**footnotetext: These authors contributed equally to this work††footnotetext:
Correspondence should be addressed to<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
# Irreversible entropy transport enhanced by fermionic superfluidity

Philipp Fabritius* Jeffrey Mohan* Mohsen Talebi  Simon Wili  Institute for
Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland
Wilhelm Zwerger  Technische Universität München, Physik Department, 85748
Garching, Germany  Meng-Zi Huang† Tilman Esslinger† Institute for Quantum
Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

The nature of the flow between two superfluids, as in the Josephson [1] and
fountain [2, 3] effects, is often understood in terms of reversible flow
carried by an entropy-free, macroscopic wavefunction [4, 5]. While this
wavefunction is responsible for many intriguing properties of superfluids and
superconductors, its interplay with excitations in non-equilibrium situations
is more subtle and less understood [6, 7]. Here, we observe large concurrent
flows–not only of particles but also entropy–through a ballistic channel
connecting two strongly interacting fermionic superfluids, both showing
similar nonlinear responses to biases in chemical potential and temperature.
We find that the entropy transported per particle is much larger than the
local entropy of the equilibrium superfluid and is robust to changes in the
channel’s geometry. In contrast, the timescales of advective and diffusive
entropy transport vary significantly with the channel geometry. In our
setting, superfluidity counterintuitively increases the speed of irreversible
entropy transport. We develop a phenomenological model describing the
nonlinear dynamics within the framework of generalised gradient dynamics [8],
an extension of Onsager’s theory of irreversible processes. The presented
approach to measure entropy currents may help uncover novel mechanisms of heat
transfer in superfluids and superconducting devices [9].

Two connected reservoirs exchanging particles and energy is a paradigmatic
system that is key to understanding transport phenomena in diverse platforms
of both fundamental and technological interest ranging from heat engines, to
superconducting qubits [9], and even heavy-ions collisions [10]. Entropy and
heat, both irreversibly produced and transported by the currents flowing
between the reservoirs, are key quantities in superfluid and superconducting
systems [11, 12]. They help to reveal microscopic information in strongly
interacting systems [13, 14] and more generally characterise far-from-
equilibrium systems [15]. Yet in traditional condensed matter systems such as
superconductors and superfluid helium, the entropy is not directly accessible
and requires indirect methods to deduce it [16, 17, 18].

In this work, we leverage the advantage of quantum gases of ultracold atoms as
naturally closed systems well-isolated from their environments to study
entropy transport and production in fermionic superfluid systems. Using the
known equation of state [19], we measure the particle number and total entropy
in each of the two connected reservoirs as a function of evolution time,
therefore directly obtaining the entropy current and production. The
hydrodynamic understanding of the bulk system within the framework of Landau’s
two-fluid model [20, 21, 22, 23, 24] can be extended to two coupled
superfluids to explain the Josephson effect [25, 26] when the normal
(entropic) current is sufficiently suppressed. In quantum gases [27] as well
as superconductors [28], this is accomplished with a low-transparency tunnel
junction, while narrow channels are used to block viscous currents in
superfluid helium [29]. On the other hand, transport through ballistic
channels [30] featuring large currents for chemical potential biases smaller
than the superfluid gap [31, 32, 33] can exhibit less intuitive behaviour
since the supercurrent no longer dominates the normal current, making the
system fundamentally non-equilibrium [34]. At low temperatures, such systems
exhibit particle currents on the order of the superfluid gap that respond non-
linearly to chemical potential biases and constitute the signature of
superfluidity, reminiscent of similar observations of weak links in
superconductors [28] and superfluid He [35, 36]. However, entropy transport in
this setting has so far only been experimentally studied indirectly and at
higher temperatures in the linear response regime where the superfluidity of
the system is ambiguous [37, 38], leaving open the question of entropy
transport between superfluids.

Here, we connect two superfluid unitary Fermi gases with a ballistic channel
and measure the coupled transport of particles and entropy between them. We
observe large sub-gap currents of both particles and entropy, indicating that
the current is not a pure supercurrent and cannot be understood within a
hydrodynamic two-fluid model. In particular, superfluidity counterintuitively
enhances irreversible entropy transport in this system by enhancing particle
current while maintaining a large entropy transported per particle. We also
observe in our parameter regime that, while the system can always thermalise
via the irreversible flow of this superfluid-enhanced normal current,
thermalisation via pure entropy diffusion is inhibited in one-dimensional
channels, giving rise to a non-equilibrium steady state previously observed in
the normal phase [37]. The observed nonlinear dynamics of particles and
entropy are captured by a phenomenological model we develop whose only
external constraints are the conservation of particles and energy and the
Second Law of Thermodynamics.

Figure 1:  Irreversible particle, energy, and entropy transport through a non-
equilibrium channel connecting two superfluids. (a) Slices along the $x=0$ and
$z=0$ planes of the calculated local degeneracy $T/T_{F}$ during transport
through the 1D channel, showing that the reservoirs are in local equilibrium
in the normal phase (red, $T/T_{F}>0.167$) over most of their volume and
superfluid (blue, $T/T_{F}\leq 0.167$) at the contacts to the channel. The
channel (green) is far from local equilibrium. Differences in the atom number
$N$, energy $U$, and entropy $S$ between the reservoirs induce currents
between them. (b) Depending on the initial conditions (filled or open circles)
and microscopically allowed processes, the reservoirs can exchange entropy
advectively and diffusively, tracing out various paths through state space
with the constraints that atom number and energy are conserved
$\derivative*{N}{t}=\derivative*{U}{t}=0$ and the entropy production is
positive definite $\derivative*{S}{t}\geq 0$. This evolution halts (stars) by
reaching either a non-equilibrium steady state ($\Delta N,\Delta S\neq 0$) or
equilibrium ($\Delta N=\Delta S=0$) where the total entropy $S=S_{L}+S_{R}$
has a global maximum.

We begin the experiment by preparing a balanced mixture of the first- and
third-lowest hyperfine ground states of 6Li at unitarity in an augmented
harmonic trap shown in Fig. 1(a); see supplement for details [39]. To induce
transport of atoms, energy, and entropy, we prepare an initial state within
the state space in Fig. 1(b) characterised by the conserved total atom number
and energy, $N=N_{L}+N_{R}$ and $U=U_{L}+U_{R}$, and the dynamical imbalances
in the atom number $\Delta N=N_{L}-N_{R}$ and entropy $\Delta S=S_{L}-S_{R}$.
The imbalances in the extensive quantities induce biases in the chemical
potential $\Delta\mu=\mu_{L}-\mu_{R}$ and temperature $\Delta T=T_{L}-T_{R}$
according to the reservoirs’ equations of state (EoS) [39] which in turn drive
currents of the extensive properties $I_{N}(\Delta N,\Delta
S)=-(1/2)\derivative*{\Delta N}{t}$ and $I_{S}(\Delta N,\Delta
S)=-(1/2)\derivative*{\Delta S}{t}$. Note that $I_{S}$ is an apparent current,
not a conserved current like $I_{N}$ and $I_{U}$, though we can place bounds
on the conserved entropy current from the apparent current and entropy
production rate
$I_{S}^{\mathrm{cons}}\in[I_{S}-(1/2)\derivative*{S}{t},I_{S}+(1/2)\derivative*{S}{t}]$
[33]. These equations of motion, together with the initial state $\Delta
N(0),\Delta S(0)$, determine the path the system traces through state space
$\Delta N(t),\Delta S(t)$ as well as the speed with which it traces this path.
The paths that we explore experimentally are shown as black lines overlayed on
top of the entropy landscape
$S=S_{L}(N_{L},U_{L})+S_{R}(N_{R},U_{R})=S(N,U,\Delta N,\Delta S)$ computed
from the EoS. The paths all exhibit a strictly positive entropy production
rate $\derivative*{S}{t}>0$, implying the transport is irreversible, until
they reach either a non-equilibrium steady state ($\Delta N,\Delta S\neq 0$)
or equilibrium ($\Delta N=\Delta S=0$) where $S$ is maximised for the fixed
$N$ and $U$. The von Neumann entropy of a closed system does not increase in
time under Hamiltonian evolution, though the measured thermodynamic entropy
$S$ can increase due to buildup of entanglement entropy shared between the two
reservoirs [40, 41, 42]. We have verified that the system is nearly closed
given the measured particle loss rate
$\absolutevalue{\derivative*{N}{t}}/N<$0.01\text{\,}\mathrm{s}^{-1}$$ and
heating in equilibrium
$\derivative*{(S/Nk_{B})}{t}<$0.02\text{\,}\mathrm{s}^{-1}$$, limited by
photon scattering of optical potentials, such that the entropy production
observed during transport is due to the fundamental irreversibility of the
transport.

We measure the system’s evolution by repeatedly preparing the system in the
same initial state, allowing transport for a time $t$, then taking absorption
images of both spin states and extracting $N_{i},S_{i}$ for both reservoirs
$i=L,R$ using standard thermometry techniques [39]. Between the end of
transport and imaging, we adiabatically ramp down the laser beams that define
the channel to image the reservoirs in well-calibrated, half-harmonic traps.
The beams do work on the reservoirs during this process and change $U_{i}$ but
$N_{i}$ and $S_{i}$ remain constant. The cloud typically contains
$N=270(30)\times 10^{3}$ atoms and $S/N=1.59(7)k_{B}$ before transport, below
the critical value of $\approx 1.90k_{B}$ for superfluidity in the transport
trap. Fig. 1(a) shows the local degeneracy $T/T_{F}$ in the $x=0$ and $z=0$
planes during transport calculated within the local density approximation [43]
using the 3D equation of state [19] to determine the local Fermi temperature
$T_{F}(x,y,z)$ [39]. The imbalance is illustrated using the representative
values $\nu_{x}=$12.4\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$,
$\Delta\mu=$75\text{\,}\mathrm{n}\mathrm{K}$\times k_{B}$, and $\Delta T=0$.
Assuming local equilibrium, the most degenerate regions in the channel would
be deeply superfluid due to the strong, attractive gate potential and reach
$T/T_{F}\approx 0.027$, $s\approx$7.2\text{\times}{10}^{-4}$k_{B}$, and
$\Delta/k_{B}T\approx 17$ where $s$ is the local entropy per particle and
$\Delta$ is the superfluid gap. When $\nu_{x}$ is small and the channel is 2D,
normal regions appear at the edges of the channel that can also contribute to
transport.

Figure 2: Observing advective entropy current from the non-exponential
evolution of imbalances in particle number and entropy. Particles (a) and
entropy (b) are transported between the reservoirs by currents that respond
nonlinearly to biases in chemical potential and temperature, evidenced by the
non- exponential decay. Both currents increase as the channel opens from 1D to
2D ($\nu_{x}$ decreasing). The net entropy (c) increases very slightly while
$\Delta S$ changes significantly, indicating that entropy is indeed
transported by the particle flow. Panel (d) shows $I_{S}=s^{*}I_{N}$ where the
entropy advectively transported per particle $s^{*}$ is nearly independent of
the channel geometry $\nu_{x}$ despite $I_{N}$ varying significantly. The
observation that $s^{*}>0$ means the nonlinear particle current is not a
superfluid current. Each data point is an average of $3-5$ repetitions and
error bars represent the standard deviation. Solid lines are fits of the
phenomenological model.

