The Joule-Thomson (JT) effect is a fundamental phenomenon in thermodynamics whereby the temperature T of a thermally isolated system changes in response to a decrease of the pressure P while the specific enthalpy h is conserved. This effect has played a momentous role in the history of thermodynamics [1] and the birth of modern cryogenics [2]. In its own right, the JT effect has attracted interest as a probe of the thermodynamics of imperfect (i.e. interacting) gases [3,4] and, more recently, in relation to black hole expansion dynamics [5][6][7].

In classical gases, the JT effect is tied to interparticle interactions. This can be illustrated with a simple equation of state (EOS) PV ¼ Nk B T þ a int N 2 =V, i.e., the Van der Waals EOS without the excluded-volume effect; N is the number of particles, V is the volume, and a int is an interaction parameter that is positive for repulsive interactions and negative for attractive ones. In the limit of weak interactions (a int N=V ≪ k B T), the Joule-Thomson coefficient μ JT ≡ ð∂T=∂PÞ h is μ JT ∝-a int =c P in this model (c P > 0 is the specific heat); thus the system heats (respectively cools) in the case of repulsive (respectively attractive) interactions.

Surprisingly, the JT effect does not require interactions. Indeed, shortly after the discovery of quantum indistinguishability, it was predicted that quantum correlations give rise to a nontrivial JT effect even in the absence of interactions [8]; in essence, Bose-Einstein particles would behave as if they were classically attracting and Fermi-Dirac particles as if they were repelling. The opposite nature of these respective quantum JT effects originate from profoundly different microscopic physics: at low temperatures, the former is driven by the Bose saturation of the single-particle excited states [9], while the latter, by Fermi hole heating [10]. Despite being a fundamental prediction of quantum statistical mechanics, only recently has the bosonic JT effect been observed [11], whereas the fermionic one has remained elusive.

In this work we measure the JT effect in Fermi systems. In the textbook presentation of the JT process, a gas is throttled through a porous plug from a high-P to a low-P compartment [see sketch in Fig. 1(a)]. In our experiment, we realize a JT rarefaction either by exploiting collisions with high-energy particles from the residual background gas (in the vacuum chamber) or by controllably transferring atoms into internal states that are essentially not interacting with the states of interest [Fig. 1(b)]. In either case, the loss process is independent of the energy per particle u ¼ U=N where U is the total internal energy. For a scale-invariant gas, whose EOS satisfies U ∝ PV, this process is thermodynamically equivalent to a JT one.

We first focus on the JT effect in the ideal Fermi gas. We prepare weakly interacting spin-1=2 Fermi gases of 6 Li atoms in a balanced mixture of the first and third lowest Zeeman sublevels (respectively labeled j1i and j3i). Our gases are confined in optical boxes so that their density and other thermodynamics quantities are spatially uniform, making the interpretation of our measurements straightforward [12,13]. Our cylindrical boxes have a radius R ¼ 77ð2Þ μm and an adjustable length L between 58 μm and 120 μm [see an example of a box in Figs. 1(b) and 1(c)]. The samples are evaporated at a bias magnetic field B ¼ 287 G, where the s-wave scattering length is a ≈ -880a 0 (a 0 is the Bohr radius). The B field is then ramped to its final value, where levitation against gravity is done with a magnetic field gradient; the difference of magnetic moments of the trapped states used in this work (<0.8%)is negligible for our experimental parameters. We typically start our experiments with a degenerate spin-1=2 Fermi gas, T ≲ E F =k B (where E F ¼ ℏ 2 =ð2mÞð6π 2 N=VÞ 2=3 is the Fermi energy); henceforth, all thermodynamic quantities (such as U, N, etc.) are defined for each spin population. We typically have N 1 ≈ N 3 ≈ 8 × 10 5 , corresponding to a Fermi temperature of T F ¼ E F =k B ≈ 300 nK.

We take advantage of the slow one-body losses due to collisions with the background gas to realize u-constant rarefactions [Fig. 2(a)], as in [11]. Here, the tunability of interparticle interactions is important as the interactions must obey conflicting requirements. On the one hand, interactions must be weak enough so that we probe essentially ideal gas physics and that PV ≈ ð2=3ÞU; furthermore, two-body energy-dependent evaporation must be suppressed on the (long) timescale of the measurements. On the other hand, interactions must be strong enough to ensure that the gas is in thermal equilibrium when the measurements are done.

