First and second sound of a unitary Fermi gas in highly elongated harmonic traps
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t First and second sound of a unitary Fermi gas in highly elongated harmonic traps
Xia-Ji Liu and Hui Hu Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia (Dated: October 18, 2018)Using a variational approach, we present the full solutions of the simplified one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas trapped in a highly elongated harmonicpotential, which is recently derived by Stringari and co-workers [Phys. Rev. Lett. , 150402(2010)]. We calculate the discretized mode frequencies of first and second sound along the weakaxial trapping potential, as a function of temperature and the form of superfluid density. We showthat the density fluctuations in second sound modes, due to their coupling to first sound modes,are large enough to be measured in current experimental setups such as that exploited by Tey et al .at the University of Innsbruck [Phys. Rev. Lett. , 055303 (2013)]. Owing to the sensitivity ofsecond sounds on the form of superfluid density, the high precision of the measured second soundfrequencies may provide us a promising way to accurately determine the superfluid density of aunitary Fermi gas, which so far remains elusive.
PACS numbers: 67.85.Lm, 03.75.Ss, 05.30.Fk
I. INTRODUCTION
Over the past few years, first and second sound of aunitary atomic Fermi gas at a broad Feshbach resonancehave received increasing attentions [1–13]. Being the in-phase density oscillations (first sound) and out-of-phasetemperature oscillations (second sound) [14, 15], thesesound modes provide a useful probe of the equation ofstate [9, 10, 16–19] and superfluid density [3–7, 11, 13]of the unitary Fermi gas. The latter quantity of super-fluid density is of particular interest, as it is notoriouslydifficult to calculate in theory. In practice, it could bemeasured through second sound wave propagation [5].Indeed, in superfluid helium He, the accurate knowledgeof its superfluid density, slightly below the lambda tran-sition, is obtained by the measurement of temperaturewaves [20].A strongly interacting unitary Fermi gas bears a lotof similarity to superfluid helium [4, 6]. Due to strongcorrelation, the first and second sound in both systemsare nearly decoupled. Yet, the weak coupling betweensounds still leads to a sizable hybridization effect andhence a measurable density fluctuation for second sound.This gives rise to a promising way of exciting and detect-ing second sound through density measurements in a uni-tary Fermi gas [5, 6]. For an isotropically trapped unitaryFermi gas, such a hybridization effect has been analyzedby Taylor et al. , using the standard dissipationless Lan-dau two-fluid hydrodynamic approach [4]. Experimen-tally, however, it is more feasible to confine a unitaryFermi gas in highly elongated traps. For this configu-ration, the viscosity and thermal conductivity terms inthe two-fluid hydrodynamic equations become importantand enable a simplified one-dimensional (1D) hydrody-namic description, as suggested by Bertaina, Pitaevskiiand Stringari [8]. In a recent milestone experiment, asecond sound wave has been excited in a highly elon-gated unitary Fermi gas, and its propagation along theweakly confined axial axis has been measured [11]. The simplified 1D hydrodynamic equations has been used toextract the superfluid density from the resulting secondsound velocity data. Unfortunately, at present the exper-imental accuracy of sound velocity is not enough to givea satisfactory determination of superfluid density [11].In this paper, we propose that the measurement of discretized mode frequencies of low-lying second soundalong the weakly confined axial direction may providean accurate means of determining superfluid