In a first experiment, we prepare an initial state $\Delta N(0),\Delta
S(0)\neq 0$ [filled circle in Fig. 1(b)] such that equilibrium is reached
within $1\text{\,}\mathrm{s}$. For the strongest confinement, $\Delta N(t)$,
shown in Fig. 2(a), clearly deviates from exponential relaxation and the
corresponding $I_{N}$ is much larger than the value $\sim\Delta\mu/h$ of a
quantum point contact (QPC) in the normal state, indicating that the sub-gap
current-bias characteristics are nonlinear (non-Ohmic) and the reservoirs are
superfluid [32, 33, 44]. When reducing $\nu_{x}$ to cross over from a 1D to 2D
channel, $\Delta N(t)$ relaxes faster ($I_{N}$ increases) and, although it is
less pronounced, the nonlinearity persists. The dynamics for the two smallest
values of $\nu_{x}$ are nearly identical, suggesting that there are additional
resistances in series with the 1D region such as the interfaces between the 3D
and 2D regions [38] or between the normal and superfluid regions [45].

Concurrently with $\Delta N(t)$, we observe non-exponential relaxation of
$\Delta S(t)$ [Fig. 2(b)] which bears a remarkable resemblance to $\Delta
N(t)$. We find that by plotting $\Delta S(t)$ against $\Delta N(t)$ in Fig.
2(d), all paths collapse onto a single line. This demonstrates that the
entropy current is directly proportional to the particle current
$I_{S}=s^{*}I_{N}$ where the average entropy advectively transported per
particle $s^{*}$ is nearly independent of $\nu_{x}$ even though $I_{N}$ itself
varies significantly. Moreover, $\derivative*{S}{t}$ [Fig. 2(c)] is barely
resolvable and is significantly smaller than $I_{S}$, meaning there is indeed
a large conserved entropy current flowing between the reservoirs. The entropy
transported per particle $s^{*}=1.18(3)k_{B}$ is near its value in the normal
phase [37] and is orders of magnitude larger than the local entropy per
particle in the channel assuming local equilibrium
$s\approx$7.2\text{\times}{10}^{-4}$k_{B}$. Because superfluidity in the
contacts enhances $I_{N}$ while only slightly suppressing $s^{*}$,
superfluidity increases $I_{S}$.

The fact that $s^{*}>0$ directly shows that the large, non-Ohmic current
between the two superfluids is itself not a pure supercurrent in the context
of a two-fluid model [5]. The observation that the flow is resistive is
insufficient alone to conclude that it is not superfluid as there are many
mechanisms for resistance to arise in a pure supercurrent [46, 4]. The
observation that $s^{*}\gg s$ suggests that the channel is far from
equilibrium and hydrodynamics breaks down as is often the case in weak link
geometries [5] unlike previous assumptions [45, 38]. We discuss in the
supplement how the degree to which hydrodynamics breaks down depends on the
preparation of the system [39]. The large entropy current suggests an
irreversible conversion process from superfluid currents in the contacts to
normal currents in the channel and back to superfluid, or the propagation of
normal currents originating in the normal regions of the reservoirs through
the superfluids while remaining normal. Moreover, the independence of $s^{*}$
from $\nu_{x}$ implies that this process is independent of the channel
geometry. There is an analogy between this observation and the central result
of Landauer-Büttiker theory that the conductance through a ballistic channel
is also independent of the geometry and depends only on the channel’s
transmission and the number of propagating modes.

Figure 3: Competition between two modes of entropy transport: nonlinear
advection and linear diffusion. A pure entropy imbalance induces currents of
particles (a) and entropy (b) which are dominated by the nonlinear advective
mode at early time and the linear diffusive mode at long times. For strong
confinement (red) the system reaches a non-equilibrium steady state which
persists beyond the experimentally accessible time, indicating the decoupling
of advective and diffusive entropy transport. Reducing the confinement
facilitates the diffusive mode, allowing equilibration and reducing the
maximal response. A fit with our phenomenological model (solid lines)
describes the data well over the explored parameter regime. The competition
between the two modes, which changes with $\nu_{x}$, determines the path
traced through state space (d). Error bars (data points) indicate the standard
deviation (average) of $3-5$ repetitions.

In a second experiment, whose results are presented in Fig. 3, we prepare the
system with a nearly pure entropy imbalance [$\Delta N(0)\approx 0,\,\Delta
S(0)\neq 0$, open circle in Fig. 1(b)]. The initial response of $\Delta N$ and
$\Delta S$ is clearly non-exponential, resembling the advective dynamics in
the first experiment with the same $s^{*}$, while the dynamics that follow are
much slower and consistent with exponential relaxation and therefore linear
response. With decreasing $\nu_{x}$, both dynamics become faster and the
maximum values of $\Delta N$ and $\Delta S$ achieved at the turning point
become smaller. For the largest value of $\nu_{x}$, the initial response is
still fast while the relaxation that follows is extremely slow and resembles a
non-equilibrium steady state. In this state, the non-vanishing imbalances
$\Delta N$ and $\Delta S$ depend on the initial state as well as the path of
the system through state space determined by $s^{*}$, i.e., the system is non-
ergodic. This shows that the non-equilibrium steady state previously observed
at higher temperatures [37] persists in the superfluid regime where the
current with which it is reached is $\gtrsim 6$ times larger and non-Ohmic.
Figure 3(d) shows the measured path in state space also illustrated in Fig.
1(b). It demonstrates that the path is determined by the competition between
the nonlinear and linear dynamics and varies with $\nu_{x}$, in contrast to
the first experiment [Fig. 2(d)].

In the following, we formulate a minimal phenomenological model to describe
our observations which are not captured by the linear response approach that
successfully describes this system in the normal state [37, 38] as it predicts
purely exponential relaxation. We therefore turn to the formalism of
generalised gradient dynamics [8], a generalisation of Onsager’s theory of
irreversible processes; see supplement [39]. While it does not provide a
microscopic theory, this formalism can describe general, irreversible, non-
equilibrium processes and provides a convenient way to impose macroscopic
constraints such as the Second Law of Thermodynamics and conservation laws for
the particle number and energy. Within this framework, we make the Ansatz

$\displaystyle\begin{split}I_{N}&=I_{\mathrm{exc}}\tanh\pqty{\frac{\Delta\mu+\alpha_{c}\Delta
T}{\sigma}}\\\ I_{S}&=\alpha_{c}I_{N}+G_{T}\Delta T/T\end{split}$ (1)

which produce entropy via the irreversible flow
$\derivative*{S}{t}=(I_{N}\Delta\mu+I_{S}\Delta T)/T$. The non-trivial result
that $\alpha_{c}$ appears in both $I_{N}$ and $I_{S}$ is a generalisation of
Onsager’s reciprocal relations to nonlinear response and is a consequence of
the irreversibility of these currents. The system exhibits two modes of
entropy transport: a nonlinear advective mode $I_{S}^{a}=\alpha_{c}I_{N}$
characterised by the excess current $I_{\mathrm{exc}}$, Seebeck coefficient
$\alpha_{c}$, and nonlinearity $\sigma$, wherein each transported particle
carries entropy $\alpha_{c}$ on average, and a linear diffusive mode
$I_{S}^{d}=G_{T}\Delta T/T$ characterised by the thermal conductance $G_{T}$
which enables entropy transport without net particle transport according to
Fourier’s law. The linear model is reproduced in the limit of large $\sigma$
with conductance $G=I_{\mathrm{exc}}/\sigma$. In a Fermi liquid, these two
modes are related by the Wiedemann-Franz law where the Lorenz number
$L=G_{T}/TG$ has the universal value $\pi^{2}k_{B}^{2}/3$. The nonlinearity
implies the breakdown of the Wiedemann-Franz law since the advective and
diffusive modes are no longer linked [37].

Figure 4: Characterising the linear and nonlinear transport modes by fitting
the phenomenological model across the 1D-2D crossover. Fitted parameters from
the first (second) experiment corresponding to Fig. 2 (Fig. 3) are plotted
with filled (open) markers here. (a) The excess current normalised to the
superfluid gap. (b) The thermal and spin conductances normalised to the non-
interacting values $G_{T}^{0}=\pi^{2}k_{B}^{2}T/3h$, $G_{\sigma}^{0}=1/h$ for
a single transverse mode (see main text). For reference, the number of
occupied transverse modes in the channel is also shown in (a) and (b). (c) The
Seebeck coefficient (advectively transported entropy per particle) extracted
directly from the fits to the first experiment. (d) The slope of the path
through state space in the first experiment and in the advective ($\square$)
and diffusive ($\diamond$) limits of the second experiment.

Fig. 4 shows the parameters of the model as functions of $\nu_{x}$ extracted
from the fits shown as curves in Figs. 2 and 3. Panel (a) shows
$I_{\mathrm{exc}}$ normalised by $\Delta/h$ along with the number of occupied
transverse modes at equilibrium $n_{m}$ [39]. Filled (open) circles were
extracted from the first (second) experiment. $I_{\mathrm{exc}}$ follows
$n_{m}\Delta/h$, increasing as $\nu_{x}$ decreases until
$\nu_{x}\approx$4\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ where it plateaus,
likely due to additional resistances in series with the 1D region. The fitted
$I_{\mathrm{exc}}$ is apparently reduced in the second experiment relative to
the first because the initial current is suppressed by the diffusive mode,
making it more difficult to fit. It is intriguing that, consistent with
previous studies [32, 33], the equilibrium superfluid gap $\Delta$ within the
local density approximation is still the relevant scale for the current,
despite the evidence that the channel region is far from equilibrium.

Fig. 4(b) shows $G_{T}$ and the spin conductance $G_{\sigma}$ separately
measured in the same system by preparing a pure spin imbalance [39] normalised
to their values for the single-mode non-interacting ballistic QPC
$G_{T}^{0}=\pi^{2}k_{B}^{2}T/3h,\,G_{\sigma}^{0}=1/h$. Both conductances are
suppressed relative to the non-interacting values and increase monotonically
with decreasing $\nu_{x}$, possibly due to the appearance of non-degenerate
transverse modes at the edges of the channel [39]. The non-equilibrium steady
state arises from the fact that $G_{T}\rightarrow 0$ in the 1D limit. In the
two-dimensional limit, $G_{\sigma}$ remains more suppressed than $G_{T}$,
suggesting that more types of excitations can contribute to diffusive entropy
transport than spin transport, e.g., both collective phonon and quasiparticle
excitations can contribute to $G_{T}$ [47] while only quasiparticle
excitations can contribute to $G_{\sigma}$ [48]. Fig. 4(c) shows the fitted
Seebeck coefficient $\alpha_{c}$ while (d) shows the slope of the path through
state space $\derivative*{\Delta S}{\Delta N}$ during the advective and
diffusive dynamics. The fitted $\alpha_{c}$ and $\derivative*{\Delta S}{\Delta
N}$ match for the purely advective transport in the first experiment, showing
that $\alpha_{c}$ is remarkably insensitive to $\nu_{x}$, while
$\derivative*{\Delta S}{\Delta N}$ more clearly shows how the two modes
compete in the second experiment to determine the net response of the system.