Consequently, we satisfy those conditions by choosing the final value of B such that a is in the range 100a 0 ≲ a ≲ 220a 0 . The specific value is picked as large as possible, while ensuring that the decay time is indistinguishable from the vacuum-limited lifetime. In Fig. 2 FIG. 2. Joule-Thomson effect of the ideal Fermi gas. (a) Sketch of an energy-independent atom loss due to collisions with highenergy background particles (red empty circle). (b) Decay of a noninteracting Fermi gas (black circles) and weakly interacting Fermi gases (colored diamonds) at different initial T=T F (see legend). The solid lines are exponential fits. The same marker color is used in (c) and (d). (c) Azimuthally averaged radial momentum distribution of Fermi gases extracted from column integrated OD after a time-of-flight expansion of duration t TOF . The distributions are normalized such that R ñðk r =k F Þð2πk r =k F Þdðk r =k F Þ¼1; here k r ¼m ffiffiffiffiffiffiffiffiffiffiffiffi ffi y 2 þz 2 p =ðℏt TOF Þ and k F is the Fermi wave number. The distributions correspond to the initial points in (b). The solid lines are fits to Fermi-Dirac distributions to extract temperatures. The black dashed line shows the momentum distribution at T ¼ 0. (d) Temperature evolution of weakly interacting Fermi gases during a JT rarefaction. The solid lines are theoretical predictions fixing ðT=T F Þ i to the experimentally measured values, with the (barely visible) bands representing the uncertainty on ðT=T F Þ i .

The dashed lines take into account the effect of technical heating in the box (see text and [14]). The dotted lines show the evolution of T=T F at constant T. (e) Sketch of Fermi hole heating at T ≪ T F . The blue (respectively brown) point represents a particle whose removal causes no temperature change (respectively causes heating). (f) Quantum-statistical interaction potential U q of ideal quantum gases in the high-T limit.

series are indistinguishable from that of a noninteracting gas (jaj ≤ 50a 0 ), τ vac ¼ 55ð2Þ s (see the black circles, the vacuum-limited lifetime in our chamber). At the same time, the two-body elastic collision rate is in the appropriate regime Γ el ≫ 1=τ vac (in our range of densities and temperatures, Γ el ≥ 0.17 s -1 ). Furthermore, for our regime of interactions and temperatures, h changes less than 0.01% during the decay, making this rarefaction an excellent approximation of a JT process [14].

We perform thermometry using time-of-flight expansions (see Appendix A and [14]). We show in Fig. 2(c) the momentum distributions for two initial conditions ðT=T F Þ i ¼ 0.27ð3Þ (blue) and ðT=T F Þ i ¼ 0.61ð4Þ (red). In Fig. 2(d), we show the temperature dynamics of the gas during rarefaction for these two cases; we plot T=T F versus N=N i , where the instantaneous T F decreases as the gas rarefies. The dotted lines correspond to T=T F ∝ ðN=N i Þ -2=3 , the expectation for constant-T rarefactions. The measurements show heating, and the main qualitative feature is that the heating is more pronounced for a more quantum-degenerate gas, i.e., the fractional change of T is larger at low T=T F for the same decrease of N=N i .

Quantitatively, we describe the temperature dynamics during this JT process using the dimensionless coefficient θ JT ≡ ½∂ logðTÞ=∂ logðPÞ h . This coefficient is related to the JT coefficient: μ JT ¼ðT=PÞθ JT . For a homogeneous gas whose EOS is universal, i.e. for which Pλ 3

T =ðk B TÞ only depends on the chemical potential μ and k B T via the ratio μ=ðk B TÞ, θ JT is a function of T=T F alone (λ T is the thermal wavelength). The evolution of T=T F follows ½∂ logðT=T F Þ=∂ logðNÞ h ¼ θ JT -2=3 [14]. In Fig. 2(d), the solid lines are the theoretical predictions derived from the EOS of the ideal Fermi gas (where ðT=T F Þ i is fixed to the experimental value). We find good agreement with the data. The small discrepancy is well accounted for by a weak technical heating in our box; the dashed lines show the theoretical predictions from the model dlogðT=T F Þ=dlogðNÞ¼ðθ JT -2=3Þð1 þð3=2Þγ tech τ vac =uÞ, where our heating rate γ tech ¼ 0.58ð7Þk B ×nK=s is characterized independently [14].

In the low-and high-T limits, simple pictures provide insights into the microscopic origin of the JT effect. First, for T ≪ T F , the state of the gas is essentially a Fermi sea [Fig. 2(e)]. In that case, the average energy per particle lost in a random (energy-independent) removal is only u loss ≈ ð3=5ÞE F ; the energy per particle that needs to be removed to keep the temperature constant, u T ≡ ð∂U=∂NÞ T;V ,i su T ≈ E F . As a result, the system heats up, a process referred to as Fermi hole heating [10].

