Abstract
Strongly interacting Fermi gases are of great current interest. Not only are fermions the most common particles in the universe, but they are also thought to have a universal thermodynamic behaviour for strong interactions^{1,2,3}. Recent experiments on ultracold Fermi gases provide an unprecedented opportunity to test universality in the laboratory^{4,5,6,7,8}. In principle this allows—for example—the interior properties of hot, dense neutron stars to be investigated on earth. Here we carry out a detailed test of this prediction. We analyse results from three ultracold fermion experiments involving two completely distinct atomic species in different kinds of atomictrap environment^{6,7,8}. The data are compared with the predictions of a recent stronginteraction theory^{9,10}. Excellent agreement is obtained, with no adjustable parameters. By extrapolating to zero temperature, we show that the experimental measurements yield a manybody parameter β≃−0.59±0.07, describing the universal energy of strongly interacting Fermi gases.
Main
Experiments on ultracold Fermi gases at microkelvin temperatures are revolutionizing many areas of physics. Their exceptional simplicity allows tests of manybody theory in areas long thought to be inaccessible. The ability to widely tune the effective interaction between fermions via a broad Feshbach resonance in gases of ^{6}Li and ^{40}K has allowed resonance models proposed in highT_{c} superconductivity theory^{11} to be implemented with fermionic ultracold atoms^{12}. Fortunately, the Pauli exclusion principle stabilizes the resulting excited molecular states against collisional damping^{13}. This has permitted the experimental observation of the smooth evolution of the Fermi gas from the attractive regime of Bardeen–Cooper–Schrieffer (BCS) superfluidity through to a regime where molecules form a Bose–Einstein condensate (BEC)^{14,15,16,17,18,19}. On the cusp of this BCS–BEC crossover, there is a strongly interacting regime—the socalled unitarity limit^{3}, which leaves the interatomic distance as the only relevant length scale.
At this point, the gas is expected to show a universal thermodynamic behaviour, independent of any microscopic details of the underlying interactions. The groundstate energy E_{0} of a homogeneous gas at zero temperature should be proportional to the free Fermi energy, E_{F}. Thus, E_{0}=E_{F}(1+β), where β is a universal manybody parameter. Substantial experimental efforts have been carried out to verify the existence of universality^{4,5}, though so far there has been no conclusive confirmation. This is mainly due to the lack of reliable thermometry in the strongly interacting regime^{10}, leading to an uncertainty in the finitetemperature corrections.
The universal parameters estimated from energies at the lowest accessible (but unknown) temperature range from β=−0.68±0.10 to β=−0.54±0.05 (refs 6,8,16,17). Because these are not at zero temperature, there is an unknown correction factor required to obtain the groundstate energy, and hence the true value of β. There is a similar range of estimated theoretical values, though a more precise value of β=−0.58±0.01 was recently obtained from zerotemperature quantum Monte Carlo simulations^{20}.
This situation has dramatically improved in the most recent thermodynamic measurements on strongly interacting Fermi gases of ^{40}K and ^{6}Li atoms^{7,8}, which allow accurate estimates of the energy in the universal regime from the fermionic cloud size^{5}. In experiments on ^{40}K carried out at JILA^{7}, an adiabatic magneticfield sweep is used to compare measurements of the cloud size in the strongly interacting and weakly interacting regimes, so that the noninteracting temperature is also known from the cloud size after the sweep. An important conceptual advance of the Duke group^{5}, who use ^{6}Li, was the realization that such sweeps give a modelindependent measurement of the entropy versus energy in the strongly interacting regime, thus allowing a precision test^{8} of theoretical predictions of universal thermodynamics. A different approach at Rice^{6}, also with ^{6}Li, makes use of the detailed density distribution to estimate temperature and entropy.
These groundbreaking investigations provide measurements accurate to the level of a few per cent, which is an exceptional accuracy in this challenging field of ultralowtemperature physics.
In this letter, we reanalyse all the available experimental data from these three laboratories, thus obtaining the measured entropy–energy relation of two completely different strongly interacting trapped Fermi gases. We compare this directly with a single universal theoretical prediction. We use a diagrammatic approach based on functional path integrals^{9,21} together with the local density approximation to treat the inhomogeneous trap. Below the superfluid transition, our calculations include pairing fluctuations, which are important in the strongly interacting regime, owing to the onset of molecule formation. This approach is described in detail elsewhere^{9}. Above threshold we use the well known Nozières–Schmidt–Rink (NSR) theory^{22,23,24}. We demonstrate a quantitative test of these thermodynamic predictions by comparing experimental results on both types of atom with a single theoretical curve, as shown in Fig. 1. There are no adjustable parameters, so this provides strong evidence for universality.
Further, by using powerlaw extrapolation to estimate finitetemperature corrections, we are able to show that all the experimental data give rise to a universal manybody coefficient of β≃−0.59±0.07. This is in excellent agreement both with the Monte Carlo results^{20}, β=−0.58±0.01, and our earlier theoretical prediction^{9} of β≈−0.599.
