# Emergent Pauli blocking in a weakly interacting Bose gas

Federica Cataldini$\dagger$1, Frederik Møller$\dagger$1, Mohammadamin Tajik1,
João Sabino1,2,3, Thomas Schweigler1,4, Si-Cong Ji1, Bernhard Rauer1,5 and
Jörg Schmiedmayer1 1 Vienna Center for Quantum Science and Technology (VCQ),
Atominstitut, TU Wien, Vienna, Austria
2 Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
3 Instituto de Telecomunicações, Physics of Information and Quantum
Technologies Group, Lisbon, Portugal
4 JILA, University of Colorado, Boulder, Colorado 80309-0440, USA
5 Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université,
Collège de France, 24 rue Lhomond, F-75231 Paris, France
$\dagger$ These authors contributed equally to this work

###### Abstract

We experimentally study the dynamics and relaxation of a single density mode
in a weakly interacting Bose gas trapped in a highly elongated potential.
Although both the chemical potential and thermal energy of the gas exceed the
level spacing of the transverse potential, we find that the observed dynamics
are accurately described by the one-dimensional, integrable theory of
generalized hydrodynamics. We attribute the absence of thermalization to the
suppression of three-dimensional excitations, as outgoing states of the
associated scattering processes obey fermionic statitistics, thus being
susceptible to the Pauli exclusion principle. The mechanisms effectively
preserves one-dimensionality, and hence integrability, far beyond conventional
limits.

The dynamics and relaxation of interacting quantum many-body systems are
highly complicated, often limiting theoretical descriptions to low-dimensional
systems. One-dimensional (1D) integrable systems offer a particularly
attractive platform, as many of their properties can be computed exactly via
the Bethe ansatz [1]. Its solutions are parameterized in terms of quasi-
momenta, the rapidities, encoding an extended set of conserved quantities. By
virtue of interactions and boundary conditions of the many-body wavefunction
only particular rapidities are possible and none can locally coincide,
effectively making them assume fermionic statistics [2]. The evolution of the
rapidities provides a description of the many-body dynamics where conservation
laws play a ubiquitous role [3, 4, 5].

(a)

(b)

Figure 1:  Illustration of dynamics. a, Geometric quench of the box trap
bottom. The gas is initialized in a 1D box trap whose bottom is shaped as the
first symmetric box eigenmode. At time $t=0$ the potential is quenched to a
flat box, initiating the dynamics of the gas. b, Schematic of emergent Pauli
blocking. Integrability breaking excitations in the transverse confining
potential can occur following collisions between quasi-particles with large,
opposite rapidities $|\theta_{2}-\theta_{1}|\geq\sqrt{8}/l_{\perp}$. The
process can be viewed as creating two particle-hole pairs with rapidities
$\theta_{1,2}^{-}$ and $\theta_{1,2}$. If all post-collision rapidities are
already occupied (the particle density $\rho_{p}$ is equal to the density of
states $\rho_{s}$), the transverse excitation can not occur by virtue of the
fermionic particle statistics.

Experimentally, a 1D Bose gas can be realized via a tight transverse
confinement whose level spacing exceeds all internal energy scales of the
system, effectively restricting it to the transverse ground state [6, 7, 8].
When these energy scales approach the transverse level spacing, excitations in
the transverse confinement are energetically possible and the system is often
referred to as quasi-1D [9]. Unlike integrable dynamics, these excitations are
not rapidity conserving and can lead to thermalization [10]. Thus,
experimental studies of dynamics in this regime are crucial for understanding
the extent at which quantum integrability survives in the presence of
perturbations.

Recent kinematic approaches offer a description of integrability-breaking
scattering processes consistent with Fermi’s golden rule [11, 12, 13, 14]. The
description permits a particular scenario owing to the fermionic nature of the
occupied rapidities: Following a scattering event, if all allowed rapidities
of the out-going state are already occupied, the process becomes Pauli
blocked. This mechanism would permit integrable dynamics to persist at much
longer time scales, and in the context of quasi-1D systems, enable one-
dimensionality to extend far beyond conventional energy scales [15].
Importantly, the mechanism is entirely different from hard-core bosons [16]
and dynamical fermionization [17], as it relies solely on the quantum
statistics of the occupied rapidities present even in weakly interacting Bose
gases.

To experimentally test the robustness of integrability, we realize a quasi-1D
weakly repulsively interacting Bose gas in a box trap. The chemical potential
and the thermal energy can be tuned to the order of, or exceeding, the
transverse level spacing. By controlling the shape of the bottom of the box
trap, we can imprint a density perturbation in the form of a single eigenmode
of the box. Following a sudden quench to a flat box, the imprinted mode
evolves and eventually relaxes, as illustrated in Fig. 1a. Reducing dynamics
to the evolution of a single density mode enables a much more detailed study
of the relaxation, making the setup an excellent probe for integrability
breaking effects.

While the dynamics of the gas following the quench is immensely complex on the
microscopic level, on large scales it exhibits emergent hydrodynamic behavior
described by the recent theory of Generalized Hydrodynamics (GHD) [18, 19].
For systems near an integrability point the theory provides a coarse grained
description for the dynamics, demonstrated by observations in 1D Bose gas
experiments [20, 21, 22]. Like the thermodynamic Bethe ansatz, in GHD the
thermodynamic properties of a local equilibrium macrostate are encoded in a
distribution of rapidities $\rho_{p}$ [23]. Occupied rapidities are considered
quasi-particles with an infinite lifetime, while unoccupied rapidities are
dubbed holes. Together they yield the local density of states $\rho_{s}$. GHD
treats dynamics as the hydrodynamic flow of quasi-particles. Their propagation
velocity is determined by a dressing of the rapidity distribution through
local interactions. Owing to these interactions, the emergent hydrodynamics of
the 1D Bose gas are very different from conventional hydrodynamics [24, 25].

In the box trap the GHD dynamics preserves the rapidity distribution, thus
inhibiting thermalization. However, if the energy of a particle collision
exceeds twice the transverse level spacing, a transverse excitation can occur
[22]. For such integrability-breaking scattering processes one can associate
the in and out states with particles and holes [11]. Here, the collision
creates two particle-hole pairs, where the rapidities of the particles are
much smaller than those of the holes, reflecting the gain in transverse
potential energy. The Pauli blocking can occur in systems either sufficiently
close to the many-body ground state or, as in our case, with a high chemical
potential. In our system all low rapidity states are filled, forming a Fermi
sea [2], leaving no states available for the out-going particles of the
transverse excitations (see Fig. 1b).