In summary, we have observed that the conceptually simple system of two
superfluids connected by a ballistic channel exhibits the highly non-intuitive
and currently unexplained effect that the presence of superfluidity increases
the rate of irreversible entropy transport between them via nonlinear
advection. This contrasts with the more familiar case of superfluid and
superconducting tunnel junctions where the reversible, entropy-free Josephson
current dominates. The entropy advectively carried per particle is nearly
independent of the channel’s geometry, while the timescales of advective and
diffusive transport depends strongly thereon, raising the question of the
microscopic origin of the observed entropy transported per particle
$s^{*}\simeq 1k_{B}$. [37, 38]. Our phenomenological model that captures these
observations, in particular the identification of advective $I_{S}^{a}\propto
I_{N}$ and diffusive $I_{S}^{d}\propto\Delta T$ modes along with the sigmoidal
shape of $I_{N}(\Delta\mu+\alpha_{c}\Delta T)$, may help guide future
microscopic theories of the system. A complete understanding of the particle
and entropy transport in superfluid systems is essential for both fundamental
and technological purposes.

## Acknowledgements

We thank Alexander Frank for his contributions to the electronics of the
experiment. We are grateful for inspiring discussions with the Madrid Quantum
Transport Group (Alfredo Levy Yeyati, Gorm Steffensen, Francisco Matute-
Cañadas, Pablo Burset Atienza, Rafael Sánchez), Panagiotis Christodoulou,
Sarang Gopalakrishnan, Anne-Maria Visuri, Shun Uchino, Thierry Giamarchi,
Alberto Montefusco, Boris Svistunov, and Selim Jochim. We thank Alex Gómez
Salvador and Eugene Demler for their productive and ongoing collaboration in
investigating this system. We acknowledge the Swiss National Science
Foundation (Grants No. 212168, UeM019-5.1, 182650, and NCCR-QSIT) and European
Research Council advanced grant TransQ (Grant No. 742579) for funding.

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## Supplemental Information

##  S1 Breakdown of hydrodynamics

The two-fluid model states that the total particle current is the sum of the
normal and superfluid (Josephson) components $I_{N}=I_{N}^{n}+I_{N}^{s}$ while
the entropy current is carried advectively by only the normal current
$I_{S}=s_{n}^{*}I_{N}^{n}$. We observe that the entropy current is
proportional to the total particle current $I_{S}=s^{*}I_{N}$ which, when
combined with the two previous equations, yields
$I_{N}^{s}=(s_{n}^{*}/s^{*}-1)I_{N}^{n}$. The normal particle current
$I_{N}^{n}$ is given by the normal particle current density $j_{n}$ integrated
over the cross section at the centre of the channel and the entropy current is
given by $j_{n}$ weighted by the local entropy per particle

$\begin{split}I_{N}^{n}&=\int\differential{x}\differential{z}j_{n}(x,y=0,z)\\\
I_{S}&=\int\differential{x}\differential{z}j_{n}(x,y=0,z)s(x,y=0,z).\end{split}$
(S2)

If the system is hydrodynamic, the local entropy per particle is given by its
value near equilibrium [49, 50], so the entropy transported per particle by
the normal current would therefore be $s_{n}^{*}=I_{S}/I_{N}^{n}\sim
s\approx$7.2\text{\times}{10}^{-4}$k_{B}$. This is several orders of magnitude
smaller than the observed entropy transported per particle
$s^{*}=1.18(3)k_{B}$, implying that $I_{N}^{s}\approx-I_{N}^{n}$, i.e., the
normal and superfluid currents flow in opposite directions and their small
difference is the observed net current. Depending on whether or not the
current-phase relation is multi-valued [1], the time-averaged supercurrent at
finite $\Delta\mu$ either vanishes by undergoing reversible, adiabatic phase
slips (AC Josephson effect) [51, 52] or flows in the same direction as the
normal current by irreversibly nucleating topological excitations such as
vortices [29]. A simple picture of this process is that the force
$\propto-\Delta\mu$ accelerates the superfluid until it reaches its critical
velocity, at which point it converts some kinetic energy to heat, then begins
to accelerate again and the process repeats [5]. This contradiction–that our
observations imply $I_{N}^{s}\approx-I_{N}^{n}$ if the system is hydrodynamic
while the equations of motion for the superfluid require
$\mathrm{sign}(I_{N}^{s})=\mathrm{sign}(I_{N}^{n})$–shows that the
hydrodynamic two-fluid model, namely that the transported entropy is
determined by the near-equilibrium local entropy in the contacts, is not valid
in the channel despite the internal equilibrium of the reservoirs sufficiently
far from the channel (Sec. S2). This argument does not rely on the precise
values of $s$ and $s^{*}$, just on the fact that $s<s^{*}$.

As shown in the main text, superfluidity in the reservoirs increased the speed
of entropy transport between them. In a bulk hydrodynamic system, entropy
transport is faster in a superfluid, where entropy propagates as waves of
second sound [24], than in a normal fluid where it propagates diffusively
[21]. Because it is a wave, second sound is in principle a reversible
mechanism of particle and entropy transport, though the damping that often
accompanies it [23] could allow it to transport entropy irreversibly as we
observe here. Because the gas in the channel is far from equilibrium, second
sound in a strict hydrodynamic sense [53] cannot propagate through it.
Nevertheless, second sound in a more general, far-from-equilibrium sense
referring to any relative motion between the normal and superfluid components
induced by fundamental excitations [54] may be present.

The channel and wall beams (Sec. S3) are on during the preparation of the
system (Sec. S4), while the gate beam is ramped on over
$10\text{\,}\mathrm{m}\mathrm{s}$ followed by $5\text{\,}\mathrm{m}\mathrm{s}$
of settling before the wall is switched off to allow transport. We have
verified that our observations of the advective mode, quantified by the values
of $I_{\mathrm{exc}}$, $\sigma$, and $s^{*}$, remain unchanged for settling
times after the gate beam ramp time up to $0.5\text{\,}\mathrm{s}$, much
longer than the advective relaxation time (cf. Fig. 1). Longer settling times
up to $3.5\text{\,}\mathrm{s}$ lead to slower, more exponential relaxation of
$\Delta N$ (smaller $I_{\mathrm{exc}}$ and larger $\sigma$) and reduction of
$s^{*}$ by up to $\sim 10\%$, suggesting that the non-equilibrium nature of
the nominally superfluid channel is crucial for the observed phenomena.
Indeed, using a high-resolution imaging system focused on the channel, we
observe that the density in the region addressed by the gate previously
referred to as pockets [37] increases very slowly after the ramp on a
timescale of several seconds, consistent with the settling times needed to
induce exponential relaxation. The fact that the observed nonlinear dynamics
are much faster than the local equilibration time of the channel and remain
unchanged over a large range of preparation times suggests that the far-from-
equilibrium state of the channel is robust and well-defined by the equilibrium
reservoirs and channel geometry.

##  S2 Generalised gradient dynamics

To ensure that our phenomenological nonlinear model satisfies some basic
properties such as the Second Law of Thermodynamics and the conservation of
particles and energy, it is useful to formulate it in terms of a dissipation
potential $\Xi$ [55, 8]. Both Onsager’s theory of irreversible processes [56]
and the more general formalism of gradient dynamics [8] identify thermodynamic
forces and fluxes which determine the entropy production rate of a system
$\dot{S}=\derivative*{S}{t}$. This production rate $\dot{S}$ can be written in
terms of thermodynamic observables from the first law of thermodynamics
$\differential{S}=\sum_{i}x_{i}\differential{X}_{i}$ which relates the change
in a system’s entropy due to changes in the extensive quantities that
characterise the system $X_{i}$ via their conjugate intensive variables
$x_{i}$ [56], e.g., chemical potential, temperature, $s$-wave scattering
length, normal and superfluid velocities, trap frequency, channel beam powers,
etc. For a spin-balanced gas at unitarity in a static trap, this is

$\differential{S}=\frac{1}{T}\differential{U}-\frac{\mu}{T}\differential{N}.$
(S3)

By dividing both sides of the equation by the time differential
$\differential{t}$, we find the entropy production rate

$\dot{S}=\derivative{S}{t}=\frac{1}{T}\derivative{U}{t}-\frac{\mu}{T}\derivative{N}{t}$
(S4)

which has the interpretation that if energy and atoms are added/removed at
rates $\dot{U}$ and $\dot{N}$ while $\mu$ and $T$ are held constant, then the
entropy increases/decreases at a rate $\dot{S}$ due to the thermalization of
non-equilibrium excitations produced by $\dot{U}$ and $\dot{N}$. Since the
total entropy of the system is the sum of the entropies of the left and right
reservoirs, the net entropy production rate

$\dot{S}=\dot{S}_{L}+\dot{S}_{R}$ (S5)

is given in terms of the production rates in the two reservoirs

$\dot{S}_{i}=\frac{1}{T_{i}}\dot{U}_{i}-\frac{\mu_{i}}{T_{i}}\dot{N}_{i}.$
(S6)

Because the total energy and atom number is conserved

$\begin{split}\dot{U}&=\dot{U}_{L}+\dot{U}_{R}=0\\\
\dot{N}&=\dot{N}_{L}+\dot{N}_{R}=0,\end{split}$ (S7)

their addition/removal rates in each reservoir are given by the conserved
currents flowing between them

$\begin{split}I_{U}&=-\frac{1}{2}(\dot{U}_{L}-\dot{U}_{R})=-\frac{1}{2}\derivative{\Delta
U}{t}\\\
I_{N}&=-\frac{1}{2}(\dot{N}_{L}-\dot{N}_{R})=-\frac{1}{2}\derivative{\Delta
N}{t}\end{split}$ (S8)

Combining this with Eq. S5 and S6, we find that $\dot{S}$ can be written
equivalently in terms of either $I_{N}$ and $I_{U}$ or, without any
approximations, $I_{N}$ and the apparent entropy current
$I_{S}=-(1/2)\derivative*{\Delta S}{t}$

$\dot{S}=I_{N}\Delta\pqty{\frac{\mu}{T}}+I_{U}\Delta\pqty{-\frac{1}{T}}=\frac{I_{N}\Delta\mu+I_{S}\Delta
T}{T}$ (S9)

where $T=(T_{L}+T_{R})/2$. This means we can formulate a theory with either
the fluxes $I_{N}$ and $I_{U}$ driven by the forces
$\Delta(\mu/T)=\mu_{L}/T_{L}-\mu_{R}/T_{R}$ and
$\Delta(-1/T)=-1/T_{L}+1/T_{R}$ or the fluxes $I_{N}$ and $I_{S}$ driven by
the forces $\Delta\mu/T$ and $\Delta T/T$ or simply $\Delta\mu$ and $\Delta
T$. We chose the latter as the entropy is more natural to measure given our
experimental sequence (the energy is not conserved by the channel beam ramps
but entropy is) and because it gives deeper insight into the nature of the
flow between the two reservoirs. The vector of state variables is therefore
$\mathbf{X}=(N,U,\Delta N,\Delta S)$ and the vector of their conjugate
quantities are
$\mathbf{x}=\partialderivative*{S}{\mathbf{X}}=(-\mu/T,1/T,-\Delta\mu/2T,-\Delta
T/2T)$, though practically they can be reduced to $\mathbf{X}=(\Delta N,\Delta
S)$ and $\mathbf{x}=(-\Delta\mu/2T,-\Delta T/2T)$ because $\dot{N}=\dot{U}=0$.
The energy and entropy currents are related by

