Interaction-Assisted Reversal of Thermopower with Ultracold Atoms
Samuel Häusler, Philipp Fabritius, Jeffrey Mohan, Martin Lebrat, Laura Corman, Tilman Esslinger
IInteraction-assisted reversal of thermopower with ultracold atoms
Samuel H¨ausler, Philipp Fabritius, Jeffrey Mohan, Martin Lebrat, Laura Corman, ∗ and Tilman Esslinger † Department of Physics, ETH Zurich, 8093 Zurich, Switzerland Department of Physics, Harvard University, Cambridge, MA 02138, USA (Dated: 20201002003100)We study thermoelectric currents of neutral, fermionic atoms flowing through a mesoscopic chan-nel connecting a hot and a cold reservoir across the superfluid transition. The thermoelectric re-sponse results from a competition between density-driven diffusion from the cold to the hot reservoirand the channel favoring transport of energetic particles from hot to cold. We control the relativestrength of both contributions to the thermoelectric response using an external optical potential inboth non-interacting and strongly-interacting systems. Without interactions, the magnitude of theparticle current can be tuned over a broad range but is restricted to flow from hot to cold in ourparameter regime. Strikingly, strong interparticle interactions additionally reverse the direction ofthe current. We quantitatively model ab initio the non-interacting observations and qualitativelyexplain the interaction-assisted reversal by the reduction of entropy transport due to pairing corre-lations. Our work paves the way to studying the coupling of spin and heat in strongly correlatedmatter using spin-dependent optical techniques with cold atoms.
I. INTRODUCTION
Transport of charge, heat, and spin are often coupledin nature. Their interplay enriches the dynamical re-sponse of materials leading to coupled transport phe-nomena such as thermoelectricity [1] and, along concep-tually similar lines, spintronics [2] and spin caloritron-ics [3]. Thermoelectricity includes two major observa-tions where an applied temperature gradient can inducea charge current (Seebeck effect) or an external voltagecan give rise to a heat current (Peltier effect). Besidestheir widespread practical applications, both are essentialto probe fundamental physics. In particular, studies ofthese effects in strongly correlated materials have allowedresearchers to identify relevant charge carriers [4–6] anddegrees of freedom [7] which have been proposed to char-acterize exotic states such as Majorana modes or anyons[8].Microscopically, thermoelectric currents in conven-tional materials originate from an electron-hole asymme-try created by an energy-dependent density of states andcarrier velocity. In the case of a temperature gradient forinstance, this asymmetry can favor the transport of high-energy particles from the hot side over the transport oflow-energy particles from the cold side. This imbalanceresults in a net carrier flow whose magnitude increaseswith asymmetry.In solid state systems, several techniques have beenexplored to engineer the thermoelectric response. First,the energy-dependence of the density of states can be en-hanced by reducing the number of free dimensions [9, 10]and using electrostatic gate potentials in low-dimensionalstructures such as quantum wires [11], point contacts [12] ∗ Current address: X-Rite Europe GmbH, 8150 Regensdorf,Switzerland, [email protected] † [email protected] and dots [13]. Furthermore, the thermoelectric responsecan be strongly modified by electron interactions as ob-served in quantum dots [13] and two-dimensional electrongases [14]. However, the interpretation of the thermoelec-tric response in solids is complicated by interactions ofthe carriers with impurities, defects and phonons.Due to the absence of these factors, ultracold atomsare well-suited to simulate the relevant physics of realmaterials. In addition, Feshbach resonances allow one tostudy the same system across a large range of interac-tion strengths under comparable conditions. These mer-its have facilitated experiments on transport phenomenain strongly interacting Fermi gases including spin diffu-sion [15, 16], sound propagation [17, 18], and heat trans-port in the form of second sound [19].As these experiments focused on bulk material prop-erties, they lacked the tunability of mesoscopic systems,however recent work in optically shaping bulk gases madeit possible to create mesoscopic cold atom ”devices” com-parable to their solid state counterparts [20–28]. In par-ticular, thermoelectric phenomena were explored in thesemesoscopic structures focusing on either weak [29] orstrong [30] interactions. Here, we exploit the ability tocompare both interaction regimes in the same structureand, by extending the accessible range of gate poten-tials, observe a striking reversal of the thermoelectriccurrent directly induced by interactions. This reversal isa novel effect in cold atoms and, to our knowledge, alsoin strongly correlated solid materials. With the gate, wecan finely tune the particle-hole asymmetry at the originof the thermoelectric current and thereby engineer bothits magnitude and direction. We can therefore smoothlyturn our system from a heat engine into a heat pump,the latter of which being an important ingredient for anefficient cooling scheme proposed for cold atoms [31].For weak interactions, the thermoelectric response canbe predicted by an ab initio Landauer model thanks tothe absence of defects and precise characterization ofour system. With strong interactions, we focus on tem- a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p peratures around the superfluid transition where a largecritical region is predicted [32] and Fermi liquid behav-ior breaks down [33]. Although the strongly correlatedregime is often challenging to understand, we interpretour observation on a fairly fundamental level based onentropy transport. Other works have studied theoreti-cally thermoelectric effects with cold atoms for bosonic[34–38] and fermionic [39–41] systems.The structure of the paper is as follows. After intro-ducing the setup in Sec. II, we explain its thermoelec-tric response with an intuitive picture [Sec. III] and dis-cuss the dynamics in the non- and strongly-interactingregimes [Sec. IV]. Based on a phenomenological modelpresented in Sec. V we extract transport properties anddiscuss their behaviors in the subsequent Sec. VI and VII.Finally, we conclude in Sec. VIII. Technical details canbe found in the Appendices. II. SETUP
Our transport setup consists of a mesoscopic channelsmoothly connected to reservoirs of degenerate fermions( Li), as sketched in Fig. 1(a) and described in previ-ous works [21, 42]. The channel is created by a repul-sive TEM -like laser beam that confines the atoms alongthe z direction to its dark nodal plane, reaching a trap-ping frequency ν cz = 4 . y ) ensures thesmooth connection of channel and reservoirs. In the x di-rection the atoms are restricted by the dipole trap provid-ing a weak confinement with frequency ν tx = 232(1) Hz.The reservoirs, denoted by left (L) and right (R), con-tain equal atom numbers with N = N L + N R = 121(2) · atoms in each of the lowest and third lowest hyper-fine states. To prepare a temperature difference we heatone side by parametrically modulating the intensity of anattractive beam inside one reservoir, while blocking thechannel. Subsequent equilibration of each reservoir leadsto a temperature difference ∆ T = T L − T R = 147(11) nKand an average value ¯ T = ( T L + T R ) / V g >
