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# Breakdown of the Wiedemann–Franz law in a unitary Fermi gas

Edited by Wolfgang Ketterle, Massachusetts Institute of Technology, Cambridge, MA, and approved July 11, 2018 (received for review February 23, 2018)

## Significance

Heat and matter currents are required to relax an out-of-equilibrium system with temperature and chemical potential gradients to thermodynamical equilibrium. The ratio of heat to particle conductance characterizes this response and takes a universal value for typical electronic materials, known as the Wiedemann–Franz law, originating in the quasi-particle nature of the excitations contributing to transport. Investigating the transport dynamics between two reservoirs of ultracold and strongly interacting Fermi gases, connected by a quantum point contact, we observe a nonequilibrium steady state, strongly violating the Wiedemann–Franz law. This cold atom version of the fountain effect, previously observed in superfluid helium superleaks, is characterized by a weak coupling between heat and particle currents that results in a nonvanishing Seebeck coefficient.

## Abstract

We report on coupled heat and particle transport measurements through a quantum point contact (QPC) connecting two reservoirs of resonantly interacting, finite temperature Fermi gases. After heating one of them, we observe a particle current flowing from cold to hot. We monitor the temperature evolution of the reservoirs and find that the system evolves after an initial response into a nonequilibrium steady state with finite temperature and chemical potential differences across the QPC. In this state any relaxation in the form of heat and particle currents vanishes. From our measurements we extract the transport coefficients of the QPC and deduce a Lorenz number violating the Wiedemann–Franz law by one order of magnitude, a characteristic persisting even for a wide contact. In contrast, the Seebeck coefficient takes a value close to that expected for a noninteracting Fermi gas and shows a smooth decrease as the atom density close to the QPC is increased beyond the superfluid transition. Our work represents a fermionic analog of the fountain effect observed with superfluid helium and poses challenges for microscopic modeling of the finite temperature dynamics of the unitary Fermi gas.

The interplay between heat and matter currents in a many-body system sheds light on its fundamental properties and the character of its excitations. Transport measurements are a particularly important probe in the presence of strong interactions and high temperatures T, when a microscopic model is absent or computationally intractable. Phenomenologically, the dependence of the currents on external biases is captured by transport coefficients, such as the particle conductance G or the thermal conductance

A cold atomic Fermi gas in the vicinity of a Feshbach resonance is a fundamental example of a strongly correlated Fermi system. Owing to the control offered by laser manipulation, its trapping potential can be shaped into custom geometries such as a two-terminal configuration, allowing one to measure transport coefficients (7). Previous studies of the unitary Fermi gas have charted out its thermodynamic properties (8, 9). Recently, transport experiments have observed dissipation processes occurring in the presence of a weak link, such as vortex nucleation in a Josephson junction and multiple Andreev reflections (10⇓–12), and heat waves in the form of second sound have been observed (13, 14).

Thermoelectric effects in cold atoms have been studied for bosonic systems (15⇓⇓–18) and within Bardeen–Cooper–Schrieffer (BCS) theory (19); however, the thermoelectric coupling between heat and particle currents in the unitary regime has not been experimentally addressed. Such a study is particularly relevant for applications to cooling protocols as well as for singling out the contribution of fermionic particles to heat flow (20⇓⇓–23). Indeed, in contrast to solid-state systems, where the lattice melts at high temperatures, the unitary Fermi gas realized in cold atoms remains free of lattice phonons at all temperatures.

In this paper, we report on measurements of heat and particle transport through a quantum point contact connecting two reservoirs of strongly correlated Fermi gases across the superfluid transition. We explore the unitary Fermi gas at temperatures slightly below the critical temperature, where the reservoirs are weakly in the superfluid regime. There we have observed a linear current–bias relation, contrary to lower-temperature regimes at a superfluid–normal–superfluid junction where nonlinearities have been shown (12). We enter the predicted large critical region of the normal-to-superfluid transition of the unitary Fermi gas (24) where the behavior of the many-body system still needs to be investigated. It is an open question whether the disappearance of the superfluid signature in particle transport is concomitant with a restored Fermi liquid behavior.

