Polarization Measurements and the Pairing Gap in the Universal Regime
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Polarization Measurements and the Pairing Gap in the Universal Regime
J. Carlson and Sanjay Reddy
Theoretical Division, Los Alamos National Laboratory Los Alamos, NM 87545, USA (Dated: November 30, 2018)We analyze recent cold-atom experiments on imbalanced Fermi systems using a minimal model inwhich a BCS-like superfluid phase coexists with a normal phase. This model is used to extract theT=0 pairing gap in the fully paired superfluid state. The recently measured particle-density profilesare in good agreement with the theoretical predictions obtained for the universal parameters fromprevious Quantum Monte Carlo (QMC) calculations. We find that the zero-temperature pairing gapis greater than 0.4 times the Fermi energy E F , with a preferred value of 0 . ± . E F . The ratioof the pairing gap to the Fermi Energy is larger here than in any other system of strongly-pairedfermions in which individual pairs are unbound. PACS numbers: 03.75.Ss
Experiments that trap and cool fermionic atoms arenow providing new insights on the nature of strongly-interacting Fermi systems with a number imbalance be-tween the interacting species [1, 2, 3, 4, 5]. Experimentsto date have studied systems containing two hyperfinestates of Li, which we label | ↑i and | ↓i for convenience.These experiments can tune both the number asymmetry(polarization) and the interaction strength and thus havethe potential to probe new phases of superfluid matterthat are expected on theoretical grounds. Furthermore,they can directly probe pairing gaps in these strongly-paired superfluid systems. In this paper we examine re-cent experimental data on polarized Fermi systems andextract the T=0 pairing gap in the BCS phase. The ratio δ of the pairing gap to the Fermi energy at unitarity is auniversal parameter and previous QMC calculations pre-dict δ ≈ .
5. Gaps in strongly-paired Fermi systems areimportant in diverse areas of physics including nuclearmatter in neutron stars and the phase structure of denseQCD.Cold-atom experiments probe the strongly-interactingregime where the superfluid properties are robust at fi-nite temperature. When the short-range interaction istuned to produce an infinite scattering length all measur-able quantities are related to their free Fermi gas coun-terparts by universal constants since the interaction doesnot present a dimensionful scale. For this reason, the sys-tem with a = ∞ is said to be in the “universal” regime.In this regime two phases are certain to exist: a super-fluid state at zero polarization and a normal state at largepolarization. The superfluid state at T=0 can be char-acterized by the ground-state energy E SF = ξ (3 / E F and the pairing gap ∆ = δE F with E F = (3 π ρ ) / / m .The normal state can be characterized by the bindingof a minority spin particle to the Fermi sea of majorityparticles, E N − E = − χE f,N , where E N is the energyof the normal state, E is the energy of non-interactingparticles, and E f,N is the Fermi energy of the majorityspin population E f,N = (6 π ρ ) / / m .The universal constants have been calculated using Quantum Monte Carlo techniques,[6, 7, 8, 9, 10] yield-ing: ξ = 0.42(01), δ = 0 . χ = 0 . . ξ using both Diffusion MonteCarlo and Auxiliary Field Monte Carlo [11] indicate that ξ may be slightly smaller, ξ = 0 . . ξ and lower bound to χ [12]. The pairing gapparameter δ is obtained from the difference between even-and odd-particle number simulations, and hence is not abound on the true value. Measurements of the parame-ters ξ and χ appear to be consistent within errors withthese calculations as shown below.These universal parameters are defined precisely at po-larization zero and one for the superfluid and normalstates, respectively. QMC calculations of the superfluidand normal state suggest that simple descriptions basedupon a quasiparticle picture and these universal parame-ters are valid over a wide range of polarizations betweenzero and one. At T = 0 the superfluid + quasiparti-cle picture appears to work reliably up to polarization σ = ( n ↑ − n ↓ ) / ( n ↑ + n ↓ ) ≤ . σ ≥ . δµ = δµ c ≃ µ where δµ = ( µ ↑ − µ ↓ ) / µ = ( µ ↑ + µ ↓ ) / µ ↑ and µ ↓ , respectively. The pairing gap is onlyslightly larger than the value required for the stability ofa gapless superfluid with finite polarization at zero tem-perature. At finite temperature, however, the superfluidquasiparticles will be thermally excited, polarizing thesuperfluid; this polarization can be exploited to extractthe pairing gap in the superfluid. The main goal of thisstudy is to extract these fundamental properties of thesystem from experiments in the universal regime.The first-order phase transition appears to have beenobserved in recent MIT [4] and Rice [2] experiments. Sig-nificant differences remain, however. In particular theRice experiment finds a transition directly from zero tofull polarization. Such a scenario cannot happen in abulk system if the binding of spin-down particles in thenormal state ( χ = 0 .
