Signature of the FFLO phase in the collective modes of a trapped ultracold Fermi gas
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Signature of the FFLO phase in the collective modes of a trapped ultracold Fermi gas
Jonathan M. Edge and N. R. Cooper
T.C.M. Group, Cavendish Laboratory, J. J. Thomson Ave., Cambridge CB3 0HE, UK. (Dated: November 7, 2018)We study theoretically the collective modes of a two-component Fermi gas with attractive inter-actions in a quasi-one-dimensional harmonic trap. We focus on an imbalanced gas in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Using a mean-field theory, we study the response of theground state to time-dependent potentials. For potentials with short wavelengths, we find dramaticsignatures in the large-scale response of the gas which are characteristic of the FFLO phase. Thisresponse provides an effective way to detect the FFLO state in experiments.
With the advent of ultracold atomic gases new statesof matter are becoming experimentally accessible. Im-portant developments have included the achievementof superfluidity in two-component Fermi gases [1], andthe study of the effects of density imbalance in thesesystems[2]. These developments open up the prospect ofthe study of unconventional superfluid phases, in partic-ular the inhomogeneous superfluid state of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [3]. This state has beenunder theoretical investigation since the 1960s, playinga central role in our understanding of superfluidity offermions both within condensed matter physics and inparticle physics [4]; yet to date no unambiguous demon-stration of the FFLO state has been achieved.While theory predicts the appearance of the FFLOphase in a bulk (3D) gas of ultracold fermions [5] theregion of parameter space that it occupies is expectedto be very small. It has been shown that this phase isgreatly stabilised in (quasi-)1D geometries [6, 7, 8, 9].Such geometries can readily be achieved in atomic gasesby optical confinement[10], so experiments on ultracoldFermi gases will soon be in the regime where the FFLOphase can be studied. In view of this, it is important andtimely to ask: what are the observable properties of theFFLO phase?Existing proposals for the detection of the FFLO phaseinclude the probing of noise correlations in an expan-sion image following release of the trap[9], and the useof RF spectroscopy to excite atoms out of the gas intoother states [11]. In this Letter we show that characteris-tic signatures of the spatial inhomogeneity of the FFLOphase can be found in the collective mode response of thetrapped gas. The method directly probes the intrinsicinhomogeneity of the FFLO phase – by the use of a per-turbation by an potential with a short-wavelength spatialperiodicity. An important feature of this method is thatthe signatures of the microscopic nature of the phase ap-pear in the response on very large length scales (the scaleof the atomic cloud). Consequently, the measurements ofthe response do not require high spatial resolution, andcould be performed in situ without expansion of the gas.We study a model of two-component fermions in 1Dwith attractive contact interactions, and unequal densi-ties. For a homogeneous system, the exact groundstates are known from the Bethe ansatz [6, 7, 12]. Dependingon the densities of the two components there are threephases that appear at T = 0: the unpolarised superfluidstate; a fully polarised (therefore non-interacting) Fermigas; and a partially polarised phase [6], associated withthe FFLO state. This is a state that, within mean-fieldtheory, has a local superconducting gap that oscillates inspace. As has been discussed in detail by Liu et al.[13],mean-field theory provides an accurate description of theexact phase diagram for the 1D system, at least withinthe weak-coupling BCS regime. We have extended themean-field theory to investigate the linear response andcollective modes of both the trapped and untrapped im-balanced Fermi gas. As we shall discuss in detail later,this approximate theory correctly captures all qualita-tive features of the collective modes that are importantfor our purposes.To effect numerical calculations, we study a dis-cretized version of the problem (the attractive Hubbardmodel), for which the mean-field Bogoliubov-de Gennes(BdG) Hamiltonian is ˆ H = − J P i,σ (cid:16) ˆ c † i +1 ,σ ˆ c i,σ + h.c. (cid:17) + P i (cid:16) ∆ i ˆ c † i, ↑ ˆ c † i, ↓ + h.c. (cid:17) + P i,σ W i,σ ˆ c † i,σ ˆ c i,σ where ˆ c ( † ) i,σ arefermionic operators for species σ = ↑ , ↓ on site i , W i,σ ≡ V exti + U h ˆ c † i, ¯ σ ˆ c i, ¯ σ i − µ σ (with V ext the external potentialand µ σ the chemical potentials), and ∆ i ≡ U h ˆ c i, ↓ ˆ c i, ↑ i is the local superfluid gap. J is the hopping parameterand U the on-site interaction strength ( U < δW i,σ ( t ), by supplement-ing the (self-consistent) solutions of the above BdG equa-tions with the random phase approximation (RPA) [14].Divergences in the response appear at the frequenciesof the collective modes of the system. The results wepresent are at sufficiently low particle density to be rep-resentative of the continuum limit. In the mapping tothe continuum, we relate site i to position x via x = ia and the mass is m = ~ Ja . The interaction strength γ ,defined as the ratio of the interaction energy density tothe kinetic energy density, is given by γ = − mg d ~ ρ [13]where g d = Ua , ρ is the total density of particles and E F ≡ π ρ J/ FIG. 1: Response [15] of a homogeneous system in the FFLOphase to a spin-asymmetric periodic potential (1). The areaof the circle is proportional to the amplitude of the response.The polarisation p ≡ k F ↑ − k F ↓ k F ↑ + k F ↓ = 0 .
