Three‐Phonon and Four‐Phonon Interaction Processes in a Pair‐Condensed Fermi Gas

Abstract

We study the interactions among phonons and the phonon lifetime in a pair-condensed Fermi gas in the BEC-BCS crossover in the collisionless regime. To compute the phonon-phonon coupling amplitudes we use a microscopic model based on a generalized BCS Ansatz including moving pairs, which allows for a systematic expansion around the mean field BCS approximation of the ground state. We show that the quantum hydrodynamic expression of the amplitudes obtained by Landau and Khalatnikov apply only on the energy shell, that is for resonant processes that conserve energy. The microscopic model yields the same excitation spectrum as the Random Phase Approximation, with a linear (phononic) start and a concavity at low wave number that changes from upwards to downwards in the BEC-BCS crossover. When the concavity of the dispersion relation is upwards at low wave number, the leading damping mechanism at low temperature is the Beliaev-Landau process 2 phonons ↔ 1 phonon while, when the concavity is downwards, it is the Landau-Khalatnikov process 2 phonons ↔ 2 phonons. In both cases, by rescaling the wave vectors to absorb the dependence on the interaction strength, we obtain a universal formula for the damping rate. This universal formula corrects and extends the original analytic results of Landau and Khalatnikov [ZhETF 19, 637 (1949)] for the 2 ↔ 2 processes in the downward concavity case. In the upward concavity case, for the Beliaev 1↔ 2 process for the unitary gas at zero temperature, we calculate the damping rate of an excitation with wave number q including the first correction proportional to q 7 to the q 5 hydrodynamic prediction, which was never done before in a systematic way.

Absorption and emission of a collective excitation by a fermionic quasiparticle in a Fermi superfluid

“…We temporarily put these source terms aside to focus on the homogeneous system inŝ k q andŷ k q . This system coincides exactly with the one obtained by [22] in a semi-classical approach [30]. We briefly recall the results obtained by this reference on the collective excitation branch.…”

“…It can be measured by Bragg spectroscopy, a technique recently applied to Fermi gases [18]. With the microscopic expression of the coupling between three bosonic excitations obtained by [22] and the present result, all three-body couplings between quasiparticles are known microscopically, which paves the way to a study of the dressing of quasiparticles by the interactions among them.…”

“…Among the existing approaches to determine the non-zero temperature equation of state, there is a phenomenological model [20] based on the thermal occupation of the bosonic and fermionic branch, and, at unitarity, effective field theories exploiting the extra symmetries of the system [5,21]. Except for the coupling between three bosonic excitations, obtained in a microscopic semi-classical approach [22], all existing descriptions of the interactions among quasiparticles rely on quantum hydrodynamics [4], a low energy effective theory limited to the leading order in temperature. In this framework, the resonant three-and four-body couplings between phonons [22,23], and between phonons and fermionic quasiparticles [24], were obtained.…”

“…Except for the coupling between three bosonic excitations, obtained in a microscopic semi-classical approach [22], all existing descriptions of the interactions among quasiparticles rely on quantum hydrodynamics [4], a low energy effective theory limited to the leading order in temperature. In this framework, the resonant three-and four-body couplings between phonons [22,23], and between phonons and fermionic quasiparticles [24], were obtained.…”

“…We present a straightforward application: the computation of the damping rate of the collective excitations of arbitrary wave number due to their inelastic coupling to the fermionic quasiparticles in the collisionless regime. This rate is the main contribution to the lifetime of the collective excitations at the usual experimental temperatures [24] when the concave collective branch [27] forbids the Beliaev-Landau damping mechanism [22]. It can be measured by Bragg spectroscopy, a technique recently applied to Fermi gases [18].…”