density. In-deed, the latest measurement of discretized first soundmode frequencies [9] indicates a very small relative error( ∼ . ), which is at least an order smaller in magnitudethan the relative error in sound velocity data [11].For this purpose, we fully solve the coupled 1D hy-drodynamic equations in the presence of a weak axialharmonic potential, and obtain the density fluctuationsof discretized low-lying second sound modes, which arisefrom to their coupling to first sound modes. We findthat these density fluctuations are significant, therebymaking second sound modes observable in current exper-iments, by modulating, for example, the weakly confinedaxial trapping potential. Our full solutions of the simpli-fied 1D hydrodynamic equations complement the earlierresults obtained by Hou, Pitaevskii and Stringari [12],where the decoupled first and second sound mode fre-quencies are calculated with simple variational ansatz fordisplacement fields. In this work, we emphasize the cor-rection to discretized mode frequencies, due to the weakcoupling between first and second sound.The rest of paper is organized as follows. In the nextsection, we briefly outline the reduced 1D thermodynam-ics, as an input for the simplified 1D hydrodynamic de-scription. In Sec. III, we show how to solve the coupled1D hydrodynamic equations by using a variational ap-proach. In Sec. IV, we first provide results for the de-coupled first and second sound solutions and then presentthe density fluctuations of some low-lying second soundmodes. Finally, in Sec. V we draw our conclusion andbriefly describe how to obtain the superfluid density ofa unitary Fermi gas from the measured low-lying secondsound mode frequencies. II. 1D REDUCED THERMODYNAMICS
We consider a unitary Fermi gas trapped in highlyanisotropic harmonic potential, V ext ( r ⊥ , z ) = 12 mω ⊥ r ⊥ + 12 mω z z , (1)with atomic mass m and trapping frequency ω z ≪ ω ⊥ .We assume that the number of atoms in the Fermicloud, typically of N ∼ in current experiments, islarge enough, so that we can safely use the local den-sity approximation and treat the atoms in the position ( r ⊥ , z ) as uniform matter with a local chemical potential µ ( r ⊥ , z ) = µ − V ext ( r ⊥ , z ) , where µ is the chemical po-tential at the trap center. In this way, we may write thelocal pressure and number density in the form, P ( r ⊥ , z ) = k B Tλ T f Dp (cid:20) µ ( r ⊥ , z ) k B T (cid:21) , (2) n ( r ⊥ , z ) = 1 λ T f Dn (cid:20) µ ( r ⊥ , z ) k B T (cid:21) , (3)where λ T ≡ p π ~ / ( mk B T ) is the thermal wavelengthat temperature T , f Dp ( t ) and f Dn ( t ) = df Dp ( t ) /dt aretwo universal functions satisfied by a unitary Fermi gasdue to its universal thermodynamics [21–28].It was shown by Bertaina, Pitaevskii and Stringari [8]that, with tight radial confinement, the standard Lan-dau two-fluid hydrodynamic equations defined in threedimensions can be greatly simplified. The key observa-tion is that, as a direct consequence of the dissipationterms (i.e., nonzero viscosity and thermal conductivity),the local fluctuations in temperature and chemical poten-tial become essentially independent on the radial coordi-nates, if we are interested in the low-energy excitationsat frequency ω z ≪ ω ⊥ . Therefore, we could integrate outthe radial degree of freedom in thermodynamic variablesand derive 1D reduced thermodynamics [12]. In particu-lar, we may obtain a reduced Gibbs-Duhem relation, δP = s δT + n δµ, (4)where the variables P , s and n are the radial integralsof their three-dimensional counterparts, namely the lo-cal pressure, entropy density and number density. Forexample, we have [12], P ( z ) ≡ ˆ dr ⊥ πr ⊥ P ( r ⊥ , z ) = 2 π ( k B T ) mω ⊥ λ T f p ( x ) , (5)where x ≡ (cid:18) µ − mω z z (cid:19) /k B T (6) and we have introduced the universal scaling