Counter-flow instability of a quantum mixture of two superfluids
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Counter-flow instability of a quantum mixture of two superfluids
Marta Abad , Alessio Recati , Sandro Stringari , and Fr´ed´eric Chevy INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, I-38123 Povo, Italy Laboratoire Kastler-Brossel, ´Ecole Normale Sup´erieure, CNRS and UPMC, 24 rue Lhomond, 75005 Paris, Francethe date of receipt and acceptance should be inserted later
Abstract.
We study the instability of a mixture of two interacting counter-flowing superfluids. For a homo-geneous system, we show that superfluid hydrodynamics leads to the existence of a dynamical instabilityat a critical value of the relative velocity v cr . When the interspecies coupling is small the critical valueapproaches the value v cr = c + c , given by the sum of the sound velocities of the two uncoupled super-fluids, in agreement with the recent prediction of [1] based on Landau’s argument. The crucial dependenceof the critical velocity on the interspecies coupling is explicitly discussed. Our results agree with previouspredictions for weakly interacting Bose-Bose mixtures and applies to Bose-Fermi superfluid mixtures aswell. Results for the stability of transversally trapped mixtures are also presented. PACS.
Recent years have witnessed the advent of ultracoldgases as a unique playground for the study of quantummany-body phenomena [2]. The wide tunability of thesesystems allowed experimentalists to engineer a broad rangeof physical situations and paved the way to the experimen-tal study of outstanding problems in condensed matterphysics or astrophysics.An important direction of research concerns the studyof quantum mixtures of two superfluids and the onsetof their instability when they move against each other(counterflow instability). In the case of weakly interact-ing Bose-Einstein condensates the dynamical counterflowinstability has been theoretically studied in several works[3,4,5,6,7,8] and has been experimentally seen to lead tothe formation of soliton trains [9,10]. The recent resultspresented in [11] for mixtures of Bosons and Fermionsdemonstrated the possibility of reaching double superflu-idity with atomic gases belonging to different statistics,opening a new scenario for the physics of superfluid mix-tures. The study of the collective excitations of the mix-ture raised, in particular, the question of the critical ve-locity of their relative superfluid motion.In [1], Castin et al. proposed a generalization of Lan-dau’s mechanism applied to a mixture with vanishinglysmall interspecies coupling, where the relative flow decaysby shedding pairs of elementary excitations in the twocomponents of the mixture. When the excitation spectraof the two superfluids are dominated by acoustic modes,the critical velocity v cr is the sum of their sound velocities: v cr = c + c . A similar result was obtained in [12] by con-sidering the lifetime of the quasi-particles of the system.This result for the critical velocity differs from the usualLandau’s prescription v cr = min( c , c ) holding for two independent superfluids. Actually the latter result holdsonly if the superfluids can exchange momentum with anexternal wall or with a moving impurity. Instead, in [1],momentum conservation is ensured by the excitation oftwo phonons with opposite momenta.In the present article, we study the stability of a su-perfluid counter-flow in the hydrodynamic approximation,discussing in an explicit way the role of the interspeciesinteraction, going beyond the small interspecies couplinglimit discussed in [1].Consider a mixture of two superfluids labeled by theindices α = 1 ,
2. In the hydrodynamic approximation, theequations of motion at zero temperature can be obtainedstarting from the energy functional E [ n , n , φ , φ ] = Z (cid:20) m n ( ∇ φ ) + 12 m n ( ∇ φ ) ++ U n + U n + e ( n ) + e ( n ) + e ( n , n ) (cid:21) d r (1)where m α is the mass of the particles of the superfluid α =1 , U α ( r ) their trapping potential. The terms e ( n ), e ( n ) and e ( n , n ) are the energy densities correspond-ing, respectively, to the interactions among particles 1,particles 2 and between particles 1 and 2. The velocitypotential, φ α , is related to the superfluid velocity of eachfluid as v α = ∇ φ α . Choice (1) for the energy functionalignores the possible coupling between the velocity fieldsof the two fluids, which lead to physical phenomena suchas the Andreev-Bashkin effect [13]. This effect is howeverexpected to be small in dilute gases. Marta Abad et al.: Counter-flow instability of a quantum mixture of two superfluids
