Abstract
The nature of particle and entropy flow between two superfluids is often understood in terms of reversible flow carried by an entropyfree, macroscopic wavefunction. While this wavefunction is responsible for many intriguing properties of superfluids and superconductors, its interplay with excitations in nonequilibrium situations is less understood. Here we observe large concurrent flows of both particles and entropy through a ballistic channel connecting two strongly interacting fermionic superfluids. Both currents respond nonlinearly to chemical potential and temperature biases. We find that the entropy transported per particle is much larger than the prediction of superfluid hydrodynamics in the linear regime and largely independent of changes in the channel’s geometry. By contrast, the timescales of advective and diffusive entropy transport vary significantly with the channel geometry. In our setting, superfluidity counterintuitively increases the speed of entropy transport. Moreover, we develop a phenomenological model describing the nonlinear dynamics within the framework of generalized gradient dynamics. Our approach for measuring entropy currents may help elucidate mechanisms of heat transfer in superfluids and superconducting devices.
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Main
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^{1} and even heavyions collisions^{2}. Entropy and heat, both irreversibly produced and transported by the currents flowing between the reservoirs, are key quantities in superfluid and superconducting systems^{3,4}. They help to reveal microscopic information in strongly interacting systems^{5,6} and more generally characterize farfromequilibrium systems^{7}. 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^{8,9,10}.
In this work, we leverage the advantage of quantum gases of ultracold atoms as naturally closed systems wellisolated from their environments to study entropy transport and production in fermionic superfluid systems. Using the known equation of state^{11}, 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. In general, the nature of these currents depends on the coupling strength between the superfluids. On the one hand, two weakly coupled superfluids exhibiting the Josephson effect^{12,13} exchange an entropyfree supercurrent described by Landau’s hydrodynamic twofluid model^{14,15,16,17,18}. In quantum gases^{19} as well as superconductors^{20}, this is accomplished with lowtransparency tunnel junctions weakly biased in chemical potential or phase, while narrow channels are used to block viscous currents in superfluid helium^{21,22}. On the other hand, superfluids strongly coupled by hightransparency channels^{23} can exhibit less intuitive behaviour since the supercurrent no longer dominates the normal current, making the system fundamentally nonequilibrium^{24,25}. In particular, the signature of superfluidity in such systems is often large particle currents on the order of the superfluid gap which respond nonlinearly to chemical potential biases smaller than the gap. This is observed in ballistic junctions between superconductors^{20}, superfluid He^{26} and quantum gases^{27,28,29}. 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^{30,31}, leaving open the question of entropy transport between strongly coupled 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 subgap currents of both particles and entropy, indicating that the current is not a pure supercurrent and cannot be understood within a hydrodynamic twofluid model. In particular, superfluidity counterintuitively enhances 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 thermalize via the irreversible flow of this superfluidenhanced normal current, thermalization via pure entropy diffusion is inhibited in onedimensional (1D) channels, giving rise to a nonequilibrium steady state previously observed in the normal phase^{30}. 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.