function, f p ( x ) ≡ ˆ ∞ dtf Dp ( x − t ) . (7)All the 1D thermodynamic variables can then be derivedfrom the reduced Gibbs-Duhem relation, such as [12] n ( z ) = (cid:18) ∂P ∂µ (cid:19) T = 2 πk B Tmω ⊥ λ T f n ( x ) , (8) s ( z ) = (cid:18) ∂P ∂T (cid:19) µ = 2 πk B Tmω ⊥ λ T (cid:20) f p ( x ) − xf n ( x ) (cid:21) , (9)where f n ( x ) ≡ df p ( x ) /dx = f Dp ( x ) according to Eq. (7).Furthermore, it is straightforward to obtain the specificheats per particle at constant linear density and pressure[12], ¯ c v ( z ) = T (cid:18) ∂ ¯ s ∂T (cid:19) n = 354 f p ( x ) f n ( x ) − f n ( x ) f ′ n ( x ) , (10) ¯ c p ( z ) = T (cid:18) ∂ ¯ s ∂T (cid:19) P = ¯ c v f p ( x ) f ′ n ( x ) f n ( x ) , (11)where ¯ s ≡ s / ( n k B ) is the entropy per particle and f ′ n ( x ) ≡ df n ( x ) /dx = f Dn ( x ) . It is also easy to check theuniversal relations, n (cid:18) ∂P ∂n (cid:19) ¯ s = 75 P , (12) (cid:18) ∂P ∂s (cid:19) n = 25 T. (13)For the local superfluid density, similarly we express itby a universal function f Ds : n s ( r ⊥ , z ) = 1 λ T f Ds (cid:20) µ ( r ⊥ , z ) k B T (cid:21) . (14)By integrating out the radial coordinate, we find the ex-pression n s ( z ) = ˆ dr ⊥ πr ⊥ n s ( r ⊥ , z ) = 2 πk B Tmω ⊥ λ T f s ( x ) , (15)where the universal scaling function f s ( x ) is defined by, f s ( x ) = ˆ ∞ dtf Ds ( x − t ) . (16)The universal function f Dp ( t ) or f Dn ( t ) of a homo-geneous unitary Fermi gas, where t ≡ µ/k B T , has beenmeasured by the MIT team with high precision [28], bothbelow and above the critical temperature for superfluidphase transition. In Fig. 1, we show the 1D universalscaling functions, calculated by using the experimentalMIT data for f Dn ( t ) , which has been smoothly extrap-olated to both low and high temperature regimes where c v /nk B (b) s / n k B , c v / n k B x = /k B Ts/nk B f p (x) f n (x) df n (x)/dx f ( x ) (a) FIG. 1: (Color online) (a) 1D universal scaling functions f p ( x ) , f n ( x ) and df n ( x ) /dx as a function of the dimension-less variable x = µ/ ( k B T ) . (b) 1D entropy ¯ s = s/ ( nk B ) and specific heat per particle ¯ c v = c v / ( nk B ) as a functionof x = µ/ ( k B T ) . The vertical grey lines indicate the criticalthreshold for superfluidity, x c ≃ . [28]. the behavior of f Dn ( t ) is known [6, 12, 29, 30]. Here-after, without any confusion, we drop the subscript “1”in all the 1D thermodynamic variables.In contrast, the universal function for superfluid den-sity f Ds ( t ) remains elusive [3, 7, 31]. In this work,we will use a phenomenological ansatz for the three-dimensional superfluid fraction ( n s /n ) D = f ( T /T c ) , fol-lowing the strategy used in Ref. [12]. Thus, recallingthat T /T c = [ f Dn ( t ) /f Dn ( t c ≃ . − / , the universalfunction f Ds ( t ) is given by, f Ds ( t ) = f Dn ( t ) f ((cid:20) f Dn ( t ) f Dn ( t c ≃ . (cid:21) − / ) . (17)In the following, we will use the phenomenological super-fluid fraction [12] f (cid:18) TT c (cid:19) = 1 − (cid:18) TT c (cid:19) , (18)unless otherwise stated. III. 1D SIMPLIFIED TWO-FLUIDHYDRODYNAMIC EQUATIONS
Using 1D thermodynamic variables in the standardLandau two-fluid hydrodynamic description [15], it is straightforward to write down the simplified 1D two-fluidhydrodynamic equations. As discussed in the previouswork [1, 3, 4, 12], the solutions of these equations withfrequency ω at temperature T can be derived by minimiz-ing a variational action, which, in terms of displacementfields u s ( z ) and u n ( z ) , is given by, S (2) = 12 ˆ dz (cid:20) mω (cid:0) n s u s + n n u n (cid:1) − (cid:18) ∂µ∂n (cid:19) s ( δn ) − (cid:18) ∂T∂n (cid:19) s δnδs − (cid:18) ∂T∂s (cid:19) n ( δs ) (cid:21) . (19)Here, n s ( z ) and n n ( z ) = n ( z ) − n s ( z ) are the re-duced 1D superfluid and normal-fluid densities. δn ( z ) ≡− ∂ ( n s u s + n n u n ) /∂z and δs ( z ) ≡ − ∂ ( su n ) /∂z are thedensity and entropy fluctuations, respectively. The effectof the weak axial trapping potential V ext ( z ) = mω z z / enters Eq. (19) via the position dependence of the equi-librium thermodynamic variables, within the local den-sity approximation.In superfluid helium, the solutions of the hydrody-namic action Eq. (19) can be well classified as den-sity and temperature waves, which are the pure in-phasemode with u s = u n and the pure out-of-phase mode with n s u s + n n u n = 0 , referred to as first and second sound, re-spectively [15]. We may use the similar characterizationfor a unitary Fermi gas. To this aim, it is useful to rewritethe action Eq. (19) in terms of the displacement fields u a = ( n s u s + n n u n ) /n and u e = u s − u n , since the den-sity and temperature fluctuations can be expressed by δn = − ∂ ( nu a ) /∂z and δT = ( ∂T /∂s ) n ∂ ( sn s u e /n ) /∂z ,respectively. Making use of standard thermodynamicidentities, we find that S (2) = 12 ˆ dz h S ( a ) + S ( e ) + S ( ae ) i , (20)where S ( a ) = m (cid:0) ω − ω z (cid:1) nu a − n (cid:18) ∂P∂n (cid:19) ¯ s (cid:18) ∂u a ∂z (cid:19) , (21) S ( e ) = mω n s n n n u e − (cid:18) ∂T∂s (cid:19) n (cid:20) ∂∂z (cid:16) sn s n u e (cid:17)(cid:21) , (22) S ( ae ) = 2 (cid:18) ∂P∂s (cid:19) n ∂u a ∂z ∂∂z (cid:16) sn s n u e (cid:17) . (23)In the absence of the coupling term S ( ae ) , it is clear thatthe first sound mode, describe by S ( a ) , is the exact solu-tion for pure density oscillations (i.e., u e = 0 or δT = 0 ),while the second sound mode given by S ( e ) correspondsto pure temperature oscillations with u a = 0 or δn = 0 .For a uniform superfluid ( V ext = 0 ), the solutions of S ( a ) and S ( e ) are plane waves of wave vector q with dispersion ω = c q and ω = c q , where mc = ( ∂P/∂n ) ¯ s and mc = k B T ¯ s ¯ c v n s n n . (24) Superfluid L P R a ti o x = /k B TNormal Gas
FIG. 2: (Color online) Landau-Placzek parameter ǫ LP as afunction of x = µ/ ( k B T ) in a highly elongated unitary Fermigas. These first and second sound velocities are the standardresults used to describe superfluid helium [15].In general, the coupling term S ( ae ) is nonzero. Ac-tually, in our case, as ( ∂P/∂s ) n = 2 T / , the first andsecond sound are necessarily coupled at any finite tem-perature. This coupling can be conveniently character-ized by the dimensionless Landau-Placzek (LP) param-eter ǫ LP ≡ γ − [6], where γ ≡ ¯ c p / ¯ c v is the ratio be-tween the specific heats per particle at constant pressureand density. Indeed, with the coupling term, the secondsound velocity c may be well approximated by [6, 12] mc = k B T ¯ s ¯ c p n s n n , (25)which differs with Eq. (24) by a factor of γ = ¯ c p / ¯ c v .Thus, the LP ratio is a useful parameter to estimate thecoupling between first and second sound.In superfluid helium, ¯ c p ≃ ¯ c v or ǫ LP ≃ , indicatingthat the first and second sound are well decoupled. For aunitary Fermi gas in highly elongated traps, we have cal-culated the LP ratio using the 1D thermodynamic data.As shown in Fig. 2, the ratio is less than / in the wholesuperfluid phase. Therefore, similar to superfluid liquidhelium, the solutions of two-fluid hydrodynamic equa-tions for a highly elongated unitary Fermi gas are wellapproximated as weakly coupled first and second soundmodes.We note that, in the presence of axial harmonic traps( V ext = 0 ), the actions S ( a ) and S ( e ) have been solvedanalytically by Hou, Pitaevskii and Stringari using sim-ple variational ansatz [12]. The coupling between firstand second sound due to S ( ae ) has also been briefly com-mented. In this work, by presenting the full variationalcalculations, we will show how the low-lying second soundmodes are affected by the coupling. In particular, we fo-cus on the density fluctuations of second sound modes,which are the key observable in real experiments [9]. A. Variational approach