15, and the interactionsstrength is γ = 1 .
5. Two gapless sound modes are seen toemerge around k ≃ k ≃ k ∗ ≡ k F ↑ − k F ↓ ). Thesimulation was done on 270 lattice sites. tem, with V ext = 0 and periodic boundary conditions.We focus on the FFLO phase, with unequal average par-ticle densities, for which the self-consistent mean-fieldsolution has an oscillating gap ∆( x ) with wavelength λ ∆ = 2 π/ ( k F, ↑ − k F, ↓ ), where k F,σ are the Fermi wavevec-tors for the non-interacting imbalanced gas. We studythe response to periodic perturbing potentials δW σ ( x, t ) = V σ sin kx cos ωt (1)We refer to the case V ↑ = V ↓ = V as “spin-symmetric”and V ↑ = − V ↓ = V as “spin-asymmetric” excitation.The response of an imbalanced gas in the FFLO phaseto a spin-asymmetric modulation is shown in Fig. 1.At low frequency and long wavelength there appear twodistinct sound modes with different velocities [16, 17].Within mean-field theory, the appearance of these twogapless modes arises from the fact that the FFLO phasebreaks both gauge symmetry and translational symme-try. (We later discuss these properties beyond mean fieldtheory.) An analysis of the two low-frequency modesshows that they are of mixed character, involving spa-tial oscillations of both density and spin-density. Thisis expected from the breaking of time reversal symmetryby the imbalance[17], and is consistent with the results(in Fig. 1) that both modes can be excited by a purelyspin-asymmetric perturbation, V ↑ = − V ↓ .In Fig. 1 it is clear that linear sound modes emerge alsoaround the point k = k ∗ ≡ k F ↑ − k F ↓ ). Within mean-field theory, this too can be understood as a consequenceof the broken translational symmetry of the FFLO phase.This leads to a Brillouin zone structure for the collectivemodes, characterised by a reciprocal lattice vector of size k ∗ . Note that this is twice the value that one would ex-pect from the translational periodicity of the gap, λ ∆ . Tounderstand this, observe that, while the gap ∆ has pe-riod λ ∆ , the densities ρ ↑ ( x ) and ρ ↓ ( x ) have periods λ ∆ / x → x + λ ∆ / , ∆ → − ∆. Ap-plying this symmetry to the RPA response calculation leads to a generalised Bloch theorem for the collectivemodes. The wavevector k is found to be conserved mod-ulo (2 π ) / ( λ ∆ /
2) = k ∗ , consistent with the response inFig. 1. For a spin-symmetric perturbation a similar qual-itative response is found. However, the response close to k = k ∗ is smaller than for the spin-asymmetric pertur-bation. We account for this by the fact that the periodicdensity of excess majority particles manifests itself morestrongly in the difference of the two densities than in thesum of the two densities. We now turn to discuss themanifestations of this response for a trapped gas.We have studied the attractive Fermi gas in a har-monic trap, with V ext ( x ) = mω x and with unequalparticles numbers N ↑ = N ↓ . According to Orso [6] a1D imbalanced trapped Fermi gas can only be in oneof two configurations: (a) A partially polarised phase inthe middle, with a fully paired phase towards the edge;(b) a partially polarised phase in the middle, with afully polarised phase towards the edge. These configu-rations appear within the self-consistent BdG mean-fieldtheory[13]. An example of the case (a) is given in Fig. 2,and of the case (b) in Fig. 3, where lengths are measuredin units of N / a ho (with N = N ↑ + N ↓ the total numberof fermions, and a ho ≡ p ~ /mω the oscillator length ofthe trap). The configurations in Figs. 2 and 3 are in theweak coupling BCS regime, having values of γ for whichBdG gives qualitatively the right density profiles [13].We have studied the response of the trapped imbalancedFermi gas in both these regimes.Perturbations with wavelength much longer than thesize of the system couple via potentials V σ ( x ) that