## Abstract

We present experimental evidence for scale invariant behaviour of the excitation spectrum in phase-fluctuating quasi-1d Bose gases after a rapid change of the external trapping potential. Probing density correlations in free expansion, we find that the temperature of an initial thermal state scales with the spatial extension of the cloud as predicted by a model based on adiabatic rescaling of initial eigenmodes with conserved quasiparticle occupation numbers. Based on this result, we demonstrate that shortcuts to adiabaticity for the rapid expansion or compression of the gas do not induce additional heating.

## Introduction

A systematic understanding of non-equilibrium dynamics in many-body quantum systems is a
longstanding goal, with far-reaching applicability for many different fields of physics.
Ultracold atom experiments offer clean implementations of systems that are tunable, well
isolated from the environment and theoretically tractable^{1,2}. In particular,
the profound understanding available for the one-dimensional (1d) Bose gas makes it an ideal
test bed for quantum many-body dynamics^{3}.

Tunable parameters in the system's Hamiltonian allow the controlled preparation of
non-equilibrium states^{4,5,6,7}. The identification of characteristic scaling
laws is an important step for the concise description of the subsequent dynamical processes.
Of particular importance are laws governing not only global parameters^{8,9,10}
but ideally the full spectrum of excitations, as studied in recent experiments with 2d
Bose^{11,12} or Tonks-Girardeau gases^{4,13}.

Recent work^{14} has shown that a general scaling property of many-body
wavefunctions holds exactly for a broad class of systems, including the weakly interacting 1d
Bose gas addressed in this Letter. The existence of such a scaling solution is a consequence
of a dynamical symmetry of the underlying Hamiltonian. For an ultracold gas, fast changes of
control parameters in the Hamiltonian generally lead to quasiparticle production and
heating^{15}. The existence of a scaling solution for the full spectrum of
quasiparticle modes implies that so-called shortcuts to adiabaticity (STA)^{16,17} can be engineered not only for the mean density profile of a 1d gas, but also for
correlation properties of the system in certain regimes of interaction strength^{18,19}.

We show in this work that the scaling solutions for a true many-body wavefunction have their counterpart in the hydrodynamic regime of our experimental system. We bring our system out of equilibrium by rapidly changing its longitudinal confinement. The subsequent system evolution gives insight into the scaling properties of the gas. This allows us to study the regimes and limits of such a manipulation, with an emphasis on STA schemes. We furthermore demonstrate for the first time that STA schemes are valid for the second-order correlation, and thereby the temperature, of weakly interacting 1d Bose gases.

## Results and Discussion

In our experiments, we investigate the scaling solutions of hydrodynamic equations and how they can be applied for the rapid control of the complete wavefunction of a many-body quantum system.

We start with a single quasicondensate of several thousand ^{87}Rb atoms in an
elongated trap on an atom chip^{20}. The initial temperatures are set between 50
nK and 150 nK and linear densities range between 50 atoms/*μ*m and 200 atoms/*μ*m.
Axially, the cloud is deeply in the Thomas-Fermi regime. Radially, the gas is described by
an interaction-broadened ground state wavefunction^{21,22,23}. For these
parameters, both the chemical potential and the average thermal energy per particle fulfil
the condition , where
denotes the radial
level spacing of the trap with frequency , so that scattering into radial excited states is
strongly suppressed and an effective 1d system is realized^{24,20,25,21}.
After evaporative cooling, we keep an RF-shield 12 kHz above the trap bottom throughout our
experiments to remove hot atoms. The cloud is probed by standard absorption imaging
techniques after a 4 ms to 10 ms long phase of time-of-flight expansion.

The geometry of the trap is governed by the current flow through a central Z-shaped wire and two U-shaped control structures on the atom chip, as shown in figure 1(a). Panels (b)–(e) show two different trapping potentials calculated for currents tuned to A and A, as well as A and A. Varying and results in traps with axial confinement ranging from Hz to Hz, and radial confinement from Hz to Hz. A rapid change of the current ratio constitutes a quench of the trapping potential and induces excitations.

In our first set of experiments we probe the dynamical scaling of the phonon ensemble in
the presence of an axial quadrupole-mode collective excitation^{26} induced by
such a quench. To this end, we employ a linear ramp from Hz to Hz, and from Hz to Hz, respectively, of duration *τ*. The ramps of the
trapping potential were designed to avoid transverse excitations. We chose to maintain a
constant transverse position to avoid inducing a corresponding sloshing of the cloud. The
ramp duration was chosen to be longer than ms so that adiabaticity with respect to the change of
transverse trap frequency is fulfilled. Axial dipole oscillations are suppressed by the
symmetric arrangement of the control wires.