We begin the experiment by preparing a balanced mixture of the first and thirdlowest hyperfine ground states of ^{6}Li at unitarity in an augmented harmonic trap shown in Fig. 1a (Methods and Supplementary Information section 3). To induce transport of atoms, energy and entropy between the left (L) and right (R) reservoirs, we prepare an initial state within the state space in Fig. 1b characterized 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 ΔN = N_{L} − N_{R} and entropy ΔS = S_{L} − S_{R}. The imbalances in the extensive quantities induce biases in the chemical potential Δμ = μ_{L} − μ_{R} and temperature ΔT = T_{L} − T_{R} according to the reservoirs’ equations of state (EoS; Methods) which in turn drive currents of the extensive properties I_{N}(ΔN, ΔS) = − (1/2)dΔN/dt and I_{S}(ΔN, ΔS) = − (1/2)dΔS/dt. 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 \({I}_{S}^{{{\;{\rm{cons}}}}}\in [{I}_{S}(1/2){{{\rm{d}}}}S/{{{\rm{d}}}}t,{I}_{S}+(1/2){{{\rm{d}}}}S/{{{\rm{d}}}}t]\) (ref. ^{29}). These equations of motion, together with the initial state ΔN(0), ΔS(0), determine the path the system traces through state space ΔN(t), ΔS(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, ΔN, ΔS) computed from the EoS. The paths all exhibit a strictly positive entropy production rate dS/dt > 0, indicating that the transport is irreversible, until they reach either a nonequilibrium steady state (ΔN, ΔS ≠ 0) or equilibrium (ΔN = ΔS = 0) where S is maximized 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^{32,33,34,35}. We have verified that the system is nearly closed given the measured particle loss rate ∣dN/dt∣/N < 0.01 s^{−1} and heating in equilibrium d(S/Nk_{B})/dt < 0.02 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 evolution of the system 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 (Methods and Supplementary Information section 5B). Between the end of transport and imaging, we adiabatically ramp down the laser beams that define the channel to image the reservoirs in wellcalibrated, halfharmonic 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) × 10^{3} atoms and S/N = 1.59(7)k_{B} before transport, below the critical value of ~1.90k_{B} for superfluidity in the transport trap (numbers in brackets represent statistical uncertainties). Figure 1a shows the local degeneracy T/T_{F} in the x = 0 and z = 0 planes during transport calculated within the local density approximation^{36} using the threedimensional (3D) equation of state^{11} to determine the local Fermi temperature T_{F}(x, y, z) (Methods). The imbalance is illustrated using the representative values of ν_{x} = 12.4 kHz, Δμ = 75 nK × k_{B} and ΔT = 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} ≈ 0.027, s ≈ 7.2 × 10^{−4}k_{B} and Δ/k_{B}T ≈ 17 where s is the local entropy per particle and Δ is the superfluid gap. When ν_{x} ≲ 7 kHz (Methods and Supplementary Information Section 3) and the channel is twodimensional (2D), normal regions appear at the edges of the channel that can also contribute to transport.
In a first experiment, we prepare an initial state ΔN(0), ΔS(0) ≠ 0 (filled circle in Fig. 1b) such that equilibrium is reached within 1 s. For the strongest confinement, ΔN(t), shown in Fig. 2a, clearly deviates from exponential relaxation and the corresponding I_{N} is much larger than the value ~Δμ/h of a quantum point contact in the normal state, where h is Planck’s constant, indicating that the subgap currentbias characteristics are nonlinear (nonOhmic) and the reservoirs are superfluid^{28,29,37}. When reducing ν_{x} to cross over from a 1D to 2D channel, ΔN(t) relaxes faster (I_{N} increases) and, although it is less pronounced, the nonlinearity persists. The dynamics for the two smallest values of ν_{x} are nearly identical, suggesting that there are additional resistances in series with the 1D region such as the viscosity of the bulk reservoirs or the interfaces between the 3D and 2D regions^{31} or between the normal and superfluid regions^{38}.
Concurrently with ΔN(t), we observe nonexponential relaxation of ΔS(t) (Fig. 2b) which bears a remarkable resemblance to ΔN(t). We find that by plotting ΔS(t) against ΔN(t) in Fig. 2d, 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 ν_{x} even though I_{N} itself varies significantly. Moreover, dS/dt (Fig. 2c) is barely resolvable and is significantly smaller than I_{S}, meaning there is indeed a large conserved entropy current flowing between the reservoirs. The dependence of S/Nk_{B} on the confinement ν_{x} in Fig. 2c has a technical origin and does not affect the system during transport (Methods). The entropy transported per particle s^{*} = 1.18(3)k_{B} is near its value in the normal phase^{30} and is orders of magnitude larger than the local entropy per particle in the channel assuming local equilibrium s ≈ 7.2 × 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, nonOhmic current between the two superfluids is itself not a pure supercurrent in the context of a twofluid model^{39}. 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^{40,41}. The observation that s^{*} ≫ s suggests that the channel is far from equilibrium and hydrodynamics breaks down as is often the case in weak link geometries^{39} unlike previous assumptions^{31,38}. We discuss in Supplementary Information section 1 how the degree to which hydrodynamics breaks down depends on the preparation of the system. 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 ν_{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.
