequency in a single well
along the lattice period is on the order of !x 2 25 kHz, whereas the transverse trapping frequency
of the waveguide is !? 2 230 Hz. Thus our system
can be described as a horizontal pile of pancakes, and the
transverse degree of freedom cannot be neglected. In [7] a
one-dimensional discrete nonlinear equation (DNL) is derived which takes into account the adiabatic change of the
wave function in the transverse direction due to the atomatom interaction. A generalized tight binding ansatz
r~; t X
r; Nj t
j tj ~
(1)
j
q
is used, with j t Nj teij t , where Nj t is the atom
number and j t is the
R phase of the jth condensate. j is
normalized to 1 (i.e., dr~2j 1) and ~r; t is normalP
ized to the total number of atoms NT (i.e., j j j j2 NT ).
The spatial real wave function j r~; Nj t is centered at
the minimum of the jth well and is time dependent through
Nt. Integrating over the spatial degrees of freedom, the
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PRL 94, 020403 (2005)
6.4 Macroscopic quantum self-trapping in periodic potentials
following DNL is obtained from the Gross-Pitaevski equation
ih
@ j
!j
@t
j
K
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PHYSICAL REVIEW LETTERS
PRL 94, 020403 (2005)
j1
j1 loc
j
j:
0
1b
(2)
K is the characteristic
tunneling energy between adjacent
R
sites. !j dr~ m2 !2k x2 2j is the on-site energy resulting
from the longitudinal trapping potential, which is negligible in the description of our experiment.
R mThe2 relevant
chemical potential is given by loc
dr~ 2 !? r2 2j j g0 j j tj2 4j with g0 4h 2 a=m (a is the scattering
length). It can be calculated approximately for our experimental situation assuming a parabolic shape in transverse
direction (Thomas-Fermi approximation) and a Gaussian
shape in longitudinal direction for j r~; Nj t (!x > !? ). This leads to loc
loc
j =h
j U1 j j tj with
v
u
u m!2 g0
(3)
U1 tp? :
2%x
Here %x =2s1=4 is the longitudinal Gaussian width
of j in harmonic approximation of the periodic potential
minima. Please note that if the local wave function j does
not depend on Nj Eq. (2) reduces to the well known
discrete nonlinear Schrödinger equation with loc
j / Nj
[3,8].
We compare the experimental and numerical results in
Fig. 2 and find very good agreement. The theory reproduces the observed features such as steepening of the edges,
the formation of the side peaks, and the final square wave
packet shape. It is important to note that all parameters
entering the theory (initial width, atom number, periodic
potential depth, and transverse trapping frequency) have
been measured independently. The observed asymmetry of
the wave packet shapes (e.g., see Fig. 2, t 50 ms) appears due to the deviation from the perfect horizontal
orientation of the wave guide ( 2 mrad) which results
from small changes in height of the pneumatic isolators of
the optical table during the measurements.
In the following we will use the numerical results to get
further insight into the self-trapping dynamics. We investigate the local tunneling dynamics and phase evolution by
evaluating the relative atom number difference Nj Nj1 Nj =Nj1 Nj and the phase difference j j1 j between two neighboring sites. In Fig. 3(a) the
wave packet shapes for t 0 and t 50 ms are shown. In
Fig. 3(b) we plot the relative atom number difference Nj
averaged over the whole propagation duration of 50 ms.