The interpretation of quantum correlations as statistical "forces" demystifies the quantum JT effect in the T ≫ T F limit. For that purpose, it is useful to consider the pair density correlation function Gðr 1 ; r 2 Þ≡ hΨ † ðr 1 ÞΨ † ðr 2 ÞΨðr 2 ÞΨðr 1 Þi=½nðr 1 Þnðr 2 Þ, where Ψ † ðr j Þ (Ψðr j Þ) is the field operator that creates (annihilates) a particle at position r j , and nðr j Þ ≡ hΨ † ðr j ÞΨðr j Þi. For an ideal homogeneous gas in the high-T (virial) limit, Gðr 1 ;r 2 Þ¼GðrÞ≈1þηexpð-2πr 2 =λ 2 T Þ, where r¼jr 1 -r 2 j, η ¼ 1 for bosons and η ¼ -1 for fermions (η ¼ 0 for the classical ideal gas) [20]. For a dilute classical gas, GðrÞ ≈ exp½-U int ðrÞ=ðk B TÞ, where U int ðrÞ is the interparticle interaction potential. By analogy, one can define an effective quantum-statistical interaction between indistinguishable noninteracting particles, U q ðrÞ ≡ -k B T log GðrÞ [21,22]. The potential U q ðrÞ is shown as yellow and purple lines in Fig. 2(f); as intuitively expected, fermions effectively "repel" while bosons "attract" each other. Furthermore, the signs of their quantum JT coefficients are consistent with their respective quantum-statistical interaction [14,23].

We now turn to the unitary Fermi gas, for which 1=a ¼ 0. Crucially, because PV -ð2=3ÞU ∝ I=a, where I is Tan's contact [24], the universal relation PV ¼ð2=3ÞU also holds for the unitary gas. This makes the unitary case another special point in the BEC-BCS crossover [25] for which a u-constant rarefaction is also a JT process. We create a unitary gas by preparing a spin-balanced mixture of atoms in states j1i and j3i that is evaporatively cooled and loaded into the optical box at B ≈ 796 G. The field is then ramped to the Feshbach resonance, B ≈ 690 G. At this stage we typically have N 1 ≈ N 3 ≈ 3 × 10 5 at T=T F ≈ 0.2 (slightly above the superfluid transition temperature T c [25]).

Just as in the ideal gas case, the two main ingredients to observe the JT effect in this setting are the realization of a u-constant rarefaction and a thermometry method. Both present new challenges compared to the weakly interacting case.

As the collision rate in the unitary gas is so high (typically Γ uni el ≥ 500 s -1 in our case), the evaporation rate is vastly higher than in the weakly interacting case, threatening the JT nature of the rarefaction. In our deepest box (U box ≳ 8E F ), the lifetime of our unitary gas is τ uni ≈ 30 s, close to but a little shorter than τ vac [26] (possibly limited by a slow residual evaporation). To mitigate this issue, we artificially increase the u-independent "loss" rate by applying a weak two-tone microwave pulse of duration t μw to transfer atoms to higher Zeeman sublevels [see Appendix B and Fig. 3(a)]. Measuring the number of atoms remaining in j1i and j3i, we find exponential decays with respective characteristic times τ μw ¼ 0.33ð1Þ sandτ μw ¼ 0.35ð1Þ s [pink and blue diamonds in Fig. 3(b)]. This timescale is such that τ uni ≫ τ μw ≫ 1=Γ uni el , i.e. the microwave-induced rarefaction is slow compared to the elastic collision rate so that the gas remains in thermal equilibrium, but fast enough so that energy-dependent losses are negligible. We validate our microwave-induced rarefaction method on the nowverified case of the weakly interacting gas; the effect of the technical heating is now negligible because the timescale of the microwave-induced rarefaction is short, see [14].

For thermometry, we use radio-frequency (rf) spectroscopy and compare it to the calibrated spectra for the unitary gas as a function of temperature measured at MIT [28,29].

We transfer atoms from state j1i to state j2i and measure the transferred fraction as a function of ω, the detuning frequency relative to the bare j1i → j2i transition frequency (see Appendix C and [14]). Specifically, we extract the temperature from the peak response frequency E p ≡ -ℏω p , whose magnitude decreases monotonically with increasing T=T F [28].

We first verify that without microwave transfers and on the timescale of the experiment, evaporation and other T dynamics are negligible. We measure the initial rf spectrum of the gas [green diamonds in Fig. 3(c)] and after a hold of 0.7 s (black diamonds), without microwave field. The spectra are essentially identical; quantitatively, we extract T=T F ¼ 0.24 þ7 -8 and T=T F ¼ 0.23 þ7 -8 from E p ¼ -0.59ð2ÞE F and E p ¼ -0.61ð2ÞE F , respectively.

