The Equation of State of a Low-Temperature Fermi Gas with Tunable Interactions

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Science  07 May 2010:
Vol. 328, Issue 5979, pp. 729-732
DOI: 10.1126/science.1187582


Interacting fermions are ubiquitous in nature, and understanding their thermodynamics is an important problem. We measured the equation of state of a two-component ultracold Fermi gas for a wide range of interaction strengths at low temperature. A detailed comparison with theories including Monte-Carlo calculations and the Lee-Huang-Yang corrections for low-density bosonic and fermionic superfluids is presented. The low-temperature phase diagram of the spin-imbalanced gas reveals Fermi liquid behavior of the partially polarized normal phase for all but the weakest interactions. Our results provide a benchmark for many-body theories and are relevant to other fermionic systems such as the crust of neutron stars.

Recently, ultracold atomic Fermi gases have become a tool of choice to study strongly correlated quantum systems because of their high controllability, purity, and tunability of interactions (1). In the zero-range limit, interactions in a degenerate Fermi system with two spin-components are completely characterized by a single parameter 1/kFa, where a is the s-wave scattering length and kF = (6π2n)1/3 is the Fermi momentum (n is the density per spin state). In cold atom gases, the value of |a| can be tuned over several orders of magnitude using a Feshbach resonance; this offers an opportunity to entirely explore the so-called BCS-BEC crossover, that is, the smooth transition from Bardeen-Cooper-Schrieffer (BCS) superfluidity at small negative values of a to molecular Bose-Einstein Condensation (BEC) at small positive values of a (1, 2). Between these two well-understood limiting situations, a diverges, leading to strong quantum correlations. The description of this system is a challenge for many-body theories, as testified by the large amount of work in recent years (1). The physics of the BEC-BCS crossover is relevant for very different systems, ranging from neutron stars to heavy nuclei and superconductors.

In the grand-canonical ensemble and at zero temperature, dimensional analysis shows that the Equation of State (EoS) of a two-component Fermi gas, relating the pressure P to the chemical potentials μ1 and μ2 of the spin components can be written as P(μ1,μ2,a)=P0(μ1)h(δ12mμ1a,ημ2μ1) (1)where P0(μ1)=1/15π2(2m/2)3/2μ15/2 is the pressure of a single-component ideal Fermi gas, m is the atom mass, is the Planck constant divided by 2π, and δ1 is the grand-canonical analog of the dimensionless interaction parameter 1/kFa. The indices 1 and 2 refer to the majority and minority spin components, respectively. From the dimensionless function h1,η), it is possible to deduce all the thermodynamic properties of the gas, such as the compressibility, the magnetization, or the existence of phase transitions. The aim of this paper is to measure h1,η) for a range of interactions (δ1) and spin imbalances (η) and discuss its physical content. Because it contains the same information as Eq. 1, the function h will also be referred to as the EoS in the rest of the text.

In situ absorption images of harmonically trapped gases are particularly suited to investigate the EoS, as first demonstrated in (3) and (4). In the particular case of the grand-canonical ensemble, a simple formula relates the local pressure P at a distance z from the center of the trap along the z axis to the doubly integrated density profiles n¯1 and n¯2 (5).P(μ1(z),μ2(z),a)=mωr22π(n¯1(z)+n¯2(z)) (2)Here, we define the local chemical potentials μi(z)=μi012mωz2z2, where μi0 is the chemical potential of the component i at the bottom of the trap, assuming local density approximation. ωr and ωz are the transverse and axial angular frequencies of a cylindrically symmetric trap, respectively, and n¯i(z)=ni(x,y,z)dxdy is the atomic density ni of the component i, doubly integrated over the transverse x and y directions. In a single experimental run at a given magnetic field, two images are recorded, providing n¯1(z) and n¯2(z) (fig. S4); the z-dependence of the chemical potentials then enables the measurement of P along a curve in the (δ1,η) plane (6). This method was validated in (4) for the particular case of the unitary limit a = ∞. Deducing the function h from the doubly integrated profiles further requires a precise calibration of ωz and the knowledge of the central chemical potentials μi0 (6).