The graph indicates two spatial regions with different
characteristic dynamics. While the average vanishes in
the central region (shaded in light gray) it has significant
amplitude in the edge region (shaded in dark gray). The
characteristic dynamics of Nj and j in the central
region is depicted in Fig. 3(c). The atom number difference
t = 0 ms
t = 50 ms
Nj
1a
0
1c
0
-1
0
20
site number j
d
40
60
0
0
10 20 30 40 50
t [ms]
0
10 20 30 40 50
t [ms]
FIG. 3. A numerical investigation of the site-to-site tunneling
dynamics. (a) The atomic distribution Nj of the wave packet for
t 0 and 50 ms. (b) The relative population difference Nj time
averaged over the expansion time indicates two regions with
different dynamics. (c) The dynamics of Nj and the phase
difference j for the marked site oscillate around zero known
as the zero-phase mode of the boson Josephson junction. (d) The
dynamics in the edge region is characterized by long time
periods where jNj j is close to 1 while at the same time j
winds up very quickly (phase is plotted modulo ) known as
‘‘running phase self-trapping mode’’ in boson Josephson junctions. Thus the expansion of the wave packet is stopped due to
the inhibited site-to-site tunneling at the edge of the wave packet.
as well as the phase difference oscillate around zero. This
behavior is known in the context of Bose-Einstein condensation in double-well potentials. It is described as the
boson Josephson junction ‘‘zero-phase mode’’ [2] and is
characteristic for superfluid tunneling dynamics if the atom
number difference stays below a critical value. At the edge
in contrast, Nj crosses the critical value during the initial
expansion (steep density edge) and locks for long time
periods to high absolute values showing that the tunneling
and thus the transport is inhibited. At the same time the
phase difference winds up. This characteristic dynamics
has been predicted within the boson Josephson junction
model for a double-well system and is referred to as the
‘‘running phase self-trapping mode’’ [2]. This analysis
makes clear that the effect of nonlinear self-trapping as
observed in our experiment is a local effect and is closely
related to boson Josephson junctions dynamics in a doublewell system.
Although the local dynamics just described is very
complex, the evolution of the root mean square width of
the wave packet, i.e., the global dynamics, can be predicted
analytically within a very simple model. In [3], a Gaussian
j 2
2
profile wave packet j t / exp&t
i 't
2 j , parametrized by the width &t (in lattice units) and the
quadratic spatial phase 't, is used as an ansatz for quasimomentum q 0 to solve the discrete nonlinear
Schrödinger equation. The time evolution of the width
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Chapter 6 Experiments with BECs in optical lattices
A surprising result of this model is the prediction of the
following scaling behavior (shown in Fig. 4):
s
&0
(4)
1 c
&1
for = c > 1, where &1 is the width of the wave packet
for t ! 1. For = c < 1 the width is not bound and thus
the system is in the diffusive regime. In the regime
= c > 1 the width is constant after an initial expansion
(see inset Fig. 4). Since = c / loc
av =K, the self-trapping
regime is reached by either reducing the initial width,
increasing the height of the periodic potential, or, as is
shown in Fig. 1, by increasing the number of atoms.
Scaling means that all data points (i.e., different experimental settings with the same = c ) collapse onto a single
universal curve. In order to confirm the scaling behavior
experimentally we measure the width of the wave packet
after 50 ms evolution for different system parameters, i.e.,
atom number, initial width of the wave packet, and depth of
the periodic potential. The experimental results shown in
Fig. 4 confirm the universal scaling dependence on = c
and follow qualitatively the prediction of the simple model.
The dashed line in Fig. 4 is the result of the numerical
integration of the discrete nonlinear equation given in
Eq. (2) evaluated at t 50 ms. It shows quantitative agreement with the experiment. The difference between the
numerical (dashed line) and analytical calculation (solid
line) is due to the initial non-Gaussian shape (numerically
obtained ground state) and the strong deviation from the
Gaussian shape for long propagation times.
Concluding, we have demonstrated for the first time the
predicted effect of nonlinear self-trapping of Bose-Einstein
condensates in deep periodic potentials. The detailed
analysis shows that this is a local effect, which occurs
due to nonlinearity induced inhibition of site-to-site tunneling at the edge of the wave packet. This behavior is
closely connected to the phenomenon of macroscopic selftrapping known in the context of double-well systems.
Furthermore, we quantitatively confirm in our experiments
the predicted critical parameter which discriminates between diffusive and self-trapping behavior.