To achieve the detailed scenario, we realize a quasi-1D gas of ultracold
bosons (${}^{87}\mathrm{Rb}$ atoms) on an atom chip [26]. The chip produces a
highly elongated magnetic trap featuring a tight transverse confinement with
trapping frequency $\omega_{\perp}=2\pi\times 1.38\,\mathrm{kHz}$ and width
$l_{\perp}=\sqrt{\hbar/m\omega_{\perp}}=0.29\,\mu\mathrm{m}$. The chemical
potential of the gas is $\mu\approx 0.8\,\hbar\omega_{\perp}$ while the
interaction strength, characterized by the dimensionless Lieb-Liniger
parameter $\gamma$ [2], is around 0.002 placing us fairly deep within the
weakly interacting quasi-condensate regime. Heating and atom losses are
negligible [27].

By using a digital mirror device (DMD) we can create an arbitrary dipole
potential along the longitudinal axis of the trap [28]. We superimpose two
hard walls on the condensate, confining it to a region of
$L=80\,\mu\mathrm{m}$. Between the walls we generate two different potentials:
A flat potential and a sinusoidal potential shaped as a symmetric eigenmode of
the box. Adjusting the amplitude of the sinusoidal potential allows us to
address the corresponding condensate mode at different strengths. By
initializing the system in one configuration and then rapidly switching to the
other we realize a geometric quench instigating dynamics of the condensate.
Following the quench, we measure the density profile of the gas, averaged over
multiple repetitions, $n(z,t)$ at various evolution durations $t$ using
absorption imaging after $2\,\mathrm{ms}$ free expansion time.

(a)

(b)

(c)

Figure 2:  Observations following geometric quench. a, Measured time evolution
of the density perturbation $\delta n(z,t)=n(z,t)-\langle n(z,t)\rangle_{t}$
for three different quenches: Low temperature with small amplitude (left
column), high temperature with small amplitude (middle column), and high
temperature with large amplitude (right column). The mean atomic densities are
68, 60 and 82 $\mu\mathrm{m}^{-1}$, respectively. For the two hot
realizations, the trap bottom is switched from the lowest symmetric mode of
the box to a flat potential. For the cold realization the order of trap
configurations is reversed. For small quench amplitudes the evolution of the
density perturbation only differs by a sign for the two quench types (see
Supplemental Material). Here the sign for the cold realization is flipped for
easier comparison. The dashed lines mark the theoretical position of the hard
walls. b, Evolution of the five lowest density modes $\delta n_{j}(t)$
obtained via a Fourier decomposition. c, Evolution of the main addressed mode
$\delta n_{1}(t)$ with GHD theory comparison (1D and quasi-1D, mostly
overlapping). The time axes of the simulations have been scaled to reflect the
slightly longer effective box-lengths in the experiment.

Figure 2a shows the evolution of the density perturbation $\delta
n(z,t)=n(z,t)-\langle n(z,t)\rangle_{t}$ for three separate quenches, where
$\langle\cdot\rangle_{t}$ denotes the average over time. For each quench the
system is prepared in a thermal state. By tuning the efficiency of the cooling
we can adjust the temperature. We prepare one cold and two hot states, with
the thermal energy scale of the hot ones close to twice the transverse energy
gap. The density perturbation can be expressed as a sum of symmetric
eigenmodes of the box trap $\delta n(z,t)=\sum_{j=1}^{\infty}\delta
n_{j}(t)\cos\left(k_{j}z\right)$, where $\delta n_{j}(t)$ is the amplitude of
the mode and $k_{j}=2\pi j/L$. Figure 2b shows the mode decomposition of the
measured density perturbations. Only the five lowest modes are depicted, where
effects of the finite resolution of our imaging apparatus are limited. In all
three cases, the $j=1$ density mode has a far greater population than any
other mode, demonstrating that we indeed can address a single mode of the
quasi-condensate with high accuracy. The small population of the higher modes
can be attributed to small imperfections in the sinusoidal potential as well
as the thermal state being a non-linear function of the potential, meaning
even without imperfections one would slightly populate higher modes. The
background $j=0$ mode exhibits weak oscillations with the same frequency as
the $j=1$ mode, but with a $\pi$ phase-shift, indicating that the density
perturbation extends slightly further than the central region of
$80\,\mu\mathrm{m}$. This is due to the finite width of the experimentally
realized walls, whereby the perturbation oscillates in a box with a slightly
longer effective length.

Figure 2c shows the evolution of the measured $j=1$ density mode compared with
GHD predictions [29]. The theory is computed for a hard-walled box of length
$L=80\,\mu\mathrm{m}$. We observe a remarkably good agreement with 1D GHD,
despite the system (in particular the two hot quenches) having internal
energies far beyond the conventional conditions for one-dimensionality. The
transverse excitations in the quasi-1D regime can be included in the GHD
evolution via a Boltzmann collision integral [22]. Although derived in the
ideal Bose gas regime, the approach has been demonstrated to capture leading
order processes in the quasi-condensate regime. However, when accounting for
the transverse excitations, we hardly observe any change in the evolution of
the density mode. The reason is the vanishing density of holes in the Fermi
sea of rapidities causes the excitation terms of the collision integral to
drop out. Note, de-excitations of thermally excited atoms in the initial state
can still occur. These events can scatter particles out of the Fermi sea, thus
weakening its effect. For the result presented here, we assume no initial
transverse excitations. Including an estimated thermal population of excited
states does lead to slightly faster relaxation of the dynamics (see
Supplemental Material). However, the observed relaxation of the mode remains
dominated by integrable processes, as the emergent Pauli blocking
significantly prolongs the time-scale of thermalization. For a single evolving
mode, GHD provides a clear interpretation of the relaxation: At finite
temperature, the excited mode has a corresponding rapidity distribution.
Following the quench, each rapidity constituent of the mode evolves at a
different velocity, which leads to a gradual dephasing between them and causes
a damping of the density mode. Higher temperatures and greater quench
amplitudes increase the spread in rapidity of the mode, leading to faster
dephasing and thus relaxation (see Fig. 5 in the Supplemental Material).
Additional sources of relaxation in the form of hydrodynamic diffusion [30]
are negligible for the quenches explored here.

Figure 3:  Relaxation time-scale of the first density mode. For both
experimental measurements (points) and GHD simulations (shaded areas), the
time-scale $\tau$ is obtained by fitting the time evolution of the first
density mode $\delta n_{1}(t)$ with the damped oscillation of Eq. (1). The
dashed lines mark $k_{B}T=\hbar\omega_{\perp}$. The experimental results are
grouped into four ranges of mode amplitudes. For each range, three different
GHD simulations have been carried out: Standard 1D GHD (red), quasi-1D GHD
accounting for the fermionic particle statistics (cyan), and quasi-1D GHD with
a classical collision integral neglecting the statistics (blue). The GHD
results are plotted as shaded areas, whose top and bottom edges mark the
smallest and largest mode amplitude within the given range, respectively. All
simulations assume a mean atomic density of 75 $\mu\mathrm{m}^{-1}$, which
deviates by up to $20\%$ for some of the measurements. We account for the
difference by scaling the experimental relaxation time for $\tau$ according to
the relative difference in speed of sound. Additionally, the experimental
results are scaled to reflect the effective box-lengths of the individual
measurements.