Our experimental setup is presented in (7). We prepared an imbalanced mixture of 6Li in the two lowest internal spin states, at the magnetic field of 834 G (where a = ∞), and trapped it in a hybrid magnetic-optical dipole trap. We then performed evaporative cooling by lowering the optical trap power, while the magnetic field was ramped to the final desired value for a. The cloud typically contained N = 2 to 10 × 104 atoms in each spin state at a temperature of 0.03(3) TF, justifying our T = 0 assumption (6). The final trap frequencies are ωz/2π ~ 30 Hz and ωr/2π ~ 1 kHz. Below a critical spin population imbalance, our atomic sample consists of a fully paired superfluid occupying the center of the trap, surrounded by a normal mixed phase and an outer rim of an ideal gas of majority component atoms (4, 7, 8).

For a given magnetic field, 10 to 20 images are taken, leading after averaging to a low-noise EoS along one line in the (δ1,η) plane. Measurements at different magnetic fields chosen between 766 G and 981 G give a sampling of the surface h1,η) in the range −1 < δ1 < 0.6 and −2 < η < 0.7 (Fig. 1). Let A1) be the limiting value of the ratio of chemical potentials μ1(z)/μ2(z) below which the minority density vanishes. At fixed δ1 and η < A1), h1,η) represents the EoS of an ideal Fermi gas of majority atoms and is equal to 1. For η > A1), it slowly rises and corresponds to the normal mixed phase, where both spin components are present. At a critical value η = ηc1), the slope of h abruptly changes (6), the signature of a first-order phase transition from the normal phase (for A < η < ηc) to a superfluid phase with a lower chemical potential imbalance (η > ηc). We notice that the discontinuity is present for all values of δ1 we investigated, and this feature is more pronounced on the BEC side.

Fig. 1

h1,η) of a zero-temperature two-component Fermi gas in the BEC-BCS crossover. (A) Samples of the data for different magnetic fields. The black (red) data points correspond to the normal (superfluid) phase and are separated at ηc1) by a clear kink in the local slope of h. Solid black lines are the predictions of the polaron ideal gas model (Eq. 8). The scattering length corresponding to each curve is (from left to right): (1.7, 3.4, ∞, and −1.3) in units of 104 a0, where a0 is the Bohr radius. (B) h1,η). The black dots are data recorded for each magnetic field value (as in Fig. 1A). The black lines correspond to the parametric curves [δ1(η),η] scanned by the density inhomogeneity in the harmonic trap (6). The red line is A1), the frontier between the fully polarized (FP) ideal gas h = 1 and the normal partially polarized (PP) phase. The green line is ηc1), marking the phase transition between the normal and superfluid (S) phases. The surface is the parametrization of h1,η) given in the text.

Let us first consider the EoS of the superfluid phase, η > ηc. Each of our in situ images has, along the z axis, values of the chemical potential ratio η(z) = μ2(z)/μ1(z) both lower and greater than ηc. In the region where η(z) > ηc, the doubly integrated density difference n¯1(z)n¯2(z) is constant within our signal-to-noise ratio (fig. S4). This is the signature of equal densities of the two species in the superfluid core, that is, the superfluid is fully paired. Using Gibbs-Duhem relation ni=Pμi, equal densities n1 = n2 imply that P12,a) is a function of μ and a only, where μ ≡ (μ1 + μ2)/2. For the balanced superfluid, we then write the EoS symmetrically.P(μ1,μ2,a)=2P0(μ˜)hS(δ˜2mμ˜a) (3)To avoid using negative chemical potentials, we define here μ˜=μEb/2, where Eb is the molecular binding energy Eb=2/ma2 for a > 0 (and 0 for a ≤ 0). hs(δ˜) is then a single-variable function. It fully describes the ground-state macroscopic properties of the balanced superfluid in the BEC-BCS crossover and is displayed in Fig. 2 as black dots.

Fig. 2

Embedded Image of the T = 0 balanced superfluid in the BEC-BCS crossover (black dots). The blue solid line is the fit Embedded Image on the BCS side of the resonance; the red solid line is the fit Embedded Image on the BEC side. The dotted (dashed) red line is the mean-field (LHY) theory (32). (Inset) Zoom on the BCS side. The dotted and dashed blue lines are the EoS, including the mean-field and LHY terms, respectively. The systematic uncertainties on the x and y axes are about 5%. The errors bars represent the standard deviation of the statistical uncertainty.