1
0.8
0.6
rms-width [µm]
/
&t is obtained analytically applying a variational principle. The result of this simple model is that the dynamics of
the wave packet width is solely determined by two global
parameters —the density of the atoms and the depth of the
periodic potential. Also, a critical parameter = c can be
deduced, which governs the transition from the diffusive to
the self-trapping regime. The transition parameter = c
for the 2D case described by Eq. (2) is obtained following
the same lines of calculation as in [3]. Assuming that the
initial width &0 1 (in the experiment typically &0 40)
we obtain
p
U N
3 9 1=4 p
1 T and
&0 :
c 2 8
2K
0.4
0.2
0
1
2
3
12
10
N = 5000
N = 3500
8
0 20 40 60 80
expansion time [ms]
4
5
FIG. 4. Experimental investigation of the scaling behavior.
The solid line shows the curve given by Eq. (4).
Experimentally the parameter = c was varied by using three
different periodic potential depths: s 10:63 (stars), 11:13
(squares), and 11:53 (diamonds). For each potential depth,
wave packets with different atom numbers and initial widths
are prepared and the width for t 50 ms is determined. The
experimental data show qualitatively the scaling behavior predicted by Eq. (4) and are in quantitative agreement with the
results of the numerical integration of the DNL (dashed line).
The inset depicts the nature of the scaling: increasing = c (by,
e.g., increasing the atom number) leads to a faster trapping and
thus to a smaller final width.
We wish to thank A. Smerzi for very stimulating discussions. This work was supported by the Deutsche
Forschungsgemeinschaft, Emmy Noether Programm, and
by the European Union, RTN-Cold Quantum Gases,
Contract No. HPRN-CT-2000-00125.
[1] F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F.
Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio,
Science 293, 843 (2001).
[2] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy,
Phys. Rev. Lett. 79, 4950 (1997); S. Raghavan, A. Smerzi,
S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999);
G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls,
Phys. Rev. A 55, 4318 (1997).
[3] A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353
(2001).
[4] For example, D. N. Christodoulides, F. Lederer, and Y.
Silberberg, Nature (London) 424, 817 (2003).
[5] B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M.
Taglieber, K.-P. Marzlin, and M. K. Oberthaler, Phys.
Rev. Lett. 91, 060402 (2003).
[6] W. Zwerger, J. Opt. B 5, S9 (2003).
[7] A. Smerzi and A. Trombettoni, Phys. Rev. A 68, 023613
(2003).
[8] D. Hennig and G. P. Tsironis, Phys. Rep. 307, 333 (1999);
P. G. Kevrekidis, K. O. Rasmussen, and A. R. Bishop, Int.
J. Mod. Phys. B 15, 2833 (2001); M. Johansson and S.
Aubry, Nonlinearity 10, 1151 (1997).
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7 Conclusion and outlook
7.1 Conclusion
This thesis describes experiments on nonlinear dynamics of 87 Rb Bose-Einstein condensates in multi well and double well potentials. Hereby, I focus on the realization of the
first single Josephson junction for BECs.
Experimental results on BECs in periodic potentials
During the first two years of my PhD period at the University of Konstanz, we investigated the dynamics of BECs in weak one-dimensional periodic potentials oriented
along a one-dimensional optical waveguide. The first experiments were concerned with
the implementation of “dispersion management” for atomic matter waves [23, 24]. This
concept, which is known from photon optics, allows to control the sign as well as the
magnitude of the dispersion of BECs in a periodic potential. The periodic potential for
matter waves is created by a pair of counter-propagating off-resonant laser beams forming a standing light wave. By detuning the frequencies of the two beams, which results
in a moving periodic potential, the BEC can be prepared at any position in quasimomentum space. This allows for an adjustment of negative effective masses, which is a
prerequisite for the generation of bright gap solitons of atoms with repulsive interaction.
Those non-spreading wavepackets which we have subsequently observed [16] form, if the
nonlinear atom-atom interaction and the anomalous dispersion cancel.