We observe the evolution of an initially imposed temperature imbalance for equal atom numbers in a two-terminal Landauer configuration (7). In general, coupled particle and heat currents tend to pull a system toward thermodynamical equilibrium. However, here the system evolves toward a nonequilibrium steady state (NESS) within the timescale of the experiment. While typically a NESS is associated to stationary states of open systems (25), here it can also describe our experiments due to the presence of dissipation and thermodynamic driving forces. Our results sharply contrast with previous experiments observing heat transport with weakly interacting atoms (26). There a single time constant was found to describe the dynamics for temperature and particle relaxation.

Here, our observations reveal a strong separation of heat and particle transport timescales, resulting in a Lorenz number much lower than the value expected for a Fermi liquid. The paradigmatic system supporting suppressed heat transport is the so-called superleak in liquid bosonic helium II. Heating one side of the superleak yields the fountain effect, where both the Seebeck coefficient and the thermal conductance vanish. Our observations represent a fermionic analog to the fountain effect, where the quantum point contact (QPC) takes the role of the superleak with very low thermal and particle conductance in the noninteracting limit. We, however, measure a finite Seebeck coefficient even in the superfluid regime, calling for a description of the transport process going beyond the standard two-fluid model.

## System

Our experiment consists of a QPC imprinted onto a cold, unitary Fermi gas of ^{6}Li atoms, as in our previous work (12, 27). We form the QPC using two far-detuned repulsive laser beams with a line of zero intensity in the center, resulting in a region of tight harmonic confinement with typical frequencies of *A*) setting an energy spacing between transverse modes of *SI Appendix*, *Transport Modes*).

We bring the system out of equilibrium by heating either of the reservoirs using an intensity-modulated laser beam focused on the reservoir, while maintaining the QPC closed. This results in a temperature difference up to *B*). These quantities can be converted to any other thermodynamic variable such as temperature T or chemical potential μ, using the unitary equation of state (EoS). Here the reduced chemical potential *SI Appendix*, *Density Distribution in the Center*). We perform a transport experiment by opening the QPC for a variable time t between 0 s and 4 s and subsequently measuring *C*.

## Dynamics

Fig. 2 presents a typical time evolution of

After a typical timescale *C*, and our observation shows that

To provide a quantitative understanding of the time evolution of the system, we use a phenomenological model based on linear response. While such an approach is known to fail in the lowest temperature regimes, where nonlinear current–bias relations have been observed (12), we find that it describes our observations well (*SI Appendix*, *Linear Model*) and allows for comparison between different QPC parameters. In this framework, the particle current

The absence of the relaxation of temperature and particle imbalance shown in Fig. 2 implies a very low heat conductance. According to the first law of thermodynamics the energy flow *A*, we find *SI Appendix*, *Transport Coefficients in a Noninteracting System*).

## Transport Coefficients

The transport parameters in mesoscopic systems strongly depend on the geometry of the channel (30). We investigate this dependency by measuring the dynamics of atom number difference and temperature difference as the channel confinement is reduced, departing from the single-mode regime. Fig. 3*A* presents the results for four different transverse confinements

We fitted the time traces with the solutions from the linear response model in Eq. **1**, which are biexponential functions where we fixed *B*). Consequently, each timescale can be mapped to the relaxation dynamics of heat (*SI Appendix*, *Linear Model*). Within this linear response solution, the direction and magnitude of the currents result from a competition between the transport properties of the channel and the thermodynamic response of the reservoirs (*SI Appendix*, *Linear Model*), a feature that was encountered already for the weakly interacting Fermi gas in ref. 26.

### Lorenz Number.

The relative weight of particle and heat conductance is captured by the Lorenz number L. Direct conversion of the fit parameters in Eqs. **3** and **4** to L is, however, not possible because the biexponential model is ill-conditioned. Instead we express L by estimating G and **1**). The conductance G is calculated for short transport times where we obtain *SI Appendix*, *Evaluation of Transport Parameters* for details).