0) and the otherattractive ( V g <
0) [Fig. 1(a)]. By tuning the gate thenumber of available modes below the chemical potential¯ µ = ( µ L + µ R ) / z direction makingthe channel quasi-two-dimensional. Note that non-zerotemperature leads to a partial occupation of modes at en-ergies above the chemical potential, which also contributeto transport.Using a broad Feshbach resonance of Li we set the in-terparticle interactions in the entire system including thechannel and the reservoirs either to zero or on resonance(unitarity). Subsequently, we enable transport for a vari-able time t and measure the differences in atom numbers µ R E f R (E)V g < 0Φ(E)V g > 0E Hot particles Cold particles µ L µ L E f L (E) Channel Cold reservoirHot reservoir V g < 0V g > 0 N R T R N L T L yxz (b)(a) FIG. 1. Concept and experimental setup. (a) Schematic viewof a quasi-two-dimensional channel connected to a hot (red)and a cold (blue) reservoir of fermionic atoms. Their unequaltemperatures T L and T R induce a net particle current thatchanges the initially equal atom numbers N L and N R overtime. The thermoelectric response is tuned by a gate poten-tial V g inside the channel that is either attractive (red) orrepulsive (green). (b) Roles of the reservoirs and channel inthermoelectric transport. The reservoirs inject particles intothe channel according to their occupations described by theFermi-Dirac distributions f L ( E ) and f R ( E ). As the hot (cold)side contains more particles at high (low) energies comparedto the other side (dotted regions), two counter-propagatingcurrents at different energies emerge (horizontal arrows). Atequal atom numbers the chemical potential µ L in the hotreservoir is lower than µ R on the cold side introducing anasymmetry between them. A particle at energy E crosses thechannel in one of the Φ( E ) transverse modes. The energydependence of their number (shaded regions) introduces anasymmetry in the channel. The gate potential energeticallyshifts the modes up ( V g >
0) or down ( V g <
0) allowing us totune the response. The picture is valid for weak interactionsas it also applies to Landau quasi-particles. ∆ N ( t ) and temperature ∆ T ( t ) from an absorption im-age. The initial conditions stated above correspond tothe non-interacting case and are given for the unitarycase in appendix A, together with additional experimen-tal details. III. INTUITIVE LANDAUER PICTURE
In the absence of interactions the response can be un-derstood in a Landauer picture [Fig. 1(b)]. The totalthermoelectric current is a result of two competing ef-fects that favor particle currents in opposite directions.(i) Since the atom numbers are equal in each reservoir thechemical potential difference only depends on the differ-ence in temperatures. Increasing temperature lowers thechemical potential as a result of the particle-hole asym-metry of the reservoir density of states around the Fermienergy. This reservoir asymmetry prefers particle cur-rents from the cold to the hot side. (ii) In order to betransferred from one reservoir to the other, a particle ofenergy E will transit via one of the transverse modesavailable in the channel, counted by the transport func-tion Φ( E ). The higher the energy, the more modes areavailable leading to a faster transfer of energetic parti-cles. This channel asymmetry favors currents from thehot to the cold reservoir, since the hot reservoir has anexcess of energetic particles compared to the cold side.The channel asymmetry is controlled by the gate po-tential inside the channel as intuitively illustrated inFig. 1(b): For the repulsive case, V g >
0, the channel pre-dominantly allows transport of the most energetic parti-cles present in the hot reservoir. Hence, the net currentflows from hot to cold. As the gate potential is madeincreasingly attractive particles from the cold reservoirstart to contribute and reduce the net current until itvanishes. The precise gate potential where the cancella-tion happens depends on the initial reservoir conditionsand on the transport function Φ( E ). At sufficiently at-tractive gate potentials, V g <
0, almost all particle en-ergies contribute and their direction from cold to hot ispurely defined by the chemical potential gradient. Weexpect the intuitive picture to be also valid for weak in-teractions as it applies to Landau quasi-particles.
IV. SYSTEM DYNAMICS
Fig. 2(a) presents the experimental evolution of theatom number and temperature difference of a non-interacting Fermi gas subject to an initial temperaturegradient for different gate strengths. Due to the com-bined effect of the asymmetries of the reservoirs and thechannel, particles flow from the hot to the cold sideand build up a negative difference in atom numbers,∆ N = N L − N R . Simultaneously, a heat flow drivesthe temperatures towards equilibrium thus weakeningthe thermoelectric current. The particle number differ-ence that builds up induces diffusion that flows againstthe thermoelectric current and it vanishes at the turn-ing point. Subsequently, diffusion brings the system toequilibrium. This evolution occurs for all gate potentials,except the most attractive one, V g = − . µ K, where thethermoelectric current vanishes. The observed influenceof the gate agrees with the intuitive Landauer picture.For more attractive potentials more transverse modes,counted by the function Φ( E ), open up and permit afaster relaxation. Moreover, the initial response reducesas quantified by the initial particle current I N (0) in theinset [Appendix B 2]. The current is normalized with theinitial bias ∆ T to remove variations in the preparation and plotted versus local chemical potential µ loc at thechannel center [Appendix D].The measurements at strong interactions are presentedin Fig. 2(b). As for the non-interacting case the initialtemperature bias is converted into a particle number dif-ference and eventually relaxes back to equilibrium. Incontrast the initial and relaxation dynamics with inter-actions are about three times faster and the normalizedinitial currents are enhanced. This is a consequence ofthe gate tuning not only the available number of modesbut also the density inside the channel. Thus from repul-sive to attractive gate potentials the gas becomes denserand is expected to eventually cross the superfluid tran-sition in the channel at µ loc = 0 . µ K (dashed line ininset) [Appendix B 4 and [43]].Strikingly, near the critical point the net initial currentreverses its direction while this effect is absent for anideal gas in the same parameter regime. In principlethe reversal may also be achieved without interactionsas predicted from the Landauer picture and is expectedto occur for more attractive gate potentials. However,strong attractive gates induce atom losses which preventthe observation of current reversal.Contrary to the present work, the current was flowingonly in one direction in previous experiments: from hotto cold for a non-interacting two-dimensional gas in theabsence of a gate [29] or from cold to hot in the strongly-interacting, quasi one-dimensional case [30].As the assumption of a Fermi liquid underlying theLandauer picture breaks down at unitarity, we employa phenomenological model that captures the transportproperties irrespective of the interaction strength. Basedon the model we reason in section VI why the reversaloccurs with interactions.