areproportional to x . We find that the response to suchpotentials is well described by two sharp modes, involv-ing the dipolar oscillations of the density and spin. Oneof these modes – the “Kohn mode” – involves the in-phase motion of both spin components, and has fre-quency ω = ω , independent of the state of the system.This exact result holds also within RPA[18], and is re-covered to high accuracy in our calculations, showingthat discretisation effects are minimal. The other mode –the “spin-dipole” mode – involves the relative motion ofthe two spin species. For attractive interactions the fre-quency lies slightly above ω (at ω sd = 1 . ω and 1 . ω for Figs. 2 and 3 respectively). While the frequency ofthe spin-dipole mode does depend on the state of thesystem, it is does not show strong features of the pres-ence of the FFLO phase. Similarly, the breathing modesare insensitive to the microscopic nature of the atomicgas[7].The most interesting features arise when one excitesthe gas by time dependent potentials (1) with short wave-lengths. We find characteristic signatures of the micro-scopic nature of the trapped gas in the response of spin-dipole mode – i.e. on length scales much larger than thewavelength of the applied perturbation.[27] In Figs 2(b)and 3(b) we show the amplitude of the response of the -0.2-0.1 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 ρ / ( N / / a ho ) , ∆ / E F x/(N a ho ) (a) ∆ρ ↑ ρ ↓ ρ ↑ - ρ ↓ (b) λ ∆ / λ/ ( N a ho ) FIG. 2: (Colour online.) Configuration with fully pairedphase towards the edge of the trap. (a): densities and thevalue of the superfluid gap ∆. (b): response of the spin-dipole mode to excitations of different wavelengths (arbitraryunits). Here p = 0 . γ = 0 .
93 (measured in centre), num-ber of particles N = 290, lattice spacing a = 3 . · − N a ho .The perturbing potential has a fixed amplitude, while thewavelength is varied. -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 ρ / ( N / / a ho ) , ∆ / E F x/(N a ho ) (a) ∆ρ ↑ ρ ↓ ρ ↑ - ρ ↓ (b) λ ∆ / λ/ ( N a ho ) FIG. 3: (Colour online.) Configuration with fully polarisedphase towards the edge. Here p = 0 . γ = 1 . a/ ( N a ho ) =4 . · − , N = 143 . Otherwise the same as fig. 2. spin-dipole mode to spin-symmetric and spin-asymmetricperiodic perturbations (1) as a function of wavelength λ = 2 π/k .For wavelengths λ that are short compared to thesize of the cloud, N / a ho , but large compared to λ ∆ , we find an oscillatory response as a functionof λ in both Figs. 2(b) and 3(b). To understandthis behaviour, we note that the spin-dipole modecan be excited when the two species experience anet difference in their acceleration. This differenceis a ( k ) = V km R [ ρ ↑ ( x ) /N ↑ ∓ ρ ↓ ( x ) /N ↓ ] cos( kx ) d x wherethe ∓ signs denote spin-symmetric/spin-asymmetric ex-citations. The difference in acceleration oscillates with k ,in a manner that is largely controlled by the sizes of theclouds. For case (b) we assume, for simplicity, parabolicdensity distributions of size x σ for the two spin compo-nents σ [28]. Defining r ≡ x ↑ + x ↓ and δ ≡ x ↑ − x ↓ , thecondition for a maximum is sin kr cos kδ = 0 for spin-asymmetric excitations, and cos kr sin kδ = 0 for spin-symmetric excitations. For δ ≪ r it follows that forspin-asymmetric (spin-symmetric) potentials maxima inthe response come from the sin kr (cos kr ) term and theantinodes in the envelope come from the cos kδ (sin kδ )term. Case (a) can be described by assuming a parabolicdensity distribution for the average density but where there is a constant difference between the two densitiesin the partially polarised region of size r p . For spin-symmetric and spin-asymmetric excitations the conditionfor a peak is given by cos kr p = 0. The modulation of thepeaks by this envelope structure allows us to distinguishthe two imbalanced trap