The dephasing longitudinal dynamics and the thermalizing transverse dynamics
present two competing time-scales of relaxation in our system. To study the
scaling of the two mechanisms, we perform a number of experimental and
theoretical quenches for a wide range of temperatures and amplitudes and
extract the relaxation time-scale $\tau$ by fitting the evolution of the $j=1$
density mode with the function

$f(t)=Ae^{-(t/\tau)^{(3/2)}}\cos(\omega t+\phi)\,.$ (1)

We have deduced empirically that the exponent $3/2$ produces a good fit to
both simulated and measured modes. The results of our study are presented in
Fig. 3. For the purely 1D GHD, we observe a faster relaxation for both higher
temperatures and greater quench amplitudes, consistent with the greater spread
in rapidity of the initial thermal state. Very similar relaxation rates are
exhibited by the quasi-1D theory, indicating that thermalization occurs at
time-scales much slower than dephasing in the system. Only at temperatures
three times larger than the transverse level spacing do we observe signs of
transverse excitations. Between a total of thirteen different experimental
quenches performed, we consistently observe good agreement when comparing to
both of the two theories. This is indicative of the highly controlled manner
in which the state is prepared and quenched as well as a testament to the
robustness of the hydrodynamic description.

To demonstrate that the slow thermalization time-scale is indeed caused by
emergent Pauli blocking, we simulate the quasi-1D GHD employing a classical
collision integral neglecting the quantum statistics of the quasi-particles.
The resulting relaxation times presented in Fig. 3 are much faster and exhibit
much weaker dependence on temperature and amplitude than the both other
theories and experiment. This suggests a different mechanism of relaxation.
Indeed, by virtue of the high chemical potential of the condensate, the system
is deep in the quasi-1D regime for even the weakest quenches. Increasing the
temperature only speeds up relaxation slightly and is, in fact, mainly caused
by dephasing effects becoming relevant. The stark difference to the dynamics
observed in the experiment demonstrates the importance of accounting for the
fermionic quasi-particle statistics, even in a weakly interacting system.

It is important to stress that the observed prolonging of the thermalization
time-scale is inherently related to the quench protocol employed: The box trap
enables the Fermi sea to be established across the entire system. Further,
switching the potential shape mostly preserves the Fermi sea. For contrast, in
protocols like the quantum Newton’s cradle hardly any of the low rapidities
are occupied following the initial quench [31, 32]. Indeed, following a
cradle-like quench in another atom chip setup, clear signs of thermalization
were observed [20].

Finally, according to effective field theory models of the Luttinger-Liquid
type, the evolving density perturbation can be viewed as the propagation of a
phononic mode. However, at the large thermal energy scales in the experiment,
the lowest order quadratic Hamiltonian of such models is unable to capture the
observed relaxation [33]. A description could be feasible by including higher
order corrections, although this falls outside the scope of this work.

To conclude, we have demonstrated that the integrable GHD accurately describes
the dynamics of a Bose gas, whose chemical potential and thermal energy far
exceed conventional limits for one-dimensionality. We attribute the apparent
lack of thermalization to an emergent Pauli blocking of integrability-breaking
scattering processes, here in the form of transverse excitations in the trap.
The mechanism relies solely on the fermionic nature of the quasi-particles,
which arises from boundary conditions and interactions in the many-body wave
function. Thus, it would also be relevant to a number of other integrable
models that also feature fermionic quasi-particles, such as the XXZ chain [19]
and sinh-Gordon model [34].

##  I ACKNOWLEDGMENTS

We thank I. Mazets, S. Erne, I. Bouchoule, and J. Dubail for helpful
discussions. This work is supported by the DFG/FWF Research Unit FOR 2724
“Thermal machines in the thermal world”, and the FQXI program on ‘Informations
as fuel’ ESQ Discovery Grant ‘Emergence of physical laws: from mathematical
foundations to applications in many body physics’ of the Austrian Academy of
Sciences (ÖAW). F.C., F.M., and J. Sabino acknowledge support by the Austrian
Science Fund (FWF) in the framework of the Doctoral School on Complex Quantum
Systems (CoQuS). S.-C.J. acknowledges support by an Erwin Schrödinger Quantum
Science and Technology (ESQ) Fellowship funded through the European Union’s
Horizon 2020 research and innovation programme under Marie Skłodowska-Curie
grant 801110. J. Sabino acknowledges support by the Fundação para a Ciência e
a Tecnologia (PD/ BD/128641/2017). T.S. acknowledges support from the Max Kade
Foundation through a postdoctoral fellowship. B.R. acknowledges support by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 888707.

##  II AUTHOR CONTRIBUTIONS

F.C. performed the experiments and F.M. performed the theory calculations.
M.T., J. Sabino, S.-C.J., T.S. and B.R. contributed to the experiments and J.
Schmiedmayer conceived the experiment and supervised the project. All authors
contributed to the interpretation of the data and to the writing of the
manuscript.

##  Appendix A SUPPLEMENTAL MATERIAL

###  A.1 Setup

We follow standard protocols of magneto-optical trapping, laser cooling and
evaporative cooling to bring our system to the degeneracy temperature and
realise a quasi-condensate of 87Rb atoms. The atom chip generates a cigar-
shape magnetic trap where the final stage of evaporative cooling is performed.
In the presented measurements the one-dimensional potential has a fixed
trapping frequency in the tight (transverse) direction
$\omega_{\perp}=2\pi\times 1.38$ kHz.