We assume the following polynomial ansatz for the dis-placement fields: u a ( z ) = N p − X i =0 A i ˜ z i , (26) u e ( z ) = N p − X i =0 B i ˜ z i , (27)where the number of the variational parameters { A i , B i } is N p , and ˜ z ≡ z/Z F is the dimensionless coordinatewith Z F being the Thomas-Fermi radius along the weaklyconfined axial direction. Inserting this ansatz into theaction Eq. (20), the mode frequencies are obtained byminimizing the resulting expression with respect the N p parameters. The precision of our variational calculationscan be improved by increasing the value of N p .In greater detail, it is easy to see that, the action isgiven by S (2) = 12 Λ † S ( ω ) Λ , (28)where Λ ≡ (cid:2) A , B , · · · , A i , B i , · · · , A N p − , B N p − (cid:3) T and S ( ω ) is a N p × N p matrix with block elements, [ S ( ω )] ij ≡ " M ( a ) ij ω − K ( a ) ij − K ( ae ) ij − K ( ae ) ji M ( e ) ij ω − K ( e ) ij . (29)Here, we have introduced the weighted mass moments, M ( a ) ij = m ˆ dz ˜ z i + j n ( z ) , (30) M ( e ) ij = m ˆ dz ˜ z i + j h n s n n n i ( z ) , (31)and the spring constants, K ( a ) ij = 75 ij ˆ dz ˜ z i + j P ( z ) /z + ω z M ( a ) ij , (32) K ( ae ) ij = 2 T i ( i − ˆ dz ˜ z i + j z − h sn s n i ( z ) , (33) K ( e ) ij = ˆ dz (cid:18) ∂T∂s (cid:19) n ∂∂z (cid:18) sn s ˜ z i n (cid:19) ∂∂z (cid:18) sn s ˜ z j n (cid:19) . (34)In deriving K ( a ) ij and K ( ae ) ij , we have used the universalrelations satisfied by the highly elongated unitary Fermigas: n ( ∂P/∂n ) ¯ s = 7 P/ and ( ∂P/∂s ) n = 2 T / . For agiven value of µ/k B T (or T /T F , see below), the weightedmass moments and spring constants can be calculatedby using local thermodynamic variables in Eqs. (5), (8),(9), (10) and (15). We note that the universal scalingfunction for superfluid density is given by Eqs. (16), (17)and (18).In practice, the minimization of the action S (2) isequivalent to solving S ( ω ) Λ = 0 . (35)Once a solution (i.e., the k -th mode frequency ω k andthe coefficient eigenvector Λ k ) is found, we calculate thedensity fluctuation of the mode, by using δn k ( z ) = − N p − X i =0 A ( k ) i ∂∂z (cid:2) n ( z ) ˜ z i (cid:3) . (36) IV. RESULTS AND DISCUSSIONS
We have performed numerical calculations for the num-ber of the variational parameter N p up to , for anygiven temperature T /T F or chemical potential µ/k B T .By recalling that the Fermi temperature T F of a three-dimensional trapped Fermi gas is given by k B T F = ~ (cid:0) N ω ⊥ ω z (cid:1) / (37)and the number of atoms N = ´ dzn ( z ) , these two pa-rameters are related by, TT F = (cid:20) √ π ˆ ∞ dt √ t f n ( µk B T − t ) (cid:21) − / . (38)In the following, we first discuss the decoupled first andsecond sound, in connection with the previous results byHou, Pitaevskii and Stringari [12]. Then, we focus on theeffect of the mode coupling. A. Decoupled first sound
In Ref. [12], the action S ( a ) has been solved by us-ing the ansatz u k =2 a ( z ) = A z + A and u k =3 a ( z ) = A z + A z for the k = 2 and k = 3 first sound modes,respectively. Here, k is the index of a mode and countsthe number of nodes ( = k + 1 ) in its density fluctuation.These are the first two solutions, whose frequency varieswith increasing temperature, due to the non-trivial tem-perature dependence of equation of state [12]. Indeed, itis easy to prove that K ( a ) ij = (cid:20) ij ( i + j −