$I_{U}=\pqty{T-\frac{\Delta T^{2}}{4T}}I_{S}+\pqty{\mu-\frac{\Delta\mu\Delta
T}{4T}}I_{N}$ (S10)

so, for small biases $\Delta T\ll T,\Delta\mu\ll\mu$, the entropy current is
approximately given by the heat current

$I_{Q}=I_{U}-\mu I_{N}\approx TI_{S}.$ (S11)

The fluxes are defined as derivatives of the dissipation potential $\Xi$ with
respect to the forces

$\displaystyle I_{N}=\partialderivative{\Xi}{\Delta(\mu/T)}\mbox{\quad
and\quad}I_{U}=\partialderivative{\Xi}{\Delta(-1/T)}$ or (S12) $\displaystyle
I_{N}=T\partialderivative{\Xi}{\Delta\mu}\mbox{\quad
and\quad}I_{S}=T\partialderivative{\Xi}{\Delta T}$

Since $\Xi$ is a single-valued scalar potential function of two scalars,
Onsager reciprocity

$\partialderivative{I_{N}}{\Delta(-1/T)}=\partialderivative{I_{U}}{\Delta(\mu/T)}\mbox{\quad
or\quad}\partialderivative{I_{N}}{\Delta
T}=\partialderivative{I_{S}}{\Delta\mu}$ (S13)

is naturally fulfilled in this formalism at any $\Delta(\mu/T),\Delta(-1/T)$
or $\Delta\mu,\Delta T$, not just in the linear response regime near
equilibrium as is usually the case when imposing this condition.

To formulate equations of motion for the evolution in state space $\Delta
N(t),\Delta S(t)$, we need to relate $\Delta\mu,\Delta T$ to $\Delta N,\Delta
S$ via the reservoirs’ equation of state (EoS). For the relatively small
imbalances used here, we can linearise the EoS in terms of thermodynamic
response functions

$\pmqty{undef}\approx\kappa\pmqty{undef}\pmqty{undef}$ (S14)

where $\kappa$ is the compressibility, $\alpha_{r}$ is the dilatation
coefficient, and $\ell_{r}$ is the “Lorenz number” of the reservoirs [57].
Within the same approximation, the energy and entropy imbalance are related by
the average chemical potential $\mu=(\mu_{L}+\mu_{R})/2$ and temperature
$T=(T_{L}+T_{R})/2$

$\Delta U\approx\mu\Delta N+T\Delta S.$ (S15)

This formalism obeys the principle that $S$ is a state variable: $S$ depends
only on $N$, $U$, $\Delta N$, and $\Delta S$ but not on the path the system
travelled in state space. In other words, the entropy produced by
equilibration is the entropy difference between the initial and final states

$\begin{split}\delta
S&=\int_{0}^{\infty}\differential{t}\dot{S}(t)=\int_{0}^{\infty}\differential{t}\frac{I_{N}(t)\Delta\mu(t)+I_{S}(t)\Delta
T(t)}{T(t)}\\\
&=\int_{0}^{\infty}\differential{t}\dot{\mathbf{X}}\cdot\mathbf{x}=\int_{\mathbf{X}_{0}}^{\mathbf{X}_{\infty}}\differential{\mathbf{X}}\cdot\partialderivative{\Xi}{\mathbf{X}}\\\
&=S(N,U,0,0)-S(N,U,\Delta N_{0},\Delta S_{0})\end{split}$ (S16)

and is independent of the microscopic processes and current-bias
characteristics of the system contained in $\Xi$. If we use the linearised
reservoir response in Eq. S14, this becomes

$\delta S\approx\frac{\Delta N_{0}^{2}}{4T\kappa}+\frac{(\Delta
S_{0}-\alpha_{r}\Delta N_{0})^{2}}{4T\ell_{r}\kappa}$ (S17)

which, together with $S(N,U,0,0)$, is the expression plotted in Fig. 1(c). The
interpretation of this result is that there is initially some potential energy
in the system due to the imbalances, like the charging energy of a capacitor,
which, due to the Joule heating from irreversible flow through the resistive
channel, is converted into heat energy, i.e., entropy. We have verified that
the system is nearly closed by ensuring the absence of particle loss
$\absolutevalue{\derivative*{N}{t}}/N<$0.01\text{\,}\mathrm{s}^{-1}$$ and
heating in equilibrium
$\derivative*{(S/Nk_{B})}{t}<$0.02\text{\,}\mathrm{s}^{-1}$$ due to, e.g.,
photon scattering from the beams, vacuum background collisions, and three-body
recombination [58]. The total duration of the sequence is kept constant,
varying only the amount of time the channel is open, so the entropy production
plotted in Fig. 2(c) and Fig. 3(c) includes the small and constant offset
coming from residual heating not due to irreversible transport.

This formalism with the state vector $\mathbf{X}=(N,U,\Delta N,\Delta S)$ only
applies when the reservoirs are in internal equilibrium, i.e., the time
evolution is quasi-stationary. If, for example, they were hydrodynamic and
their breathing modes were excited, then the amplitudes and phases of these
modes would have to be added to the state vector $\mathbf{X}$ to completely
characterise the state of the system. Once the complete set of experimentally
distinguishable degrees of freedom are added to $\mathbf{X}$, this formalism
applies again even away from equilibrium. The fact that our observations are
consistent with a description using only these four state variables of the
reservoirs confirms the assumptions that the atom number and energy are
conserved and that the reservoirs are in internal equilibrium during
transport. It is therefore natural to ask whether this formalism still applies
when the channel is far from equilibrium and in principle contains many
degrees of freedom that must be added to $\mathbf{X}$. Indeed it does because
the state of the channel is fixed by its boundary conditions imposed by the
equilibrium reservoirs. Therefore, each additional degree of freedom of the
channel is a function of only the four state variables in $\mathbf{X}$ and
need not be added for a complete description.

Generalised gradient dynamics can only account for irreversible dynamics,
however a phase bias $\Delta\phi$ between the superfluids can in principle
drive a reversible Josephson supercurrent $I_{N}\sim I_{c}\sin(\Delta\phi)$
which transports and produces no entropy [59]. Our observations show that the
dynamics are irreversible in nature since the particle current both transports
and produces entropy and we observe no undamped oscillations characteristic of
the Josephson effect in a finite system [25, 26]. It is nevertheless somewhat
surprising that we do not observe this effect, however this is likely a
consequence of the high transmission of the channel. In a ballistic quantum
point contact (QPC), the critical supercurrent
$I_{c}^{\mathrm{QPC}}=\Delta/\hbar$ [60] and the excess normal current
$I_{\mathrm{exc}}^{\mathrm{QPC}}=16\Delta/3h$ [61] are comparable. This normal
current can flow at small bias $\Delta\mu/\Delta$ in ballistic channels but
not tunnel junctions [62, 61] and can therefore shunt the supercurrent and
damp reversible oscillations [44]. If both irreversible and reversible
dynamics are significant, then they can both be incorporated into a
phenomenological theory in the form of a general equation for non-equilibrium
reversible-irreversible coupling (GENERIC) [63, 8].

In the normal system where linear response theory is valid [37, 38], the
dissipation potential and currents it generates are

$\begin{split}\Xi_{n}&=\frac{G}{2T}(\Delta\mu+\alpha_{c}\Delta
T)^{2}+\frac{G_{T}}{2}\pqty{\frac{\Delta
T}{T}}^{2}=\frac{G}{2T}[(\Delta\mu+\alpha_{c}\Delta T)^{2}+L\Delta
T^{2}]=\Xi_{n}^{a}+\Xi_{n}^{d}\\\
\pmqty{undef}&=G\pmqty{undef}\pmqty{undef}.\end{split}$ (S18)

This model exhibits a linear advective mode of entropy transport
$I_{S}^{a}=\alpha_{c}G(\Delta\mu+\alpha_{c}\Delta T)=\alpha_{c}I_{N}$
characterised by the conductance $G$ and Seebeck coefficient $\alpha_{c}$, and
a linear diffusive mode $I_{S}^{d}=G_{T}\Delta T/T$ characterised by the
thermal conductance $G_{T}$ or equivalently the Lorenz number $L=G_{T}/TG$.

In the superfluid system, the experimental observation $I_{S}=s^{*}I_{N}$
simplifies the problem of identifying $\Xi$ since it implies the existence of
an advective mode $\Xi_{s}^{a}$. Eq. S2 imposes that $\Xi_{s}^{a}$ must
satisfy $(\partialderivative*{\Xi_{s}^{a}}{\Delta
T})_{\Delta\mu}=s^{*}(\partialderivative*{\Xi_{s}^{a}}{\Delta\mu})_{\Delta
T}$, which has the unique solution $\Xi_{s}^{a}(\Delta\mu,\Delta
T)=\Xi_{s}^{a}(\Delta\mu+s^{*}\Delta T)$. Based on our observations that
$I_{N}$ is a sigmoidal function of $\Delta\mu$ [32, 33], we make the Ansatz

$\Xi_{s}^{a}=\frac{\sigma
I_{\mathrm{exc}}}{T}\log\bqty{\cosh\pqty{\frac{\Delta\mu+s^{*}\Delta
T}{\sigma}}}\,.$ (S19)

In fact, we find that the simplest form that describes our observations from
both the first and second experiment is the sum of the nonlinear advective and
linear diffusive modes $\Xi=\Xi_{s}^{a}+\Xi_{n}^{d}$, which generates the
currents

$\displaystyle\begin{split}I_{N}&=I_{\mathrm{exc}}\tanh\pqty{\frac{\Delta\mu+\alpha_{c}\Delta
T}{\sigma}}\\\ I_{S}&=\alpha_{c}I_{N}+G_{T}\Delta T/T\end{split}$ (S20)

where the Seebeck coefficient $\alpha_{c}$ takes the place of the entropy
advectively transported per particle $s^{*}$. It is readily verifiable that
$\Xi_{n}^{a}$, $\Xi_{n}^{d}$, and $\Xi_{s}^{a}$ satisfy the six formal
criteria of dissipation potentials [8]. Eq. S20 shows that entropy transport
in the superfluid system has the same form as the normal system but the
advective mode now responds nonlinearly to the thermodynamic force
$\Delta\mu+\alpha_{c}\Delta T$ and is characterised by the excess current
$I_{\mathrm{exc}}$ [33], originally defined as the additional current above
the normal, Ohmic current in superconducting tunnel junctions [28], and the
sharpness of the nonlinearity $\sigma$. Ballistic QPCs of superconductors and
superfluid 3He exhibit similar current-bias characteristics where the
nonlinearity is determined by the highest order multiple Andreev reflection
process the QPC can support, i.e., the maximum number of Cooper pairs that can
be transported in a single coherent process
$\sigma\sim\Delta/n_{\mathrm{pair}}$ [28, 35]. The nonlinearity implies the
breakdown of the Wiedemann-Franz law since the advective and diffusive modes
are no longer linked. The linear model (Eq. S18) is reproduced in the limit of
large $\sigma$ with $G=I_{\mathrm{exc}}/\sigma$. Our procedure to fit this
model to the data is described in Sec. S6. Our fits are also consistent with
the coexistence of both linear and nonlinear advective modes
$\Xi=\Xi_{s}^{a}+\Xi_{n}^{a}+\Xi_{n}^{d}$ so we cannot exclude a finite linear
advective mode $\Xi_{n}^{a}$ but find that the nonlinear advective mode
$\Xi_{s}^{a}$ adequately describes our observations, yielding reduced chi-
square statistics slightly below 1.