The box potential is realised by superimposing on the elongated trap a dipole
potential generated by a blue-detuned laser of wavelength $660$ nm. The dipole
trap only affects the potential in the longitudinal direction and can be
arbitrarily designed by means of a digital micromirror device (DMD) [28]. A
simple scheme of the setup is illustrated in Fig.4. The great advantage of
using the DMD lies in its accuracy since it allows us to engineer complicated
pattern with a very small resolution: the DMD pixel size corresponds to
0.417$\mu$m on the plane of the atoms. In the presented experiments we
imprinted onto the atomic cloud hard walls at a distance of $L=80\mu$m from
each other, and we shaped the bottom of the trap either as a homogeneous
potential or a sinusoidal modulation with a specific $k=2\pi j/L$, with
$j=1,2$. In other words we can prepare the system either in a flat box
potential or in a symmetric eigenmode of the box itself. For every type of
trap we want to realize we implement an optimization procedure, where the DMD
pattern is iteratively adjusted until the observed density profile reproduces
the desired shape. Given the complex optimization process and experimental
limitations, the walls of the box are not perfectly sharp but have a finite
width which makes the effective box length slightly longer. To estimate the
walls width more precisely, we fit the density profile with two hyperbolic
tangent functions. The wall width in the present measurements ranges from
2$\mu$m to 5$\mu$m each. Moreover, we can tune with high accuracy the
amplitude of the cosine mode such that we can explore a wide range of
perturbation strengths. In our case the amplitude of the addressed mode varies
between 10$\%$ and 40$\%$ of the mean homogeneous density.

Figure 4:  Imprinting the dipole trap onto the atoms. The blue-detuned light
beam is sent through the DMD and superimposed on the magnetic trap generated
by the chip wires. Given the presence of optical elements between the DMD and
the condensate, an optimization process is required such that the effective
potential acting on the atoms matches the desired target potential. In the
picture two hard walls are imprinted on the atomic cloud. Here, x is the
direction of propagation of the dipole light, y is the vertical direction and
z is the longitudinal one. Figure adapted from [28].

###  A.2 Out of equilibrium protocol

For every experiment we need to optimize two potentials, corresponding to the
state of the system before and after the quench. For most of the measurements
presented in the main text, the thermal state is prepared in the eigenmode
potential and subsequently quenched to a flat box. The switch between the two
potentials occurs in a few microseconds and it is much faster than the
timescale of the macroscopic dynamics of the system. To keep track of the
dynamics we extract the density of the quasi-condensate at different evolution
times via absorption picture, recorded after 2 ms of time-of-flight. Due to
the strong confinement in the transverse direction, once the atoms are
released from the trap interactions in the gas drop down on a time scale much
shorter than the considered TOF. While the condensate greatly expands
transversely, the expansion in the longitudinal direction is so small that the
average density profile after 2 ms safely reproduces the in situ density
distribution.
During the 160 ms of evolution the atom loss rate is about 2 atoms/ms; it
arises from three-body recombination, collisions with the background gas
particles and technical noise. The heating of the cloud during this long
evolution time is negligible.

##  Appendix B Data analysis

For each measurement the described protocol is repeated several times, usually
between 100 and 200 realizations for each data set. Hence for all the relevant
observables we compute the statistical average. A post-selection process is
applied to the atom number in order to have, at every time step, a variation
$\Delta N$ of about (10-15)$\%$ around the mean value.
Thermometry. The temperature of the gas in the thermal state can be estimated
from the analysis of the density ripples emerging after a long time-of-flight
[35]. Hence for the initial state, together with the density profile after 2ms
of TOF, we record the density ripples pattern after 11.2 ms of free expansion.

##  Appendix C Thermodynamic Bethe ansatz of the Lieb-Liniger model

The degenerate gas of $N$ bosonic atoms of mass $m$ trapped in a highly
elongated confinement is described by the second-quantized Hamiltonian

$\hat{H}=\int
d\mathbf{r}\left\\{\frac{\hbar^{2}}{2m}\left(\nabla\hat{\Psi}^{\dagger}\right)(\nabla\hat{\Psi})+\left[U(z)+V_{\perp}(x,y)\right]\hat{\Psi}^{\dagger}\hat{\Psi}+\frac{2\pi
h^{2}a_{s}}{m}\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger}\hat{\Psi}\hat{\Psi}\right\\}$
(2)

where $\hat{\Psi}=\hat{\Psi}(\mathbf{r})$ is the atom annihilation operator,
$a_{s}$ is the s-wave scattering length, $U(z)$ is the loose trapping
potential in the longitudinal direction, and $V_{\perp}(x,y)$ is the tight
transverse trapping potential. We assume that $V_{\perp}(x,y)$ is harmonic and
axially symmetric, $\omega_{\perp}$ being its fundamental frequency and
$l_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$ being the corresponding length
scale.

Upon confining the system to 1D, Eq. (2) reduces to the seminal Lieb-Liniger
Hamiltonian, which in the absence of the longitudinal potential is exactly
solvable via the Bethe ansatz [2]. In 1D, the interaction strength is
parameterized by the parameter $\gamma=mg/\hbar^{2}n$, where
$g=2\hbar\omega_{\perp}a_{s}\left(1-1.03a_{s}/l_{\perp}\right)^{-1}$ is the 1D
contact interaction and $n$ is the 1D atomic density.

Following the thermodynamic Bethe ansatz, for finite temperatures the local
thermodynamic state of the Lieb-Liniger model can be fully encoded in a
density of quasi-particles $\rho_{p}(\theta)$, with each quasi-particle
uniquely labeled by its rapidity $\theta$ [23]. Similarly, one can introduce a
density of unoccupied rapidities, or holes, $\rho_{h}(\theta)$ and the density
of states $\rho_{s}(\theta)$ obeying the relation

$\rho_{s}(\theta)=\rho_{p}(\theta)+\rho_{h}(\theta)=\frac{1}{2\pi}+\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>\varphi(\theta-\theta^{\prime})\rho_{p}(\theta^{\prime})\;,$
(3)

with the differential two-body scattering phase given by

$\varphi(\theta-\theta^{\prime})=\frac{2mg/\hbar}{(mg/\hbar)^{2}+(\theta-\theta^{\prime})^{2}}\;.$
(4)

Given the quasi-particle distribution, one can compute local thermodynamic
expectation values of all conserved quantities of the system. In our case, we
are only interested in the expectation value of the atomic density, which is
given by

$n=\int_{-\infty}^{\infty}\mathrm{d}\theta\>\rho_{p}(\theta)\;.$ (5)

Further, one can introduce the filling function
$\vartheta(\theta)=\rho_{p}(\theta)/\rho_{s}(\theta)$ describing the fraction
of allowed rapidities occupied. Since the Bethe ansatz quasi-particles of the
Lieb-Liniger model obey fermionic statistics, a thermal state can be
calculated following

$\vartheta(\theta)=\frac{1}{1+e^{\epsilon(\theta)\beta}}\;,$ (6)

where $\beta=1/k_{B}T$ is the inverse temperature and the pseudo-energy
$\epsilon(\theta)$ is given by the relation

$\epsilon(\theta)=\frac{\hbar^{2}\theta^{2}}{2m}-\mu+\frac{1}{2\pi\beta}\int_{\infty}^{\infty}\mathrm{d}\theta^{\prime}\>\varphi(\theta-\theta^{\prime})\ln\left(1+e^{\epsilon(\theta^{\prime})\beta}\right)\;.$
(7)

In the presence of an external potential $U(z)$ one can locally shift the
chemical potential accordingly $\mu(z)=\mu_{0}-U(z)$ under the local density
approximation.

