Landau-Khalatnikov phonon damping in strongly interacting Fermi gases

We derive the phonon damping rate due to the four-phonon Landau-Khalatnikov process in low temperature strongly interacting Fermi gases using quantum hydrodynamics, correcting and extending the original calculation of Landau and Khalatnikov [ZhETF, 19 (1949) 637]. Our predictions can be tested in state-of-the-art experiments with cold atomic gases in the collisionless regime. Introduction. – Phonons, sound waves, low energy normal modes or gapless collective excitations are ubiquitous in physics. In uniform weakly-excited quantum many-body systems with short-range interactions, they are described as quasiparticles characterized by a dispersion relation approximately linear at low wavenumber, ωq∼cq with c the speed of sound, and by a damping rate much smaller than the angular eigenfrequency Γq ≪ ωq. Phonon damping plays a central role in transport phenomena such as thermal conduction in dielectric solids, and in hydrodynamic properties such as temperature dependent viscosity and attenuation of sound in liquid helium [1, 2]. It is also crucial for macroscopic coherence properties, since it determines the intrinsic coherence time of bosonic and fermionic gases in the condensed or paircondensed regime [3–5]. In the absence of impurities the damping of low-energy phonons is determined by phononphonon interactions that conserve energy and momentum and it crucially depends on the curvature of the phonon dispersion relation [6,7]. For a concave dispersion relation, 1 ↔ 2 Beliaev-Landau processes involving three phonons are not resonant and the 2 ↔ 2 Landau-Khalatnikov process involving four quasiparticles dominates at low q. In this paper we consider an unpolarized gas of spin-1/2 fermions prepared in thermal equilibrium at a temperature T below the critical temperature, where a macroscopic coherence between pairs of opposite spin fermions appears. Compared to other many-body fermionic systems, atomic gases offer the unique possibility to tune the interaction strength with an external magnetic field close to a so-called Feshbach resonance. This allows experimentalists to explore the crossover between the Bose-Einstein Condensate (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes [8–16]. The dispersion relation of low energy excitations, describing the collective motion of the pair center of mass, has a phononic start at small wavenumbers [17–22] and changes from convex to concave in the BEC-BCS crossover, close to the strongly interacting unitary limit [20, 22]. Therefore, the damping caused by the 2 ↔ 2 processes should be directly observable in cold Fermi gases, contrarily to weakly-interacting Bose gases where the convex Bogoliubov dispersion relation supports Landau-Beliaev damping. On the theoretical side, the original study by Landau and Khalatnikov of the 2 ↔ 2 damping rate [1] is limited to the case where one of the colliding phonons has a small wavenumber compared to the other and it performs as we shall see an unjustified approximation on the coupling amplitude. Here, we give the general expression of the phonon damping rate in the concave dispersion relation regime at low temperature, where it is dominated by the 2 ↔ 2 processes, correcting and extending the original calculation of reference [1]. In the whole paper we restrict to the so-called collisionless regime where the phonon angular frequency times the typical collision time in the gas is much larger than one, ωqτc ≫ 1 [23, 24]. This is in general the case in superfluid gases at low temperature . 1One can estimate τc ≃ 1/Γqth where ~cqth = kBT . Then for excitation frequencies scaling as kBT , as in eq. (18), the condition ωq ≫ Γq, satisfied for a weakly-excited gas, implies ωq ≫ Γqth