In a second experiment, whose results are presented in Fig. 3, we prepare the system with a nearly pure entropy imbalance (ΔN(0) ≈ 0, ΔS(0) ≠ 0, open circle in Fig. 1b). The initial response of ΔN and ΔS from t = 0 to when ΔN reaches its maximum value is clearly nonexponential, 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 ν_{x}, both dynamics become faster and the maximum values of ΔN and ΔS achieved at the turning point become smaller. For the largest value of ν_{x}, the initial response is still fast while the relaxation that follows is extremely slow and resembles a nonequilibrium steady state over experimentally accessible timescales: the relaxation time of this state is 8(2) s while it is reached from the initial state in only ~0.2 s. In this state, the nonvanishing imbalances ΔN and ΔS depend on the initial state as well as the path of the system through state space determined by s^{*}, that is, the system is nonergodic. This indicates that the nonequilibrium steady state previously observed at higher temperatures^{30} persists in the superfluid regime where the current with which it is reached is ≳6 times larger and nonOhmic. Figure 3f shows the measured path in state space also illustrated in Fig. 1b. It demonstrates that the path is determined by the competition between the nonlinear and linear dynamics and varies with ν_{x}, in contrast to the first experiment (Fig. 2d).
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^{30,31} as it predicts purely exponential relaxation. We therefore turn to the formalism of generalized gradient dynamics^{42}, a generalization of Onsager’s theory of irreversible processes (Methods and Supplementary Information section 2). While it does not provide a microscopic theory, this formalism can describe general, irreversible, nonequilibrium 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
which produce entropy via the irreversible flow dS/dt = (I_{N}Δμ + I_{S}ΔT)/T. The nontrivial result that α_{c} appears in both I_{N} and I_{S} is a generalization 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 \({I}_{S}^{\,\rm{a}}={\alpha }_{\rm{c}}{I}_{N}\) characterized by the excess current I_{exc}, Seebeck coefficient α_{c} and nonlinearity σ, wherein each transported particle carries entropy s^{*} = α_{c} on average, and a linear diffusive mode \({I}_{S}^{\,\rm{d}}={G}_{\rm{T}}\Delta T/T\) characterized 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 σ with conductance G = I_{exc}/σ. 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 \({\uppi }^{2}{k}_{\rm {B}}^{2}/3\). The nonlinearity implies the breakdown of the Wiedemann–Franz law since the advective and diffusive modes are no longer linked^{30}. The excess current I_{exc} is the particle current with the nonlinearity saturated, as in superconducting weak links^{40}.
Figure 4 shows the parameters of the model as functions of ν_{x} extracted from the fits shown as curves in Figs. 2 and 3. Panel a shows I_{exc} normalized by the fermionic superfluid gap Δ/h along with the number of occupied transverse modes at equilibrium n_{m} (Methods). Filled (open) circles were extracted from the first (second) experiment. I_{exc} follows n_{m}Δ/h, increasing as ν_{x} decreases until ν_{x} ≈ 4 kHz where it plateaus, likely due to additional resistances in series with the 1D region. The fitted I_{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^{28,29}, the equilibrium superfluid gap Δ within the local density approximation is still the relevant scale for the current, despite the evidence that the channel region is far from equilibrium.
Figure 4b shows G_{T} and the spin conductance G_{σ} separately measured in the same system by preparing a pure spin imbalance (Methods) normalized to their values for the singlemode noninteracting ballistic quantum point contact \({G}_{\rm{T}}^{0}=2{\uppi }^{2}{k}_{\rm{B}}^{2}T/3h,\,{G}_{\sigma }^{0}=2/h\). Both conductances are suppressed relative to the noninteracting values and increase monotonically with decreasing ν_{x}, possibly due to the appearance of nondegenerate transverse modes at the edges of the channel (Methods and Supplementary Information section 3). The nonequilibrium steady state arises from the fact that G_{T} → 0 in the 1D limit. The relative increase of G_{T} with decreasing ν_{x} is larger than that of G_{σ}, suggesting that more types of excitations can contribute to diffusive entropy transport than spin transport, for example, both collective phonon and quasiparticle excitations can contribute to G_{T} (ref. ^{43}) while only quasiparticle excitations can contribute to G_{σ} (ref. ^{44}).