V. PHENOMENOLOGICAL MODEL
To extract transport properties from the time evolu-tions we apply a phenomenological model in linear re-sponse. It relates currents of particles I N and entropy I S to the differences in chemical potential ∆ µ and temper-ature ∆ T [44, 45]. (cid:18) I N I S (cid:19) = G (cid:18) α c α c L + α c (cid:19) · (cid:18) ∆ µ ∆ T (cid:19) (1)As a consequence of the Onsager reciprocal relations,the channel properties are described by only three co-efficients: the particle conductance G , the heat conduc-tance G T , and the Seebeck coefficient or thermopower α c . The thermopower quantifies the coupling betweenparticle and entropy currents and the Lorenz number L = G T / ¯ T G measures the relative strength of particleand heat conductances.Thermopower is essential to understand how the com-peting asymmetries of reservoirs and channel appear inthe model. At the initial time, the reservoir asymmetry P a r t i c l e i m b a l a n ce N ( ) (a) Non-interacting 2010010(b) Strongly-interacting0100 V g = -1.06µK loc = 1.07µK0100 V g = 0.13µK loc = -0.10µK0100 V g = 0.24µK loc = -0.22µK0 1 2 3 4Time t (s)0100 V g = 0.37µK loc = -0.36µK 0100 V g = -1.06µK loc = 1.05µK0100 V g = 0.13µK loc = -0.09µK0100 V g = 0.24µK loc = -0.23µK0.0 0.5 1.0 1.5Time t (s)0100 V g = 0.37µK loc = -0.35µK0 1 loc (µK)010 I N ( ) T ( k B h ) loc (µK)50050 I N ( ) T ( k B h ) T e m p e r a t u r e i m b a l a n ce T ( n K ) FIG. 2. Tuning the thermoelectric response. Temporal evolution of the differences in atom number ∆ N = N L − N R andtemperature ∆ T = T L − T R between the reservoirs of (a) a non- and (b) a strongly-interacting Fermi gas at different gatepotentials V g . Each data point represents a mean over ∼ G , thermopower α c and Lorenznumber L [Sec. V]. For all estimations conductance is fixed by a separate measurement (solid lines), except for the dashed lineas it failed to converge otherwise. Insets: Initial particle current I N (0) normalized to the prepared temperature difference ∆ T versus local chemical potential µ loc at the channel center. In the strongly-interacting regime the vertical dashed line shows thesuperfluid transition inside the channel while the reservoirs remain uncondensed [Appendix B 4]. translates a difference in temperature ∆ T to a differ-ence in chemical potential ∆ µ = − α r ∆ T described bythe dilatation coefficient α r = − ( ∂µ/∂T ) N . Togetherwith Eq. (1) the initial response is I N (0) = Gα ∆ T (2)where the effective thermopower α = α c − α r determinesthe outcome of the competition between the asymmetriesof the channel and the reservoirs, captured by α c and α r ,respectively.At strong interactions and lowest temperatures thisphenomenological model is expected to fail due to non-linearities caused by superfluid effects in the reservoirs[46]. However at our elevated temperatures it capturesthe behavior well as observed in [30].The transfer of particles and entropy between the reser- voirs via the associated currents I N ( t ) = − ∂ t ∆ N ( t ) / I S ( t ) = − ∂ t ∆ S ( t ) / N and∆ T towards equilibrium [44] [Appendix B 2],∆ N ( t ) = { e + ( t ) + Ae − ( t ) } ∆ N + Be − ( t )∆ T , (3)∆ T ( t ) = { e + ( t ) − Ae − ( t ) } ∆ T + Ce − ( t )∆ N , (4)with e ± ( t ) = (e − t/τ + ± e − t/τ − ) / τ ± . Both timescales and the coefficients A,B and C depend on the transport properties conduc-tance G , thermopower α c and Lorenz number L and onthe thermodynamical properties of the reservoirs throughcompressibility, heat capacity and dilatation coefficient α r . The values of the reservoir properties are taken fromthe measurement [Appendix B 1 c].We directly extract the transport coefficients from thetransients by fitting ∆ N ( t ) and ∆ T ( t ) simultaneously,normalized by their statistical uncertainty. To improveaccuracy we reduce the number of free parameters byfixing conductance to a separately measured value [Ap-pendix B 3]. The fits are shown as lines in Fig. 2. For thedashed curve conductance was left as a free parameter asit failed to converge otherwise. We discuss the deter-mined transport coefficients in the subsequent chapters. VI. THERMOPOWER
We first discuss conductance and then present and an-alyze the results for thermopower. Fig. 3(ab) displaysthe measured conductances versus local chemical poten-tial µ loc in the non- (dark blue) and strongly-interacting(orange) regime. Conductance is increased by interac-tions, as previously observed [42] and is a factor of 13(1)larger at the most attractive gate strength [Fig. 3(b)]. Inthe non-interacting regime the solid line represents an abinitio prediction that reproduces the measurement well.The model is based on Landauer’s theory that followsthe idea that both reservoirs inject particles according totheir occupations and the channel transmits them in oneof the available modes [47] [Fig. 1(b) and Appendix E].The light blue point indicates the fitted conductance cor-responding to the dashed curve in Fig. 2(a) and is con-sistently higher than model and separate measurement.The behavior of the thermopower is summarized inFig. 3(c). In the non-interacting regime the thermopower α c of the channel (blue points) reduces with increasinglocal chemical potential towards the dilatation coefficient α r (horizontal blue line). As expected the reductionoriginates from a suppression of the channel asymme-try with more attractive gate strengths as illustratedin Landauer’s picture. Moreover, the dilatation coef-ficient is a property of the reservoirs and thus unaf-fected by the gate that locally acts on the channel. TheLandauer prediction is indicated as a solid black curveand reproduces well the extracted thermopower in thenon-interacting case, except for the most attractive gatestrength ( µ loc = 1 . µ K). There theoretically the rever-sal of the thermoelectric current is anticipated ( α c < α r ),while experimentally it is absent ( α c (cid:38) α r ) which mightoriginate from the detailed shape of the confining po-tential not captured in the theory, such as anharmonic-ities. Overall, the tunability of the thermopower in thenon-interacting regime demonstrates almost full controlof the thermoelectric response. In contrast, at unitaritythe thermopower α c becomes smaller than the dilatationcoefficient α r for sufficiently attractive gate potentialsand additionally α r and α c are reduced compared to thenon-interacting case.The interaction-assisted reduction and reversal can be qualitatively understood from an interpretation based onentropy. On the one hand, the role of the reservoirsis captured by the dilatation coefficient that can be ex-pressed as the entropy content to add a particle isother-mally, α r = ( ∂S/∂N ) T . On the other hand, rewritingEq. (1) as I S = α c I N + G T ∆ T ¯ T (5)allows for reinterpreting the thermopower of the chan-nel as the average entropy that is reversibly transportedby one particle, while the second term captures the irre-versible entropy exchange between the reservoirs. At uni-tarity, pairing correlations reduce the entropy in the spinsector and account for the decrease of α c and α r com-pared to the non-interacting case, as visible in Fig. 3(c)and in [43]. Because the attractive gate increases the den-sity and therefore the interaction effects at the center, weexpect further reduction of the thermopower α c .This argument suggests a way to estimate where thecurrent reverses in the presence of interactions. Sincethe interparticle collision rate is enhanced at unitarity,we assume the gas to be locally in equilibrium at thecenter [Appendix C]. This allows us to think of the ther-mopower α c as the dilatation coefficient inside the chan-nel at temperature ¯ T and chemical potential µ loc leadingto a reversal expected at µ loc ∼ − . µ K. The value isrelatively close to the observed location despite the sim-plicity of the estimation and the neglect of the transversemode structure.