After moving to the University of Heidelberg, we addressed investigation of Josephson
junction arrays for Bose-Einstein condensates. They are experimentally implemented by
a deep standing light wave superimposed on a two-dimensional harmonic confinement.
The nonlinear dynamics in this system can be theoretically described in the framework
of the tight-binding approximation [106], in which nearest neighbours are coupled via
interwell tunneling. Two distinct dynamical regimes are theoretically predicted and could
be experimentally observed [27]. For small atom numbers, the atomic cloud expands
infinitely after turning off the harmonic trapping potential along the optical lattice. A
totally different and counterintuitive situation arises if the atom number is increased
above a critical value. In this case, the expansion of the wavepacket stops after a short
evolution time. This effect, which is caused by the atom-atom interaction is referred to
as “macroscopic quantum self-trapping”.
105
Chapter 7 Conclusion and outlook
Josephson junctions for BECs
A Josephson junction generally consists of two weakly linked macroscopic wavefunctions
separated by an energy barrier. A weak link between two Bose-Einstein condensates is
provided by an external double well potential. The tunneling dynamics in this new experimental system constitutes the nonlinear generalization of the well-known Josephson
tunneling in superconductors and superfluids.
A simple and intuitive theoretical description in terms of only two physical quantities
- population imbalance and relative phase between the two wells - is provided in the
framework of the two-mode approximation [29], where the tunneling dynamics is mapped
onto the motion of a nonrigid pendulum1 . In this analogy, the relative phase corresponds
to the angle and the population imbalance to the angular momentum of the pendulum.
For small initial population imbalances of the wells, the atoms tunnel left and right
over time, corresponding to an oscillatory motion of the pendulum around the stable
equilibrium position. If the initial population is chosen above a critical value, resonant
tunneling between the two wells is inhibited because the difference of the interaction
energy between the two localized modes exceeds the tunneling energy splitting. In this
case, the population imbalance becomes self-locked and the relative phase winds up
linearly over time.
Experimental setup for the observation of Josephson tunneling dynamics
The experimental apparatus to reach the quantum degenerate regime is based on standard cooling methods. After loading a laser-cooled atomic sample into a time orbiting
potential magnetic trap, a first stage of forced evaporative cooling to a phase space density of Ω ≈ 5 · 10−3 is performed. Bose-Einstein condensation is subsequently reached
by evaporative cooling in a three-dimensional optical dipole trap created by two crossed
off-resonant laser beams. A BEC consisting of approximately 1000 atoms is prepared in
an effective double well potential by adiabatically ramping up a one-dimensional standing light wave, which is created by two beams crossing at a relative angle of 9◦ . This
optical lattice has a periodicity of 5.18 µm and a potential depth which is adjustable by
controlling the beam intensities. The resulting double well potential has a well-spacing
of d = 4.4 µm and a typical potential barrier height of Vb = h · 260 Hz. The realization of such small lattice spacings below 10 µm and atom numbers below 3000 atoms is
necessary in order to make both dynamical regimes experimentally accessible.
A relative shift of the harmonic confinement with respect to the optical lattice is used
to create an asymmetric double well potential. The degree of asymmetry defines the initial population imbalance between left and right well. A relative position shift of only
350 nm results in a population imbalance corresponding to the self-trapping threshold.
In order to be able to clearly experimentally distinguish between Josephson tunneling
and self-trapping regime, a relative position stability of below 100 nm is required. This
demands high passive stability of the mechanical setup and requires an active stabilization of the phase of the optical lattice. After the preparation, the Josephson tunneling
dynamics is initiated by non-adiabatically switching to a symmetric double well potential.
1
The tunneling dynamics of superconducting Josephson junctions can be mapped onto the dynamics
of a rigid pendulum in the framework of the RCSJ model [43].