##  Appendix D GHD of the Lieb-Liniger model

In GHD, the system is treated as a continuum of mesoscopic fluid cells in
space-time, each in local equilibrium. The macroscopic flow between
neighboring fluid cells occurs at a rate slower than the local microscopic
relaxation, such that local thermodynamic equilibrium in the fluid cells is
maintained at all times. Thus, each fluid cell is characterized by the
thermodynamic Bethe ansatz resulting in a time and space dependent quasi-
particle distribution $\rho_{p}(\theta,z,t)$ [18, 19]. Note, for brevity we
omit all spacial and temporal arguments in the following.

Hence, the complicated dynamics of the interacting Bose gas can be solved
using a single continuity equation for the analogous system of quasi-particles

$\partial_{t}\rho_{p}+\partial_{z}(v^{\mathrm{eff}}\rho_{p})-\hbar^{-1}\partial_{\theta}(\partial_{z}U\,\rho_{p})=0\;,$
(8)

where $U(z)$ is the longitudinal potential and $v^{\mathrm{eff}}(\theta)$ is
the propagation velocity of a quasi-particle with rapidity $\theta$

$v^{\mathrm{eff}}(\theta)=\frac{\hbar\theta}{m}+\int_{-\infty}^{\infty}d\theta^{\prime}\varphi(\theta-\theta^{\prime})\rho_{p}(\theta^{\prime})\left[v^{\mathrm{eff}}\left(\theta^{\prime}\right)-v^{\mathrm{eff}}(\theta)\right]\,.$
(9)

The propagation velocity encodes the Wigner delay time associated with the
quantum mechanical phase shifts occurring upon elastic collisions of the
interacting atoms.

##  Appendix E Transverse excitations and collision integral

In the quasi-1D regime excited states of the transverse confinement
$V_{\perp}(x,y)$ can be populated through high energy collisions. We restrict
our treatment to the three lowest states of the transverse potential and
assume that a majority of atoms remain in the transverse ground state, whereby
the transverse state of a quasi-particle can be treated as a pseudo-spin
degree of freedom [36, 37]. By virtue of parity two possible excitation (and
de-excitation) events are possible: (i) Two atoms in the ground state collide
and both are excited to the first excited state, or (ii) two atoms in the
ground state collide and one is excited to the second excited state. Thus, the
total collision energy must exceed $2\hbar\omega_{\perp}$ for transverse
excitations to occur.

In the quasi-particle framework of integrable models, the collisions can be
identified as scattering processes whose in an out states are two holes and
two particles, respectively. The $2\hbar\omega_{\perp}$ gain in transverse
potential energy following an excitation is reflected in much lower rapidities
of the particles compared to the holes, as the single-particle energy is given
by $\varepsilon(\theta)=\hbar^{2}\theta^{2}/2m$.

In response to the creation of holes or particles in an interacting integrable
model, all other rapidities are locally rearranged causing a shift in both the
total energy and momentum described by the so-called backflow function [38].
Thus, the total collision energy is the sum of the single-particle energies of
the colliding particle and the energy shift following the backflow. As a
result, the rapidities of the in- and out-going scattering states are shifted,
and the collision integral is perturbed [11]. However, the fermionic
statistics must still be respected.

In the ideal Bose gas phase of the Lieb-Liniger model, where the collision
integral of Ref. [22] originally was derived, the backflow is negligible.
Thus, the in-going rapidities of the scattering events must simply fulfill the
relation $|\theta_{2}-\theta_{1}|\geq\sqrt{8}/l_{\perp}$ for a transverse
excitation to be energetically possible. In the quasi-condensate regime
contributions from the backflow may start becoming relevant, however, the
”bare” (ideal Bose gas) collision integral should still represent leading
order processes. Indeed, applying the ”bare” collision integral to the
experimental results of Ref. [20] (which featured a system comparable to ours)
resulted in a better agreement with observations than purely 1D GHD [22].

For the ideal Bose gas, the collision integral reads

$\mathcal{I}(\theta)=\sum_{n=1}^{2}\frac{1}{2}\left(\left[\mathcal{I}_{h}^{+}(\theta)-\mathcal{I}_{p}^{-}(\theta)\right]+\left[\mathcal{I}_{h}^{-}(\theta)-\mathcal{I}_{p}^{+}(\theta)\right]\nu_{n}^{\beta_{n}}\right)$
(10)

where $\nu_{n}$ is the probability for an atom to be in the $n$’th transverse
excited state, and $\beta_{1}=2$ and $\beta_{2}=1$ are the number of atoms
changing state via the collisions. The terms in the first square brackets
describe the change in rapidities following transverse excitations, while the
second square brackets hold the terms describe de-excitations. The terms are
defines as

$\mathcal{I}_{\alpha}^{\pm}(\theta)=\frac{(2\pi)^{2}\hbar}{m}\int_{\mathcal{R}_{\pm}}d\theta^{\prime}\,|\theta-\theta^{\prime}|P_{\updownarrow}(|\theta-\theta^{\prime}|,\,|\theta_{\pm}-\theta^{\prime}_{\pm}|)\;\rho_{\alpha}(\theta)\rho_{\alpha}(\theta^{\prime})\rho_{\bar{\alpha}}(\theta_{\pm})\rho_{\bar{\alpha}}(\theta^{\prime}_{\pm})\;,$
(11)

where $\bar{\alpha}=h$ for $\alpha=p$ and vice versa,
$P_{\updownarrow}(\theta_{1},\,\theta_{2})=4c^{2}\theta_{1}\theta_{2}/[\theta_{1}^{2}\theta_{2}^{2}+c^{2}(\theta_{1}+\theta_{2})^{2}]$
is the scattering probability with $c=2a_{s}/l_{\perp}^{2}$, while
$\theta_{\pm}=\frac{1}{2}(\theta+\theta^{\prime})+\frac{1}{2}(\theta-\theta^{\prime})\sqrt{1\pm
8/[(\theta-\theta^{\prime})l_{\perp}]^{2}}$ and
$\theta_{\pm}^{\prime}=\frac{1}{2}(\theta+\theta^{\prime})-\frac{1}{2}(\theta-\theta^{\prime})\sqrt{1\pm
8/[(\theta-\theta^{\prime})l_{\perp}]^{2}}$ are the rapidities after a
collision leading to excitation or de-excitation of the transverse states,
respectively. The integration ranges in Eq. (11) are the following:
$\mathcal{R}_{+}$ is the whole real axis, and $\mathcal{R}_{-}$ is comprised
of those real values of $\theta^{\prime}$, which yield real $\theta_{-}$ and
$\theta_{-}^{\prime}$, i.e.
$\mathcal{R}_{-}=\\{\theta^{\prime}:\theta^{\prime}<\theta-\sqrt{8}/l_{\perp}\\}\cup\\{\theta^{\prime}:\theta^{\prime}>\theta+\sqrt{8}/l_{\perp}\\}$.
Neglecting any heating effects, the excitation probabilities $\nu_{n}(t)$
follow the simple rate equations

