# Characterising transport in a quantum gas by measuring Drude weights

Philipp Schüttelkopf<EMAIL_ADDRESS>Mohammadamin Tajik
Nataliia Bazhan  Federica Cataldini  Si-Cong Ji  Jörg Schmiedmayer  Frederik
Møller<EMAIL_ADDRESS>Vienna Center for Quantum Science and
Technology (VCQ), Atominstitut, TU Wien, Vienna, Austria

###### Abstract

Transport properties play a crucial role in defining materials as insulators,
metals, or superconductors. A fundamental parameter in this regard is the
Drude weight, which quantify the ballistic transport of charge carriers. In
this work, we measure the Drude weights of an ultracold gas of interacting
bosonic atoms confined to one dimension, characterising the induced atomic and
energy currents in response to perturbations with an external potential. We
induce currents through two distinct experimental protocols; by applying a
constant force to the gas, and by joining two subsystems prepared in different
equilibrium states. By virtue of integrability, dynamics of the system is
governed by ballistically propagating, long-lived quasi-particle excitations,
whereby Drude weights almost fully characterise large-scale transport. Indeed,
our results align with predictions from a recently developed hydrodynamic
theory, demonstrating almost fully dissipationless transport, even at finite
temperatures and interactions. These findings not only provide experimental
validation of the hydrodynamic predictions but also offer methodologies
applicable to various condensed matter systems, facilitating further studies
on the transport properties of strongly correlated quantum matter.

Drude weights are fundamental transport coefficients of condensed matter
physics, often used for classifying materials according to their conductivity
properties [1]. Initially formulated for solid state systems, where it is also
referred to as charge stiffness, the Drude weight measures the ratio of the
density of mobile charge carriers to their mass [2]. In the context of quantum
many-body systems, Drude weights quantify the ballistic transport of charge
carriers and, for non-vanishing values, signal the presence of persistent
dynamical correlations. Hence, the scaling behavior of Drude weights in the
thermodynamic limit is an essential criterion for distinguishing metallic and
insulating phases in many-body systems; in the absence of impurities, metals
exhibit a finite Drude weight, whereas it vanishes for insulators [3].

At finite temperatures, Drude weights of interacting systems are generally
expected to vanish, however, there are notable exceptions with anomalous
transport properties. Among the most studied examples are graphene [4, 5, 6],
whose linear dispersion allows thermally excited carriers to contribute to
conductivity, and integrable models [7, 8], where a macroscopic number of
conservation laws prevents the complete decay of currents. Recently, the
development of a hydrodynamic theory for integrable systems, dubbed
Generalized Hydrodynamics (GHD) [9, 10], has advanced the study of transport
and dynamics in several paradigmatic models of quantum many-body physics [11,
12, 13]. In these models, conserved charges are carried by interacting quasi-
particles representing collective excitations with infinite lifetimes, even at
finite temperatures, which propagate mainly ballistically with sub-leading
diffusive corrections [14, 15]. Therefore, Drude weights may fully
characterise the large-scale dynamics of integrable systems. However, despite
theoretical progress, direct measurements of transport coefficients have yet
to be performed, even though several experiments have shown hydrodynamic
evolution consistent with the theoretical predictions [16, 17, 18, 19].

((a))

((b))

((c))

Figure 1:  Schematic of the experimental protocol. a, An Atom Chip creates a
tight transverse confinement (along the $x$ and $y$ axes), confining a gas of
ultracold bosons to one dimension. Along the longitudinal $z$-axis, arbitrary
potentials can be realised using a Digital Micromirror Device (DMD). b, An
initially homogeneous system with chemical potential
$\bar{\mu}=g_{1\mathrm{D}}\bar{n}$ is realised by confining the 1D gas in a
box trap. At time $t=0$ the bottom of the box trap $V(z)$ is tilted, leading
to an accelerating current of atoms across the center $z=0$. c, An initial
chemical potential imbalance $\delta\mu$ is achieved by creating a box trap
with a step-function bottom. At time $t=0$ the trap bottom is quenched to
flat, leading to two counter-propagating waves originating from the center.

In this work, we measure Drude weights of an ultracold gas of interacting
bosonic atoms confined to one dimension, making the system integrable.
Analogous to their solid state counterparts, the Drude weights of the gas are
extracted by measuring the current of particles and energy in response to
controlled perturbations. We accomplish this through the implementation of two
distinct protocols: First, probing the response of the gas to a constant,
external force and, second, measuring the flow of charges between two
subsystems in different equilibrium states. In accordance with theoretical
predictions, we find that the Drude weights almost entirely characterise the
observed dynamics of the gas, despite its finite temperature and interaction
dominated nature. Further, the methodologies that we have employed can be
generalised to a number of condensed matter systems in a straightforward
manner, thus offering a means to probing the emergent transport behavior of
strongly correlated quantum systems.