configurations (a) for which thetwo clouds have equal size x ↑ = x ↓ and there is no mod-ulation and (b) for which x ↑ = x ↓ and a modulationappears, see figs. 2(b) and 3(b).As the wavelength is further reduced, we find dramaticsignatures of the presence of the FFLO phase, in boththe cases (a) and (b) described above. Specifically, wefind an unusually large response of the amplitude of thespin-dipole mode when the wavelength of the excitationbecomes λ = λ ∆ . This response can be understood interms of coupling to the gapless modes at k = πλ ∆ = k ∗ of the homogeneous system (see Fig.1). A small devia-tion from k ∗ (of the order the inverse system size) allowsmixing of this mode to the spin-dipole mode, causing theresponse at k ∗ to be apparent in the dipolar motion ofthe atomic cloud.While the calculations we have presented are withinRPA mean-field theory, as we now argue, the qualitativeconclusions are valid more generally. Essentially, our re-sults rely on the fact that in the unpolarised FFLO phase,there exists low-frequency (of order ω ) response that issharply peaked at a spatial wavevector k ∗ . In the above,this response was accounted for in terms of the brokentranslational symmetry of the mean-field state. As iswell known, in a true 1D quantum system no contin-uous symmetries are broken. Thus, the broken (phaseand translational) symmetries of the mean-field FFLOstate can lead only to power-law decay of the respec-tive correlation functions[19]. However, the qualitativefeatures described above are the same as those expectedfrom an exact treatment of the system. The transitionfrom the (unpolarised) superconducting phase to the par-tially polarised phase is marked by the closing of thespin-gap, leading to a second gapless sound mode. Thiscan be viewed as a Luttinger liquid representing the ex-cess fermions[20]. These excess particles have density ρ ↑ − ρ ↓ , and thus a Fermi wavevector ( k F ↑ − k F ↓ ). Thus,there are two gapless collective modes, arising from thefully paired particles, and the liquid of excess majorityspin particles. Furthermore, as in the general theory ofLuttinger liquids[21] the spectral function of these ex-cess majority spin particles will show gapless responseat multiples of twice their Fermi wavevector. This is2( k F ↑ − k F ↓ ) = 4 π/λ ∆ which is precisely the wavevector k ∗ at which one finds the gapless response in mean fieldtheory. Thus, the qualitative features of the collective ex-citation spectrum obtained in mean-field theory are fullyconsistent with those expected for the exact system.Any distinction between the exact results and those ofmean-field theory will be further reduced in the finite-sizesystems on which experiments can be performed, wherethe distinction between long-range and power-law cor-relations becomes blurred. Indeed, one can expect that density inhomogeneity will play a more significant rolethan will the power-law decay of correlations. Since thedensities of spin-up and spin-down particles are inhomo-geneous, so too is (the local value of) λ ∆ , so one canexpect smearing of the condition λ = λ ∆ /
2. However,even for the relatively small systems studied in Figs. 2and 3, the collective mode response has a sharp onsetat the value λ = λ ∆ / λ ∆ / λ ∆ , which decreases slightly towardsthe edge of the FFLO phase.We propose that the signatures of Figs. 2 and 3 pro-vide a convenient way to detect the FFLO phase in ex-periment. The sharp feature at λ ∆ / k B T ≃ . E F [22]).In order to observe these signatures in experiments oneneeds to be able to create a variable wavelength opticallattice. This can be done using the technique used bySteinhauer et al. [23]. While the signal appears for a spin-symmetric perturbation, the response is larger for spin-asymmetric. Thus, any spin-dependence of the opticallattice will enhance the ability to distinguish the peaksassociated with the oscillation of ∆ from the backgroundpeaks. Optical lattices with spin-dependence have beencreated by Mandel et al. [24], and would be ideally suitedto this purpose. There are two natural ways to find theresponse of the spin-dipole mode. One way is to fol-low precisely the approach of the calculation, and makethe strength of the optical lattice time dependent, as inRef.