$\frac{d\nu_{n}}{dt}=\zeta_{n}\beta_{n}\left[\Gamma_{h}^{+}-\Gamma_{p}^{+}\nu_{n}^{\beta_{n}}\right],$
(12)

where
$\Gamma_{\alpha}^{+}=(2N)^{-1}\int_{-\infty}^{\infty}dz\int_{-\infty}^{\infty}d\theta\,\mathcal{I}_{\alpha}^{+}(\theta)$,
$\alpha=p,\,h$.

In the case where all allowed rapidities are occupied
$\rho_{p}(\theta)=\rho_{s}(\theta)$, the density of holes vanishes for most
incoming rapidities $\theta$ and $\theta^{\prime}$ resulting in
$\Gamma_{h}^{+}\approx 0$, meaning no transverse excitations can occur.
Meanwhile, for a non-degenerate 1D Bose gas where the quasi-particle
statistics become insignificant, and we let
$\rho_{p}(\theta)\ll\rho_{h}(\theta)\approx 1/(2\pi)$ in Eq. (11), whereby the
collision integral in Eq. (10) takes the classical (Boltzmann) limit.

##  Appendix F Solving the GHD equations in a box

For a box of length $L$, centered on $z=0$, the hard walls results in the
following boundary conditions

$\displaystyle\rho_{p}(z=-L/2,\theta)$
$\displaystyle=\rho_{p}(z=-L/2,-\theta)$ (13a)
$\displaystyle\rho_{p}(z=L/2,\theta)$
$\displaystyle=\rho_{p}(z=L/2,-\theta)\;,$ (13b)

which can be interpreted as the quasi-particles having their rapidity
reflected upon colliding with the walls. Here we modify the solution by
characteristics method featured in the iFluid numerical framework [29] in
order to account for the rapidity reflection. Within iFluid, one propagates
the filling function $\vartheta$, rather than the quasi-particle density
itself. Thus, the propagation step $t\to t^{\prime}$ is performed using the
implicit solution by characteristics

$\vartheta(t^{\prime},z,\theta)=\vartheta(t,\mathcal{U}(t^{\prime},t),\mathcal{W}(t^{\prime},t))\;,$
(14)

where the position and rapidity characteristics are given by

$\displaystyle\mathcal{U}(t^{\prime},t)$
$\displaystyle=z-\int_{t}^{t^{\prime}}\mathrm{d}\tau\,v_{\tau}^{\mathrm{eff}}\left(\mathcal{U}(\tau,t),\mathcal{W}(\tau,t)\right)$
(15) $\displaystyle\mathcal{W}(t^{\prime},t)$
$\displaystyle=\theta-\int_{t}^{t^{\prime}}\mathrm{d}\tau\,a_{\tau}^{\mathrm{eff}}\left(\mathcal{U}(\tau,t),\mathcal{W}(\tau,t)\right)\;,$
(16)

respectively. Here, the subscript $\tau$ denotes the dependence on the state
at said time. Further, $a_{\tau}^{\mathrm{eff}}=-\hbar^{-1}\partial_{z}U(z)$
for box-potentials with a non-flat bottom. For the numerical simulation the
time axis was discretized in steps of $\Delta t$ and the characteristics were
approximated to first order. Note, we omitted the dependencies on $z$ and
$\theta$ for a more compact notation.

We can account for the rapidity reflection due to the hard walls by imposing
the following boundary conditions on the rapidity characteristic

$\mathcal{W}_{\mathrm{box}}(z,\theta,t^{\prime},t)=\begin{cases}-\mathcal{W}&\text{for
$\mathcal{U}-L/2>0$}\\\ -\mathcal{W}&\text{for $\mathcal{U}+L/2<0$}\\\
\mathcal{W}&\text{otherwise.}\end{cases}$ (17)

Here, the first line captures collision with the right wall at $z=L/2$, and so
on. Further, for finite step sizes, the additional travel distance of a quasi-
particle after reflection off a wall can be accounted for via

$\mathcal{U}_{\mathrm{box}}(z,\theta,t^{\prime},t)=\begin{cases}L-\mathcal{U}&\text{for
$\mathcal{U}-L/2>0$}\\\ -L-\mathcal{U}&\text{for $\mathcal{U}+L/2<0$}\\\
\mathcal{U}&\text{otherwise.}\end{cases}$ (18)

After imposing the boundary conditions on the characteristics, the state can
be propagated following Eq. (14).

##  Appendix G Relaxation of the mode via dephasing

In a hard-walled box the Euler-scale GHD equation preserves the rapidity
distribution, whereby no thermalization can occur. Instead, the system relaxes
through dephasing to a non-thermal stationary state, which can be described by
a generalized Gibbs ensemble. Only in the presence of integrability breaking
mechanisms can thermalization happen, whereby the equilibrium state is
characterized by the conventional Gibbs ensemble.

To illustrate how the relaxation of a single mode in a hard-walled box-trap
occurs via dephasing, we employ a linearized version of GHD [39]. First we
split the time-dependent filling function into a stationary background and an
evolving perturbation
$\vartheta(\theta,z,t)=\underline{\vartheta}(\theta)+\delta\vartheta(\theta,z,t)$.
The background can be identified as the zeroth order mode of the filling,
while the perturbation contains all higher modes. If
$\delta\vartheta\ll\underline{\vartheta}$, we can neglect interactions during
evolution within the perturbation itself and only treat the interactions
between the perturbation and the stationary background. Thus, the GHD equation
in the hard-walled box can be simplified as (neglecting the walls of the box
for now)

$\partial_{t}\delta\vartheta(\theta,z,t)+v_{\underline{\vartheta}}^{\mathrm{eff}}(\theta)\,\partial_{z}\delta\vartheta(\theta,z,t)=0\;.$
(19)

Here, the effective velocity is computed using only the background state, as
signified by the subscript. Since the background state, and by extension the
velocity field, is homogeneous each Fourier mode evolves independently.
Indeed, plugging a single mode
$\delta\vartheta_{k}(\theta,t)=\delta\vartheta(\theta,z,t)e^{ikz}$ into Eq.
(19) yields