Our experimental system is a one-dimensional (1D) ultracold Bose gas trapped
below an Atom Chip [20] (Fig. 1(a)). The chip produces a cigar-shaped magnetic
trap with a trapping frequency of $\omega_{\perp}=2\pi\times
1.38\,\mathrm{kHz}$ along the two transverse directions, strong enough to
confine the atoms to a single dimension. We prepare a thermal equilibrium
state by cooling an atomic cloud of 87Rb atoms with mass $m$ using standard
techniques of laser and evaporative cooling, resulting in temperatures
typically in the range 10-50 nK. The interaction strength is quantified by the
dimensionless Lieb-Liniger parameter
$\gamma=mg_{1\mathrm{D}}/\hbar^{2}\bar{n}$ [21], where $g_{1\mathrm{D}}$ is
the effective atomic coupling strength [22] and $\bar{n}$ is the mean atomic
density. In our experiment, $\gamma\approx 0.002$, which, combined with the
low temperatures, places the gas deeply in the quasi-condensate regime. Hence,
the energy density is dominated by interactions
$\varepsilon\gtrsim\frac{1}{2}g_{1\mathrm{D}}n^{2}g^{(2)}(0)$, where
$g^{(2)}(0)\approx 1$ is the local two-body autocorrelation function [23], and
the atomic density is approximately linearly proportional to the chemical
potential $\mu=g_{1\mathrm{D}}n$. To manipulate the gas, we utilize a Digital
Micromirror Device (DMD) to generate fully customised optical dipole
potentials [24], which are superimposed onto the magnetic trap along the
longitudinal $z$-axis, see Fig. 1(a). In all of our experiments, two hard
walls are imposed on the condensate, confining it to a region of typical
length $L=100\,\mu\mathrm{m}$. The system length $L$ sets the maximal duration
of the experiments, as atoms reflected off the walls move against the induced
current. Finally, dynamics are instigated by quenching the potential between
the walls, denoted as $V(z)$; the subsequent evolution of the system is probed
by measuring the atomic density following different evolution times $t$ using
absorption imaging.

The conductivity of a system $\sigma_{ab}$ characterises the linear response
of the current of the operator $b$ when coupling an external field to the
operator $a$. For a homogeneous system, its real part reads [25]

$\mathrm{Re}\>\sigma_{ab}(\omega)=2\pi\mathfrak{D}_{ab}\delta(\omega)+\sigma_{ab}^{\mathrm{reg}}(\omega)\;,$
(1)

where $\sigma_{ab}^{\mathrm{reg}}(\omega)$ denotes the regular frequency-
dependent part, while the singular part indicates a ballistic
(dissipationless) contribution, whose magnitude is specified by the Drude
weight $\mathfrak{D}_{ab}$. From the perspective of hydrodynamics, the regular
or dissipative part of the conductivity encodes diffusive processes, which in
integrable systems arise from thermal fluctuations [14, 15] or weak breaking
of integrability [26]. The former can generally be excluded for quasi-
condensates [27], as diffusion is irrelevant on experimental timescales
(Methods). Further, previous studies of our system show that mechanisms of
integrability breaking are weak on experimental time scales [18, 19]. Hence,
Drude weights fully characterise the transport of the condensate. The
conventional way of expressing $\mathfrak{D}_{ab}$ is via the Kubo formula
[28] using the time-averaged current-current autocorrelation function
(Methods). However, this definition is inapproachable from an experimental
standpoint; the destructive nature of absorption imaging makes measuring
dynamical autocorrelation functions highly challenging, thus necessitating
protocols where the Drude weight can be accessed otherwise.

((a))

((b))

Figure 2:  Particle flow in response to constant external force. a, Measured
difference of atoms between the left and right halves of the system for two
experiments with different mean densities and potential gradients, where
$\delta\mu=\phi_{n}L/2$. Quadratic fits of the imbalances are plotted as solid
lines, while shaded areas indicate results of GHD simulations accounting for
the uncertainty in $\bar{n}$ and $\delta\mu$. b, Atomic currents obtained from
the time derivative of the fitting functions. The measured current is
approximated as $\Delta N/t$. Error bars indicate the standard error of the
mean and numbers in brackets represent statistical uncertainties over 40
repititions.

A key signature of the Drude weight is a dissipationless current, whereby it
may alternatively be defined as the time-asymptotic growth rate of the local
current in response to a small, time-independent external field [29]

$\mathfrak{D}_{ab}=\lim_{\phi_{a}\to
0}\lim_{t\to\infty}\frac{j_{b}(z,t)}{t\phi_{a}}\;,$ (2)

where $\phi_{a}$ is the gradient of the field coupling to the operator $a$. In
our first experiment, we implement such a protocol starting from a flat
potential between the walls, thus preparing a homogeneous gas; at time $t=0$
the flat potential is switched to a constant gradient $V(z)=\phi_{n}z$,
equivalent to creating a linear electric field across a charge carrier (Fig.
1(b)). We extract the local atomic current in the center of the system by
measuring the imbalance of atoms between the left and right side of the system
$\Delta N(t)=N_{L}(t)-N_{R}(t)$, such that
$j_{n}(z=0,t)=\frac{1}{2}\frac{\mathrm{d}\Delta N(t)}{\mathrm{d}t}$, repeating
the experiment for various mean densities $\bar{n}$ and potential gradients
$\phi_{n}$.