6) is so large, and hence the observedpolarization must be related to the unique geometry ofthe trap. The MIT experiment has been performed forlarger numbers of particles and more spherical traps, andhence the simple local density approximation employedhere is better suited for an analysis of these experiments.Our picture of the two states at very low temperaturesare presented below. The calculated universal parame-ters then yield precise density distributions for the mi-nority and majority species in the trap.Superfluid state with a small polarization: In the uni-versal regime the energy density, pressure and the chemi-cal potential of the unpolarized superfluid state are givenby ǫ SF = ξ k F / (10 π m ), P SF = ξ k F / (15 π m ) and µ = ξ k F / m , respectively. Here k F is the Fermi mo-mentum k F = (3 π ρ ) / . In Ref. [9] we calculated theexcited state quasi-particle (qp) dispersion relation byintroducing additional fermions to the unpolarized su-perfluid state. To leading order in the momentum ex-pansion, the dispersion curve for qp’s measured relativeto µ is given by ω qp ( k ) = ∆ p (1 + a ( x − x ) + O [( x − x ) )] , (1)where x = k /k F , x ≃ . a ≃ . δµ c ∼ ∆ at the first-order transi-tion [9]. Thus even when T ≪ E F spin-up quasiparticlesare thermally excited in the superfluid in the vicinity ofthe transition. Further, zero-temperature QMC calcula-tions suggests that interactions between quasi-particlesin the superfluid are weak at low polarization. Hence thenumber density and pressure of thermally excited quasi-particles are taken to be n qp = Z d k (2 π ) (cid:18) (cid:18) ω qp ( k ) − δµT (cid:19)(cid:19) − , (2) P qp = T Z d k (2 π ) Log (cid:20) (cid:18) ω qp ( k ) − δµT (cid:19)(cid:21) , (3)respectively. The superfluid phonons with a dispersionrelation ω = c k , where c = p ξ/ k F /m , are negligiblecompared to spin-up quasi-particles except at very lowtemperature when T ≪ (∆ − δµ ).We anticipate that the gap will decrease due to bothfinite temperature and polarization. For the presentcalculations we assume that for low temperature thepairing gap decreases slowly according to ∆( T ) =∆(0) p − ( T /t c E F ) ). The zero-temperature pairing gap is a constant fraction δ of the Fermi energy: ∆(0) = δE F . The coefficient t c controls the temperature depen-dence of the gap and depends on ∆(0) and the ratio δµ/µ .Its calculation is beyond the scope of this work and amean-field treatment of the polarized finite temperaturesuperfluid phase is discussed in Ref.[13]. Here we treat t c as a parameter in the range 0 . − . E ↓ ( k ) = k m − χ k F ↑ m (4)with χ = 0 . E ↑ ( k ) = k m − χ k F ↓ m ˜ k F ; E ↓ ( k ) = k m − χ k F ↑ m ˜ k F (5)where ˜ k F = ( k F ↑ + k F ↓ ) / and χ = 0 .