$\partial_{t}\delta\vartheta_{k}(\theta,t)+ikv_{\underline{\vartheta}}^{\mathrm{eff}}(\theta)\delta\vartheta_{k}(\theta,t)=0\;,$
(20)

which has the time-dependent solution

$\delta\vartheta_{k}(\theta,t)=\delta\vartheta_{k}(\theta,0)\,e^{-iktv_{\underline{\vartheta}}^{\mathrm{eff}}(\theta)}\;.$
(21)

From Eq. (21) we can understand how a single mode $k$ can dephase, leading to
the apparent relaxation of the system. At finite temperature, the initial
perturbation $\delta\vartheta_{k}(\theta,0)$ is a distribution of occupied
rapidities determined by the thermodynamic Bethe ansatz. Upon evolution of the
perturbation, each of its rapidity constituents will evolve at different rates
given by the corresponding effective velocity
$v_{\underline{\vartheta}}^{\mathrm{eff}}(\theta)$. Importantly, the effective
velocity is a continuous, monotonically increasing function of rapidity,
whereby no two rapidity constituents of the mode can evolve at the same rate,
ultimately leading to dephasing between the individual rapidity components, as
illustrated in Fig. 5. Thus, the relaxation time-scale for a given mode $k$ is
determined by its spread in rapidity. As temperature increases, higher
rapidities will become increasingly occupied leading to a larger variance in
propagation velocity of the modes constituents. Likewise, a greater
perturbation will also feature a larger spread in rapidity.

(a)

(b)

(c)

Figure 5:  Dephasing of rapidities. a, Evolution of the
$\delta\vartheta_{1}(\theta,t)$ perturbation of the filling following Eq. (21)
for two of the quenches presented in the main text. Both quenches have similar
amplitudes, whereby the difference in rapidity spread of the perturbation is
determined by temperature. Note, the filling is symmetric in rapidity and only
the positive rapidities are plotted. b, Same evolution as plotted in a but
only for select rapidities, highlighting their different oscillation rates. c,
Evolution of the $j=1$ density perturbation for reference. As the different
rapidities of the filling function dephase, the amplitude of the density mode
decreases.  Figure 6:  Relaxation time-scale of the first density mode (1D
GHD). Time-scale $\tau$ obtained by fitting the first density mode $\delta
n_{1}(t)$ obtained via 1D GHD simulations with the damped oscillation of Eq.
(1). The fitted time-scale is shown as a function of temperature $T$ and mode
amplitude $A$.

##  Appendix H Damping in the Luttinger Liquid theory

Another approach in the description of the dynamics of one-dimensional many-
body systems is provided by mean-field theories. For a weakly interacting 1D
quasi-condensate, an extension of the Bogoliubov theory has been made [40]. In
this framework the field operator can be expressed in the phase-density
representation as:

$\hat{\Psi}(z,t)=e^{i\hat{\phi}(z,t)}\sqrt{n_{0}(z)+\delta\hat{n}(z,t)},$ (22)

where $\hat{\phi}$ and $\delta\hat{n}$ are conjugated variables, the first
being the fluctuating phase of the condensate and the latter the density
fluctuations relative to the background linear density $n_{0}(z)$. The low-
energy description of the 1D dynamics is provided by the Luttinger liquid
model, whose Hamiltonian is quadratic with respect to $\hat{\phi}$ and
$\delta\hat{n}$:

$\hat{H}_{LL}=\int
dz\Big{[}\frac{g}{2}\delta\hat{n}^{2}+\frac{\hbar^{2}n_{0}}{2m}(\partial_{z}\hat{\phi})^{2}\Big{]},$
(23)

here $g$ is the one-dimensional coupling constant, and $m$ the mass of 87Rb
atoms. Solving the problem in a box potential of length $L$ implies that
Neumann boundary conditions have to be fulfilled and therefore allows for an
expansion of the phase and the density fluctuations in terms of their cosine
modes, whose wave number is $k=\frac{\pi}{L}j$, with $j=1,2,3,...$

$\delta\hat{n}(z)=\sqrt{\frac{2}{L}}\sum_{k}\delta\hat{n}_{k}\cos(kz),\quad\quad\hat{\phi}(z)=\sqrt{\frac{2}{L}}\sum_{k}\hat{\phi}_{k}\cos(kz).$
(24)

Furthermore the Hamiltonian can be diagonalised by defining for each mode a
creation and annihilation operator $\hat{b}_{k}^{\dagger}$, $\hat{b}_{k}$,
such that the mode amplitudes can be expressed as a plane-wave of frequency
$\omega_{k}=ck$

$\displaystyle\delta\hat{n}_{k}=\sqrt{\frac{n}{2}\frac{\varepsilon_{k}}{\mu}}\Big{(}\hat{b}_{k}e^{-i\omega_{k}t}+\hat{b}_{k}^{\dagger}e^{i\omega_{k}t}\Big{)},$
(25)
$\displaystyle\hat{\phi}_{k}=-i\sqrt{\frac{1}{2n_{0}}\frac{\mu}{\varepsilon_{k}}}\Big{(}\hat{b}_{k}e^{-i\omega_{k}t}-\hat{b}_{k}^{\dagger}e^{i\omega_{k}t}\Big{)}.$
(26)

After introducing $\hat{b}_{k}^{\dagger}$ and $\hat{b}_{k}$ the Hamiltonian
takes the form

$\hat{H}_{LL}=\sum_{k}\hbar\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}.$ (27)

The energy of the k-mode is therefore $\varepsilon_{k}=\hbar\omega_{k}$ and we
recall that the dispersion relation is linear with respect to the speed of
sound $c=\sqrt{gn_{0}/m}$:

$\omega_{k}=ck=c\frac{\pi}{L}j,\quad\quad j=1,2,3,....$ (28)