Fig. 2(a) shows the measured imbalance dynamics of two such experiments; as
expected, larger densities and gradients lead to a more rapid growth of
$\Delta N$. To extract the current, we fit the measured imbalances with a
second order polynomial.Examples of the imbalance fits and their derivatives
are shown in Figs. 2(a) and 2(b), respectively. Given a linear growth of the
current, it may alternatively be expressed as $j_{n}=\Delta N/t$, which, for
the measured imbalance, is plotted in Fig. 2(b) and aligns with the fit,
particularly at late times. At short timescales, the measured current is
somewhat irregular owing to the initial density profile not being completely
homogeneous (Methods). Since the energy of the condensate is dominated by
interactions, the local energy current can be expressed in terms of the atomic
current as $j_{\varepsilon}=g_{1\mathrm{D}}nj_{n}$, thus enabling us to
calculate the corresponding Drude weight via Eq. (2). The results, plotted in
Fig. 4, will be discussed after explaining the second protocol.

((a))

((b))

((c))

((d))

((e))

((f))

((g))

Figure 3:  Extracting bipartition currents with Physics Informed Neural
Networks (PINNs). a, The PINN has two components: (i) A neural network that
outputs local atomic density $n$, atomic current $j_{n}$, and energy current
$j_{\varepsilon}$ given input $z,t$, and (ii) Partial differential equations
for mass $f_{n}$ and energy $f_{\varepsilon}$ continuity. By minimizing the
mean square error (MSE) of the continuity equations, boundary conditions, and
the difference between measured and output density, the currents are inferred.
b,c,d, Measured mean atomic density over 65 repetitions used for training (b),
along with predicted atomic current (c) and atomic continuity equation (d)
obtained from averaging the output of ten independently trained networks.
e,f,g, At large scale, the atomic density profile (e), atomic current profile
(f), and energy current profile (g) reach a self-similar solution along
$\xi=z/t$. The PINN fits, averaged over intermediate times $t$ and ten models,
are compared to GHD simulations.

Our second experiment implements the bipartition protocol, in which two
subsystems, initially in equilibrium with a chemical potential imbalance
$\delta\mu$, are joined at a point contact, inducing particle flow between
them. Unlike the flow of charges through a 1D channel connecting two
imbalanced reservoirs, which has been studied in experiments with ultracold
Fermions [30, 31], the dynamics of the bipartiton protocol is determined
solely by the bulk transport properties of the system. The extraction of Drude
weights from the ensuing dynamics is facilitated by the recently developed
theory of Generalized Hydrodynamics [9, 10], whose inception has heralded
significant progress in understanding one-dimensional transport (Methods).
Following hydrodynamic evolution in the bipartition, a local quasi-stationary
state emerges along each ray $\xi=z/t$; the Drude weight is given in terms of
the total asymptotic current via [32]

$\mathfrak{D}_{nb}=\lim_{\delta\mu\to
0}\frac{1}{2\>\delta\mu}\int\mathrm{d}\xi\>j_{b}(\xi)\;.$ (3)

To realise the bipartition protocol, we first prepare a thermal state in the
potential $V(z)=\delta\mu\,\Theta(z)$, where $\Theta(z)$ is the Heaviside step
function, such that the left and right half sides of gas have different
densities. At time $t=0$ ms we quench to a flat potential and observe in the
ensuing dynamics two outwards counter-propagating waves (Fig. 1(c)).

((a))

((b))

((c))

Figure 4:  Comparison of Measured Drude Weights with hydrodynamic theory. a,b,
Drude weights characterising the linear response of the atomic current (a) and
energy current (b) to a field coupled to the atomic density. The
experimentally measured coefficients from the constant force and bipartition
protocols are derived from Eqs. (2) and (3), respectively, while the
theoretical predictions for a thermal states at $T=50\>\mathrm{nK}$ are
calculated from Eq. (18). Results are plotted as a function of the background
atomic density $\bar{n}$. c,d, Residuals of the measurements compared to the
GHD theory are shown as a function of temperature $T$ (c) and perturbation
$\delta\mu$ (d), where $\delta\mu=\phi_{n}L/2$ for the constant force
measurements. Error bars indicate standard error.

In order to obtain both the atomic and energy currents from the sparsely
measured atomic density, we employ a Physics Informed Neural Network (PINN)
[33]. Briefly, given sparse or noisy measurements, PINNs leverage knowledge of
the governing physical equations, here continuity of mass and energy, to
accurately infer the underlying fields. By ensuring consistency with the
conservation laws, an enhanced fidelity of the reconstructed currents compared
to methods based on finite difference can be achieved. See Methods for further
details. An illustration of our PINN model is depicted in Fig. 3(a); given an
input set of coordinates $(z,t)$, the model predicts the local atomic density
$n$, atomic current $j_{n}$, and energy current $j_{\varepsilon}$. Figs.
3(b-d) show the measured density of one bipartition experiment used for
training, the corresponding atomic current predicted by the model, and the
mass continuity of the predictions, respectively. Despite lacking direct
knowledge of the currents beyond their boundary conditions, the model infers
them with high accuracy, as evidenced by adherence to the conservation laws
and agreement with GHD simulations [34, 35] of the dynamics (Methods) shown in
Figs. 3(e-g). Across multiple realisations of the bipartition protocol, a
slight discrepancy in the amplitude of the quasi-stationary currents along the
ray $\xi=z/t$ consistently emerges between the GHD and PINN predictions. We
attribute this deviation to imperfections in the optical potential, which
introduce density variations in the initial state and disorder in the flat
potential [30]; both not accounted for in the GHD simulations.