VII. LORENZ NUMBER
The Lorenz number L = G T / ¯ T G compares the abil-ity of systems to conduct heat and particles and in-dicates whether Fermi liquid behavior is present. Inthis case transport is described by Landau quasiparticlesthat carry both charge and energy leading to a constantLorenz number, L WF = π / · k B . This is known as theWiedemann-Franz law.Fig. 4 displays the fitted Lorenz number versus localchemical potential in the non- (blue dots) and strongly-interacting (orange dots) case. Without interactions theextracted values are consistently higher than L WF andalso than the Landauer theory (solid line). It consid-ers our mesoscopic geometry at finite temperature andapproaches L WF with increasing degeneracy. Deviationsfrom the Wiedemann-Franz law are found in other sys-tems with either increased [48] or decreased [49–51] num-bers. Here, we partly attribute the inconsistency to smallsystematic shifts in the measured temperature differencethat mostly affect the Lorenz number [Appendix B 3].Despite the challenging absolute estimation the Lorenznumber is reduced by one order of magnitude whenincreasing the interactions to unitarity. As the fittedheat conductance is relatively insensitive to interactionstrength, this decrease can be mostly attributed to the loc (µK)04080 G ( / h ) (a) Non-interacting 0 1 loc (µK)05001000(b) Strongly-interacting0.3 0.0 0.3 0.6 0.9 1.2Local chemical potential loc (µK)234 T h e r m o p o w e r c ( k B ) rr (c) FIG. 3. Controlling thermopower with a gate potential. (ab)Conductance G versus local chemical potential µ loc in the (a)non- and (b) strongly-interacting regimes. Separately mea-sured values are indicated as dark points [Appendix B 3] andthe light blue one shows the fitted conductance correspondingto the dashed curve in Fig. 2(a). (c) Fitted thermopower α c without (blue points) and with strong (orange points) inter-particle interactions and the corresponding dilatation coeffi-cients α r of the reservoirs (horizontal lines). Shaded regionsindicate one standard deviation to each side. Solid black linesshow an ab initio prediction based on Landauer theory in theabsence of interactions [Appendix E]. The vertical dashed linelocates the superfluid transition inside the channel while thereservoirs remain uncondensed [Appendix B 4]. No measure-ment was taken at µ loc ∼ . µ K in the non-interacting case. enhancement of the conductance seen in Fig. 3(ab). Alsothe smaller uncertainties on the Lorenz number stemfrom the enhanced conductances and give us confidencein thinking that the Wiedemann-Franz law is violatedhere, as experimentally observed in a one-dimensionalgeometry [30] and theoretically supported in [40, 41].
VIII. DISCUSSION
Mesoscopic transport properties are influenced by thegeometry of the structure [52] thus it is instructive tocompare our findings at unitarity with a one-dimensionalchannel measured in similar conditions [30]. There, thenarrow geometry blocked irreversible heat currents lead-ing to a non-equilibrium steady state with a finite tem-perature difference. Relaxation was restored by wideningthe channel thanks to an enhanced heat exchange. Thisagrees with our results for a wide quasi-two-dimensionalchannel where we observe relaxation to equilibrium anda heat conductance that is similar to the non-interacting loc (µK)0123456 L o r e n z nu m b e r L ( k B ) L WF FIG. 4. Lorenz number L versus local chemical potential µ loc at the channel center. Fitted values in the non- andstrongly-interacting regimes are shown as blue and orangepoints, respectively. The theoretical prediction based on Lan-dauer theory is indicated by a solid black curve [Appendix E]and the Wiedemann-Franz value L WF = π / · k B valid fordegenerate Fermi liquids by a blue horizontal line. In thestrongly-interacting case the vertical dashed line locates thesuperfluid transition inside the channel while the reservoirsremain uncondensed [Appendix B 4]. case. In both works the Lorenz number is reduced withinteractions by an order of magnitude. However the rea-sons are different: In [30] it stems from a reduced heatconductance while here it is almost purely an effect ofenhanced particle conductance. Thermopower is reducedby interactions in our wide quasi two-dimensional geome-try as qualitatively explained by pairing correlations thatrestrict average entropy transfer per particle. In contrast,in the quasi one-dimensional channel it shows the surpris-ing behavior of following the non-interacting prediction,which so far eluded any explanation [40].In summary, we control magnitude and direction ofthermoelectric currents through a mesoscopic structurein the presence of weak and strong interparticle inter-actions. In our parameter regime at weak interactionsparticles are flowing consistently in one direction whileat unitarity we observe a striking interaction-assisted re-versal. We explain the reversal by a competition of reser-voir and channel parameters which are affected by pairingcorrelations that are expected in the large critical regionaround the superfluid transition [32]. Indeed, the reversaloccurs before the normal-to-superfluid transition and itsprecise location depends on the geometry of the systemand the reservoirs conditions via thermopower and di-latation coefficient. Unitary Fermi gases are not the onlysystem where thermopower can be affected by interac-tions: In a two-dimensional Bose gas the Seebeck coef-ficient changes its sign close to the superfluid transition[53] and in two-dimensional electron gases interactionsare predicted to reduce the thermopower and potentiallyreverse its sign [54].The option to induce thermoelectric currents in eitherdirection is appealing when considering our dynamics asan open thermodynamic cycle. In our system, an atomicflow from hot to cold acts against the chemical poten-tial bias and converts heat into work as a thermoelec-tric engine. Inversely, the system acts as a thermoelec-tric cooler when the flow transfers heat from cold to hot.The initial direction of the current therefore determineswhich mode of operation takes place before the other: athermoelectric engine in the channel-dominated regimeor a cooler in the reservoir-dominated regime. In bothmodes the conversion efficiency of the cycles is charac-terized by the figure of merit ZT = α c /L [45]. Con-trary to a non-interacting system interactions stronglyreduce the Lorenz number while thermopower remainsat a similar order of magnitude. Overall, we estimatethat interactions improve the figure of merit by a fac-tor of 7(3) showing the relevance of strongly correlatedquantum materials for thermoelectric applications.Our system readily allows to probe the thermoelectricresponse of more complicated structures. Drawing fromthe technique to imprint local effective Zeeman shifts[55] it opens new perspectives on coupling spin and heattransport [3]. By adding strong correlations intriguingthermoelectric effects could be observed [56, 57]. ACKNOWLEDGMENTS
We thank Thierry Giamarchi, Leonid Glazman, Ren´eMonnier, Shun Uchino, Anne-Maria Visuri for fruitfuldiscussions, Alexander Frank for electronic support andMohsen Talebi for careful reading of the manuscript. Wethank Daniel Kestner for support and source code regard-ing the rank order fitting method. We acknowledge theSwiss National Science Foundation (Grants No. 182650and No. NCCR-QSIT) and European Research Counciladvanced grant TransQ (Grant No. 742579) for funding.L.C. is supported by an ETH Zurich Postdoctoral Fel-lowship, the Marie Curie Actions for People COFUNDprogram, and the European Union Horizon 2020 MarieCurie TopSpiD program (Grant No. 746150).