##  S3 Potential energy landscape

The total potential energy landscape is a combination of the trap, the channel
beams, and the spatially-varying zero point energy of the channel beams’
confinement
$V(\mathbf{r})=V_{\mathrm{trap}}(\mathbf{r})+V_{\mathrm{ch}}(\mathbf{r})+V_{\mathrm{ZPE}}(y)$.
Our trap is a combination of a Gaussian optical dipole trap with wavelength
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$ and a magnetic field oriented
along $z$ with harmonic curvature, giving a combined trap potential

$\begin{split}V_{\mathrm{trap}}(\mathbf{r})&=V_{\mathrm{mag}}(\mathbf{r})+V_{\mathrm{dip}}(\mathbf{r})\\\
V_{\mathrm{mag}}(\mathbf{r})&=\frac{1}{2}m(2\pi\nu_{\mathrm{mag},y})^{2}(x^{2}+y^{2}-2z^{2})\\\
V_{\mathrm{dip}}(\mathbf{r})&=V_{\mathrm{dip}}^{0}\quantity{1-\frac{w_{\mathrm{dip},x,0}w_{\mathrm{dip},z,0}}{w_{\mathrm{dip},x}(y)w_{\mathrm{dip},z}(y)}\exp\bqty{-2\frac{x^{2}}{w_{\mathrm{dip},x}^{2}(y)}-2\frac{z^{2}}{w_{\mathrm{dip},z}^{2}(y)}}}\end{split}$
(S21)

where $w_{\mathrm{dip},x/z}(y)$ follows the usual hyperbolic Gaussian beam
width divergence,
$\nu_{\mathrm{mag},y}=$28.30(2)\text{\,}\mathrm{H}\mathrm{z}$$,
$w_{\mathrm{dip},x}(0)=w_{\mathrm{dip},z}(0)=$80.7(1)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$V_{\mathrm{dip}}^{0}=$1.33(1)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$\times
k_{B}$. When $\mu,k_{B}T\ll V_{\mathrm{dip}}^{0}$, the trap is approximately
harmonic with confinement frequencies in all three spatial directions
$\nu_{\mathrm{trap},x}=$171(1)\text{\,}\mathrm{H}\mathrm{z}$$,
$\nu_{\mathrm{trap},y}=$28.31(2)\text{\,}\mathrm{H}\mathrm{z}$$,
$\nu_{\mathrm{trap},z}=$164(1)\text{\,}\mathrm{H}\mathrm{z}$$ and the average
trap frequency
$\bar{\nu}_{\mathrm{trap}}=$92.7(5)\text{\,}\mathrm{H}\mathrm{z}$$.

A pair of repulsive $\text{TEM}_{01}$-like beams propagating along $x$ and
$z$, which we call the lightsheet (LS) and wire respectively, intersecting at
the centre of the trapped cloud separate it into two reservoirs connected by a
channel. The transverse confinement frequencies at their centre are
$\nu_{z}=$9.42(6)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$
($k_{B}T/h\nu_{z}=0.21$) and $\nu_{x}$ ranges from
$0.606(1)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ to
$12.4(2)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
($k_{B}T/h\nu_{x}\in[0.16,3.3]$) depending on the power of the beam such that
we can explore the crossover from one dimension (1D) to two dimensions (2D) in
the channel. An attractive Gaussian beam propagating along $z$ acts as a gate
potential in the channel. The combined potential energy landscape of these
three beams is

$\begin{split}V_{\mathrm{ch}}(\mathbf{r})&=V_{\mathrm{LS}}(\mathbf{r})+V_{\mathrm{wire}}(\mathbf{r})+V_{\mathrm{gate}}(\mathbf{r})\\\
V_{\mathrm{LS}}(\mathbf{r})&=f_{\mathrm{LS}}(y)\quantity{V_{\mathrm{LS}}^{0}+\frac{\pi^{3}m(\nu_{\mathrm{LS},z}w_{\mathrm{LS},z})^{2}}{2}\bqty{\mathrm{erfi}(z/w_{\mathrm{LS},z})}^{2}f_{\mathrm{LS}}(z)}\\\
V_{\mathrm{wire}}(\mathbf{r})&=f_{\mathrm{wire}}(y)\quantity{V_{\mathrm{wire}}^{0}+m(\pi\nu_{\mathrm{wire},x}w_{\mathrm{wire}}^{\mathrm{notch}})^{2}f_{\mathrm{wire}}(x)\bqty{1-f_{\mathrm{wire}}^{\mathrm{notch}}(x)}}\\\
V_{\mathrm{gate}}(\mathbf{r})&=f_{\mathrm{gate}}(y)f_{\mathrm{gate}}(x)V_{\mathrm{gate}}^{0}\end{split}$
(S22)

The envelope function for each beam $f_{b}(y)\propto
P_{b}e^{-2(y/w_{b,y})^{2}}$ determine their spatial profile along the
respective direction and are proportional to the power of each beam $P_{b}$.
The finite potential of the LS and wire at the origin
$V_{\mathrm{LS}/\mathrm{wire}}^{0}$ arise from optical aberrations that causes
their nodal planes to not be perfectly dark, but this is a small correction on
the order of 0.1% of the peak potential of each beam. The LS and wire are made
with repulsive $532\text{\,}\mathrm{n}\mathrm{m}$ light while the gate with
attractive $766.7\text{\,}\mathrm{n}\mathrm{m}$. The beam waists are
$w_{\mathrm{LS},y}=$30.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{LS},z}=$9.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{wire},x}=$78\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{wire},y}=$6.82\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{wire}}^{\mathrm{notch}}=$1.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{gate},x}=$30.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
$w_{\mathrm{gate},y}=$31.8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$,
and the peak confinement frequencies are
$\nu_{\mathrm{LS},z}=$9.42(6)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ and
$\nu_{\mathrm{wire},x}=$12.4(2)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ at
the reference beam powers used for calibration where
$f_{\mathrm{LS}/\mathrm{wire}}(0)=1$. The peak gate potential is
$V_{\mathrm{gate}}^{0}=-$2.17(1)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$\times
k_{B}$.

The confinement of the channel beams along $x$ and $z$ raise the zero point
energy (ZPE) of the atoms, effectively producing a new potential energy along
$y$ relative to the ZPE of the trap

$V_{\mathrm{ZPE}}(y)=\frac{h}{2}\left(\nu_{x}(y)+\nu_{z}(y)-(\nu_{\mathrm{trap},x}+\nu_{\mathrm{trap},z})\right)$
(S23)

with the transversal confinement
$\nu_{x}(y)=\sqrt{f_{\mathrm{wire}}(y)\nu_{\mathrm{wire},x}^{2}+f_{\mathrm{gate}}(y)\nu_{\mathrm{gate},x}^{2}+\nu_{\mathrm{trap},x}^{2}}$
and
$\nu_{z}(y)=\sqrt{f_{\mathrm{LS}}(y)\nu_{\mathrm{LS},z}^{2}+\nu_{\mathrm{trap},z}^{2}}$
and the confinement frequency of the gate beam given by
$\nu_{\mathrm{gate},x}=\sqrt{-V_{\mathrm{gate}}^{0}/4\pi^{2}mw_{\mathrm{gate},x}^{2}}$.
The higher lying modes defined by the transverse confinement that can
contribute to transport have energies determined by the quantum numbers of the
transverse harmonic oscillator states $n_{x/z}=0,1,\dots$ Their effective
potential energy landscape above the trap potential is

$V_{\mathrm{eff}}(y,n_{x},n_{z})=V_{\mathrm{ZPE}}(y)+V_{\mathrm{ch}}(0,y,0)+h\nu_{x}(y)n_{x}+h\nu_{z}(y)n_{z}$
(S24)

which determines number of occupied modes that contribute to transport

$n_{m}=\sum_{n_{x},n_{z}=0}^{\infty}\min_{y}\frac{1}{1+\exp\quantity{[V_{\mathrm{eff}}(y,n_{x},n_{z})-\mu]/k_{B}T}}.$
(S25)

The minimum occupation of each mode is used to account for modes that may be
occupied in the centre of the channel but are unoccupied elsewhere and
therefore non-propagating [38]. The particle conductance of a non-interacting
gas through a contact with perfect transparency is $G=n_{m}/h$ [64].

The wall beam, which is only on during preparation and imaging and not during
transport, is similar to the wire beam but without the notch cut out

$V_{\mathrm{wall}}(\mathbf{r})=f_{\mathrm{wall}}(y)V_{\mathrm{wall}}^{0}e^{-2(x/w_{\mathrm{wall},x})^{2}}$
(S26)

where the barrier height $V_{\mathrm{wall}}^{0}$ is larger than $\mu$ and
$k_{B}T$ to completely block transport. Its width is large enough to
completely suppress tunnelling [65].

The complete potential energy landscape $V(\mathbf{r})$ was used to produce
Fig. 1(a) via the local density approximation for the density
$n(\mathbf{r})=n[\mu-V(\mathbf{r}),T]$ [19, 43] that determines the local
Fermi temperature
$k_{B}T_{F}(\mathbf{r})=\hbar^{2}[3\pi^{2}n(\mathbf{r})]^{2/3}/2m$. For weak
wire beam powers, the transverse confinement frequency at the centre of the
channel $\nu_{x}=\nu_{x}(y=0)$ is small and the barriers at the edges of the
cloud disappear. In this case, the equilibrium degeneracy varies slows along
$x$ from deeply degenerate and superfluid in the centre of the channel to
weakly degenerate and normal at the edges of the channel to vacuum far from
the centre. Fig. S1(a) shows how the degeneracy varies with $x$ (in units of
the thermal de Broglie wavelength $\lambda_{T}=h/\sqrt{2\pi
mk_{B}T}=$2.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$) around
$\mathbf{r}=0$ for various powers of the wire beam. The horizontal dashed line
is the critical degeneracy at the superfluid transition
$(\mu/k_{B}T)_{c}=2.49$ [19]. This shows that in the 1D channel, the only
occupied modes are tightly confined and expected to be superfluid at
equilibrium, while in the 2D channel, weakly confined and normal transport
modes appear at the edges. The 2D contacts to the 1D channel are the most
degenerate regions in the system in equilibrium, see Fig. 1(a). We estimate
the superfluid gap here using the local density approximation applied to a
calculation of the gap in a homogeneous system $\Delta(\mu_{c},T)$ [66] where
$\mu_{c}=\max_{\mathbf{r}}[\mu-V(\mathbf{r})]$.