We summarize the experimental procedures, as typified by the JILA work using ^{40}K atomic gas^{7}. Here, the strongly interacting gas is prepared in a harmonic trap at the Feshbach resonance, and the potential energy is measured from the observed radius. Next, the magnetic field is swept adiabatically to a zeroscatteringlength field, and the potential energy at this field is again measured, which gives the noninteracting temperature. From these data, we obtain the entropy of the interacting gas^{7}, as the energy–entropy relation of a noninteracting Fermi gas is known. The total energy is also obtained, as it is twice the potential energy at the resonance, owing to the virial theorem^{5}. Figure 2a gives the predicted potential energy ratio as a function of the noninteracting temperature in the presence of a harmonic trap, together with the experimental measurements. The observed reduction of the potential energy in the strongly interacting regime is theoretically reproduced. Converting the noninteracting temperature and the potential energy into the total entropy and energy, respectively, we obtain the entropy–energy relation for ^{40}K gas in Fig. 2b. We find an excellent agreement between the experimental data and theoretical predictions below threshold. There is a small discrepancy just above the critical temperature of (T/T_{F})^{0}≈0.25 or the critical entropy of S_{c}≃2.2Nk_{B}, where we expect that the above threshold NSR theory may be less reliable. This effect is clearly visible in Fig. 2a, which gives the original experimental measurements. The nonlinear transformation used to obtain the entropy–energy relation in Fig. 2b means that conventional rectangular error bars give only a qualitative indication of the uncertainties in this figure.
Next, we discuss in greater detail the ^{6}Li experiment at Duke^{8}, which has greater accuracy but involves some extra corrections owing to the anharmonic trap used, and residual interactions in the weakly interacting cloud. The strongly interacting Fermi gas of N=1.3(2)×10^{5} atoms is prepared in a shallow gaussian (anharmonic) trap at a magnetic field B=840 G, slightly above the resonance position B_{0}=834 G. Here V_{0} gives the trap energy scale, m is the atomic mass, and ω_{z} are the transverse and axial trap frequencies and ρ and z are the transverse and axial coordinates of the spheroidal trap geometry. The coupling constant k_{F}a=−30.0, where k_{F} is the Fermi wavevector and a is the swave interatomic scattering length, is sufficiently large to ensure the onset of the universal thermodynamic behaviour. Experimentally, the entropy of the gas is measured by an adiabatic passage to a weak interacting field B=1,200 G, where k_{F}a=−0.75 and the entropy and temperature are known from the cloud size after the sweep. The energy E is determined model independently from the mean square radius of the strongly interacting fermion cloud 〈z^{2}〉_{840} measured at 840 G, according to the virial theorem^{5,8}, where is the Fermi energy for an ideal harmonically trapped gas at the trap centre, and z_{F}^{2} is defined by 3mω_{z}^{2}z_{F}^{2}≡E_{F}. The correction factor 1−κ accounts for the anharmonicity in the shallow trapping potential V_{0}≃10E_{F}.
Figure 3a shows the bare experimental data on the ratio of the mean square axial cloud size at 1,200 G to that at 840 G, as a function of the energy at 840 G, as compared with the theoretical simulations with the same realistic parameters, except that we use a resonance field B_{0} for the strongly interacting gas. Good agreement is found, with no free parameters. As before, there is a small discrepancy between the raw data and theoretical predictions, just above the critical energy. We have recalculated the entropy corrections due to residual interactions in the 1,200 G cloud to improve the accuracy at the 1% level, by using an abovethreshold NSR theory. Calibration of the entropy from the measured mean square axial cloud size at 1,200 G using the theoretically predicted dependence of the entropy on the size (inset in Fig. 3b) leads to the comparison for the entropy–energy relation, as shown in Fig. 3b. The agreement is even more impressive.
We can now describe the procedures used to obtain Fig. 1, which illustrates the universal thermodynamic behaviour of a strongly interacting Fermi gas. Here we have plotted all the measured data in a single figure, and compared them with our prediction for the entropy dependence of the energy of a harmonically trapped, strongly interacting Fermi gas, as well as that of an ideal Fermi gas. The slight shift of the experimental data in the Duke experiment due to the anharmonicity of the trap has been corrected, by subtracting the (small) theoretical difference between a shallow gaussian trap and a harmonic trap for the energy at the same entropy. We also plot a single data point from the ^{6}Li experiment at Rice^{6} for the energy at their lowest entropy. The agreement between theory and experiment is excellent for almost all the measured data. Exactly the same theory is used in all cases, with results from three different laboratories^{6,7,8}. The universal thermodynamics of a strongly interacting Fermi gas is strikingly demonstrated, independently of which atomic species we compare with.
Just above the critical entropy S_{c}≃2.2Nk_{B}, for the superfluid–normalfluid phase transition, there is a suggestion of a discrepancy between theoretical predictions and these precise measurements. At this point the abovethreshold NSR theory is least accurate^{10}. The data here may even indicate a firstorder superfluid phase transition. We note that the exact order of phase transition for a strongly interacting Fermi gas is still an open question^{25}, and merits further investigation.