1) + 1 (cid:21) ω z M ( a ) ij . (39)Thus, if i = 0 or j = 0 , we have K ( a ) ij = ω z M ( a ) ij . To-gether with the fact that K ( ae ) i =0 ,j = 0 or K ( ae ) j =0 ,i = 0 , itis clear that the k = 0 first sound mode with varia-tional ansatz u k =0 a ( z ) = A is an exact solution of thetwo-fluid hydrodynamic equations. In fact, it is pre-cisely the undamped dipole oscillation, with invariantfrequency ω = ω z . Similarly, in the case of i = 1 or (b) k = / z T/T F / z T/T F (a) FIG. 3: (Color online) (a) Temperature dependence of thefirst sound mode frequencies. (b) Enlarged view for the fourthfirst sound mode frequency. The red dashed line shows theresult with the ansatz u a ( z ) = A z + A z , obtained earlierby Hou, Pitaevskii and Stringari [12]. The vertical grey linesshow the critical temperature, T c ≃ . T F . j = 1 , K ( a ) ij = (12 / ω z M ( a ) ij , revealing that the k = 1 first sound mode - the breathing mode - is another exactsolution of the two-fluid hydrodynamic equation with in-variant frequency ω = p / ω z [12, 32].In Fig. 3(a), we report the variational results for firstsound mode frequencies with N p = 24 . In agreementwith the observation by Hou, Pitaevskii and Stringari[12], we find that u k =2 a ( z ) and u k =3 a ( z ) provide excellentvariational ansatz for the third and fourth modes. Asshown in Fig. 3(b), the higher-order correction, for ex-ample, for the k = 3 mode, is at the order of . inrelative and can only be seen in the vicinity of the criti-cal temperature. B. Decoupled second sound
In Fig. 4(a), we present the results for the mode fre-quency of decoupled second sounds. With increasingtemperature, the mode frequency initially increases andreaches a maximum before finally dropping to zero atthe superfluid phase transition [12]. In sharp contrastto first sound modes, the convergence of the polynomialansatz for second sound modes appears to be slow. Forthe lowest dipole second sound mode, our variational ap-proach only converges at N p ≥ , as can be seen fromFig. 4(b). Moreover, compared with the fully convergedresult, a constant displacement field u e (i.e., N p = 1 ) canlead to a relative error as large as close to the super-fluid phase transition. For the higher order second soundmodes, we observe that the convergence of the polyno-mial ansatz becomes even slower. / z (a)(b) N p = 24 N p = 16 N p = 8 N p = 1 l o w e s t / z T/T F FIG. 4: (Color online) (a) Temperature dependence of thesecond sound mode frequencies with N p = 24 . The modefrequencies vanish right at superfluid phase transition. How-ever, our numerical calculations become less accurate slightlybelow the transition and can not produce correctly the zerofrequency exactly at T c . (b) Enlarged view for the lowest sec-ond sound mode frequency. The mode frequency convergeswith increasing the number of variational ansatz N p . Theresult with N p = 1 (green dot-dashed line) corresponds to aconstant displacement field u e [12]. The vertical grey linesshow the critical temperature. / z T/T F FIG. 5: (Color online) Temperature dependence of the fulltwo-fluid hydrodynamic mode frequencies (blue circles). Forcomparison, we show also the mode frequencies of the decou-pled first and second sounds, respectively, by black solid andred dashed lines. The vertical grey line indicates the criticaltemperature. / z T/T F (a) CB full solution first sound second sound(b) T/T F A FIG. 6: (Color online) Blow-up of the full two-fluid hydro-dynamic mode frequencies (blue circles), near the k = 3 firstsound mode (a) and the lowest second sound modes (b). In(a), the experimental data from the Innsbruck experiment [9]are shown by brown solid squares with error bars. C. Full solutions of 1D two-fluid hydrodynamics