When the microwave induces rarefaction, the rf spectra significantly change [see magenta diamonds in Fig. 3(c), corresponding to a rarefaction of N=N i ¼ 0.28ð6Þ]. In fact, we observe that the magnitude of E p =E F continuously decreases with rarefaction [upper panel of Fig. 3(d)], indicating qualitatively that the quantum degeneracy decreases. In the lower panel of Fig. 3(d) we show the evolution of T=T F in a unitary-gas JT process. For an initial condition ðT=T F Þ i ¼ 0.24 þ7 -8 (blue diamonds), the data show that the unitary gas experiences a less pronounced heating compared to the ideal Fermi gas (purple dashdotted line); the data are in very good agreement with the prediction based on the experimentally measured EOS (blue solid line) [27]. The band represents the uncertainty window arising from the uncertainty in ðT=T F Þ i . We took an additional data set at a lower initial degeneracy, corresponding to ðT=T F Þ i ¼ 0.35ð4Þ, and observe weaker heating (see [14] for details).

Despite the theoretical challenge in describing the strongly interacting Fermi gas, its JT effect is relatively simple to interpret in both the low-T (T ≪ T c ) and high-T (T ≫ T F ) limits. In the low-T limit, the unitary gas should exhibit a strong JT heating as θ JT ∝-ðT=T F Þ -4 (which originates from both its nonvanishing ground state energy in the thermodynamic limit and its low-lying phononic excitations [30]). In the high-T limit, the unitary gas exhibits an effective interaction [∝-log GðrÞ] that is attractive [31]; it should thus cool during a JT process [14], akin to the ideal Bose gas. From the EOS, we expect that there exists an inversion temperature, i.e. the temperature at which the JT effect changes from heating to cooling, at T inv ≈ 0.9T F . In the intermediate range of T=T F explored in this work, we observe weak heating, in between the expectations for the ideal Bose and Fermi gases [yellow and purple dot-dashed lines in Fig. 3(d)].

In conclusion, we realized JT processes in the essentially ideal Fermi gas and the unitary Fermi gas by exploiting scale invariance and implementing u-constant rarefactions. In the range of temperature explored, we observed JT heating in both cases and the effect is lessened when the (c) Radio-frequency (rf) thermometry. The cartoon shows the internal states used in the rf spectroscopy: the gas, initially in a balanced mixture of j1i-j3i, is driven on the transition j1i → j2i; the states that are imaged are marked with the lightning symbols.

Green and magenta diamonds are the spectra at t μw ¼ 0 s and t μw ¼ 0.5 s, and black diamonds correspond to the spectrum after 0.7 s no-microwave hold. Dot-dashed vertical lines mark the peak response. (d) Degeneracy of a unitary Fermi gas during isenthalpic rarefaction. The peak frequency response E p and T=T F are shown along the rarefaction N=N i , respectively in the upper and lower panel. The green and magenta diamonds correspond to the spectra in (c). The blue solid line is the prediction based on the EOS [27], fixing ðT=T F Þ i to the experimental values. The blue band is the uncertainty arising from the uncertainty on ðT=T F Þ i . The dotted line shows the evolution of T=T F at constant T. The purple (respectively yellow) dot-dashed line is the theoretical temperature evolution of an ideal Fermi (respectively Bose) gas during JT process; T F is defined with the density of the corresponding (Bose or Fermi) gas.

PHYSICAL REVIEW LETTERS 132, 153402 (2024)

153402-4 repulsive quantum-statistical force is either weakened by the loss of degeneracy or counterbalanced by attractive interparticle forces.

In the future, it would be interesting to study the JT effect in other many-body platforms, such as dipolar gases [32], low-dimensional gases, and Hubbard systems [33,34]. The JT effect can encode interesting physics such as complex P-T diagrams delineated by boundaries called inversion curves-where μ JT changes sign [35]. These diagrams are essentially unknown for strongly correlated quantum systems and they could provide valuable new information in settings where the interplay between interactions and quantum statistics is essential.

More specifically, it would be interesting to extend the study of the JT effect to the BEC-BCS crossover, where one expects a continuous transition from bosonic to fermionic behavior. It would be particularly intriguing to understand how the sign change of the JT effect at low T relates to the transition point where the nature of the single-particle excitations turns from bosonic to fermionic in the crossover (i.e. where μð1=k F aÞ¼0) [25]. While the absence of scale invariance poses interesting experimental challenges on how to realize a JT process in that system, the JT coefficient could also be extracted from the isothermal compressibility [14,27,36].