Figure S1: 1D-2D crossover of the channel. The equilibrium local degeneracy at
$y=z=0$ vs. $x$ in units of the thermal de Broglie wavelength
$\lambda_{T}=h/\sqrt{2\pi
mk_{B}T}=$2.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ for various
powers of the wire beam shows that for high powers (large $\nu_{x}$), the
channel is tightly confining and there are few occupied modes (regions with
degeneracy $>0$) while for low powers (small $\nu_{x}$), the confinement is
very weak and there are many occupied modes. Since, in the 2D limit, the
potential varies over a much larger length scale than $\lambda_{T}$, we can
apply a local density approximation picture and identify regions at large $x$
(the edges of the channel) which are occupied and can contribute to transport
but below the critical degeneracy to be superfluid (horizontal dashed line).

##  S4 Preparation of the initial state

As part of the experimental sequence, we ramp up the channel beams to separate
the two reservoirs then performed forced optical evaporation before allowing
transport by opening the wall. Using a magnetic field gradient along $y$, we
can shift the centre of the magnetic trap with respect to the channel beams
before separation to prepare $\Delta N(0)$ before shifting the centre back to
coincide with the channel beams for transport, which ensures that $\Delta N=0$
at equilibrium $\Delta\mu=0$. By shifting the centre after separation to a
different position during evaporation, we can compress one reservoir and
decompress the other, thereby changing their evaporation efficiencies and
inducing a controllable $\Delta S(0)$ as well. The average efficiency changes
only slightly with the imbalances such that the net entropy per particle
$S/Nk_{B}$ varies by $<9\%$ over the large range of $\Delta N(0)$ and $\Delta
S(0)$ we prepare. This method achieves much lower entropy–therewith enabling
the study of entropy transport in the superfluid phase–than a previously used
method [67, 37, 38] where evaporation was performed in the same trap as
separation, but additional entropy was injected into one reservoir by focusing
the gate beam into the reservoir and modulating the power.

Between the end of transport and the start of imaging, we ramp down the
channel beams while keeping the wall on. Typical images of the cloud at each
of these stages are shown in Fig. S2. For the experiments at all $\nu_{x}$
that contain only the advective modes (Fig. 2), we prepare a density imbalance
$\Delta N(0)$ and the corresponding entropy imbalance $\Delta S(0)$ such that
the system relaxes to equilibrium in the 1D channel
$\nu_{x}=$12.4(2)\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ at long times and
not the NESS. For the experiments with both the advective and diffusive modes
(Fig. 3), we fix $\Delta N(0)=0$ and prepare as large $\Delta S(0)$ as
possible without losing atoms.

To measure the spin conductance $G_{\sigma}$, we employ a similar method as in
[68] though instead of quenching the gradient along $y$, we modulate it at the
average trap frequency for the two spin states for half the period of the trap
frequency difference to induce a larger initial spin imbalance. We use the
recently computed equation of state of the finite-temperature polarised
unitary fermi gas [69] to compute a spin susceptibility $\chi\approx
0.32\kappa$ [33] in agreement with measurements on a harmonically-trapped
unitary fermi gas at low temperatures [70].

Figure S2: Absorption images of the system at various stages of the
experiment: just after separation into two reservoirs, during evaporation,
during transport without the gate beam, and during imaging. Each image is an
average over 10 shots of spin $\ket{\uparrow}$ and the magnified pixel size is
$4.3\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

##  S5 Reservoir thermodynamics

###  S5.1 Effect of the potential energy landscape

The potential energy landscape $V(\mathbf{r})$ described in Sec. S3 determines
the equilibrium thermodynamic properties of the reservoirs. In the local
density approximation [43], the grand canonical free energy of the full system
(both spins in both reservoirs) is given in terms of the universal pressure
equation of state of the unitary fermi gas [19]

$\Omega(\mu,T)=-\frac{k_{B}T}{\lambda_{T}^{3}}\int\differential[3]{r}f_{P}\bqty{\frac{\mu-V(\mathbf{r})}{k_{B}T}}\,.$
(S27)

While $\Omega$ takes a simple form for a harmonic trap, the trap is highly
anharmonic during transport due to the channel beams (especially the
lightsheet) and is slightly anharmonic during imaging due to the wall and
gaussian dipole trap. It is therefore a non-universal function of $\mu$, $T$,
and the powers of the various beam
$\Omega(\mu,T,P_{\mathrm{dip}},P_{\mathrm{LS}},P_{\mathrm{wire}},P_{\mathrm{gate}},P_{\mathrm{wall}})$.
The magnetic trap frequency $\nu_{y,\mathrm{mag}}$ is fixed at unitarity by
the geometry of the Feshbach coils. In a given trap defined by these powers,
the extensive quantities and response functions are given by derivatives of
$\Omega$

$\begin{split}N&=-\partialderivative{\Omega}{\mu}\\\
S&=-\partialderivative{\Omega}{T}\\\ U&=\Omega+\mu N+TS\\\
\end{split}\quad\quad\begin{split}\kappa&=-\frac{1}{4}\partialderivative[2]{\Omega}{\mu}\\\
\alpha_{r}&=-\frac{4}{\kappa}\partialderivative{\Omega}{\mu}{T}\\\
\ell_{r}&=-\frac{4}{\kappa}\partialderivative[2]{\Omega}{T}-\alpha_{r}^{2}.\end{split}$
(S28)

The factor of $1/4$ in $\kappa$ converts the compressibility of the whole
cloud in equilibrium $-\partialderivative*[2]{\Omega}{\mu}$ into the effective
compressibility of two half clouds (reservoirs) in parallel [71].

As explained in Sec. S3 and shown in Fig. S2, the potential energy landscape
$V(\mathbf{r})$ differs between transport, where we wish to know the response
coefficients, and imaging, where we measure $N_{i},S_{i}$, so the EoS of the
gas $\Omega(\mu,T)$ is different in these two conditions. However, the two are
connected because $N_{i},S_{i}$ are the same in both the transport and imaging
conditions as a consequence of the adiabaticity of the channel beam ramps.
This fact can be used to determine the chemical potential and temperature in
each reservoir during transport $\mu_{i}^{\mathrm{tr}},T_{i}^{\mathrm{tr}}$
from their values extracted in the imaging condition
$\mu_{i}^{\mathrm{im}},T_{i}^{\mathrm{im}}$ using the EoS of each reservoir in
the two configurations
$\Omega^{\mathrm{tr}}(\mu_{i}^{\mathrm{tr}},T_{i}^{\mathrm{tr}})/2$ and
$\Omega^{\mathrm{im}}(\mu_{i}^{\mathrm{im}},T_{i}^{\mathrm{im}})/2$ that can
be computed from the known $V^{\mathrm{tr}}(\mathbf{r})$ and
$V^{\mathrm{im}}(\mathbf{r})$. The response coefficients can then be computed
from $\mu^{\mathrm{tr}}=(\mu_{L}^{\mathrm{tr}}+\mu_{R}^{\mathrm{tr}})/2$,
$T^{\mathrm{tr}}=(T_{L}^{\mathrm{tr}}+T_{R}^{\mathrm{tr}})/2$, and
$\Omega^{\mathrm{tr}}(\mu^{\mathrm{tr}},T^{\mathrm{tr}})$.

The lightsheet has the strongest influence on the reservoirs’ EoS since it is
large in comparison to the cloud as seen in Fig. S2. Its primary effect is to
increases the chemical potential by pushing the cloud to the edges of the trap
where the potential energy is larger. We observe this directly by imaging the
cloud in the transport configuration for various lightsheet beam powers and
see that, while the atom number in the system remains constant, atoms are
pushed away from the central region. The gate has a weaker but still
significant effect at the strong lightsheets used here: it draws about 7% of
the total number of atoms into the channel independently of $\nu_{x}$. The
local density approximation with the 3D EoS $f_{P}$ overestimates the fraction
of atoms drawn into the channel since the system is in fact quasi-2D or
quasi-1D in most of the channel where the density is lower at the same $\mu$
and $T$ [72, 73], though this has a small effect on the overall thermodynamics
as the 2D and 1D regions are small.

Using the procedure outlined above, we estimate the ratio of several
thermodynamic properties during transport to their values during imaging. We
estimate that $T^{\mathrm{tr}}$ is within 1% of $T^{\mathrm{im}}$ while
$\mu^{\mathrm{tr}}$ is 24% larger than $\mu^{\mathrm{im}}$. Furthermore,
$\kappa^{\mathrm{tr}}$ is within 1% of $\kappa^{\mathrm{im}}$ but
$\alpha_{r}^{\mathrm{tr}}$ and $\ell_{r}^{\mathrm{tr}}$ are respectively 3.4
and 2.6 times larger than their values during imaging. In short, $T$ and
$\kappa$ are essentially the same during transport as during imaging, with the
other parameters, especially the response functions, are much more sensitive
to the trap potential. This is consistent with our observation that the
conductance of the weakly-interacting gas is quantised in units of $1/h$ which
relies on an accurate value of $\kappa$ without incorporating these these
corrections [64]. Motivated by this finding and the fact that the potential
energy landscape we estimate is idealised and not exact (there are clear
asymmetries in the lightsheet in Fig. S2), we fix $T$ and $\kappa$ to the
measured values but fit $\alpha_{r}$ and $\ell_{r}$ along with the other
parameters of the model (Sec. S6).

###  S5.2 Thermometry from absorption images

It is not straightforward to apply standard thermometry techniques [74] to
absorption images of the reservoirs in the transport configuration including
all the channel beams (see Fig. S2) since it is difficult to precisely
characterise the complex potential energy landscape that determines the
reservoirs’ EoS. We therefore ramp the channel beams down between the end of
transport and the beginning of imaging such that the potential energy
landscape is well-known and harmonic. This ramp is performed adiabatically
such that the atom number and entropy in each reservoir remains constant even
though their energies change due to the work done by the time-varying beams.
This was confirmed by ensuring that $N_{i}$ and $S_{i}/N_{i}k_{B}$ are the
same for slower ramps.