We now include the mode coupling term S ( ae ) . In Fig.5, we report the full variational results with N p = 24 by blue circles. For comparison, we show also the de-coupled first and second sound mode frequencies, by us-ing black lines and red dashed lines. As anticipated, atthe qualitative level, the full variational predictions canbe well approximated by the decoupled results, confirm-ing our previous idea that in highly elongated harmonictraps, the solutions of two-fluid hydrodynamics of a uni-tary Fermi gas can indeed be viewed as weakly coupledfirst and second sound modes.At the quantitative level, the first sound mode is nearlyunaffected by the coupling term S ( ae ) . This is evident inFig. 6(a), where we present an enlarged view for the k = 3 first sound mode. The mode frequency has beenpushed up by about . at T ∼ . T F by the coupling.Experimentally, the frequency of the k = 3 first soundmode has been measured very recently [9, 10]. In thefigure, we show the experimental data by solid squareswith error bars. It is known that the data systematicallylie above the variational result with u k =3 a ( z ) [9, 10, 12].Our full variational predictions seem to agree better withthe experimental data. However, the improvement is tooslight to account for the discrepancy.On the other hand, the frequency of second soundmodes is notably pushed down by the coupling, as shownin Fig. 6(b). The maximum correction is up to when the temperature is about . T F . Therefore, for aquantitative prediction of second sound modes over thewhole temperature regime, we must fully solve the cou-pled Landau two-fluid hydrodynamic equations. -1.0 -0.5 0.0 0.5 1.0-2-1012 -1.0 -0.5 0.0 0.5 1.00.00.51.0 A B C (b) n z/Z F k=2 k=3 n s n / n F n tot (a) T = 0.18T F FIG. 7: (Color online) (a) Density distribution (solid line) andsuperfluid density distribution (dashed line) at T = 0 . T F ,in units of the peak linear density of an ideal Fermi gas atthe trap center ( n F ). (b) Density fluctuations of the k = 2 and k = 3 first sound modes (thin lines) and of the threelowest second sound modes (thick lines), at the frequenciesindicated in Fig. 6(b) by A, B and C. The amplitude of thesecond-sound density fluctuations is about / of that of thefirst-sound density fluctuations. D. Density measurement of discretized secondsound modes
The sizable correction in mode frequencies strongly in-dicates that, the density fluctuation δn k ( z ) of a secondsound mode, as a result of its coupling to first soundmodes, could also be significant. In Fig. 7(b), we showthe density fluctuations of the lowest three second soundmodes at the temperature T = 0 . T F (thick lines), inrelative to the density fluctuations of the k = 2 and k = 3 first sound modes (thin lines). The absolute amplitudeof density fluctuations depends on the detailed excitationscheme used in experiments.In a recent investigation by the Innsbruck group [9, 10],the density fluctuations of the k = 2 and k = 3 first soundmodes have been excited and measured, to a reasonableaccuracy. For the detailed resonant excitation scheme,we refer to the experimental papers of Refs. [9] and [10].We anticipate that the similar excitation procedure worksalso for second sound modes, by carefully choosing theposition and size of excitation laser beam, which providebest mode matching. As shown in Fig. 7(b), it is re-markable that the amplitude of the second sound densityfluctuations is at the same order as that of the k = 2 and k = 3 first sound modes, over a useful range of temper- atures. As we shall see below from the analysis of thedensity response function, this implies that the low-lyingsecond sound mode could be observed by looking at itsdensity fluctuation, after a proper excitation.To obtain the density response function, we consideradding a density perturbation of the form δV ( z, t ) = λf ( z ) e − i Ω t to the equilibrium two-fluid equations, where λ is the strength of the perturbation and f ( z ) is a nor-malized shape function (i.e., ´ dzf ( z ) = 1 ). This leadsto a density fluctuation δn ( z, t ) with its amplitude pro-portional to λ and in turn gives the response function χ nn (Ω; f ) defined by λχ nn (Ω; f ) e − i Ω t = ˆ dzf ( z ) δn ( z, t ) . (40)Note that in general the response function