The equations for the density and velocity fields, n α ( r , t )and v α ( r , t ), can be derived by taking the velocity poten-tial and the density as conjugate variables obeying Hamil-ton’s equations, ∂n α /∂t = δE/δφ α and ∂φ α /∂t = − δE/δn α .They take the form m α (cid:0) ∂ t v α + ∇ v α / (cid:1) = − ∇ ( U α + µ α ) (2) ∂ t n α + ∇ ( n α v α ) = 0 , (3)where µ α ( n , n ) is the chemical potential of each speciesat rest, given by µ ( n , n ) = ∂e ( n ) ∂n + ∂e ( n , n ) ∂n (4) µ ( n , n ) = ∂e ( n ) ∂n + ∂e ( n , n ) ∂n (5)We address first the case of a homogeneous system( U α = 0). The stationary solutions correspond to uniformdensities and velocity fields, with n α = n (0) α and v α = v (0) α .We consider the general case v (0)1 = v (0)2 corresponding toa non vanishing counter-flow velocity v = v (0)1 − v (0)2 . Ifthe system is weakly perturbed with respect to the station-ary configuration, we can look for solutions in the linearregime: n α ( r , t ) = n (0) α + n (1) α e i ( q · r − ωt ) (6) v α ( r , t ) = v (0) α + w α e i ( q · r − ωt ) . (7)Expanding the hydrodynamic equations to first order inthe perturbation yields − im α (cid:16) ω − q v (0) α · u (cid:17) w α = − i q X β ∂µ α ∂n β n (1) β (8) i (cid:16) ω − q v (0) α · u (cid:17) n (1) α = i q (cid:16) n (0) α w α (cid:17) . (9)with u = q /q a unitary vector in the direction of thequasimomentum q . Using Eq. (8) to eliminate w α fromEq. (9) we obtain m α ( ω − q v (0) α · u ) n (1) α = q n (0) α X β ∂µ α ∂n β n (1) β (10)By writing the frequency ω of the solution as ω = cq , wefind that the sound velocity c should satisfy the condition h ( c − v (0)1 · u ) − c i h ( c − v (0)2 · u ) − c i = c . (11)In the above equation we have introduced the quantities c α = n (0) α m α ∂µ α ∂n α (12) c = s n (0)1 n (0)2 m m ∂µ ∂n ∂µ ∂n (13)Notice that the hydrodynamic formalism is valid in thelong wavelength limit, that is when q →
0, and for sound-like excitations. In the following we will always consider -2 -1 0 1 2c-202 P ( c ) c c c c c c* -2 -1 0 1 2c-202 P ( c ) c c c c c Fig. 1.
Left: Typical graph of P ( c ). c i, ± are the roots of P and c ∗ is the abscissa of the maximum of P . For c < P ( c ∗ )there are four real solutions of P ( c ) = c , hence no imaginarysolution. Right: For c > P ( c ∗ ), then the equation P ( c ) = c has two complex roots, leading to an instability. The instabilitythreshold condition is therefore given by P ( c ∗ ) = c . that the system is dynamically stable at rest, that is c Critical relative velocity, v/c , as a function of theinterfluid coupling ∆ , for various values of c /c . Having understood the behavior of the solutions ofthe hydrodynamic equations for small values of the inter-species coupling and c we can now investigate the effectfor arbitrary values of c and c by solving numericallythe relevant Eq. (11). To this purpose it is convenient torewrite (11) in dimensionless units, dividing it by c c andexpressing all the velocities in units of c : c → ˜ c = c/c , c → ˜ c = c /c , v (0) α → ˜ v (0) α = v (0) α /c . The solutions for˜ c then depend on ˜ v (0) α , ˜ c and ∆ . As justified above, theonset of dynamical instability depends on ˜ v = ˜ v (0)1 − ˜ v (0)2 as well as ˜ c and ∆ .In Fig. 2 we show the results for the critical velocity,˜ v cr , associated with the emergence of a complex solutionof ˜ c as a function of ∆ , for different choices of the ratio˜ c . Some comments are in order here.i) For ∆ → v cr = c + c found by Castin et al. and discussedabove. In general we find that the critical value v cr givenby the condition Eq. (19) for small values of ∆ provides agood approximation to the exact solution up to ∆ ∼ . ∆ → c c → c ), corresponding to the onset of dynamic instability inthe absence of moving fluids discussed above.iii) In addition to the two previous cases, a simple ana-lytical solution can be obtained for c = c ≡ c for whichwe find c ( v ) = 12 r ( v · u ) + 4 c ± q ( v · u ) c + c . (22)The condition for dynamical instability is | ( v · u ) − c | =4 c , which leads to the critical velocities v ± cr = 2 c √ ± ∆ . (23)The system is thus dynamically unstable in the range v − cr < | v · u | < v + cr , whereas remains dynamically stableoutside this region.Let us make a remark here on the upper bound of theinstability zone, v + cr . This result emerges naturally in hy-drodynamics where the dispersion is linear in q . However, Marta Abad et al.: Counter-flow instability of a quantum mixture of two superfluids inclusion of dispersive terms in the excitation spectrumcan give rise to a different scenario. For instance, in thecontext of bosonic mixtures, where the excitation spec-trum is known for all q , it can be shown that for | v · u | > v + cr