Appendix A: Experimental details
A cloud of fermionic lithium atoms is prepared in abalanced mixture of the lowest and third lowest hyper-fine state in a hybrid trap. In the transverse directions( x , z ) the trap is formed by a far-detuned laser beamat a wavelength of 1064 nm and longitudinally ( y ) bya quadratic magnetic field. We evaporatively cool theelongated cloud by reducing the transverse confinementeither on a broad Feshbach resonance at 689 G whenpreparing a unitary gas, or at 319 G when preparing anon-interacting gas. Subsequently, the magnetic field iskept on resonance or ramped to the zero-crossing of thescattering length at 568 G. Finally, the optical trap is re-compressed leading to the confinement frequencies dur-ing transport ν tx = 232(1) Hz and ν tz = 212(1) Hz in Quantity Non-interacting Unitary N (10 ) 121(2) 107(4) T L (nK) 281(8) 220(5) T R (nK) 134(7) 129(3) µ L ( k B nK) − µ R ( k B nK) 344(12) 238(8)¯ T (nK) 208(6) 174(2)¯ µ ( k B nK) 151(16) 152(9)∆ T (nK) 147(11) 90(7)∆ µ ( k B nK) − − transverse direction and longitudinally ν ty of 26 . . π phase plate intoa TEM mode and focused onto the center of the cloud.This compresses the cloud in vertical z direction resultingin a harmonic confinement of ν cz = 4 . /e waists are in longitudinal direction w cy = 30 . µ m andtransversally w cz = 9 . µ m.We prepare the initial conditions of the two reservoirsin two steps. First, the cloud is centered on the trans-port channel using a magnetic gradient to make the atomnumbers on both sides equal, and then we split the reser-voirs with an elliptical repulsive laser beam. Second, atemperature difference is created by heating one side witha red-detuned beam at 767 nm focused into one reser-voir and modulating its intensity parametrically. The ex-perimentally optimized modulation frequency is 547 Hz,which is on the order of the transverse confinement fre-quencies ν tx and ν tz . The beam position is controlledwith a piezo-mirror and can be directed into either reser-voir. After letting the reservoirs thermalize for 10 ms wereach the initial conditions reported in Table I.The attractive gate potential is created with the samebeam used for heating and requires a careful alignmentonto the channel center. Its waists are w gx = 34 . µ mand w gy = 33 . µ m. The repulsive one is formed by anelliptical beam at 532 nm that is focused onto our channelwith waist w gx = 53 . µ m in transversal directionand along transport w gy = 8 . µ m. Besides acting asa gate the same beam enables and blocks transport atlarge powers in a controlled way. After the time t thereservoirs are again separated and an absorption pictureis taken after a short time of flight of 1 ms. This reducesthe central densities and allows to image with intensitieswell below saturation. From the absorption image weextract the thermodynamical properties as described inB 1 c. Appendix B: Data analysis1. Thermodynamics
Across the appendix, the equation of state of a non-interacting and unitary Fermi gas are used in the homo-geneous and trapped cases. This section summarizes therelevant thermodynamical relations. a. Homogeneous unitary Fermi gas
The universality hypothesis states that the homoge-neous unitary Fermi gas is described by the interatomicdistance and the thermal wavelength only [58]. This re-stricts the equation of state to the form [59] nλ T = f n ( q ) (B1)with the particle density n , the thermal wavelength λ T = (cid:112) π (cid:126) /mk B T and the dimensionless function f n depending on the reduced chemical potential q = βµ .The scaling function f n was measured around the su-perfluid transition [43] and theoretically extended in thedegenerate and thermal limits with a phonon model anda third order virial expansion, respectively [59][60]. Us-ing the equation of state the normalized temperature isgiven as TT F = (cid:18) √ πf n ( q ) (cid:19) / (B2)with the Fermi energy E F = k B T F = (cid:126) / (2 m )(6 π n ) / [59]. This links the normalized temperature T /T F andthe chemical potential µ/E F = T /T F · q to each other.Below the critical temperature T c = 0 . · T F the gasbecomes superfluid as observed in [43]. b. Harmonically trapped Fermi gas In a harmonic trap the equilibrium properties are cap-tured by the geometric mean ¯ ν = ( ν x ν y ν z ) / of the con-finement frequencies, the Fermi energy E F = h ¯ ν (6 N ) / and the reduced chemical potential q = βµ at the trapcenter. For our work the total atom number N in eachhyperfine state, the compressibility κ = ( ∂N/∂µ ) T , thedilatation coefficient α r = − ( ∂µ/∂T ) N , the specific heat C N = T ( ∂S/∂T ) N at fixed atom number and the inter-nal energy U are relevant. Following reference [61] they are given in the non-interacting case by N = (cid:18) k B Th ¯ ν (cid:19) F ( q ) , (B3) κN = 1 k B T F ( q ) F ( q ) , (B4) α r k B = 3 F ( q ) F ( q ) − q , (B5) C N N k B = 12 F ( q ) F ( q ) − F ( q ) F ( q ) , (B6) UN E F = 36 / F ( q ) F ( q ) / , (B7) TT F = (6 F ( q )) − / , (B8)with F j symbolizing the complete Fermi-Dirac integralof order j [62, Eq. (25.12.14)]. At unitarity the relationsare stated in [63] and read as, N = 4 √ π (cid:18) k B Th ¯ ν (cid:19) N ( q ) , (B9) κN = 12 k B T N ( q ) N ( q ) , (B10) α r k B = 6 N ( q ) N ( q ) − q , (B11) C N N k B = 8 N ( q ) N ( q ) − N ( q ) N ( q ) , (B12) UN E F = (cid:16) π (cid:17) / N ( q ) N ( q ) / , (B13) TT F = (cid:18) √ π N ( q ) (cid:19) − / , (B14)with the integral N j ( q ) = (cid:82) ∞ r j f n ( q − r ) d r involvingthe scaling function f n defined in section B 1 a.To describe the properties of a single reservoir we needto divide the extensive quantities for the full trap by two.Here, this concerns the atom number N , the compress-ibility κ , the specific heat C N and the internal energy U . c. Extracting thermodynamical properties The basis to extract thermodynamical properties ofeach reservoir r from the absorption images is the virialtheorem [64]. It holds for non-interacting and uni-tary [65] Fermi gases and relates the internal energyper particle U r /N r = 3 mω ty (cid:104) y (cid:105) to the second moment (cid:104) y (cid:105) = (cid:82) y n r ( y ) d y/N r in transport direction. The one-dimensional density n r ( y ) can be directly deduced fromthe absorption images by summing over the transversedirection. Together with the total atom number N r we obtain the Fermi energy E F,r and subsequently withEq. (B7) or (B13) the reduced chemical potential βµ at the trap center. Then, the temperature T r and allother thermodynamical properties of the reservoirs canbe derived from the formulae in appendix B 1 b.