At the end of each shot, we obtain the column density
$n_{i\sigma}^{\mathrm{col}}(y,z)$ of both reservoirs $i=L,R$ and both spin
states $\sigma=\downarrow,\uparrow=\ket{1},\ket{3}$ from two absorption images
$225\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ apart taken in situ along
the $x$ axis with a calibrated imaging system. The atom number in each spin
state and reservoir is determined by integrating the column density over the
half-plane of each reservoir

$\begin{split}N_{L\sigma}&=\int_{-\infty}^{0}\differential{y}\int_{-\infty}^{\infty}\differential{z}n_{L\sigma}^{\mathrm{col}}(y,z)\\\
N_{R\sigma}&=\int_{0}^{\infty}\differential{y}\int_{-\infty}^{\infty}\differential{z}n_{R\sigma}^{\mathrm{col}}(y,z).\end{split}$
(S29)

The centre $y=0$ is fixed to the centre of a gaussian fit to the density
profile in the wall region. We fit the degeneracy
$q_{i\sigma}=\mu_{i\sigma}/k_{B}T_{i\sigma}$ and temperature $T_{i\sigma}$ of
both reservoirs for each spin state using the EoS of the harmonically-trapped
gas

$n_{i\sigma}^{\mathrm{col}}(y,z)=\lambda_{Ti\sigma}^{-3}\int_{-\infty}^{\infty}\differential{x}f_{n}\bqty{q_{i\sigma}-\pqty{\frac{x}{R_{xi\sigma}}}^{2}-\pqty{\frac{y}{R_{yi\sigma}}}^{2}-\pqty{\frac{z-z_{0}}{R_{zi\sigma}}}^{2}}$
(S30)

where $\lambda_{Ti\sigma}=h/\sqrt{2\pi mk_{B}T_{i\sigma}}$ is the thermal de
Broglie wavelength, $f_{n}$ is the universal density EoS of a single spin in a
balanced unitary Fermi gas [19], and
$R_{ji\sigma}^{2}=2k_{B}T_{i\sigma}/m(2\pi\nu_{\mathrm{trap},j})^{2}$ is the
Gaussian thermal length given by the known trap frequencies and the fitted
temperature. In practice, we do not fix $R_{z}$ to $R_{y}$ via the trap
frequencies and the common temperature, but rely only on the calibrated and
almost harmonic magnetic trap frequency along $y$. We exclude the region
$\approx$60\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ wide containing
the wall from the fit, fix both reservoirs and spin states to have a common
centre $y=0$ fixed to the fitted centre of the wall, and fit $z_{0}$ with a
common value for the two reservoirs. We also fit a small rotation angle
between the camera’s CCD grid and the harmonic trap principle directions,
which we find to be $0.45(5)^{\circ}$. We estimate that, at the same $\mu$ and
$T$, the gas with the wall of finite width has $S/Nk_{B}$ that is $<2\%$
larger than the half-harmonic approximation wherein the wall is assumed to
have zero width. While the atom numbers match for the two spin states, we see
that the fitted $q$ for the second spin state to be imaged (regardless of
which spin is imaged first) is systematically larger, likely due to off-
resonant photon scattering from the first pulse and energy transfer from
collisions with the the first spin. We therefore use the thermodynamics of the
first spin state. This thermometry method based on directly fitting
$q_{i\sigma}$ and $T_{i\sigma}$ gives similar results as a previously-used
method using the second spatial moment of the reservoirs’ column density
distributions to extract thermodynamic properties [37].

###  S5.3 Calibration of absorption images

The absorption images of the column density are determined from raw image
containing atoms $A(y,z)$ and a “bright” reference image $B(y,z)$

$\frac{\sigma_{0}}{\alpha}n_{\mathrm{col}}(y,z)=\mathrm{OD}(y,z)=\log\frac{B(y,z)}{A(y,z)}+\frac{B(y,z)-A(y,z)}{\chi
t}$ (S31)

where $\sigma_{0}$ is the resonant absorption cross section, $\alpha$ is a
correction of the absorption cross section, $\chi$ is the imaging transition’s
saturation intensity $I_{\mathrm{sat}}$ in units of the camera count rate, and
$t$ is the pulse duration [75, 76]. For each shot, we use the optimal
reference image $B$ [77] to suppress photon shot noise and technical noise,
though the latter contribution is negligible. The duration of the imaging
pulses was $t=$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$ to avoid any
Doppler shifts, which we verified by ensuring that the resonance frequency
does not depend on the pulse power. We calibrated $\chi$ by ensuring the
measured atom number does not depend on the imaging intensity $I$ [78]; we
ultimately used used $I/I_{\mathrm{sat}}\approx 0.75$.

Figure S3: Calibration of the absorption images via Bose-Einstein condensation
of molecules. (a) Average of 5 absorption images of a single spin acquired in
situ in the harmonic trap at unitarity with a relatively deep dipole trap at
the end of the forced optical evaporation to prepare a non-degenerate cloud.
(b) The same system prepared in (a) after an adiabatic sweep of the magnetic
field to the BEC side of the Feshbach resonance followed by a short time of
flight. The density distribution is well-fitted by a gaussian and is therefore
above the critical point for Bose-Einstein condensation. (c) Same as (a) but
with a lower dipole trap depth at the end of evaporation to prepare a more
degenerate cloud. (d) Same as (c) after the magnetic field sweep and time of
flight, displaying the bimodal density distribution that indicates Bose-
Einstein condensation of Feshbach molecules. (e) The condensate fraction
measured on the BEC side vs. the entropy per fermion measured at unitarity.
The solid line shows the expected behaviour for a non-interacting Bose gas,
the orange circles show the measurement with $\alpha=1$, and the green squares
show the measurement when correcting the absorption cross section with
$\alpha=2.15$ to match the measured and theoretical critical points.

The correction $\alpha$ is typically calibrated by imaging a very degenerate
cloud and fixing the amplitude of the OD using the equation of state [79, 75],
however the clouds that we prepare are not sufficiently degenerate to robustly
fit $q$ and $\alpha$ independently. We therefore calibrated $\alpha$ by
measuring the onset of Bose-Einstein condensation after an adiabatic sweep of
the magnetic field to the BEC side of the Feshbach resonance [74] as shown in
Fig. S3. We prepared harmonically trapped gases (no channel beams) at varying
levels of degeneracy by varying the depth of the optical dipole trap at the
end of evaporation and then either imaged in situ to perform the thermometry
procedure described in S5.2 or adiabatically swept the magnetic field over
$50\text{\,}\mathrm{m}\mathrm{s}$ to the BEC regime and imaged the cloud after
a $6\text{\,}\mathrm{m}\mathrm{s}$ time of flight. The magnetic field at the
end of the sweep is $611.1\text{\,}\mathrm{G}$ where the scattering length
between fermions of opposite spin is $a=904a_{0}$ [80], $1/k_{F}a=7.4$, and
the scattering length between the Feshbach molecules is $0.6a$ [74]. For non-
degenerate clouds (a,b), the density distribution after the expansion is
gaussian and therefore not condensed, however for degenerate clouds (c,d), the
density is bimodal and therefore the entropy per fermion is below the critical
value $(S/Nk_{B})_{c0}=1.801$ for a non-interacting Bose gas in a harmonic
trap [53]. The condensate fraction $N_{0}/N$ measured on the BEC side, which
is independent of $\alpha$ and $\chi$, vs. the entropy per particle measured
at unitarity (e) shows the critical point as a kink where the condensate
fraction is flat to where it monotonically increases as $S/Nk_{B}$ decreases.
The cloud is distorted due to the sweep of the magnetic trap frequency, so a
finite condensate fraction is always fitted, and the critical entropy per
fermion is set by the kink rather rather then the extrapolated $x$-intercept.
Assuming the theoretical absorption cross section ($\alpha=1$), we find the
measured critical entropy per particle $(S/Nk_{B})_{c}$ to be larger than
$(S/Nk_{B})_{c0}$, indicating that the true entropy per particle is lower than
our uncorrected thermometry indicates. We scale the cross section $\alpha$ to
2.15 such that $(S/Nk_{B})_{c}=(S/Nk_{B})_{c0}$ at the onset of condensation.
We have verified that slower sweeps, sweeping further into the BEC regime with
smaller scattering length, and longer time of flight do not change the
measured condensate fraction. The reduced slope of $N_{0}/N$ vs. $S/Nk_{B}$
relative to the theoretical prediction may be due to residual weak repulsive
interactions [53] or heating and three-body losses during this sweep. Both
these effects also shift the transition to a lower $(S/Nk_{B})_{c}$, so the
true $S/Nk_{B}$ is likely slightly smaller than this calibration procedure
indicates. The direct fit of $\alpha$ with the EoS is consistent with 2.15,
but the systematic uncertainty is larger due to the low degeneracy of the
clouds. We have also cross-checked this value of $\alpha$ calibrated at
unitarity by measuring conductance quantisation to $1/h$ in the non-
interacting gas, which relies on accurate thermometry to compute the
compressibility [64].

The calibrated value $\alpha=2.15$ is similar to previously reported values
[79, 75] though the origin of this large correction factor is not completely
clear. It is important to note that miscalibration of the imaging system’s
magnification and in $\chi$ is absorbed into $\alpha$, though we expect these
corrections to be small: the 1% (4%) relative uncertainty in magnification
($\chi$) leads to 3% (3%) relative uncertainty in $S/Nk_{B}$. $\chi/\alpha$
can be independently calibrated with the camera’s magnified pixel size,
quantum efficiency, gain, and the transmission of the imaging system [75, 76]
though we find that this value is 50% larger than the above calibration
procedure predicts. It is therefore likely that there are still some non-ideal
effects that are not explicitly accounted for in Eq. S31 but are approximately
absorbed into $\chi$ and $\alpha$. Indeed, as in [75], we measure higher atom
numbers at lower intensities, which would reduce the value of $\alpha$
required to match $(S/Nk_{B})_{c}=(S/Nk_{B})_{c0}$, and we observe the speckle
pattern of the beam being partially visible in the OD at both low and high
intensities. We can reasonably rule out density effects on the absorption
cross section, e.g. from multiple scattering of photons [81], by verifying
that the measured atom number is independent of the time of flight after
release from the optical trap which significantly varies the density of the
gas at fixed atom number. Imperfections of the beam’s polarisation can
increase $\alpha$, though this effect should be small as we measured with a
polarisation analyser that the polarisation is within $1^{\circ}$ of the
expected horizontal polarisation before and after the vacuum chamber. Even
though we use short pulses, the Doppler shift can still be significant: Not
only do the atoms acquire an average Doppler shift due to the absorbed photon
recoil, they also develop an increasingly broad distribution due to the
stochasticity in the direction of the re-emitted photons [82]. We estimate
that this can increase $\alpha$ by 10-20%. For the short pulses we use, the
atoms are not in the steady internal state that is assumed by Eq. S31; we find
by solving the Bloch equations for the 6 ground states and the 18 excited
states in the $D_{1}$ and $D_{2}$ lines, we find that the number of scattered
photons and therefore $\sigma$ is reduced by about 5% relative to the steady
state solution. The laser spectrum, both its lorentzian line and broad
background typical of diode lasers, can also increase $\alpha$ by 5-10%.
Optical effects such as aberrations and refraction by the second spin state
are also present but their effect on absorption imaging is rarely considered.
A complete explanation of the deviation of $\alpha$ and $\chi$ from their
expected values is beyond the scope of this study and we rely on the
condensate fraction as an absolute calibration of $S/Nk_{B}$.