We probe phononic excitations in the quasicondensate using a thermometry scheme based on
the analysis of density correlations in free expansion^{27,28}, as shown in the
inset of figure 2(a). To extract the temperature we compare the
measured density correlation functions with the results of a stochastic model^{29}. Our analysis accounts for the effects of the collective excitation on the free expansion
(see methods section below), and for the finite resolution of our imaging system.

Figure 2 summarises our temperature measurements following a quench.
We show data for ramp times of 10 and 30 ms and mean atom numbers of 11000 and 16000,
compared to the behaviour expected from a scaling model building upon the results of
Ref^{14}.

The scale invariance of the underlying Hamiltonian allows to calculate time-dependent
correlation functions: In the Thomas-Fermi regime, the density profile exhibits self-similar
scaling described bywith a time-dependent scale factor . Here, and denote the initial Thomas-Fermi radius and peak density, respectively,
is the Heaviside
function and *z* represents the axial coordinate. The scale factor obeys an
Ermakov-like equation^{31}Using the rescaled mean-field density (1), we can write
the linearised hydrodynamic equations for density and velocity fluctuations and , disregarding the quantum pressure term,
asandTo solve these equations we introduce an ansatz of rescaled eigenmodes for density
and phase fluctuations. This approach yields a set of uncoupled equations and hence no
mixing of modes, finally predicting an adiabatic time evolution of the corresponding
occupation numbers. For a thermal state, the initial phonon occupation numbers are given by
a Bose distributionAdiabaticity results in a constant ratio . The spectrum at is given by^{32} with mode index *l*
and initial sound velocity . For , it
scales as , due to the
time-dependence of the sound velocity and radius . Hence, for an initial state in thermal equilibrium, we obtain the
temperature scalingThe density correlations in free expansion that our thermometry scheme
relies on are governed by the coherence function. For a thermal state with homogeneous
density, as realised in the vicinity of the cloud center, it has the form^{20,32}:where denotes
the density at time
and the Boltzmann
constant. Based on our model, the coherence function is expected to scale as

Figure 3 summarizes the first central result of our experiments: The
inset shows absolute temperatures plotted against measured Thomas-Fermi radii. If the
measured temperatures are scaled to the initial temperature and plotted against the scale
parameter , the
datasets collapse onto a single line. This illustrates a scaling behaviour that is universal
in sense that it is independent of absolute temperature, density or quench time. To validate
our results we furthermore performed numerical simulations based on a stochastic
Gross-Pitaevskii equation (SGPE)^{33,34,35,36}, showing excellent agreement
with the scaling model (fig. 3).

So far, we considered the dynamics induced by a linear ramp of the trapping potential. In
the following, we demonstrate the conservation of phonon occupation numbers during shortcuts
to adiabaticity^{31,18,17} for the rapid expansion and compression of a 1d
quasi-BEC. To implement these shortcuts, we make use of an optimal control approach that is
in spirit similar to the method proposed in ref. 37. We
numerically solve the time-dependent 1d GPE with a suitable parametrisation of the trap
which is subject to a global optimization procedure based on a genetic algorithm^{38,39}. The ramp speed is limited by the requirement of adiabaticity in the
transverse degree of freedom. This constraint also guarantees that the gas remains in the 1d
hydrodynamic regime, and that the interaction strength varies slowly with time. The
properties of the ultracold gas therefore remain consistent with the conditions necessary
for the validity of the microscopic scaling laws^{14} throughout the ramp.

The upper panel in figure 4 shows a comparison between simulation and
experiment for a linear and a shortcut ramp performing a decompression within 30 ms from a
trap with frequencies
Hz and Hz to Hz and Hz. The subsequent
dynamics is observed throughout a period of 170 ms, each picture taken after a short free
expansion time of 5 ms, showing excellent agreement with simulations. It is interesting to
note that our shortcut ramps are similar to theoretical results derived from a
counter-diabatic driving method reported recently^{19}.

For the STA, we expect an adiabatic state change, defined by . The temperature measurements, corrected for the measured heating rate, are in good agreement with the adiabatic prediction of for the implemented decompression shortcut, confirming that there is no additional heating during the applied procedure.