From eq. (27) and (28) it is clear that in this picture the low-energy
dynamics is governed by the propagation of non-interacting phononic
excitations, which, as expected for integrable models, are supposed to live
forever.
The experimental damping observed in our experiment is therefore in striking
contrast with the theoretical prediction coming from the Luttinger liquid
model. While the oscillation frequencies from fig. 9c are compatible with the
expected $\omega_{k}$, suggesting that the dynamics is certainly dominated by
the propagation of the imprinted phononic mode, the origin of the damping is
hard to understand in this framework. Given the statistical nature of our
measurements, the only source of damping comes from the shot to shot density
fluctuations that leads to a variation in the speed of sound. As mentioned
before, the heating of the system over time as well as the global atom loss
due to three-body recombinations and other technical noise (about 2 atoms/ms),
are too small to play a significant role in the dynamics. In other words,
assuming that the phononic modes are non-interacting, the only explanation of
the damping would be a dephasing arising from the atom number flucutation,
since at every shot the excited mode would travel at a slightly different
speed of sound. The larger the deviation from the mean atom number, the faster
the damping.
To quantify the effect of the statistics on the dynamics, we compute the
damping from the same measurement in two different configurations. First
assuming a variation in the atom number $\Delta N$ of about 20$\%$ around the
mean N = 5700, and afterwords by post selecting a much tighter distribution
with about $3\%$ of variation around the same N. We compare the experimental
results with the Luttinger liquid prediction. The results are presented in
Fig.7a. The theoretical prediction clearly shows a significant difference
between the two configurations, while surprisingly the damping that emerges
form the experimental data does not seem to be affected by the width of the
atom number distribution. An additional study can be made by considering other
two data sets: from the same measurement we now consider the same atom number
variation, but centered around the tails of the original atom number
distribution. This leads to a different oscillation frequency between the two
cases but similar damping, as shown in Fig.7b. This suggests again that the
quadratic Luttinger-Liquid theory is not feasible to explain the observed
dynamics in the explored range of temperature and perturbation strength.

Figure 7:  Comparison of the damping between experimental data and Luttinger
liquid theory. (a) Comparison of the damping at two different atom number
variation. We extract the mode oscillation for two different atom number
distribution around the same mean N = 5700 atoms. The continuous lines
represents the theoretical damping deriving from the Luttinger liquid model,
while the dots points are the experimental data. The blue data set consider a
variation of the $\Delta N=3\%$ around N, while the red one correspond to
$\Delta N=20\%$. In the first case (blue data set) the LL damping comes from
the dephasing of the phononic mode due to a large variation in the speed of
sound. In the red data set instead the damping is essentially absent. In
striking contrast, the experimental damping is unchanged. (b) Comparison of
the damping between two different mean atom number. We consider two data sets.
The first (in red) is centered around N = 4650 atoms, while the second set (in
blue) is centered around N = 6781 atoms. In both cases $\Delta N=6\%$ of N. In
this case one can notice a different frequency in the two oscillations, which
is consistent with the different mean N. Again there is a considerable
difference between the theoretical and the experimental damping.

##  Appendix I Extended data

We present an additional summary plot of all the measurements where the
dynamics of the first symmetric mode was studied. Further the data are
extended to experiments where the second symmetric eignemode was addressed.
Finally, we show that in the explored regime of temperature and mode
amplitude, a quench from an eigenmode to a flat box is symmetric to the
protocol where the two potential are swapped.

Nr. | 1 | 2 | 3 | 4 | 5 |  |  |
---|---|---|---|---|---|---|---|---
$T$ (nK) | 46(4) | 48(8) | 64(10) | 68(8) | 120(10) |  |  |
$\langle n\rangle$ ($\mu\mathrm{m}^{-1}$) | 68(2) | 69(2) | 62(2) | 72(2) | 83(2) |  |  |
$A\;(\mu\mathrm{m}^{-1})$ | 7.05(37) | 11.51(52) | 9.15(76) | 25.04(1.10) | 25.69(1.35) |  |  |
Nr. | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
$T$ (nK) | 120(8) | 88(12) | 126(21) | 132(16) | 124(13) | 180(22) | 154(18) | 192(19)
$\langle n\rangle$ ($\mu\mathrm{m}^{-1}$) | 60(1) | 74(1) | 84(2) | 72(1) | 82(2) | 74(1) | 83(1) | 77(1)
$A\;(\mu\mathrm{m}^{-1})$ | 8.38(57) | 14.21(84) | 15.91(76) | 26.88(98) | 24.87(97) | 22.42(2.97) | 34.58(1.39) | 34.44(1.80)

Figure 8:  Relaxation time-scale of the first density mode. For both
experimental measurements (points) and GHD simulations (shaded areas), the
time-scale $\tau$ is obtained by fitting the time evolution of the first
density mode $\delta n_{1}(t)$ with the damped oscillation of Eq. (1). Four
different GHD simulations have been carried out: Standard 1D GHD, quasi-1D GHD
assuming no initial transverse excitations, quasi-1D GHD with an estimated
thermal population of excited states, and quasi-1D GHD with a classical
collision integral. The extend of the shaded areas reflect the measurement
uncertainties in temperature and mode amplitude (see the table). The
uncertainty on the experimental data points is the $95\%$ confidence interval
from the fit with Eq. (1). All simulations assume a mean atomic density of 75
$\mu\mathrm{m}^{-1}$ in the box, while the mean atomic densities between the
walls of the experimental realizations $\langle n\rangle$ can be found in the
table. We account for the difference by scaling the experimental relaxation
time for $\tau$ according to the relative difference in speed of sound.
Additionally, the experimental results are scaled to reflect the effective
box-lengths of the individual quenches.

(a)

(b)

(c)

Figure 9:  Observations following geometric quench of the second symmetric
mode. a, Measured time evolution of the density perturbation $\delta
n(z,t)=n(z,t)-\langle n(z,t)\rangle_{t}$ for three different quenches. The
mean atomic densities are 55, 63 and 65 $\mu\mathrm{m}^{-1}$, respectively. In
each quench the trap bottom is switched from the second symmetric mode of the
box to a flat potential. The dashed lines mark the theoretical position of the
hard walls. b, Evolution of the five lowest density modes $\delta n_{j}(t)$
within the walls for each of the quenches. c, Evolution of the main addressed
mode $\delta n_{2}(t)$ with GHD theory comparison (both 1D and quasi-1D, here
fully overlapping). The time axes of the GHD simulations have been scaled to
reflect the slightly longer effective box-lengths of the experiment.

Figure 10: Symmetry of geometric quench -  (a, b, c): GHD simulations for a
system of $N=5470$ atoms at a temperature of $T=46\,\mathrm{nK}$. a, Evolution
of density perturbation $\delta n(z,t)$ following a quench from the first
symmetric eigenmode of the box trap to a flat potential between the hard
walls. b, Evolution following the reversed quench order. c, Evolution of the
$j=1$ density mode. The blue curve corresponds to the quench in a. The red
curve corresponds to the quench in b but with a flipped sign. (d, e, f):
Experimental data at high temperature. Here the cloud has on average $N=7800$
atoms and the initial state has a temperature $T=152$ nK. In the two 2D plots
is presented the evolution of the density perturbation $\delta n(z,t)$ after a
quench from the first symmetric mode to flat box d, and viceversa e. Figure f
shows the evolution of the excited mode $\delta n_{1}(t)$. The blue data
points correspond to the quench in d, while the red ones correspond to the
quench in e but with opposite sign.

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