A key feature of current ultracold Fermigas experiments is that the lowest attainable entropy is around S=0.7Nk_{B}, which corresponds to a temperature of 0.10–0.15T_{F} at unitarity. This nonzero entropy or temperature affects the precise determination of the universal manybody parameter β. To remove the temperature dependence, we assume that in the lowentropy regime (below the phase transition) there is a powerlaw dependence of the energy on the entropy: E−E_{0}∝S^{α}, as anticipated theoretically. For noninteracting Fermi and Bose gases, the powerlaw exponents are 2 and 4/3, respectively. The thermodynamics of a unitary gas should lie between an ideal Fermi gas and an ideal Bose–Einstein condensate. The fitting procedure leads to E_{0}/(NE_{F})=0.48±0.03 and E_{0}/(NE_{F})=0.48±0.04, for the Duke and JILA experiments, respectively, with a similar powerlaw exponent α=1.7±0.4. Here the error bar accounts only for the fitting error. Using the relation E_{0}/N=(3/4)(1+β)^{1/2}E_{F} for a harmonic trap^{4}, this gives rise to β≃−0.59±0.07: which agrees fairly well with the most accurate quantum Monte Carlo simulations^{20}, β=−0.58±0.01, and our theoretical prediction^{9}, β≈−0.599. Our theoretical powerlaw prediction is α=1.5 in the LDA regime, which also agrees with experiment.
Methods
We briefly explain our analytic theory. This is an approximate method using perturbation theory summed to all orders, because no exact results are known. In the homogeneous gas case, it relies on the manybody Tmatrix approximation to account for the effects of collective Bogoliubov–Anderson modes, and extends the standard NSR approach to the brokensymmetry state^{9}. This amounts to considering the contributions of gaussian fluctuations around the meanfield saddle point to the thermodynamic potential (with Nambu notation), where are respectively the diagonal and offdiagonal parts of the pair propagator. Here, Q=(q,iν_{n}), K=(k,iω_{m}), and (q and k are wavevectors, ν_{n} and ω_{m} bosonic and fermionic Matsubara frequencies, respectively), m is the fermion mass, T the temperature, and ε_{k}=ℏ^{2}k^{2}/2m, G_{11} and G_{12} are BCS Green functions with a variational order parameter Δ. Together with the meanfield contribution with excitation energy E_{k}=[(ε_{k}−μ)^{2}+Δ^{2}]^{1/2} and Fermi distribution function f(x)=1/(1+e^{x/kBT}), we obtain the full thermodynamic potential Ω=Ω_{mf}+Ω_{pf}. All the observables are calculated straightforwardly following the thermodynamic relations, once the chemical potential μ and the order parameter Δ are determined. Number conservation in the form n=−∂Ω/∂μ is strictly satisfied, yielding an exact identity for the pressure P and energy density U of a unitarity gas: P=2U/3 (refs 3,5). For simplicity, in our calculations we determine the order parameter at the level of the mean field, using the gap equation ∂Ω_{mf}/∂Δ=0. Part of our approach was also previously derived using a functional integral method^{21}. In the case of the normal Fermi liquid with vanishing order parameter, the usual NSR formalism is recovered^{22}.
This type of perturbation theory with bare BCS Green functions in the pair propagators constitutes the simplest description of the BCS–BEC crossover, including the essential pair fluctuations. More sophisticated approximations with dressed Green functions in the pair propagators, that is, the GG_{0} (ref. 26) and the fully selfconsistent GG schemes^{27,28}, have also been proposed. In ref. 10, we carried out a comparative study of these different approximation schemes for a unitarity gas in the normal state. A related calculation in the superfluid phase has also been carried out recently^{25}. Compared with the latest pathintegral Monte Carlo simulations^{29,30}, our analytic perturbation scheme seems to be the optimal choice for the calculations of the type required for the entropy–energy relation. However, there is a small region around the critical temperature where none of the current calculations are reliable, and we see indications of this in the data.
To include the effects of the trap, we use the localdensity approximation by assuming that the system can be treated as locally uniform, with a positiondependent local chemical potential μ(r)=μ−V (r), where V (r) is the trapping potential. The local entropy and energy, calculated directly from the local thermodynamic potential using thermodynamic relations, are then summed to give the total entropy and energy. We note that in the presence of a harmonic trap the exact identity P=2U/3 yields the virial theorem^{5}, which states that the potential energy of the gas is a half of its total energy.
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Acknowledgements
We are extremely grateful to J. E. Thomas and B. Clancy for many helpful discussions, and for sharing their data before publication. We also thank D. S. Jin and J. T. Stewart for communications on their data, and R. G. Hulet et al. for explaining their temperature measurements. This work was supported by an Australian Research Council Center of Excellence grant, the National Natural Science Foundation of China Grant No. NSFC10574080, and the National Fundamental Research Program Grants Nos 2006CB921404 and 2006CB921306.
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Department of Physics, Renmin University of China, Beijing 100872, China
 Hui Hu
ARC Centre of Excellence for QuantumAtom Optics, Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia
 Hui Hu
 , Peter D. Drummond
 & XiaJi Liu
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Correspondence to Peter D. Drummond.
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