The estimates of the Lorenz number are presented in Fig. 3*C*, together with the expected value for a noninteracting QPC obtained through Landauer theory with equivalent chemical potential, temperature, and channel properties (28). For all values of

### Seebeck Coefficient.

The NESS observed in Fig. 2 allows us to relate chemical potential and temperature differences to the Seebeck coefficient. Therefore we express the vanishing particle current as a competition between a current driven by **1**) at the times where particle current *A* for time traces) and convert them to

For the specific case of *B*). This confirms that the linear model Eq. **1** constitutes an adequate description of our system. The linear relation yields a Seebeck coefficient *SI Appendix*, *Transport Coefficients in a Noninteracting System*). We further investigate the Seebeck coefficient by increasing the attractive gate potential *i*) It probes the single-particle energy dependence of the transport parameters by increasing the number of available modes in the QPC and (*ii*) the density in the vicinity of the QPC is modified by tuning the chemical potential, locally increasing the superfluid gap. We measure *B* and deduce

Fig. 4*C* shows *C*) and is explained there by an increase of the number of 1D channels available for the transport of single particles. This similarity is surprising as transport coefficients in this regime close to the superfluid transition have shown order of magnitude deviations from the Landauer model (12, 34). The residual deviation from the noninteracting curve—manifested in the faster decrease with

## Discussion

The coexistence of a vanishing Lorenz number and a finite Seebeck coefficient leading to a NESS at finite

The nonvanishing *SI Appendix*, *Efficiency*).

Our fountain effect setting with fermions provides a conceptual link between the thermoelectric transport witnessed in electronic devices and the bosonic fountain effect observed with helium II. Its anomalous features—exceptionally small Lorenz number and finite Seebeck coefficient—shed light on the out-of-equilibrium properties of the unitary Fermi gas and portend potential applications to ultracold atoms, such as the realization of novel cooling schemes. They also underline the necessity of a better understanding of strongly correlated systems at finite temperatures. Here, measuring the spin degree of freedom could yield additional information (20, 21), but the inhomogeneous nature of the system makes the interpretation even more challenging. A complementation to our present study would be the probing the full Bose–Einstein condensate (BEC)-BCS crossover. This, however, requires knowledge of the full finite temperature EoS, which has not been measured yet.

## Materials and Methods

### Preparing the Cloud and QPC.

We prepare an elongated cloud of fermionic ^{6}Li atoms in a balanced mixture of the lowest and third lowest hyperfine state in a hybrid configuration of a far-detuned 1,064-nm dipole trap and a harmonic magnetic trap, confining the atoms along the transverse (

### Transport.

An amplitude-modulated beam at a wavelength of 767 nm is directed on one of the reservoirs, parametrically heating it up. The modulation frequency is optimized experimentally to

### Thermodynamic Properties of the Reservoirs.

From the density profiles we deduce the atom number in each reservoir as well as their internal energy *B*.

Along with the known equation of state of the unitary Fermi gas, these two quantities define all of the thermodynamic parameters of the individual reservoirs, including their temperatures *SI Appendix*, *Thermodynamic Properties of the Reservoirs*).

## Acknowledgments

We thank H. Aoki, A. Georges, T. Giamarchi, L. Glazman, D. Papoular, S. Pershoguba, S. Uchino, and W. Zwerger for discussions; B. Frank for providing the data on the superfluid gap; and B. Braem and P. Fabritius for careful reading of the manuscript. We acknowledge financing from the Swiss National Science Foundation under division II (Project 200020_169320 and National Centre of Competence in Research-Quantum Science and Technology); the Swiss State Secretary for Education, Research and Innovation Contract 15.0019 (Quantum Simulations of Insulators and Conductors); European Research Council (ERC) Advanced Grant TransQ (Mass, Heat and Spin Transport in Interlinked Quantum Gases) (Project 742579); and Army Research Office Multidisciplinary University Research Iniviative program Non-equilibrium Many-body Dynamics Grant W911NF-14-1-0003 for funding. J.-P.B. is supported by the ERC Starting Grant DECCA (Devices, Engines and Circuits: Quantum Engineering with Cold Atoms) (Project 714309) and the Sandoz Family Foundation–Monique de Meuron Program for Academic Promotion. L.C. is supported by an ETH Zurich Postdoctoral Fellowship and the Marie Curie Actions for People Cofunding of Regional, National and International Programmes.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: lcorman{at}phys.ethz.ch.

Author contributions: D.H., M.L., S.H., J.-P.B., L.C., and T.E. designed research; D.H., M.L., S.H., and L.C. performed research; D.H., M.L., S.H., and L.C. analyzed data; and D.H., M.L., S.H., J.-P.B., L.C., and T.E. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1803336115/-/DCSupplemental.

Published under the PNAS license.

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