106
7.2 Outlook
Experimental results on Josephson dynamics
The experimental results show, that both dynamical regimes - Josephson oscillations
and self-trapping - are experimentally accessible. In order to completely characterize the
ongoing dynamics, the population imbalance as well as the relative phase is determined
as a function of the evolution time in the symmetric double well potential. The population imbalance is obtained from absorption images, which are taken after increasing the
distance of the two atomic clouds by exciting a collective dipole oscillation inside the
individual wells. The high optical resolution 2.7(2) µm of our imaging system allows to
clearly distinguish the two wavepackets. This allows for a direct in-situ observation of
tunneling massive particles for the first time.
The relative phase of the two modes is obtained by releasing the BEC from the double
well potential. After time-of-flight, clear atomic double slit interference patterns form
unveiling the relative phase of the wavepackets. The knowledge of both dynamical variables allows to determine the complete phase plane trajectories in the different tunneling
regimes.
We observe Josephson oscillations for an initial population imbalance z(0) = 0.28(6),
which means that 64% of the atoms are initially localized in the left well. This regime
is characterized by an oscillating population imbalance and a relative phase around a
zero mean value. The observed tunneling timescale of 40(2) ms is much shorter than the
tunneling period 500(50) ms expected for non-interacting atoms in the realized potential.
This reveals the important role of nonlinear atom-atom interaction in bosonic Josephson
junction experiments. The experimental results are in excellent agreement with the
numerical solution of the non-polynomial nonlinear Schrödinger equation [73]. It is
important to note, that all parameters entering the simulation have been independently
calibrated.
At an initial population imbalance of z(0) = 0.62(6) the system evolves in the selftrapped regime. Here, the population is stationary within the experimental errors and
the relative phase evolves unbound in time (running phase modes).
The transition between the two regimes is in agreement with the prediction of the
numerical solution of the nonlinear non-polynomial Schrödinger equation, which predicts
a critical population imbalance2 zc,N P SE = 0.375. However, the two-mode approximation
predicts the transition at zc,CT M = 0.23, which is inconsistent with the experimental
findings. Therefore, recently an extended two-model has been developed [80], which is
capable to explain our experimental data quantitatively.
The experimental results on Josephson junctions for Bose-Einstein condensates have
been published in Physical Review Letters [100].
7.2 Outlook
The preparation of Bose-Einstein condensates in the groundstate of an external double
well potential implies a zero initial phase difference between the two localized modes.
Therefore, π-oscillations [30], which are tunneling oscillations around the mean value
hφit = π of the relative phase, are so far inaccessible experimentally. We are currently
setting up an experimental system, which allows to imprint an arbitrary phase on one of
2
The full solution of the three dimensional Gross-Pitaevskii equation yields zc,T DGP E = 0.39.
107
Chapter 7 Conclusion and outlook
the two modes with a spatial resolution of below 2 µm . The details of the experimental
implementation will be presented in the diploma thesis of Jonas Fölling.
The experimental results show, that self-trapped states decay to the symmetrically
populated state with a timescale of approximately 300 ms. This decay is most likely due
to the influence of the residual thermal cloud and should allow for a determination of
the temperature of the atomic sample. However, this implies a solution of the GrossPitaevskii equation for a BEC in the double well potential taking into account of a
coupling to the thermal bath, which has not been implemented so far. We are currently
investigating the effect of thermal and quantum decoherence on the fluctuations of the
relative phase. The visibility α = hcos φi of the relative phase is measured as a function of
the barrier height. Thermal effects have been shown to significantly affect the transition
from the coherent to the incoherent regime even at temperatures far below the critical
temperature [101]. This should provide a new thermometry method especially for small
BECs. The preliminary results suggest, that the lowest achievable temperatures in our
setup are approximately 10 nK, which has to be compared to the critical temperature
TC ≈ 55 nK. This would imply that the thermal cloud consists of only about 10 atoms.
Furthermore, our experimental setup could be used to study the tunneling dynamics
of a BEC in a double well potential with an oscillating energy barrier. It has been shown
by Luca Salasnich [107], that the dynamics of this system critically depends on the
driving frequency ω of the oscillating barrier3 . In the regime of macroscopic self-trapping,
where only a small fraction of the atoms tunnels with a frequency ωJ , the dynamics is
strongly affected at the parametric resonance condition ω = 2ωJ . At a sufficiently large
perturbation, the system eventually escapes from the self-trapped state. The oscillating
barrier can easily be implemented in the current setup by simply periodically varying
the height of the potential barrier.