## Conclusion

In summary, we have characterised the temperature of the phonon ensemble in a breathing quasi-1d Bose gas for different initial conditions, and used it to test the predicted dynamical scale invariance in the excitation spectrum of a quasi-1d Bose gas. Following these scaling laws, we have experimentally demonstrated rapid adiabatic expansion and compression of a 1d Bose gas in the hydrodynamic regime, allowing fast transformation of the trapped cloud without additional heating.

Our work is only the beginning for studies of many-body scaling solutions and shortcuts to
adiabaticity. The existence of scaling solutions has been proposed for a large class of cold
atom systems^{14}. In principle, this opens up the interesting possibility to
apply the techniques applied here to a variety of settings, such as fermionic systems or the
1d Bose gas with intermediate or strong interactions. We expect that studying the effect of
quasiparticle interactions on the implementation of shortcuts to adiabaticity will shed new
light on the complex many-body dynamics in these systems, in addition to providing novel
tools for their controlled manipulation.

We expect that such extensions to studies in regimes of greater interaction strength, and to systems out of thermal equilibrium, will benefit from the tools presented in this work.

## Methods

### Condensate preparation and detection

We employ standard cooling and magnetic trapping techniques^{40} to prepare
ultracold quasi-one-dimensional samples of 87Rubidium atoms in the state on an atom chip^{20,41}. Atom chips feature microfabricated wire structures to create fields for
atom trapping and manipulation^{42}. The structures used in our experiments
are produced by masked vapor depositon of a 2 *μm* gold layer on a silicon substrate,
with a width of both trapping and control wires of 200 *μm*. For detection, we employ
resonant absorption imaging^{43} using a high quantum-efficiency CCD camera
(Andor iKon-M 934 BR-DD) and a diffraction-limited optical imaging system characterised by
an Airy radius of 4.5 *μm*. The RF shield at 12 kHz above the bottom of the trap is
used to limit the number of atoms in the thermal background cloud populating transverse
excited states of the trap, which would otherwise adversely affect our thermometry scheme
by reduction of interference contrast in free expansion.

### Characterization of the breathing mode

We characterise the breathing mode excited by a linear trap frequency ramp from Hz to Hz, and Hz to Hz, respectively, in
figure 5. As an example, the upper panel shows the time evolution
of the cloud radius after a ramp with duration *τ* = 12.5 ms. Fitting data as
presented here allows us to extract frequencies, damping rates and amplitudes of the
breathing mode. The frequency is influenced by the total atom number in the trap, and is expected
to vary with the axial trap frequency between in the 1d limit, and representing the elongated 3d regime^{26}. The amplitude strongly depends on the duration and shape of the trap
frequency ramp. The lower panel in figure 5 shows a comparison of
measured breathing amplitudes for different ramp times between 2 ms and 100 ms with
results calculated with a 1d Gross-Pitaevskii equation (GPE), taking into account
corrections to the interaction term relevant in the 1d/3d crossover regime^{44}, and shows good agreement in the chosen parameter range.

### Thermometry

In this work we use the thermometry scheme proposed and demonstrated in ref. 28,27 based on the analysis of density
correlations in freely expanding phase-fluctuating quasi-1d condensates and comparison
with numerically calculated density profiles^{29}.

Breathing contributes a velocity field characterized by the derivative of the scale
parameter , leading
to an additional axial compression or expansion of the density profile during free
expansion. This effect can be accounted for by an additional phase factorin the
numerics, where *b* and are determined by fits to the measured breathing oscillations. The
error on the temperature measurements is estimated by a bootstrapping method as outlined
in ref. 30.