[25]. One can then selectively excite the spin-dipolemode by bringing the temporal oscillation of the lattice inresonance with the spin-dipole mode. Another way (sim-ilar to the approach used in Ref. [26]) is to apply a staticperiodic potential to the system, allow it to equilibrate,and then switch off this potential abruptly. This willexcite collective modes of many frequencies from whichthe response of the spin-dipole mode can be obtained byFourier transform.In either case, we emphasise that the required mea-surements of particle density are on the length scale ofthe size of the cloud . Thus, our proposed method allowsthe probing of the microscopic physics by perturbing thesystem with short wavelengths while requiring only themeasurement of densities on the length scale of the cloud.We acknowledge useful comments from F. Essler, C.Lobo, G. Orso and C. Salomon and useful discussionswith M. K¨ohl and N. Tammuz on the experimental real-isability. This work was supported by EPSRC. [1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod.Phys. , 885 (2008); S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. , 1215 (2008).[2] G. Partridge, W. Li, R. Kamar, Y. Liao, and R. Hulet,Science , 503 (2006); M. Zwierlein, A. Schirotzek,C. Schunck, and W. Ketterle, Science , 492 (2006).[3] P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964);A. Larkin and Y. Ovchinnikov, Zh. Eksp. Teor. Fiz. ,1136 (1964), [Sov. Phys. JETP , 762 (1965)].[4] R. Casalbuoni and G. Nardulli, Rev. Mod. Phys. , 263(2004).[5] For a review, see D. E. Sheehy and L. Radzihovsky, An-nals of Physics , 1790 (2007).[6] G. Orso, Phys. Rev. Lett. , 070402 (2007).[7] H. Hu, X.-J. Liu, and P. D. Drummond, Phys. Rev. Lett. , 070403 (2007).[8] A. E. Feiguin and F. Heidrich-Meisner, Phys. Rev. B , 220508(R) (2007); M. M. Parish, S. K. Baur, E. J.Mueller, and D. A. Huse, Phys. Rev. Lett. , 250403(2007); G. G. Batrouni, M. H. Huntley, V. G. Rousseau,and R. T. Scalettar, Phys. Rev. Lett. , 116405 (2008).[9] A. L¨uscher, R. M. Noack, and A. M. L¨auchli, Phys. Rev.A , 013637 (2008).[10] H. Moritz, T. St¨oferle, M. K¨ohl, and T. Esslinger, Phys.Rev. Lett. , 250402 (2003).[11] M. R. Bakhtiari, M. J. Leskinen, and P. T¨orm¨a, Phys.Rev. Lett. , 120404 (2008).[12] M. Gaudin, Phys. Lett. A , 55 (1967).[13] X.-J. Liu, H. Hu, and P. D. Drummond, Phys. Rev. A , 043605 (2007).[14] G. M. Bruun and B. R. Mottelson, Phys. Rev. Lett. ,270403 (2001).[15] The response is defined as rP i “ δρ i, ↑ + δρ i, ↓ ” . The di-vergent response at ω = 0 is not shown.[16] A. E. Feiguin and D. A. Huse, Phys. Rev. B ,100507(R) (2009).[17] H. Frahm and T. Vekua, Journal of Statistical Mechanics:Theory and Experiment , P01007 (2008).[18] Y. Ohashi, Phys. Rev. A , 063613 (2004).[19] M. Rizzi, M. Polini, M. A. Cazalilla, M. R. Bakhtiari,M. P. Tosi, and R. Fazio, Phys. Rev. B , 245105 (2008).[20] K. Yang, Phys. Rev. B , 140511(R) (2001).[21] F. D. M. Haldane, J. Phys. C , 2585 (1981).[22] X.-J. Liu, H. Hu, and P. D. Drummond, Phys. Rev. A , 023601 (2008).[23] J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Phys.Rev. Lett. , 120407 (2002).[24] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W.H¨ansch, and I. Bloch, Phys. Rev. Lett. , 010407 (2003).[25] T. St¨oferle, H. Moritz, C. Schori, M. K¨ohl, andT. Esslinger, Phys. Rev. Lett. , 130403 (2004).[26] A. Altmeyer, S. Riedl, M.J. Wright, C. Kohstall,J.H. Denschlag, and R. Grimm, Phys. Rev. A76