Finally, having extracted the atomic and energy currents from both the
constant force and bipartition protocols, we compute the corresponding Drude
weights via Eqs. (2) and (3), respectively. Formally, these expressions are
exact only in the limits of asymptotic time and infinitesimal perturbations.
The former limit ensures that only dissipationless currents are accounted for.
However, due to the negligible diffusion in the quasi-condensate, transport at
accessible time scales is entirely ballistic. Meanwhile, the latter limit
reflects the linear response of the conductivity; although the experimental
perturbations are finite, GHD simulations confirm that the responses remain
within the linear regime such that Eqs. (2) and (3) accurately yield the Drude
weights under experimental conditions (see Appendix).

The Drude weights obtained from experimental measurements are presented in
Figs. 4(a,b) as a function of mean density $\bar{n}$, alongside theoretically
predicted values for homogeneous thermal states at 50 nK, derived from the
Kubo formalism using hydrodynamic projections [36] (Methods). Analogously to
solid state systems, where the Drude weight measures the density of mobile
charge carriers relative to their mass, we find that $\mathfrak{D}_{nn}$,
which characterizes the atomic current response to a density perturbation, is
proportional to $\bar{n}/m$, consistent with theoretical predictions. This
surprisingly simple scaling behavior, particularly in an interacting system at
finite temperature, arises from a uniform and interaction-independent coupling
of the external potential to the quasi-particle excitations of the gas [37].
Moreover, $\mathfrak{D}_{n\varepsilon}$, which quantifies the ballistic energy
current induced by coupling to the atomic density, also shows agreement with
theory, notwithstanding that the theory accounts for the total energy, whereas
only the interaction energy current is measured. At lower atomic densities,
the interaction energy scales quadratically with $\bar{n}$ density, while at
higher densities, the effective 1D coupling $g_{1\mathrm{D}}$ acquires a weak
density dependence [38], leading to the energy becoming more linear in its
dependence on $\bar{n}$.

For both protocols studied, the experimentally extracted Drude weights are
generally slightly lower than theoretical predictions. Plotting the residuals
$R_{nn}=\left(\mathfrak{D}_{nn}^{\mathrm{meas}}-\mathfrak{D}_{nn}^{\mathrm{theory}}\right)m$
as a function of temperature and perturbation amplitude reveal no dependence
on either parameter (see Figs. 4(c,d)). Since the GHD prediction is calculated
for a fixed temperature of 50 nK, the former demonstrates that, within quasi-
condensate regime, the probed responses of the gas are independent of
temperature. Meanwhile, the latter confirming the linear response of the
system for the given perturbations. Thus, we attribute the observed
discrepancies in the measured Drude weights to imperfections in the optical
dipole potential, which effectively reduce the conductivity of the system.

In summary, we have measured the Drude weights of a 1D quasi-condensate,
quantifying the ballistic transport of mass and energy when coupling to the
atomic density via an external potential. The simplicity of the 1D Bose gas
makes it an ideal platform for verifying GHD predictions of transport
coefficients [36, 32], however, our methodologies can be easily adapted to
other systems, providing a foundation for future studies of quantum matter
transport properties. Integrable models such as the XXZ spin-1/2 chain and the
sine-Gordon model exhibit anomalous transport [39] with Drude weights showing
fractal-like dependence on interactions [40, 41]. Additionally, GHD extends to
non-thermal equilibrium states supported by integrable models [42, 43], whose
transport properties remain experimentally unexplored. Finally, introducing
non-integrable perturbations has shown to significantly alter long-term
dynamics, transitioning from integrable to conventional hydrodynamics and
resulting in a dynamical phase transition in transport properties [44, 18,
45].

## Acknowledgements

We thank Sebastian Erne and Igor Mazets for helpful discussions. This research
was supported by the European Research Council: ERC-AdG “Emergence in Quantum
Physics” (EmQ) under Grant Agreement No. 101097858 and the Austrian Science
Fund (FWF) “Strongly correlated Quantum Fields out of equilibrium” [grant DOI:
10.55776/P36656].