2. Parameter extraction
The following section explains first the linear phe-nomenological model [44] and second its application toextract the transport parameters from the measured evo-lutions [Fig. 2]. a. Phenomenological model
The evolutions in the differences in atom number andtemperature are phenomenologically modeled by the re-sponse matrices of the channel and the reservoirs. Thechannel reacts to the applied biases with currents of par-ticles and entropy [Eq. (1)] described by (cid:18) I N I S (cid:19) = G (cid:18) α c α c L + α c (cid:19) · (cid:18) ∆ µ ∆ T (cid:19) (B15)with the transport coefficients conductance G , ther-mopower α c and Lorenz number L . The reservoir ther-modynamics are described in linear response by (cid:18) ∆ N ∆ S (cid:19) = κ (cid:18) α r α r l + α r (cid:19) · (cid:18) ∆ µ ∆ T (cid:19) . (B16)The involved thermodynamical quantities of each reser-voir are the compressibility κ , the dilatation coefficient α r and the reservoir analogue of the Lorenz number l = C N / ¯ T κ that depends on the heat capacity C N .Within linear response they are constant throughout theevolution and are evaluated at the equilibrium chemicalpotential ¯ µ and temperature ¯ T [Appendix B 1 b]. Thetwo sets of equations are related via I N ( t ) = − ∂ t ∆ N ( t ) / I S ( t ) = − ∂ t ∆ S ( t ) / N ( t ) = { e + ( t ) + Ae − ( t ) } ∆ N + Be − ( t )∆ T , (B17)∆ T ( t ) = { e + ( t ) − Ae − ( t ) } ∆ T + Ce − ( t )∆ N , (B18)with e ± ( t ) = (e − t/τ + ± e − t/τ − ) / τ ± = τ /λ ± and τ = κ/ G . The eigenvalues of the evolutionmatrix describing the system of differential equations are λ ± = 12 (cid:18) L + α l (cid:19) ± (cid:115) (cid:18) L + α l (cid:19) − Ll . (B19) The coefficients A, B and C depend on the transportand reservoir properties A = 1 − ( L + α ) /lλ + − λ − (B20) B = 2 καλ + − λ − (B21) C = 2 ακl ( λ + − λ − ) (B22)with the effective thermopower α = α c − α r . Note thatthese equations correct typographical errors present in[44]. b. Data fitting We extract the transport parameters by fitting simul-taneously the measured evolutions to the phenomenolog-ical model using the standard least squares method. Itminimizes the following sum of the squared residuals: χ = (cid:88) i (cid:32) ∆ ˜ N ( t i ) − ∆ ˜ N i σ N (cid:33) + (cid:32) ∆ ˜ T ( t i ) − ∆ ˜ T i σ T (cid:33) (B23)The simultaneous fitting requires normalizing the residu-als with the respective uncertainties σ N and σ T whichwe determine as an average over the standard devia-tions at the individual times. The measured differences∆ ˜ N i and ∆ ˜ T i at the time t i include small offsets dueto imperfections in the calibrations. We account forthese by shifting the model evolutions ∆ N ( t ) and ∆ T ( t )by constants and obtain ∆ ˜ N ( t ) = ∆ N ( t ) + ∆ N off and∆ ˜ T ( t ) = ∆ T ( t ) + ∆ T off , respectively.In the model the reservoir properties κ , α r and l areset to the values directly extracted from the absorptionimages [Appendix B 1 c]. Moreover, the conductance G is fixed by a separate measurement [Appendix B 3] whichreduces the number of free parameters and improves thefit.We determine ∆ N and ∆ T from the correspondingmeasured initial values. The offset ∆ N off follows eitherfrom an average over points where the evolution is re-laxed or is taken to be the same as the initial value. Incontrast, the offset ∆ T off cannot be easily determined forevolutions that do not completely relax within our mea-surement. Thus we fit it consistently for all dynamics andchecked with evolutions where temperature completelyrelaxes that the results do not depend on whether theoffset ∆ T off is fixed or left free. In summary the parame-ters α , L and ∆ T off are free in the model. Exceptionally,for the non-interacting curve at V g = 0 . µ K conduc-tance is additionally left free as otherwise the fit failedto converge [Fig. 2].To analyze the parameter estimation we exemplify thesum of the squared residuals χ versus thermopower α and Lorenz number L for the gate potential V g = 0 . µ K0[Fig. 5(ab)]. For both interaction strengths there existsa single, isolated global minimum that is found by theLevenberg-Marquardt algorithm (gray point). Its uncer-tainty only includes the standard deviations σ N and σ T and assumes the fixed parameters to be precisely known.However, their uncertainties will lead to a change in theoptimal values for α and L . Fig. 5(cd) shows a plot ofthese optimal values when one fixed parameter is variedwithin its uncertainty. These variations are mostly sym-metrical and thus leave the values unaffected but increasethe overall error bar in thermopower and Lorenz number.It is known that estimating parameters of exponentialmodels is challenging as small changes in the data canstrongly influence the parameters [66]. Thus in additionto the least squares method we used a technique basedon rank order that is robust against outliers and shownto improve the estimation for exponential models [67,68]. Applying this method led to comparable results andhence we conclude that for our model and data the leastsquares method performs well. As the estimation withthe rank order fit is numerically demanding due to manylocal closely spaced minima [67] we report the values withthe least squares method which gives a single well-definedminimum.Based on the fitted results the initial response I N (0) / ∆ T = Gα is shown in the insets of Fig. 2, thethermopower of the channel α c = α + α r in Fig. 3 andthe Lorenz number L in Fig. 4.
3. Conductance measurement
To measure conductances we prepare the reservoirs atdifferent particle numbers and equal temperatures andstudy its exponential decay towards equilibrium. We fitthe characteristic time τ and together with the com-pressibility κ of each reservoir [details in appendix B 1]conductance follows from the relation τ = κ/ G analo-gous to a discharging capacitor [24]. The conductancesare measured at the same average chemical potential ¯ µ and temperature ¯ T as the thermoelectric responses toallow comparison between them. The conductances areshown in Fig. 3(ab).The conductances are extracted under the assumptionthat both reservoirs are at equal temperatures. However,we suspect that in our measured temperature differencesan offset is present, as visible in Fig. 2 when the system isrelaxed. During preparation we heat both reservoirs upto seemingly the same temperature ¯ T while the measure-ment offset leads to a physical temperature difference. Inturn it induces a thermoelectric current with value I N = G ∆ Nκ (cid:18) κα ∆ T ∆ N (cid:19) , (B24)which follows from Eq. (B15) and (B16) and the ther-mopower α = α c − α r . The second summand in thebrackets quantifies the relative deviations that are around10 % estimated for the non-interacting case at a gate potential V g of 0 . µ K. We used the compressibility κ = 22 s / h, the thermopower α = 0 . k B , the particlenumber difference ∆ N = 48 · and the residual tem-perature bias ∆ T ∼ −
15 nK.A systematically increased conductance mostly reducesthe fitted Lorenz number and only slightly decreases thethermopower, as visible in Fig. 5(cd). The Lorenz num-ber is reduced to around 4 . k B for a gate potential of V g = 0 . µ K and similarly for the other non-interactingconditions. Thus the systematic shifts in the conductancecan partly explain the deviations from the Wiedemann-Franz value L WF = π / · k B in the non-interacting mea-surements.