To extract relevant physical quantities, such as beyond mean-field corrections, it is convenient to parametrize our data with analytic functions. In this pursuit, we use Padé-type approximants (6), interpolating between the EoS measured around unitarity and the well-known mean-field expansions on the BEC and BCS limits. The two analytic functions, hSBCS and hSBEC, are respectively represented in blue and red solid lines in Fig. 2 and represent our best estimate of the EoS in the whole BEC-BCS crossover.

On the BCS side, (δ˜)<0, hSBCS yields the following perturbative expansion of the energy in series of kFaE=35NEF(1+109πkFa+0.18(2)(kFa)2+0.03(2)(kFa)3+) (4)where N is the total number of atoms, EF is the Fermi energy, and where by construction of hSBCS, the mean-field term (proportional to kFa) is fixed to its exact value 10/9π. We obtain beyond mean-field corrections up to the third order. The term proportional to (kFa)2 agrees with the Lee-Yang (9, 10) theoretical calculation 4(11−2log2)/21π2 ≅ 0.186. The third-order coefficient also agrees with the value 0.030 computed in (11).

Around unitarity, the energy expansion yieldsE=35NEF(ξsζ1kFa+) (5)We find the universal parameter of the unitary T = 0 superfluid, ξs = 0.41(1) with 2% accuracy. This value is in agreement with recent calculations and measurements (1). Our thermodynamic measurement ζ = 0.93(5) can be compared with a recent experimental value ζ = 0.91(4) (12), as well as the theoretical value ζ = 0.95 (13), both of them obtained through the study of the pair correlation function. This experimental agreement confirms the link between the macroscopic thermodynamic properties and the microscopic short-range pair correlations, as shown theoretically in (14).

In the BEC limit, the energy of the superfluid is that of a weakly interacting Bose-Einstein condensate of molecules (9, 15)E=N2Eb+Nπ2add2m×n(1+12815πnadd3+...) (6)where add = 0.6a is the dimer-dimer scattering length (1) and n is the dimer density. The term in nadd3 is the well-known Lee-Huang-Yang (LHY) correction to the mean-field interaction between molecules (9, 15). Signatures of beyond mean-field effects were previously observed through a pioneering study of collective modes (16) and density profile analysis (17), but no quantitative comparison with Eq. 6 was made. Fitting our data in the deep BEC regime with Eq. 6, we measure the bosonic LHY coefficient 4.4(5), in agreement with the exact value 128/15π4.81 calculated for elementary bosons in (9) and recently for composite bosons in (15).

Having checked this important beyond mean-field contribution, we can go one step further in the expansion. The analogy with point-like bosons suggests that the next term should be written as [83(4π33)nadd3(log(nadd3)+B)] (6, 18, 19). Using hSBEC(δ˜) (Fig. 2) (6), we deduce the effective three-body parameter for composite bosons B = 7(1). Interestingly, this value is close to the bosonic hard-sphere calculation B = 8.5 (20) and to the value B ≈ 7.2 for point-like bosons with large scattering length (19).

Our measurements also allow direct comparison with advanced many-body theories developed for homogeneous gases in the strongly correlated regime. As displayed in Fig. 3A, our data are in agreement with a Nozières-Schmitt-Rink approximation (21) but show significant differences from a quantum Monte-Carlo calculation (22) and a diagrammatic approach (23). The measured EoS strongly disfavors the prediction of BCS mean-field theory.

Fig. 3

Comparison with many-body theories. (A) Direct comparison of Embedded Image with a quantum Monte-Carlo calculation [red open circles (22)], a diagrammatic method [green open squares (23)], a Nozières-Schmitt-Rink approximation [blue open triangles (21)], and the BCS mean-field theory (solid blue line). (Inset) Zoom on the BCS side. (B) EoS in the canonical ensemble ξ(1/kFa) (solid black line) deduced from the Padé-type approximants to the experimental data Embedded Image and Embedded Image plotted in Fig. 2. Fixed-Node Monte-Carlo theories: red squares (24), blue circles (25), and green triangles (26).

Comparison with Fixed-Node Monte-Carlo theories requires the calculation of the EoS ξ(1/kFa) in the canonical ensembleξ(1kFa)EN2Eb35NEF (7)that is deduced from hSBCS(δ˜) and hSBEC(δ˜) (6). As shown in Fig. 3B, the agreement with theories (2426) is very good.