Figure 4c shows the fitted Seebeck coefficient α_{c}, while Fig. 4d shows the slope of the path through state space dΔS/dΔN during the advective and diffusive dynamics. The fitted α_{c} and dΔS/dΔN match for the purely advective transport in the first experiment, showing that α_{c} is remarkably insensitive to ν_{x}, while dΔS/dΔN more clearly shows how the two modes compete in the second experiment to determine the net response of the system. Figure 4d shows that, while both modes are generally present in the system’s dynamics, their relative prevalence depends on ν_{x} as well as the initial state: the initial state in the first experiment was carefully chosen to allow the system to relax to equilibrium via the advective mode alone by preparing ΔS(0) = α_{c}ΔN(0) while the initial state in the second was chosen to contain both modes.
In summary, we have observed that the conceptually simple system of two superfluids connected by a ballistic channel exhibits the highly nonintuitive 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, entropyfree 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^{*} ≃ 1k_{B}. Our phenomenological model that captures these observations, in particular the identification of advective \({I}_{S}^{\,\mathrm{a}}\propto {I}_{N}\) and diffusive \({I}_{S}^{\,\mathrm{d}}\propto \Delta T\) modes along with the sigmoidal shape of I_{N}(Δμ + α_{c}ΔT), may help guide future microscopic theories of the system. While extensive research has been conducted on the entropy producing effects of topological excitations of the superfluid order parameter^{8,12,21,39,41,45,46,47,48,49}, less attention has been given to their influence on entropy transport and the possible pairbreaking processes they can induce. Early studies of superconductors found that mobile vortices can advectively transport entropy by carrying pockets of normal fluid^{50} with them^{24,51,52,53}. More generally, entropycarrying topological excitations, which give rise to a finite chemical potential bias according to the Josephson relation Δμ = hN_{v}/dt (refs. ^{39}), where N_{v} is the number of vortices, can result from a complex spatial structure of the order parameter^{49}. Alternatively, it is possible that an extension of microscopic theories of multiple Andreev reflection^{37,54}, which reproduce the finding that the excess current scales linearly with the number of channels and the gap^{28,29}, may explain our observations. Clearly, a proper microscopic theory of this system is a challenge for the future. A complete understanding of the particle and entropy transport in superfluid systems is essential for both fundamental and technological purposes.
Methods
Transport configuration
The atoms are trapped magnetically along y and optically by a reddetuned beam along x and z, with confinement frequencies ν_{trap,x} = 171(1) Hz, ν_{trap,y} = 28.31(2) Hz and ν_{trap,z} = 164(1) Hz. A pair of repulsive TEM_{01}like beams propagating along x and z, which we call the lightsheet (LS) and wire respectively, intersect at the centre of the trapped cloud and separate it into two reservoirs connected by a channel. The transverse confinement frequencies at the centre are set to ν_{z} = 9.42(6) kHz (k_{B}T/hν_{z} = 0.21) and ν_{x} = 0.61…12.4(2) kHz (k_{B}T/hν_{x} = 0.16…3.3) with the powers of the beams. An attractive Gaussian beam propagating along z with a similar size as the LS beam acts as a gate potential in the channel. The peak gate potential is \({V}_{{{{\rm{gate}}}}}^{\;0}=2.17(1)\,\upmu {{{\rm{K}}}}\times {k}_{\rm{B}}\). We use a wall beam, which is thin along y and wide along x, during preparation and imaging to completely block transport with a barrier height \({V}_{{{{\rm{wall}}}}}^{\;0}\) much larger than μ and k_{B}T. The repulsive LS, wire and wall are generated using bluedetuned 532 nm light while the attractive gate is created with reddetuned 766.7 nm light.
The effective potential energy landscape along y at x = z = 0 (Supplementary Information section 3) is approximately V_{eff}(y, n_{x}, n_{z}) ≈ hν_{x}(y)(n_{x} + 1/2) + hν_{z}(y)(n_{z} + 1/2), where ν_{x} and ν_{z} vary along y due to the beams’ profile and n_{x}, n_{z} are the quantum numbers of the harmonic potential in x and z directions. The number of occupied transverse modes n_{m} (Fig. 4) is calculated via the Fermi–Dirac occupation with local chemical potential set by V_{eff}(y, n_{x}, n_{z}),
where the minimum occupation of each mode is used to account for modes that are not always occupied throughout the channel (Supplementary Information section 3).