4. Superfluid transition at unitarity
Superfluidity inside the reservoirs or the channel canstrongly influence the transport properties and thus it isimportant to characterize the gas in these locations inthe strongly-interacting regime.The transition is captured by the thermodynamics ofthe gas and characterized by the degeneracy
T /T F aspresented in Fig. 6 versus the local chemical potential.At the channel center (black points) the gas is in a hy-drodynamic regime thanks to the strong interparticle in-teractions [Appendix C] and thus it is represented by thelocal equilibrium described by the average temperature¯ T = 172 nK and the local chemical potential µ loc [Ap-pendix D]. Thus with increasing chemical potential thedegeneracy increases as stated by the thermodynamics ofa homogeneous unitary Fermi gas [Appendix B 1 a] andeventually crosses the critical value T c /T F = 0 . µ loc = 0 . µ K (gray dotted lines) [43].The hot (red points) and cold (blue points) reservoirsare not affected by the gate potential and hence their de-generacy remains constant. They are defined by a half-harmonic potential where the densest point at the centerspecifies the transition at a critical value T c /T F = 0 . T F .In summary the reservoirs are both above the criticaltemperature while the channel center transitions froma non-condensed to a superfluid state depending on thechemical potential. Thus in our system the gas may be-come superfluid inside the channel. Appendix C: Boltzmann approach at unitarity
Inside the channel interatomic collisions may alter thetransport regime from ballistic to hydrodynamic depend-ing on how frequently they scatter while crossing the con-striction. To assess the regime at unitarity we follow theBoltzmann approach outlined in [69] and compare theresulting mean free path with the length of the channel.First we focus on the center where the gas is homoge-neous and then extend the argument for the full channel.1
500 1000 1500 2000 (a) Non-interacting0.0 0.5 1.0 1.5Thermopower (k B )2468 L o r e n z nu m b e r L ( k B ) -0.09 0 0.09 (k B )-0.5900.59 L ( k B ) -0.09 0 0.09 (k B ) 300 600 900 1200 (b) Strongly-interacting0.5 0.0 0.5 1.0Thermopower (k B )0.10.20.30.40.5 L o r e n z nu m b e r L ( k B ) -0.07 0 0.07 (k B )-0.0500.05 L ( k B ) -0.07 0 0.07 (k B )(c) (d) FIG. 5. Transport parameter estimation. (ab) Sum of squared residuals χ versus effective thermopower α and Lorenz number L in the (a) non- and (b) strongly-interacting regimes at V g = 0 . µ K. Contour lines highlight the behavior at logarithmicallyspaced level values indicated by vertical black lines in the colorbar. The offset ∆ T off in temperature difference is fixed to thefitted value. The optimal fit parameters and its uncertainties are indicated by the point and the error bar, respectively. (cd)Variations of the optimal thermopower and Lorenz number when one fixed fit parameter is changed within its uncertainty(colored lines). Left panel: Modifications in conductance (orange) and compressibility (blue). Right panel: Modifications ininitial values ∆ N (solid green), ∆ T (solid violet) and the offsets ∆ N off (dashed green) and ∆ T off (dashed violet). Changes inthe thermodynamic analogue of the Lorenz number l are too small to be visible.
1. Scattering time
Within the Boltzmann theory the scattering rate γ fol-lows from the collision integral and reads [69] γN = mπ (cid:126) (cid:90) d E d E d E d E δ ( E + E − E − E ) × (1 − f )(1 − f ) f f · H ( E, E m ) , (C1)where N denotes the number of particles in each hyper-fine state. The integral describes two particles with ener-gies E and E that exchange energy during the collisionand leave with energies E and E . As the collision iselastic the total particle energy E = E + E = E + E is conserved, as visible from the delta function. The prod-uct of the Fermi-Dirac distributions f i = 1 / (1 + e (cid:15) i ) with (cid:15) i = ( E i − µ ) /k B T ensures that the initial states areoccupied and the final ones unoccupied, as dictated byPauli’s principle. The factor H is the integrated densityof states weighted by the collision cross section σ ( k ). Inour situation the gas is locally homogeneous at the centerwhich leads to H ( E, E m ) = 2 πmV (2 π (cid:126) ) (cid:90) P + P − d P σ ( k ) , (C2)with the volume V and the integral over total momen-tum P whose bounds are P ± = (cid:112) m ( E − E m ) ±√ mE m with the minimal energy E m = min( E , E , E , E ). Atunitarity the collision cross section is σ ( k ) = 4 π/k ex-pressed with the relative wavevector k given by 4( (cid:126) k ) = P + P − − P .2 loc (µK)10 T / T F hot reservoircold reservoirchannel FIG. 6. Prepared degeneracy
T /T F inside the hot and coldreservoirs and the channel center versus local chemical po-tential µ loc . The gray dotted lines locate the normal-to-superfluid transition inside the locally homogeneous channelat T c /T F = 0 .
167 and µ loc = 0 . µ K, respectively. Theviolet horizontal line indicates the critical degeneracy in thereservoirs. Error bars represent the standard deviation over5 repetitions prepared under the same conditions.
We rewrite the integral by realizing that it is symmet-ric under exchange of all four energies. Hence, we canchoose E to be the lowest energy E m and include theother cases by a factor four. To ensure that E is thesmallest and that the total energy is conserved we trans-form the variables E i to the non-negative x i and normal-ize with the Fermi energy E F = (cid:126) / (2 m )(6 π n ) / withthe particle density n = N/V . E /E F = x (C3) E /E F = x + x (C4) E /E F = x + x (C5) E /E F = x + x + x (C6)The resulting scattering rate γ = γ I ( T /T F ) consistsof a prefactor γ and a dimensionless integral I ( T /T F ).For the prefactor we choose the classical collision rate ina homogeneous gas at temperature T F γ = n · √ v · σ ( k F ) = 8 √ π / E F (cid:126) , (C7)with the average relative velocity √ v between two par-ticles expressed through their individual mean speed¯ v = (cid:112) E F /πm . The dimensionless integral reads I (cid:18) TT F (cid:19) = 9 √ π (cid:90) ∞ d x d x d x f ( x + x ) f ( x + x ) × [1 − f ( x )][1 − f ( x + x + x )] × F (2 x + x + x , x ) , (C8) with f ( x ) = 1 / (1 + e ξ ) and ξ = ( T F /T )( x − µ/E F ) wherethe normalized temperature T /T F and chemical poten-tial µ/E F are related in appendix B 1 a. The function F ( x, x m ) depends on the normalized total x = E/E F and minimal energy x m = x as F ( x, x m ) = 1 √ x log (cid:32) x/x m + 2 (cid:112) x/x m x/x m − (cid:112) x/x m (cid:33) . (C9)This form is unsuited for numerical evaluation as itcontains a divergence at x = 0 which we lift with thesubstitution w = x , y = ( x + x ) / x and z = ( x − x ) / x leading to the final result I (cid:18) TT F (cid:19) = 92 (cid:114) π (cid:90) ∞ d y log (cid:18) y/ √ y y/ − √ y (cid:19) × (cid:90) ∞ d w [1 − f ( w )][1 − f ( w (1 + 2 y ))] (cid:115) w y × (cid:90) y − y d z f ( w (1 + y − z )) f ( w (1 + y + z )) . (C10)Instead of scattering rates we use its inverse, the aver-age time between interparticle collisions τ = τ /I ( T /T F ).
2. Mean free path
Based on the average scattering time calculated inthe previous section we estimate the mean free path l mfp = τ v F a particle with velocity v F = (cid:112) E F /m trav-els between collisions. For the estimation we need thelocal degeneracy ¯ T /T F and the Fermi energy E F at thecenter. They both follow from the temperature ¯ T andchemical potential ¯ µ via the local chemical potential µ loc ,as detailed in appendix D, and the thermodynamics of ahomogeneous unitary Fermi gas in section B 1 a.