6. We note thatthe dispersion relation in Eq. 4 and the symmetric formin Eq. 5 both provide a fair description of the QMC datanear unit polarization and differ only by a few percent at σ = 0 . µ ↑ = µ + δµ and µ ↓ = µ − δµ . The chemical potential µ = λ − V Trap ( r ),where V Trap ( r ) = 0 . ~ ω ( r/r ) and r = ~ c/ √ ~ ω mc while δµ is constant since the trap does not distinguishbetween the different fermion species. Using the notationof reference [5] distances are scaled relative to the radius R ↑ at which the zero-temperature spin-up density van-ishes. The radius R ↓ is similarly defined as the radius atwhich the zero-temperature spin-down density vanishes.The temperature T ′ is scaled to the Fermi energy of anon-interacting Fermi gas at r = 0 with the same den-sity profile as the exterior cloud. Similarly the densitiesare measured in units of n , where n is the density ofa non-interacting Fermi gas at r = 0 with the same den-sity profile as the exterior cloud. At low temperature thedensity profiles predicted by the theory discussed aboveare shown in Fig. 1. The inner superfluid region and theouter normal region separated by a first-order transitionare clearly seen. The transition is characterized by σ s and σ n – the polarizations in the superfluid and normalphase, respectively. As mentioned earlier, at low tem-perature σ N ≃ . σ s increases rapidly from zero N o r m a li z ed D en s i t y n up n down σ =(n up -n down )/n total n ref δ (r) Phase Boundary σ N σ S R c R FIG. 1: Spin-up and spin-down densities and polarizationversus radius predicted by theory for δ = 0 .
43 at T ′ = 0(dashed) and T ′ = 0 .
03 (solid) are shown. The universalparameters ξ and χ are taken as 0.4 and 0.6, respectively.For T ′ = 0 .
03 the local value of δ shown by the dot-dashedcurve was obtained using t c = 0 . at zero-temperature with increasing temperature due tothe low excitation-energy of spin-up quasiparticles in thevicinity of the transition. In our implementation of theequation of state for the normal phase, thermal effects onthe density profiles are negligible except close to r = R ↓ and r = R ↑ .The density distribution of the spin-up particles (ma-jority species) between R ↓ and R ↑ provides a measure ofthe temperature and the chemical potential of the spin-up particles. Similarly, the density distribution of theunpolarized superfluid state at the origin provides an in-dependent measure of the average chemical potential µ assuming the calculated value of ξ . Since χ = 0 . R ↓ can independently provide a measure of the spin-down chemical potential. Thus, the two radii, the densityat large radii, and the central density measure the twochemical potentials and the temperature and provide aconsistency check between the calculated values of ξ and χ in extracting µ and δµ .In order to compare with experimental results, we as-sume a specific ratio of the density at the origin to themaximum density of a cloud of free fermions with thesame density distribution for r > R ↑ , which is equiva-lent to assuming a specific average chemical potential µ .The values of the ratio R ↓ /R ↑ , the total polarization P tot in the trap, the transition radius R c , and the polariza-tion as a function of radius are then predicted for variousvalues of the pairing gap and the temperature. We findthat present measurements provide strong constraints onthe pairing gap even though the temperature is not veryprecisely determined in the lowest temperature measure- N o r m a li z ed D en s i t y n MIT datan MIT datann FIG. 2: Spin-up and spin-down densities theory and experi-ment for δ = 0 . ξ = 0 . χ = 0 . T ′ = 0 . ments.Fig. 2 compares the calculated spin-up and spin-downdensities as a function of radius. We used a normalizedtemperature T ′ = 0 .
03 and a normalized density at theorigin n ↑ ( r = 0) /n = n ↓ ( r = 0) /n = 1 .
72 consistentwith the experiment. This simple model reproduces theradius of the transition, the radius where the spin-downdensity goes to zero, and the overall polarization in thetrap. The calculated polarization is 0.44 for these valuesof the parameters, and the measured value is 0.44(0.04)[5].In Fig. 3 we compare the measured and calculatedpolarization at T ′ = 0 .