##  Appendix A Experimental protocol

Using standard techniques of laser cooling, magnetic trapping, and evaporative
cooling we load $\mathrm{Rb}^{87}$ atoms in a tight transversal confinement
with trapping frequency $\omega_{\perp}=2\pi\times 1.38\,\mathrm{kHz}$ with
$\mu,k_{B}T<\hbar\omega_{\perp}$ on an Atom Chip [20]. The initial thermal
states are realised by evaporatively cooling the atoms while trapped in the
combined magnetic and optical dipole potential. By tuning the evaporative
cooling, we can realise a gas with number of atoms $N$ in the range
$(3-10)\times 10^{3}$ at temperatures (10-50) nK, the latter measurable via
density-ripples thermometry [46]. Typical shot to shot density variation after
post selection is $7-10\%$, while heating and atom losses ($\sim 0.8\%$) are
negligible over the time scales of our protocols. Nonetheless, we rescale the
mean density of every measure evolution time $t>0$ to equal that at $t=0$.

Using a DMD we can create an arbitrary blue detuned optical dipole potential
along the longitudinal z-axis of the trap [24]. To this end, we employ an
optimization algorithm, which toggles individual DMD pixels on or off until a
desired profile is reached. Due to technical limitations of this approach
[24], the optimized density profiles feature systematic imperfections from the
desired profile with typical standard deviation $\sigma_{n}=0.05\bar{n}$.
Further, because the optical potential needs to compensate the underlying
harmonic confinement created by the chip, the length of the boxes created is
limited by our available laser power. To instigate dynamics, we rapidly quench
(on time scales $\sim 20\mu$s) between pre-optimised patterns, then hold the
condensate for a desired evolution time. Finally, the 1D density profiles are
measured using absorption imaging after 2 ms free expansion time; we repeat
the experiment for different evolution times and average the measurements over
multiple realizations to obtain $n(z,t)$ and $\bar{n}$. The gradients of the
constant force fields $\phi_{n}$, the initial chemical potential imbalances
$\delta\mu$ of the bipartition, and the condensate temperature $T$ are all
extracted from separate measurements, taken of thermal equilibrium states in
the corresponding potential configurations.

##  Appendix B Lieb-Liniger model

Following confinement to one dimension, a gas of $N$ bosons with mass $m$ and
contact interactions is well described by the integrable Lieb-Liniger
Hamiltonian [21] plus an additional external longitudinal potential $V(z)$

$\mathcal{H}=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{N}\partial_{z_{i}}^{2}+g_{1\mathrm{D}}\sum_{i<j}\delta\left(z_{i}-z_{j}\right)+\sum_{i=1}^{N}V\left(z_{i}\right)\;.$
(4)

The 1D contact interaction is depends on the transverse trapping confinement
via
$g_{1\mathrm{D}}=2\hbar\omega_{\perp}a_{s}\varpi\left(1-1.03a_{s}/l_{\perp}\right)^{-1}$
[22], with $a_{s}$ being the 3D scattering length and
$l_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$. The factor
$\varpi=\frac{1}{2}\frac{2+3a_{s}n}{\left(1+2a_{s}n\right)^{3/2}}$ accounts
for a reduction of the 1D coupling following a transverse broadening of the
bosonic wavefunction as $na_{s}$ approaches unit [38]. Technically, the
density dependence of coupling spoils the integrability of the system.
However, we find that employing the integrable theory using an effective
contact interaction evaluated for the mean density of the gas $\bar{n}$
accurately describes our experimental observations. Similar observations have
been made in bosonic 1D dipolar quantum gases near equilibrium, where a
description using the Lieb-Lininger Hamiltonian is still feasible, with the
integrability-breaking dipole interactions simply leading to an effective
modification of the contact interaction [47].

##  Appendix C Extracting currents with Physics Informed Neural Networks

The neural network of our PINN model consists of two input neurons
representing the coordinates $(z,t)$ and three output neurons corresponding to
the atomic density $n$, atomic current $j_{n}$, and energy current
$j_{\varepsilon}$, as illustrated in Fig. 3(a). The input and output layers
are connected by 6 hidden layers of 30 neurons each. Following conservation of
mass and energy, the dynamics of the 1D Bose gas in our experiment is subject
to the continuity equations

$\displaystyle\partial_{t}n+\partial_{z}j_{n}=0\;,$ (5)
$\displaystyle\partial_{t}\varepsilon+\partial_{z}j_{\varepsilon}=0\;,$

where the energy density is approximated as the quasi-condensate interaction
energy density $\varepsilon=\frac{1}{2}g_{1\mathrm{D}}n^{2}$.

The model is trained to satisfy the given training data as well as the imposed
governing equations (5) and boundary conditions [33]. The derivatives of the
output fields with respect to input coordinates are obtained via Automatic
Differentiation [48]. Thus, given experimental measurements of the atomic
density profile, arranged as the set $\\{z_{i},t_{i},n_{i}\\}_{i=1}^{M}$, we
minimise the mean squared error (MSE) loss

$\displaystyle\mathrm{Loss}=$
$\displaystyle\frac{1}{M}\sum_{i=1}^{M}\left(n(z_{i},t_{i})-n_{i}\right)^{2}$
(6)
$\displaystyle+\frac{1}{M}\sum_{i=1}^{M}\left(|f_{n}(z_{i},t_{i})|^{2}+|f_{\varepsilon}(z_{i},t_{i})|^{2}\right)+\mathrm{MSE}_{\mathrm{b.c.}}\;,$

where $f_{n}=\partial_{t}n+\partial_{z}j_{n}$ and
$f_{\varepsilon}=g_{1\mathrm{D}}n\partial_{t}n+\partial_{z}j_{\varepsilon}$
encode the continuity equations. The final term of Eq. (6) enforces boundary
conditions of the currents, namely that they must vanish at the edges of the
box trap, which is necessary to ensure that their solution is unique.