### Derivation of the temperature scaling

The general conditions for the existence of a scaling solution are stated in
reference^{14}. For the 1d Bose gas, they are fulfilled in the presence of
contact interactions, as well as a harmonic, linear or vanishing axial trapping potential.
Given that our system is a 1d quasicondensate, and the trapping potential is harmonic, we
can derive the corresponding hydrodynamic scaling relations for correlation functions. Our
starting point is the self-similar scaling of the density profile: denotes the Heaviside function, the initial
Thomas-Fermi radius and *b* the scale parameter. Similar to the discussion of the
corresponding equilibrium problem^{32}, a scaling solution in terms of
eigenmodes for density and velocity fluctuations and can be formulated asandwith the Legendre
polynomials , the
interaction constant *g*, rescaled coordinates , and time-dependent amplitudes and . denotes the frequency of the oscillation between the
quadratures of the mode *l*. Correspondingly, the initial equilibrium spectrum scales
asSubstituting
and into the
linearised Euler equationswhere we have disregarded the quantum pressure term,
yieldsSince the characteristic inverse time scale of the breathing mode is small compared to
the characteristic frequencies of the phonon modes with , we can average over rapid oscillations of and to reduce these
expressions to and
, and the phonon
modes are expected to scale adiabatically. Then the initial number of phonons in a thermal
state,is conserved, resulting inThis leads to the observed temperature scaling .

The decay of the coherence function of a quasicondensate is dominated by phase noise^{45}. We can express phase fluctuations in terms of velocity fluctuations using
the relationwhere

Therefore the relation between initial and time-dependent modes readsThe time-dependent
one-body reduced density matrix can be expressed aswithas well as and . Using , we can write the
density matrix (20) in terms of the modes given in equation (19).
Substituting and following the steps in reference^{45}, we find that near the
cloud center, where the density is practically uniform and we can use trigonometric
approximations for ^{46},with a coherence length . This corresponds to a transformation of
the formas predicted in reference^{14}, with the difference that the spatial
coordinates scale with instead of . This difference is a consequence of the the Thomas-Fermi approximation. In the
hydrodynamic regime, scale invariance therefore holds even if the interaction strength is
kept constant. In contrast, reference^{14} assumes a suitable tuning of the
interaction constant, thereby yielding an exact solution valid for arbitrary values of the
Lieb-Liniger parameter.

### Heating

The temperature scaling satisfies the equationIn our experiment we observe heating during evolution
times of several hundreds of milliseconds. We find that all our measurements are
compatible with a linear increase of temperature over time, which can be represented by
adding a constant heating term to the equation:This equation is solved by , with given by The
integral can be calculated numerically and *α* corresponds to the regular heating
rate in units of the initial temperature for constant *b*.

### Finite temperature simulations

We solve a stochastic 1d Gross-Pitaevskii equation (SGPE)^{33,34,35,36} of
the formwhereHere *μ* denotes an external chemical potential, and is a damping
coefficient that is coupled to the *δ*-correlated noise term *η* via a
fluctuation-dissipation theorem:Repeated solution of the SGPE yields a set of
independent wave functions representing a thermal state. We use this state as initial
condition for propagation with a time-dependent Gross-Pitaevskii Hamiltonian without any
noise or damping terms. Such an approach has previously been applied to model condensate
formation in atom chip traps^{47} and is very similar to other classical field
methods based on stochastic sampling of initial conditions^{48,49}. The
simulation results are analysed with the same procedures as the experimental data.

## Change history

### Updated online 13 July 2015

A correction has been published and is appended to both the HTML and PDF versions of this paper. The error has not been fixed in the paper.

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## Acknowledgements

We are grateful to M. Wilzbach, D. Heine and B. Hessmo for initial work building the experimental apparatus. We thank J-F. Schaff, N. Proukakis, P. Grišins and B. Rauer for fruitful discussions. This work was supported by the Austrian FWF through the Wittgenstein Prize, the FFG project PLATON, the Doctoral Programme CoQuS (W1210), and the EU through the projects QuantumRelax (ERC-ADG-320975) and SIQS. I.E.M. acknowledges the financial support from the FWF (project P22590-N16).

## Author information

## Affiliations

### Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria

- W. Rohringer
- , D. Fischer
- , F. Steiner
- , I. E. Mazets
- , J. Schmiedmayer
- & M. Trupke

### Ioffe Physical-Technical Institute of the Russian Academy of Sciences, 194021 St. Petersburg, Russia

- I. E. Mazets

### Wolfgang Pauli Institute, 1090 Vienna, Austria

- I. E. Mazets

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### Contributions

W.R. and D.F. performed the experiments and analysed the data. F.S. contributed to building the experiment and provided help with the data analysis. I.E.M. provided important advice for the execution of the scaling measurements and developed the theoretical model. W.R. devised the optimal control scheme and performed numerical simulations. J.S. and M.T. provided scientific guidance and funding for the experiment. All authors contributed to the interpretation of the data and the writing of the manuscript.

### Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to W. Rohringer.

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