The complete potential energy landscape V(r), where r = (x, y, z), was used to produce Fig. 1a via the local density approximation for the density n(r) = n[μ − V(r), T] (refs. ^{11,36}) that determines the local Fermi temperature \({k}_{\rm{B}}{T}_{\rm{F}}({{{\bf{r}}}})={\hslash }^{2}{[3{\uppi }^{2}n({{{\bf{r}}}})]}^{2/3}/2m\), where m is the atomic mass. The superfluid gap Δ assuming local equilibrium is estimated using the calculation in a homogeneous system Δ(μ_{c}, T) (ref. ^{55}) where \({\mu }_{\rm{c}}=\mathop{\max }\limits_{{{{\bf{r}}}}}[\mu V({{{\bf{r}}}})]\) is the maximum local chemical potential in the system. The crossover between 1D and 2D regimes (ν_{x} ≈ 7 kHz) of the channel is estimated by comparing the local degeneracy along x (at y = z = 0) to the superfluid transition. In the 2D limit, nonsuperfluid modes can pass through the edges of the channel while in the 1D limit the degeneracy across the channel is below the superfluid transition (Supplementary Information section 3).
Transport preparation
To prepare imbalances ΔS(0), ΔN(0) we ramp up the channel beams to separate the two reservoirs followed by forced optical evaporation. Using a magnetic field gradient along y, we shift the centre of the magnetic trap with respect to the channel beams before separation to prepare ΔN(0). By shifting the trap centre during evaporation, we can compress one reservoir and decompress the other, thereby changing their evaporation efficiencies and inducing a controllable ΔS(0). See Supplementary Information Section 4 for more details. To measure the spin conductance G_{σ} (Fig. 4b), we prepare a ‘magnetization’ imbalance ΔM = ΔN_{↑} − ΔN_{↓}. To do this, we ramp down the magnetic field before separating the reservoirs at 52 G where the spins’ magnetic moments are different and modulate a magnetic gradient along y until the two spins oscillate out of phase. We then separate the two reservoirs and ramp back the magnetic field.
Imaging and thermometry
Between the end of transport and the start of imaging, we ramp down the channel beams while keeping the wall on. At the end of each run, we obtain the column density \({n}_{i\sigma }^{{{{\rm{col}}}}}(\;y,z)\) of both reservoirs i = L, R and both spin states σ = ↓, ↑ (first and thirdlowest states in the ground state manifold) from two absorption images taken in quick succession in situ. We fit the degeneracy q_{iσ} = μ_{iσ}/k_{B}T_{iσ} and temperature T_{iσ} of both reservoirs for each spin state using the EoS of the harmonically trapped gas^{11}. However, we use the fitted temperature from the first image (↓) for both spins since the density distribution in the second image is slightly perturbed by the first imaging pulse. The thermometry is calibrated using the critical S/N of the condensation phase transition on the BEC side of the Feshbach resonance. See Supplementary Information section 5B for more details.
Generalized gradient dynamics
To ensure that the phenomenological nonlinear model satisfies basic properties such as the second law of thermodynamics, it is formulated it in terms of a dissipation potential Ξ (ref. ^{42}). The thermodynamic fluxes are defined as derivatives of the dissipation potential Ξ with respect to the forces I_{N} = T∂Ξ/∂Δμ and I_{S} = T∂Ξ/∂ΔT. In this formalism, Onsager reciprocity and the conservation of particles and energy are fulfilled. Our model (equation (1)) is the result of the following dissipation potential, which is constructed based on the experimental observation I_{S} = s^{*}I_{N} and that I_{N} follows a sigmoidal function of Δμ (ref. ^{28}),
where the first part describes the advective and the second part the diffusive transport mode. See Supplementary Information section 2 for more details.