depends onthe form of the perturbation f ( z ) . In greater detail, thedensity perturbation generates an additional term δS (2) in the two-fluid hydrodynamic action, S (2) = 12 Λ † Ω S (Ω) Λ Ω + δS (2) , (41)where δS (2) = λ N p − X i =0 ˆ dzf ( z ) ∂∂z (cid:2) n ( z ) ˜ z i (cid:3) A (Ω) i . (42)Here we have used the index “ Ω ” to distinguish Λ Ω from the eigenvector Λ obtained by diagonaliz-ing the action matrix Eq. (29). By introducing avector F = [ f , , f , , · · · , f N p − , T , where f i ≡ ´ dzf ( z ) ∂ [ n ( z ) ˜ z i ] /∂z , we obtain Λ Ω = − λ S − (Ω) F (43)by minimizing the perturbed action. Substituting thisresult into the expression for the density fluctuation Eq.(36), we find that, χ nn (Ω; f ) = F † S − (Ω) F, = X k Z k ω k (cid:18) − ω k −
1Ω + ω k (cid:19) , (44)where Z k = (cid:12)(cid:12) F † Λ k (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ dzf ( z ) δn k ( z ) (cid:12)(cid:12)(cid:12)(cid:12) (45)is the residue of the response function for the k -th col-lective mode with eigenvector Λ k and frequency ω k [33].Using Eq. (45), it is clear that a sizable density fluctua-tion δn k ( z ) of second sound modes - as reported in Fig. 7- implies that a significant residue in the density responsefor second sound can be achieved by optimizing the unit shape function f ( z ) such that f ( z ) ∝ δn k ( z ) . There-fore, discrete second sound could be excited in a similarway as first sound, without imposing a large strength λ for the density perturbation δV ( z, t ) . n s /n = 1 (T/T c ) n s /n = (1 T/T c ) / z T/T c FIG. 8: (Color online) Sensitivity of the lowest second soundmode on the superfluid density. The critical temperature T c ≃ . T F . E. Dependence on the superfluid fraction
We so far restrict ourselves to the phenomenologicalsuperfluid fraction Eq. (18). In Fig. 8, we report thedependence of the lowest second sound frequency on theform of superfluid fraction. The sensitive dependenceindicates that practically the unknown superfluid densityof a unitary Fermi gas could be accurately determined bymeasuring the mode frequency of low-lying second soundmodes.
V. CONCLUSIONS
In conclusion, using a variational approach, we havefully solved the one-dimensional simplified Landau two-fluid hydrodynamic equations, which describe the collec-tive excitations of a unitary Fermi gas in highly elongatedharmonic traps. Resembling the superfluid helium, the solutions are well characterized by weakly coupled firstand second sound modes. Discretized first and secondsound mode frequencies have been accurately predicted.Though the coupling between first and second sound isweak, it still induces significant density fluctuations forsecond sound modes, suggesting that second sound couldbe observed by measuring the density fluctuations afterproperly modulating the axial harmonic trapping poten-tial. Owing to the very high precision in the frequencycalibration, the experimental measurement of discretizedsecond sound mode frequency provides a promising wayto accurately determining the superfluid density of a uni-tary Fermi gas, which remains elusive to date.Ideally, we anticipate that the relative error in the mea-surement of the second sound mode frequency is about . . For the low-lying modes, whose mode frequency issmaller than ω z , the damping rate might be reasonablysmall. By assuming a superfluid fraction in the form, f (cid:18) TT c (cid:19) = (cid:18) − TT c (cid:19) / (cid:20) a + a (cid:18) TT c (cid:19) + · · · (cid:21) , (46)which correctly reproduces the critical behavior near su-perfluid phase transition, we may determine the param-eters { a , a , a , · · · } by fitting the experimental data tothe full variational predictions for the discretized low-lying second sound mode frequencies. Acknowledgments
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