3. Transport regime
To discuss the transport regime we first focus on thescattering time and mean free path at the channel cen-ter [Fig. 7]. Versus local chemical potential the classicalscattering time τ (dashed line) monotonically decreaseswhich indicates more frequent collisions. Although thecross section σ ( k F ) ∼ /E F reduces as particles collideat higher relative velocities the simultaneous increase indensity n ∼ E / F and mean particle speed ¯ v ∼ E / F dominates [compare with Eq. (C7)]. In the quantum case(solid line) the time τ behaves the same at small chem-ical potentials. In contrast at larger values the gas ismore degenerate and Pauli’s principle limits collisions asit requires the final states to be empty. Eventually thiseffect dominates and leads to a longer time between colli-sions. The local minimum in scattering time appears lesspronounced in the mean free path as the Fermi velocitymonotonically increases [Fig. 7(b)].3 S ca tt e r i n g t i m e ( µ s ) (a) loc (µK)0204060 F r ee p a t h l m f p ( µ m ) (b) FIG. 7. Boltzmann approach for a homogeneous unitaryFermi gas at the channel center. (a) Interparticle scatter-ing time in classical (dashed curve) and quantum (solid line)regimes and (b) mean free path versus local chemical poten-tial. The horizontal line indicates the channel length 2 w cy in terms of the waist w cy of the creating laser beam. Thesuperfluid transition (vertical dashed line) is estimated in ap-pendix B 4. The predictions are valid in the normal regimeabove the critical region. In the critical region we expect thereal values to be lower (see text) and in the superfluid regimethe theory fails. The predictions based on Boltzmann’s approach arevalid in the normal Fermi liquid phase located abovethe critical and superfluid regimes. In the critical regionpairing correlations are relevant and modify the scatter-ing rate. In a harmonic trap they were found to almostcompensate Pauli blocking giving rates that follow theclassical prediction [70]. Hence, we expect in the homo-geneous case that in the critical region Pauli blocking isreduced by correlations giving effectively lower scatteringtimes and mean free paths. Above the superfluid transi-tion (vertical dashed line) the mean free path is boundedby Boltzmann’s theory and is at most 9 µ m, shorter thanthe channel length of 2 w cy = 60 µ m.Next, we extend the argument to the entire channelwhose variations are captured by the effective potential[appendix D]. If the gas is locally non-superfluid through-out the constriction the mean free path is bounded byBoltzmann’s theory that predicts hydrodynamic trans-port. This is the case above the transition indicated inFig. 2, 3, 4 and 7 [appendix B 4]. Appendix D: Effective potential
In two-terminal setups particles move along one direc-tion and are confined in the others. As a result of theconfinement the transverse motion is quantized into dif- ferent modes labelled by quantum numbers. As long asthe confinement varies adiabatically through the channelparticles remain in the same mode which is called adia-batic approximation. Then, the confinement energy actsas an additional potential as it is invested in the trans-verse direction and is missing for the longitudinal motion.In the following we present the effective potential of ourchannel that is useful to calculate the local chemical po-tential at the center and to deduce transport parameters[appendix E].Harmonically approximating our transverse confine-ment ( x , z ) at each longitudinal position ( y ) leads tothe potential V ( x, y, z ) = 12 mω x ( y ) x + 12 mω z ( y ) z + V g ( y ) . (D1)The harmonic frequencies ω x ( y ) and ω z ( y ) include anearly constant contribution from the dipole trap due tothe long Rayleigh length of 20 mm and a spatially vary-ing one from the gate potential and the one creating thechannel, respectively. They are given by ω x ( y ) = ω tx + ω gx e − y /w gy (D2) ω z ( y ) = ω tz + ω cz e − y /w cy (D3)with the dipole trap frequencies ω tx/tz = 2 π · ν tx/tz ,the frequencies at the center created by the gate ω gx = − V g /mw gx and channel beam ω cz . Their waists in lon-gitudinal direction are w gy of 8 . µ m for the repulsive and33 . µ m for the attractive gate beam and w cy = 30 . µ mfor the channel beam. Besides modifying the trappingfrequency the gate beam creates an additional potential V g ( y ) = V g e − y /w gy . (D4)As at each longitudinal position y the transverse po-tential is quadratic and shifted by the energy V g ( y ) theeigenenergies are E n ( y ) = (cid:126) ω x ( y )( n x + 1 /
2) + (cid:126) ω z ( y )( n z + 1 /
2) + V g ( y )(D5)with the quantum number n = ( n x , n z ). Fig. 8 displaysthe eigenenergies in transport direction for different gatestrengths V g .
1. Local chemical potential
To characterize the gas at the channel center we usethe local density approximation with the effective poten-tial V eff ( V g ) = E (0) in the ground state. This resultsin a local chemical potential µ loc = ¯ µ − V eff ( V g ) with V eff ( V g ) = (cid:126) ( ω x (0) + ω z (0)) / V g . Note that ω x (0)implicitly depends on V g in a square root fashion. Thelocal chemical potential is used to display transport coef-ficients [Fig. 3 and 4], to locate the superfluid transition[section B 4] and to discuss its scattering properties [ap-pendix C].4
2. Transport function Φ( E ) From the reservoirs to the center the channel narrowsand tends to increase the mode energies, visible in Fig. 8in the absence of the gate ( V g = 0). Here, a particle atenergy E (horizontal line) can cross the channel in anymode indicated in blue. An additional repulsive poten-tial ( V g >
0) pushes them up and further peaks them atthe center within the size of the beam. In the attrac-tive case ( V g <
0) the energies are pulled down and somemodes might be energetically allowed at the center whileaway from it they are above (red lines). Only the modeswhose energies are below the particle energy throughoutthe channel are relevant for transport. Their number iscounted with the transport function Φ( E ) directly fromthe effective potential and is used in appendix E to cal-culate transport parameters. Appendix E: Landauer-B¨uttiker theory
In this appendix we detail how we model the trans-port parameters in the non-interacting situation. In theLandauer framework particles come from the reservoirsfollowing its Fermi-Dirac distribution and cross the chan-nel with a transmission probability. In linear response thetransport coefficients are evaluated at the mean chemicalpotential ¯ µ and temperature ¯ T imposed by the reservoirs.They read as follows [44, 71, 72] G = 1 h (cid:90) + ∞−∞ Φ( E ) (cid:18) − ∂f ( (cid:15) ) ∂E (cid:19) d E, (E1) Gα c = 1 h ¯ T (cid:90) + ∞−∞ Φ( E )( E − ¯ µ ) (cid:18) − ∂f ( (cid:15) ) ∂E (cid:19) d E, (E2) G ( L + α c ) = 1 h ¯ T (cid:90) + ∞−∞ Φ( E )( E − ¯ µ ) (cid:18) − ∂f ( (cid:15) ) ∂E (cid:19) d E, (E3) with the Fermi-Dirac distribution f ( (cid:15) ) = 1 / (1 + e (cid:15) ) andthe normalized particle energy (cid:15) = ( E − ¯ µ ) /k B ¯ T .In the classical regime the transmission Φ( E ) throughthe channel reduces to the number of transverse modesbelow energy E . We count their number directly fromthe effective potential discussed in appendix D. Then bynumerically evaluating the Landauer integrals the trans-port coefficients follow and are indicated in Fig. 3 and4.
1. Benchmarking
To benchmark the method we compare it with themeasured conductance for a non-interacting gas shown inFig. 3(ab) and find good agreement. Note that countingavailable modes at the channel center leads to a wrongprediction that increases roughly quadratically with lo-cal chemical potential, in contrast to the observed linearbehavior.
2. Validity of linear response
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