Reservoir thermodynamics
To formulate equations of motion in state space (ΔN, ΔS), we relate Δμ, ΔT to ΔN, ΔS in terms of thermodynamic response functions
where κ is the compressibility, α_{r} is the dilatation coefficient and ℓ_{r} is the ‘Lorenz number’ of the reservoirs^{56} (Supplementary Information section 5A). To obtain the spin conductance, we use ΔM = (χ/2)Δb, where Δb = (Δμ_{↑} − Δμ_{↓})/2. The spin susceptibility χ ≈ 0.32κ, following the computed EoS of a polarized unitary Fermi gas^{57}.
The potential landscape V(r) during the transport experiment deviates from simple harmonic potential due to the confinement beams as well as the anharmonicity of the optical dipole trap. We estimate from numeric simulations based on our knowledge of the V(r) that T and κ agree within 1% to those determined from absorption imaging in nearharmonic traps while μ is 24% higher during transport. However, α_{r} and ℓ_{r} are more sensitive to the trap potential and can deviate by a factor of 3. We therefore fit these response coefficients in our model. See Supplementary Information section 5A for more details.
The total entropy S, being a state variable (contour plot in Fig. 1b), depends on the imbalances ΔN(t) and ΔS(t) but not the currents I_{N} and I_{S}. With the linearized reservoir response, the entropy produced by equilibration is given by
where S_{eq} is the maximum entropy at equilibrium given fixed total N and U. The increase in S/Nk_{B} with decreasing ν_{x} in Figs. 2c and 3c is caused by switching on the wall beam to block transport. For lower ν_{x}, there are more atoms in the channel to be perturbed by this process.
Fitting procedure
We fit the phenomenological model (equation (1)) with linear reservoir responses (equation (4)) to each dataset—the set of different transport times at fixed ν_{x}—independently for both the first and second experiment. We do this by solving the initial value problem for ΔN(t) and ΔS(t) given the parameters α_{c}, I_{exc}, σ and G_{T} along with the reservoir response functions κ, α_{r} and ℓ_{r}. From these solutions, we also compute the total entropy S(t) as a function of time (equation (5)). We fit ΔN(t), ΔS(t) and S(t) simultaneously to the data using a leastsquares fit. For the first experiment, only I_{exc}, σ, α_{c}, ΔS(0) and S_{eq} are free parameters. Other parameters are fixed to their theoretical values for better fit stability. For the second experiment, we fit σ, ΔS(0), S_{eq}, G_{T}, α_{r}, ℓ_{r} and an offset in ΔS to account for drifts in alignment. α_{c} is fixed to the averaged value obtained in the first experiment. See Supplementary Information section 6 for more details. The slopes shown in Fig. 4d are obtained by simple linear fits in the state space ΔS versus ΔN (Figs. 2d and 3f). The advective and diffusive modes in the second experiment are separated in time at the maximum ΔN(t) (Fig. 2a).
Data availability
All data files are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
Code availability
Source codes for the data processing and numerical simulations are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank A. Frank for his contributions to the electronics of the experiment. We are grateful for inspiring discussions with the Madrid Quantum Transport Group (A. Levy Yeyati, G. Steffensen, F. MatuteCañadas, P. Burset Atienza, R. Sánchez), P. Christodoulou, S. Gopalakrishnan, A.M. Visuri, S. Uchino, T. Giamarchi, A. Montefusco, B. Svistunov and S. Jochim. We thank A. Gómez Salvador and E. Demler for their productive and ongoing collaboration in investigating this system. We acknowledge the Swiss National Science Foundation (212168, UeM0195.1 and TMAG2_209376) and European Research Council advanced grant TransQ (742579) for funding.
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P.F., J.M., M.T., S.W. and M.Z.H. upgraded the experiment and performed the measurements. J.M. and P.F. analysed the data and were in charge of writing the paper while all authors discussed the findings and contributed to the text and other aspects of the manuscript. M.Z.H. and T.E. supervised the project.
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Fabritius, P., Mohan, J., Talebi, M. et al. Irreversible entropy transport enhanced by fermionic superfluidity. Nat. Phys. 20, 1091–1096 (2024). https://doi.org/10.1038/s41567024024833
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DOI: https://doi.org/10.1038/s41567024024833
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