We now discuss the EoS of the partially polarized normal phase (black points in Fig. 1). At low concentrations, we expect the minority atoms to behave as noninteracting quasiparticles, the fermionic polarons (27). The polarons are dressed by the majority Fermi sea through a renormalized chemical potential μ2A11 (28) and an effective mass m*(δ1) (26, 29, 30). Following a Fermi liquid picture, we propose to express the gas pressure as the sum of the Fermi pressure of the bare majority atoms and of the polarons (4).h(δ1,η)=1+(m(δ1)m)3/2(ηA(δ1))5/2 (8)Our measured EoS agrees with this model at unitarity and on the BEC side of the resonance (Fig. 1), where for m*(δ1) we use the calculations from (30, 31). On the BCS side of the resonance, however, we observe at large minority concentrations an intriguing deviation to Eq. 8. In the BCS regime, the superfluid is less robust to spin imbalance. Consequently, the ratio of the two densities n1/n2 in the normal phase becomes close to unity near the superfluid/normal boundary ηc. The polaron ideal gas picture then fails.

Alternatively, we can let the effective mass m* be a free parameter in the model in Eq. 8 in the fit of our data around η = A. We obtain the value of the polaron effective mass in the BEC-BCS crossover (Fig. 4).

Fig. 4

Effective mass m*/m of the polaron in the BEC-BCS crossover (black dots). The blue dashed line is a calculation from (29), red open squares (30), green dot-dashed line (26), and blue solid line (31). Measurements at unitarity through density profile analysis [blue triangle (3)] and collective modes study [brown empty circle (7)] are also displayed. (Inset) Phase diagram of a zero-temperature imbalanced Fermi gas in the BEC-BCS crossover. The blue line is the theoretical value of A (26, 29, 30) that sets the separation between the partially polarized (PP) and the fully polarized (FP) phases. Black dots are the measured values of ηc (as in Fig. 1A), which set the separation between the superfluid (S) phase and the partially polarized phase. The red line is the calculation of ηc using our EoS of the superfluid and the model (Eq. 8) for the normal phase. The green squares are lower bounds of ηc given by the values of the gap measured in (33); see (6).

An important consistency check of our study is provided by the comparison between our direct measurements of ηc1) (from Fig. 1, black dots in the inset of Fig. 4) and a calculated ηc1) from Eq. 8 and the EoS of the superfluid hS. Assuming negligible surface tension, the normal/superfluid boundary is given by equating the pressure and chemical potential in the two phases. This procedure leads to the solid red line in the inset of Fig. 4, in excellent agreement with the direct measurements. In addition, by integrating our measured EoS of the homogeneous gas over the trap, one retrieves the critical polarization for superfluidity of a trapped gas, in agreement with most previous measurements (6).

We have measured the equation of state of a two-component Fermi gas at zero temperature in the BEC-BCS crossover. Extensions of our work include exploring the thermodynamics of the far BEC region of the phase diagram where a new phase associated with a polarized superfluid appears (17, 26), mapping the EoS as a function of temperature, and investigating the influence of finite interaction range, which is playing a key role in higher-density parts of neutron stars.

Supporting Online Material

Materials and Methods

Figs. S1 to S4


References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. In the BEC limit, the grand-canonical EoS expands as hS(δ˜) ≅ 15πa/4addδ˜ − 162; the first term is the mean-field interaction and the second is the LHY correction asymptotic behavior.
  3. We thank K. Jiang for participation in the early phase of the experimental work. We are grateful to X. Leyronas, C. Mora, Y. Castin, F. Werner, R. Combescot, J. Dalibard, F. Gerbier, and G. Shlyapnikov for stimulating discussions and critical comments on the manuscript. We thank S. Giorgini, P. Drummond, J. Drut, R. Haussmann, and W. Zwerger for providing us with their data. We acknowledge support from European Research Council, European Science Foundation (Euroquam), SCALA (Scalable Quantum Computing with Light and Atoms), Agence Nationale de la Recherche FABIOLA (Fermions and Bosons in Optical Lattices), Région Ile de France Institut Francilien de Recherche sur les Atomes Froids, and Institut Universitaire de France.
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