We find that measurement noise causes training of our model to take an
excessive amount of time to converge, whereby the density profiles used for
training have been convolved with a Gaussian, whose width is twice the
resolution of our imaging system. This reduces noise, resulting in a faster
convergence, without affecting the observed large scale dynamics. Further, for
each data set, we independently train ten model starting from different,
random parameters of the neural network; the reported currents are averaged
over the different models, while errors are given by the standard error of the
mean.

Having extracted the currents, their solution along the ray $\xi=z/t$ is
obtained via interpolation for evolution times $t$ within an particular
interval: The self-similar solution is reasonably approached after 4 ms,
whereby $\xi$ is evaluated for only later times. Further, when quenching trap
configurations from the step to the flat-bottomed box, a small perturbation is
launched from the wall on the low-density side. To exclude its contribution to
the solution on the ray, only evolution times $t$ before the perturbation
reaches the front of the current profile are employed (see Fig. 5 for
example).

((a))

((b))

((c))

((d))

Figure 5:  Example of current fitting using PINNs. a, Measured evolution of
atomic density $n$ in the bipartition protocol. The dashed lines indicate the
coordinates $(z_{i},t_{i})$ at which $n$ is cropped to create the training
data. b, Training data for the PINN, after Gaussian smoothening. c,d, Fitted
atomic current (c) and density(d), obtained by averaging the outputs of ten
PINNS trained independently on the data in (b). The output is evaluated at
coordinates with ten times the resolution of the training data. The dash-
dotted lines indicate the time interval for which the self-similar solution
along the ray $\xi=z/t$ is obtained.

The PINN model, which we have employed, can easily be extended to arbitrary
regimes of the Bose gas (where the energy is not dominated by interactions) by
including the momentum-continuity equation
$\partial_{t}p+\partial_{z}j_{p}=0$, where the momentum current is equal to
the pressure $j_{p}=\mathcal{P}$ [49]. The momentum density is given by the
atomic current $p=mj_{n}$ following Galilean invariance of the Lieb-Liniger
model, thus need not be measured. However, knowledge of the pressure near the
walls of the box trap is required to fix the boundary conditions of the PINN,
as the pressure does not vanish at the boundaries of the box trap.

##  Appendix D Generalized Hydrodynamics of the 1D Bose gas

Viewed from the perspective of the Bethe Ansatz solution, the 1D gas of
neutral bosons is actually not so far from electronic solid state systems; the
elementary collective excitations are fermionic quasi-particles and holes,
which carry not electrical, but quanta of a macroscopically large set of
conserved charges/quantities. Each particle is uniquely labeled by its
rapidity, or quasi-momentum, $\theta$, while the local distribution of quasi-
particles $\rho_{\mathrm{p}}(\theta)$ fully encodes the local thermodynamic
state [21]. Introducing the density of allowed rapidities $\rho_{\mathrm{s}}$
and the occupations function
$\vartheta(\theta)=\rho_{\mathrm{p}}(\theta)/\rho_{\mathrm{s}}(\theta)$, a
thermal distribution is given by

$\vartheta(\theta)=\frac{1}{1+e^{\epsilon(\theta)\beta}}\;,$ (7)

following the fermionic statistics of the quasi-particles. Here,
$\beta=1/k_{B}T$ is the inverse temperature, and the pseudo-energy
$\epsilon(\theta)$ is given by the Yang-Yang equation [49]

$\epsilon(\theta)=\frac{\hbar^{2}\theta^{2}}{2m}-\mu+\frac{1}{2\pi\beta}\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>\Delta\left(\theta-\theta^{\prime}\right)\ln\left(1+e^{\epsilon(\theta^{\prime})\beta}\right)\;,$
(8)

where $\Delta$ is the rapidity derivative of the two-body scattering phase
given by

$\Delta\left(\theta-\theta^{\prime}\right)=\frac{2mg_{1\mathrm{D}}/\hbar^{2}}{(mg_{1\mathrm{D}}/\hbar^{2})^{2}+(\theta-\theta^{\prime})^{2}}\;.$
(9)

In the presence of an external potential $V(z)$ one can locally shift the
chemical potential accordingly $\mu(z)=\mu_{0}-V(z)$ under the local density
approximation, thus yielding a spatially dependent quasi-particle
distribution.