Most of the experiments with Bose-Einstein condensates performed so far can be
well described by the Gross-Pitaevskii equation. However, the ground state of a BEC
of repulsive atoms in a double well potential of a specific shape has been found to be
fragmented, meaning that at least two natural orthogonal orbitals are macroscopically
occupied [108, 109]. Therefore, Bose-Einstein condensates of low atom numbers in double well systems can provide a test of the validity of the mean-field description, which
assumes a single macroscopically occupied wavefunction.
Furthermore, the combination of a double well potential and an additional onedimensional optical lattice should allow for the creation of two independent atomic gapsolitons. By applying an additional harmonic external potential, it should be possible to
investigate soliton collisions in a very reproducible way. A detailed investigation of this
possible future experiment can be found in the diploma thesis of M. Taglieber [110].
During the last years, Bose-Einstein condensates in double well potentials have
attracted enormous theoretical and experimental interest. The first experimental implementation presented in this thesis brings the observation of many of the predicted
effects into reach. Therefore, one can be anxious to future publications.
3
108
The time-dependency is modeled by a time-dependent scaled on-site interaction energy Λ.
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Acknowledgements
During the realization of the presented work, I was collaborating and getting help from
many colleagues and friends. To these people I would like to say “thank you”:
• I am deeply grateful to my supervisor Markus Oberthaler for his guidance and
support throughout my PhD period. Markus is not only an excellent scientist who
shared with me his enthusiasm for physics, but also a compassionate human with
a concern for my family situation. This was the basis for having a great time in
his group. Thanks for everything, Markus!
• Together with Bernd Eiermann and Thomas Anker, who already worked on the
experiment when I arrived, I have spent innumerable enjoyable (and from time to
time successful) hours in the lab in Konstanz and Heidelberg. Bernd’s attention
to details, his patience in explaining physical details and helping whenever he was
asked for, have made him an invaluable partner. Tom, a brilliant scientist and
dancing machine, has always taken care of the suitable music in the lab. We had
a lot of fun (not only with physics)! Both of them have continued to contribute to
my intellectual and personal development. I am grateful for their friendship!
After the move to Heidelberg, Rudolf Gati, an excellent scientist and a great asset
to the lab, joined our experiment. He made big contributions to the success of the
Josephson junction experiments and got never tired during the “datathlon”. I wish
him all the best for the rest of his PhD period. “Mille grazie” to Matteo Cristiani,
who worked in our lab during the first three months spent on the Josephson junction
experiment. He is a truly brilliant guy and has been a great source of knowledge
and ideas.
• Special mention and thanks go to our diploma students Jonas Fölling, Börge
Hemmerling, Stefan Hunsmann, Matthias Taglieber and Philipp Treutlein for their
substantial contributions to the success of the experiment. I enjoyed sharing the
lab and office, and discussing physical and private problems with great people.
• I also want to thank the other group members who are or were working on other
projects: Ramona Ettig, Alex Greiner, Martin Göbel, Anja Habenicht, Thomas
Hörner, Igor Mourachko, Dirk Jürgens, Lisa Kierig, Martin Störzer and Ralf
Stützle. They contributed substantially to the great atmosphere in the group and
to many enjoyable hours during the coffee breaks and the numerous pub-crawls.
• The department of physics of the universities of Konstanz and Heidelberg, who
have provided an excellent professional environment for my research.
117
Bibliography
• I would like to thank my parents, my grandparents and my brother for their unconditional never-ending support.
• Finally, I would like to thank my wonderful wife Karen, who performs the most
challenging job of taking care of our daughters Clara and Luca. It is through their
love, support and encouragement that I have made it through all the steps during
the last ten years to reach this point in life.
118
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