the dynamical instability arises at finite values of q [3,8].Therefore even if two superfluids moving at a relative ve-locity | v · u | > v + cr are dynamically stable against the exci-tation of long wavelength phonons, they can be unstableagainst the excitation of shorter wavelength modes.Let us finally comment that our results apply to bothBose-Bose and Bose-Fermi mixtures (the case of Fermi-Fermi mixtures will not be discussed here), in the weakas well as in the strong coupling limit. Let us discuss theexplicit form for the various quantities entering the calcu-lation of the critical velocity in the two cases:a) Weakly interacting Bose-Bose gases . In thiscase the sound velocities are given by the Bogoliubov form m α c α = g α n α , with the coupling constant fixed by thes-wave scattering length according to g α = 4 π ~ a α /m α .The dimensionless interspecies coupling ∆ is given by thedensity-independent result ∆ = g / √ g g , with g =2 π ~ a /m r , where we have introduced the reduced mass m − r = m − + m − . Our results agree with those previ-ously reported (see [3,4,5,6,7,8,10]) in the limit q → Dilute Bose gas interacting with a unitaryFermi gas . In this case the sound velocity of the Fermigas (hereafter called c ) is given by c = v F p ξ/ 3, with v F the Fermi velocity and ξ the Bertsch parameter [14],while the constant ∆ takes the density-dependent form ∆ = (cid:18) π (cid:19) / ( m + m ) ξm m a ( n ) / a . (24)In the case of Li- Li superfluid mixtures, recently ex-perimentally implemented in [11], the interaction param-eters can be tuned in a rather flexible way thanks tothe occurrence of various Feshbach resonances, so thatthese mixtures are excellent candidates to explore in de-tail the mechanisms of dynamic instability discussed inthe present paper. Let us also notice that the results pre-sented in this work are based on the assumption that thetwo fluids are miscible at rest. According to Viverit et al. [15], for positive values of the interspecies coupling con-stant g , phase separation in a Bose-Fermi mixture canactually occur before the onset ∆ = 1 of dynamical insta-bility.Let’s now turn to the case of a transversally trappedsystem where the external potentials U α depend on thetransverse coordinate ρ = ( x, y ). The stationary densityprofiles are then given by the Local Density Approxima-tion (LDA) condition µ α ( n (0)1 ( ρ ) , n (0)2 ( ρ )) + U α ( ρ ) = µ (0) α .From the LDA results for the 3D density one can calcu-late the double integrated (1D) densities ¯ n α = R d ρ n (0) α of the two fluids, in terms of the chemical potentials µ and µ . The knowledge of the equations of state µ (¯ n , ¯ n )and µ (¯ n , ¯ n ) then permits to derive the 1D hydrody-namic equations, following the same procedure employedin the first part of the paper, provided the wave vector q of the sound wave is smaller than the radial size of the mixtures. The simplest case is when the coupling term isvanishingly small. In this case one finds that the systembecomes unstable for v cr = ¯ c + ¯ c , in analogy with the 3Dresult, where m α ¯ c α = ∂µ α /∂ ¯ n α are the sound velocitiesof the two independent fluids in the 1D-like configura-tions. In the Bose case one finds µ Bose ∝ ¯ n / and hence¯ c Bose = 1 / √ c Bose [16,17]. In the unitary Fermi gas oneinstead finds µ Fermi ∝ ¯ n / yielding ¯ c Fermi = p / c Fermi [18,19]. In the above equations c Bose and c Fermi are thesound velocities calculated, for a uniform Bose and Fermigas, respectively, at the central density. In the experimentreported in [11], ∆ ≃ − . The effects of the interactionsare then negligible and the measured critical velocity isindeed very close to the prediction v cr = ¯ c Fermi + ¯ c Bose . Itis also worth noticing that our results apply to any twospecies Luttinger liquid, in particular to spin-1/2 Fermigases confined to one-dimension. In the latter case therelative velocity breaks the spin-charge separation of thesystem and according to our result produces a dynamicalinstability if it is large enough.In conclusion, using a hydrodynamic approach, we havederived explicit results for the emergence of dynamic in-stability in a mixture of two superfluids, by calculating thecritical velocity associated with the relative motion of thetwo components of the mixture. Our results hold also forfluids belonging to different quantum statistics and gener-alize previous results derived for mixtures of Bose-Einsteincondensates. For relative velocities larger than the criticalvalue the solutions of the hydrodynamic equations exhibitan imaginary component in the sound velocity which is re-sponsible for decay processes. 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