Integrable systems abide to an infinite number of conservation laws
$\partial_{t}q_{a}+\partial_{z}j_{a}=0$, where $q_{a}$ is density of the
$a$’th conserved quantity $Q_{a}=\int\mathrm{d}z\>q_{a}$, while $j_{a}$ is its
associated current. Note, for the sake of generality, we will in the following
let the subscript $a$ denote the index of the conserved charge.
Conventionally, the indices $a=0,1,2$ correspond to particle number, momentum,
and energy, respectively, such that $q_{0}\equiv n$ and
$q_{2}\equiv\varepsilon$. Their local thermodynamic expectation values are
given in terms of the quasi-particle distribution via [9, 10]

$\displaystyle q_{a}$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}\theta\>\rho_{\mathrm{p}}(\theta)h_{a}(\theta)\;,$
(10) $\displaystyle j_{a}$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}\theta\>v^{\mathrm{eff}}(\theta)\rho_{\mathrm{p}}(\theta)h_{a}(\theta)\;.$

Here, $h_{a}$ is the single-particle eigenvalue of the $a$’th conserved
quantity, which for atom number and energy read $h_{0}=1$ and
$h_{2}=\hbar^{2}\theta^{2}/2m$, respectively, while $v^{\mathrm{eff}}(\theta)$
is the local propagation velocity of a quasi-particle with rapidity $\theta$,
given by

$v^{\mathrm{eff}}(\theta)=\frac{\hbar\theta}{m}+\int_{-\infty}^{\infty}d\theta^{\prime}\Delta\left(\theta-\theta^{\prime}\right)\rho_{\mathrm{p}}(\theta^{\prime})\left[v^{\mathrm{eff}}\left(\theta^{\prime}\right)-v^{\mathrm{eff}}(\theta)\right]\,.$
(11)

The propagation velocity is modified by interactions with other particles,
which manifests as the Wigner delay time [50] associated with the quantum
mechanical phase shifts occurring upon elastic collisions of the atoms.

The infinite number of continuity equations of integrable systems makes a
conventional hydrodynamics treatment intractable; GHD overcomes this by
expressing the dynamics via a collisionless Boltzmann equation for the quasi-
particles [9, 10]

$\partial_{t}\rho_{\mathrm{p}}+\partial_{z}(v^{\mathrm{eff}}\rho_{\mathrm{p}})-\hbar^{-1}\partial_{\theta}(\partial_{z}V\>\rho_{\mathrm{p}})=0\;.$
(12)

Eq. (12) facilitates numerical simulation of the large-scale dynamics of the
system; all GHD simulations performed for this work solve Eq. (12) using
backwards semi-Lagrangian schemes [35], implemented in the iFluid library
[34]. Further, when comparing GHD simulations to experimental measurements, we
account for the additional density dynamics during the 2 ms free expansion
time.

((a))

((b))

Figure 6:  Drude Weights extracted from GHD simulations. a,b, Drude weights
characterising the linear response of the atomic current (a) and energy
current (b) to a field coupled to the atomic density. The coefficients are
extracted from GHD simulations of the experiment using Eqs. (2) and (3): For
symbols labeled GHD, the currents are calculated directly from the quasi-
particle distribution using Eq. (10), while for those labeled imbalance, the
current is calculated via $\Delta N$. The former account for the full energy
density, whereas the latter only consider the interaction energy density. The
theoretical predictions for a thermal states at $T=50\>\mathrm{nK}$ are
calculated from Eq. (18) and plotted as a solid line. All results are plotted
as a function of the background atomic density $\bar{n}$. Errorbars are
obtained from simulations including the standard error of $\bar{n}$ and
$\delta\mu$.

In Fig. 6 the results of simulating the experiment with GHD and extracting the
Drude weights using Eqs. (2) and (3) are shown. Although the external field
perturbation $\delta\mu$ is finite, we find that the results agree very well
with the analytic prediction of Eq. (18) based on quasi-particle propagation
in a homogeneous system.

##  Appendix E Conductivity and Drude weights of integrable models

The conductivity matrix $\sigma_{ab}$ quantifies the linear response of the
current $j_{b}$ to an external perturbation of dynamics coupled to the charge
density $q_{a}$ [25]. Given an external force applied to a one-dimensional
many-body system with equilibrium Hamiltonian $\mathcal{H}_{0}$, such that
$\mathcal{H}(t)=\mathcal{H}_{0}-\phi_{a}(t)\int\mathrm{d}z\>zq_{a}$, where the
second term represents a linear potential with time dependent strength
$\phi_{a}(t)$, the response of the current $j_{b}$ is

$j_{b}(t)=\int_{-\infty}^{t}\mathrm{d}t^{\prime}\>\sigma_{ab}(t-t^{\prime})\phi_{a}(t^{\prime})\;.$
(13)

Taking the inverse Fourier transform of the conductivity (1), we see that the
Drude weight gives a constant contribution
$\sigma_{ab}(t)=\mathfrak{D}_{ab}+\ldots$, while the contribution of the
regular part, which we neglect for now, may be time dependent. Thus, given a
constant force $\phi_{a}(t)=\phi_{a}$, the ballistic response of the current
$j_{b}$ is a linear growth in time

$j_{b}(t)=\mathfrak{D}_{ab}\phi_{a}t+\ldots$ (14)