##  S6 Fitting procedure

We fit the phenomenological model to each data set–the set of different
transport times at fixed $\nu_{x}$–independently for both the first and second
experiment. We do this by solving the initial value problem for $\Delta N(t)$
and $\Delta S(t)$

$\begin{split}\derivative{\Delta N(t)}{t}&=-2I_{N}[\Delta\mu(t),\Delta
T(t)]=-2\quantity{I_{\mathrm{exc}}\tanh\bqty{\frac{\Delta\mu(t)+\alpha_{c}\Delta
T(t)}{\sigma}}+G[\Delta\mu(t)+\alpha_{c}\Delta T(t)]}\\\ \derivative{\Delta
S(t)}{t}&=-2I_{S}[\Delta\mu(t),\Delta
T(t)]=-2\quantity{\alpha_{c}I_{N}[\Delta\mu(t),\Delta T(t)]+G_{T}\Delta
T(t)/T}\\\
\pmqty{undef}&=\frac{1}{\kappa\ell_{r}}\pmqty{undef}\pmqty{undef}\end{split}$
(S32)

given the parameters $G$, $\alpha_{c}$, $I_{\mathrm{exc}}$, $\sigma$, $G_{T}$
along with the reservoir response functions $\kappa$, $\alpha_{r}$, $\ell_{r}$
and average temperature $T$ during transport and the initial values $\Delta
N(0)$, $\Delta S(0)$. From these solutions, we also compute the total entropy
as a function of time

$S(t)=S(0)+\frac{\Delta N^{2}(t)}{4T\kappa}+\frac{[\Delta
S(t)-\alpha_{r}\Delta N(t)]^{2}}{4T\ell_{r}\kappa}$ (S33)

where $S(0)$ is the fitted initial total entropy. From the data–the set of
times $t_{i}$, relative particle imbalances $\Delta N_{i}/N_{i}$, relative
entropy imbalances $\Delta S_{i}/N_{i}k_{B}$, and relative total entropies
$S_{i}/N_{i}k_{B}$–we perform a least-squares fit by minimising the reduced
chi-squared statistic

$\chi^{2}=\frac{1}{\nu}\sum_{i}\pqty{\frac{\Delta N_{i}/N_{i}-\Delta
N(t_{i})/N}{\sigma_{\Delta N/N}}}^{2}+\pqty{\frac{\Delta
S_{i}/N_{i}k_{B}-\Delta S(t_{i})/Nk_{B}}{\sigma_{\Delta
S/Nk_{B}}}}^{2}+\pqty{\frac{S_{i}/N_{i}k_{B}-S(t_{i})/Nk_{B}}{\sigma_{S/Nk_{B}}}}^{2}.$
(S34)

In other words, we simultaneously fit the particle imbalance, entropy
imbalance, and total entropy. $\nu$ is the number of degrees of freedom: the
number of data points minus number of fit parameters. For each quantity
$x=\Delta N/N,\Delta S/Nk_{B},S/Nk_{B}$, we use the same uncertainty for each
data point defined as the average over each point’s statistical uncertainty
computed from 3-5 shots $\sigma_{x}=\sum_{i}\sigma_{x,i}/n_{\mathrm{points}}$
in order to improve the stability of the fits. We use the relative quantities
normalised by the atom number (e.g. $\Delta N/N$ instead of $\Delta N$) to
minimise the effect of $\sim 10\%$ shot-to-shot fluctuations of the total atom
number.

For both experiments, we set $G=0$ as allowing $G$ to be a fit parameter
underconstrains the fit ($G$ is strongly correlated to
$I_{\mathrm{exc}}/\sigma$) and enforcing $G=n_{m}/h$ increases $\chi^{2}$ by
$\sim 20\%$. We also fix
$\kappa=\kappa^{\mathrm{tr}}(\mu^{\mathrm{tr}},T^{\mathrm{tr}})$ to the value
computed from the equation of state in the transport trap (see Sec. S5.1). For
the first experiment, we fit $I_{\mathrm{exc}}$, $\sigma$, $\alpha_{c}$,
$\Delta N(0)$, $\Delta S(0)$, and $S(0)$, and fix $G_{T}=0$ (the dynamics are
dominated by the advective mode),
$\alpha_{r}=\alpha_{r}^{\mathrm{tr}}(\mu^{\mathrm{tr}},T^{\mathrm{tr}})$, and
$\ell_{r}=\ell_{r}^{\mathrm{tr}}(\mu^{\mathrm{tr}},T^{\mathrm{tr}})$. For the
second experiment, we fit the same parameters as in the first experiment as
well as $G_{T}$, $\alpha_{r}$, and $\ell_{r}$, and we fix $I_{\mathrm{exc}}$
to the initial current $I_{N}(0)$ determined by a linear fit to all points in
the initial response with $\Delta N/N\leq 0.07$. Without fixing
$I_{\mathrm{exc}}$ in this way, it is strongly correlated to $\sigma$ and
$G_{T}$ and yields similar values of $\chi^{2}$. Fixing $\alpha_{r}$ and
$\ell_{r}$ to their theoretical values prevents the fit from reproducing the
large $\Delta N$ that is induced by the initial entropy imbalance $\Delta
S(0)$ in the second experiment.

With this method, the resulting $\chi^{2}\approx 1$ for all data sets. The
fitted values of the reservoir response coefficients from the second
experiment are shown in Fig. S4. While $\alpha_{r}$ is relatively near the
expected value for the anharmonic trap, $\ell_{r}$ is significantly smaller.
There can be many reasons for these systematic biases, for example
miscalibration of the absorption cross section and magnification, the contacts
defined by the gate beam and wire forming smaller reservoirs contacted to the
larger ones which develop their own effective thermodynamic response
functions, or generally deviations of the true equation of state from the
computed one. Despite this deviation from our expectations, this result does
not change the overall form of the phenomenological model nor the central
result of the large advective entropy current. We find that fixing
$\alpha_{r}$ and $\ell_{r}$ in the first experiment yield good fits while they
must be allowed to vary in the second experiment to fit the data. Allowing
them to vary in the first experiment does not change the extracted parameters
$I_{\mathrm{exc}}$ and $\alpha_{c}$. The average fitted nonlinearity from all
the data sets is $\sigma=$13(2)\text{\,}\mathrm{n}\mathrm{K}$\times
k_{B}=$8(1)\text{\times}{10}^{-3}$\Delta$.

Figure S4: Fitted reservoir response coefficients for each configuration of
the second experiment normalised to their values in the anharmonic transport
trap.

We can derive an expression for the instantaneous slope of the path through
state space $\Pi=\derivative*{\Delta S}{\Delta N}$ by combining the expression
for $I_{S}$ in Eq. S20 and the reservoir responses in Eq. S14

$\Pi=\derivative{\Delta S}{\Delta N}=\frac{\derivative*{\Delta
S}{t}}{\derivative*{\Delta N}{t}}=\alpha_{c}+\frac{2G_{T}(\alpha_{r}\Delta
N-\Delta S)}{T\kappa\ell_{r}\derivative*{\Delta N}{t}}$ (S35)

In the limit where either the advective or diffusive mode dominates, the path
is linear $\Delta S(t)\approx\Pi[\Delta N(t)-\Delta N_{0}]$ [Figs. 1(d) and
3(d)]. When combined with the previous expression, this gives

$\Pi\approx\frac{2G_{T}\alpha_{r}\Delta
N+T\kappa\ell_{r}\alpha_{c}\derivative*{\Delta N}{t}}{2G_{T}(\Delta N-\Delta
N_{0})+T\kappa\ell_{r}\derivative*{\Delta N}{t}}$ (S36)

If the transport is fast as is the case for the superfluid-induced advective
mode, we observe that $\Pi$ is independent of $\Delta N$, implying that the
$\derivative*{\Delta N}{t}$ terms dominate and the slope simplifies to

$\Pi_{a}\approx\alpha_{c}.$ (S37)

We observe in the second experiment that after the advective mode has quickly
relaxed and the diffusive mode is dominant, then the time evolution is
exponential $\derivative*{\Delta N}{t}\approx-\Delta N/\tau_{d}$, indicating
that the response becomes linear in this regime
$I_{\mathrm{exc}}\tanh[(\Delta\mu+\alpha_{c}\Delta T)/\sigma]\approx
I_{\mathrm{exc}}(\Delta\mu+\alpha_{c}\Delta T)/\sigma$.

Linearising tanh is equivalent to applying the linear model
$\Xi=\Xi_{n}^{a}+\Xi_{n}^{d}$ with effective conductance
$\tilde{G}=I_{\mathrm{exc}}/\sigma+G$ and Lorenz number $L=G_{T}/T\tilde{G}$

$\derivative{t}\pmqty{undef}\approx-2\tilde{G}\pmqty{undef}\pmqty{undef}=-\frac{2\tilde{G}}{\kappa\ell_{r}}\pmqty{undef}\pmqty{undef}\pmqty{undef}=-\mathbf{\Lambda}\pmqty{undef}.$
(S38)

In this limit, the path the system traces though state space is a line passing
through equilibrium and can be written

$\pmqty{undef}=\Delta N(0)e^{-t/\tau_{d}}\pmqty{undef}$ (S39)

so the exponential timescale $\tau_{d}$ and the slope of the path $\Pi_{d}$
are then given by the smaller eigenvalue and corresponding eigenvector of the
matrix $\mathbf{\Lambda}$

$\displaystyle\begin{split}\tau_{d}^{-1}&=\frac{\tilde{G}}{\kappa\ell_{r}}\quantity{L+\ell_{r}+(\alpha_{c}-\alpha_{r})^{2}-\sqrt{[L+\ell_{r}+(\alpha_{c}-\alpha_{r})^{2}]^{2}-4L\ell_{r}}}\\\
&=\frac{2\tilde{G}}{\kappa\ell_{r}}[\ell_{r}+(\alpha_{r}-\alpha_{c})(\alpha_{r}-\Pi_{d})]\\\
\Pi_{d}&=\frac{L-\ell_{r}+\alpha_{c}^{2}-\alpha_{r}^{2}-\sqrt{[L+\ell_{r}+(\alpha_{c}-\alpha_{r})^{2}]^{2}-4L\ell_{r}}}{2(\alpha_{c}-\alpha_{r})}\\\
&=\frac{2\tilde{G}[\ell_{r}+\alpha_{r}(\alpha_{r}-\alpha_{c})]+\kappa\ell_{r}\tau_{d}^{-1}}{2\tilde{G}(\alpha_{r}-\alpha_{c})}.\end{split}$
(S40)

In the non-equilibrium steady state, $\tau_{d}^{-1}\rightarrow 0$ which
implies that $[L+\ell_{r}+(\alpha_{c}-\alpha_{r})^{2}]^{2}\gg 4L\ell_{r}$,
i.e., $L\rightarrow 0$. Taylor expanding $\tau_{d}^{-1}$ and $\Pi_{d}$ to
first order in $L$ yields

$\displaystyle\begin{split}\tau_{d}^{-1}&\approx\frac{2\tilde{G}L}{\kappa[\ell_{r}+(\alpha_{c}-\alpha_{r})^{2}]}\\\
\Pi_{d}&\approx\alpha_{r}+\frac{\ell_{r}}{\alpha_{r}-\alpha_{c}}\bqty{1-\frac{L}{\ell_{r}+(\alpha_{r}-\alpha_{c})^{2}}}.\end{split}$
(S41)

As first observed in [37], we see that $L$ decreases with increasing $\nu_{x}$
from Fig. 3 where the diffusive timescale $\tau_{d}$ becomes longer and from
Fig. 4d where $\Pi_{d}$ increases.