03 and T ′ = 0 .
05. Results fordifferent values of δ and t c are shown. At T ′ = 0 . .
12 at the interfaceat r/R ↑ = 0 .
43, consistent with the experimental results.The qualitative and quantitative features of the measuredpolarization at T ′ = 0 .
03 are captured by the normalphase at r/R ↑ > ∼ .
45 and a thermally polarized super-fluid phase for r/R ↑ < ∼ . T ′ = 0 .
05 suggeststhat the superfluid extends further out. Extrapolationof the polarization predicted by theory in the superfluidstate (dotted curve) to p ≃ . T /E F and polarization.For a fixed central density and R ↑ , our minimal modelpredicts that the phase-boundary R c moves outward inthe trap with increasing temperature. This behavior isvery sensitive to the thermal properties of both phasesat low temperature. In our analysis, which is expectedto hold only for small temperature and polarization, thethermal response of the superfluid phase in the vicinity of r/R P o l a r i z a t i on MIT data: T’=0.03 δ =0.5, t c =0.25 δ =0.5, t c =0.05 δ =0.38, t c =0.25 δ =0.43, t c =0.07 δ (r) P o l a r i z a t i on MIT data: T’=0.05 δ =0.5, t c =0.25 δ =0.46, t c =0.1 δ =0.38, t c =0.25 δ (r) FIG. 3: Polarization versus radius, theory and experiment, fordifferent values of δ and t c at T ′ = 0 .
03 and T ′ = 0 .
05. Thedashed curves show the local finite temperature gap. Theresults indicate that the data provide both an upper and alower bound on the gap: 0 . ≥ δ ≥ . the transition is stronger than that of the normal phase– driven entirely by the fact that spin-up quasiparticlesare easy to excite and have a large density of states.The comparison in Fig. 3 provides compelling lowerand upper bounds for the superfluid gap. Even if thetemperature was extracted incorrectly from the exper-iment, the extracted gap cannot be too small. A gapsmaller than ≈ . E F would produce a shell of polarizedsuperfluid before the transition even at zero temperature.Furthermore, the radial dependence of this polarizationwould be quite different than observed experimentally,rising abruptly from the point where ∆ = δµ and beingconcave rather than convex. A gap larger than ≈ . E F would be unable to produce the observed polarization inthe superfluid phase.We have also examined the dependence of our results on the universal parameters ξ and χ . Both of these areexpected to be uncertain by 0 .
02. These uncertainties,as well as the uncertainties in the superfluid quasiparticledispersion relation do not significantly alter the extractedbounds on the superfluid gap. If the transition tempera-ture and gap decrease significantly at finite temperaturewith increasing polarization, the extracted gap could beslightly higher.We note that the pairing gap extracted here is signifi-cantly larger than that obtained in simple analysis of RFspectroscopy. These experiments measure the responseto a probe tuned to a transition between a minority spin,for example, and a third hyperfine state [14, 15, 16].RF spectroscopy analysis, however, is complicated by thestrong final-state interactions that reduce the average en-ergies of the transition[17].In summary, it is possible to extract the pairing gapfrom measurements of polarized Fermi gases in the uni-tary regime. These systems have an extremely large gapof almost one-half the Fermi energy – the value extractedin this work is clearly the largest gap measured in anyFermi system. Precise measurements of the particle den-sities in the unitary regime, and measurements extendinginto the BCS regime would be valuable, in particular asdirect experimental tests of the pairing gap in low-densityneutron matter which is relevant in neutron stars.We would like to thank M. Alford, A. Gezerlis and Y.Shin for useful comments on the manuscript. The work ofS.R. and J.C. is supported by the Nuclear Physics Officeof the U.S. Department of Energy and by the LDRDprogram at Los Alamos National Laboratory. [1] G. B. Partridge et al.,
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