Furthermore, the conductivity of a system is connected to spreading of
correlations in response to a local change in the initial state. Consider the
connected two-point functions of local conserved densities

$S_{ab}(z,t):=\left\langle q_{a}(z,t)q_{b}(0,0)\right\rangle^{c}\;.$ (15)

The late-time hydrodynamic spreading of $S_{ab}$ under such a local
perturbation is governed by separate ballistic and diffusive contributions
following [12]

$\frac{1}{2}\int\mathrm{d}z\>z^{2}\left(S_{ab}(z,t)+S_{ab}(z,-t)\right)=\mathfrak{D}_{ab}t^{2}+\mathfrak{L}_{ab}t+O(t)\;,$
(16)

given finite Drude weights $\mathfrak{D}_{ab}$ and Onsager matrix
$\mathfrak{L}_{ab}$. The Onsager matrix is another fundamental transport
coefficient, which accounts for diffusive contributions arising from thermal
fluctuations of the thermodynamic state and is defined as the zero-frequency
limit of the regular part of the conductivity
$\mathfrak{L}_{ab}=\lim_{\omega\rightarrow
0}\sigma_{ab}^{\mathrm{reg}}(\omega)$ [12].

The response function, and thus the Drude weight, of a finite temperature
system is given in terms of the time-averaged current-current autocorrelation
function via the Kubo formula [28], yielding

$\mathfrak{D}_{ab}=\lim_{t\to\infty}\frac{\beta}{2t}\int_{-\infty}^{\infty}\mathrm{d}z\int_{0}^{t}\mathrm{d}t^{\prime}\left\langle
j_{a}(z,t^{\prime})j_{b}(0,0)\right\rangle^{c}\;.$ (17)

Following GHD, analytic expressions for transport coefficients can be derived
using the method of hydrodynamics projections, which in essence describes the
emergent dynamics by projecting onto the subspace of slowly varying conserved
densities and their gradients, enables evaluation of the current-current
autocorrelation function of Eq. (17). Thus, the Drude weights can be expressed
in terms of the thermal distribution of quasi-particles via [36]

$\mathfrak{D}_{ab}=\beta\int_{-\infty}^{\infty}\mathrm{d}\theta\>\rho_{\mathrm{p}}(\theta)\left(1-\vartheta(\theta)\right)\left(v^{\mathrm{eff}}(\theta)\right)^{2}h_{a}^{\mathrm{dr}}(\theta)h_{b}^{\mathrm{dr}}(\theta)\;,$
(18)

where the superscript ’dr’ denotes the dressing of a function, defined as

$h_{a}^{\mathrm{dr}}(\theta)=h_{a}(\theta)-\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\Delta\left(\theta-\theta^{\prime}\right)\vartheta\left(\theta^{\prime}\right)h_{a}^{\mathrm{dr}}\left(\theta^{\prime}\right).$
(19)

Like the Drude weight, an analytic expression for the Onsager matrix can be
obtained via form factor expansion over the elementary excitations of the
integrable model yielding [14]

$\displaystyle\mathfrak{L}_{ab}=$
$\displaystyle\int\frac{\mathrm{d}\theta_{1}\mathrm{~{}d}\theta_{2}}{2}\rho_{\mathrm{p}}\left(\theta_{1}\right)\left(1-\vartheta(\theta_{1})\right)\rho_{\mathrm{p}}\left(\theta_{2}\right)\left(1-\vartheta(\theta_{2})\right)$
(20)
$\displaystyle\times\left|v^{\text{eff}}\left(\theta_{1}\right)-v^{\text{eff}}\left(\theta_{2}\right)\right|\left(\frac{\Delta^{\mathrm{dr}}\left(\theta_{1},\theta_{2}\right)}{2\pi}\right)^{2}$
$\displaystyle\times\bigg{(}\frac{h_{a}^{\mathrm{dr}}\left(\theta_{2}\right)}{\rho_{\mathrm{s}}\left(\theta_{2}\right)}-\frac{h_{a}^{\mathrm{dr}}\left(\theta_{1}\right)}{\rho_{\mathrm{s}}\left(\theta_{1}\right)}\bigg{)}\bigg{(}\frac{h_{b}^{\mathrm{dr}}\left(\theta_{2}\right)}{\rho_{\mathrm{s}}\left(\theta_{2}\right)}-\frac{h_{b}^{\mathrm{dr}}\left(\theta_{1}\right)}{\rho_{\mathrm{s}}\left(\theta_{1}\right)}\bigg{)}\;.$

In integrable systems, diffusive contributions are generally subleading to
ballistic, particularly in the quasi-condensate regime of the 1D Bose gas
[27]. Knowing the large-scale spreading of correlations via Eq. (16), we can
estimate a time until which diffusion is dominant over ballistic spreading as
$t_{ab}^{\star}=\mathfrak{L}_{ab}/\mathfrak{D}_{ab}$. For density-density
correlations in a typical thermal state of the experiment $t^{\star}\sim
10^{-40}\>\mathrm{s}$, whereby diffusion can be fully neglected.

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