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arXiv:1611.01954 · Annalen der Physik

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

Hadrien Kurkjian, Yvan Castin, Alice Sinatra

Published Jul 13, 2017• 32 citations

Abstract

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 q7 to the q5 hydrodynamic prediction, which was never done before in a systematic way.

DOI: 10.1002/andp.201600352

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Physical Review•Aug 15, 1962•7 citations

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 q7 to the q5 hydrodynamic prediction, which was never done before in a systematic way.

A condensate of pairs in an isolated, homogeneous, unpolarised, finite-size spin 1/2 Fermi gas at low nonzero temperature T, undergoes with time a phase change with a random component, due to coupling to the gas thermal phonons. With the quantum second Josephson relation relating the derivative of the condensate phase operator to the phonon occupation numbers, and linearised kinetic equations giving the evolution of occupation number fluctuations, we access the behaviour of the phase change variance at times much longer than the phonon collision time. The case of a convex phonon branch is similar to the Bose gas case: the leading collisional processes are the Beliaev-Landau 3-phonons processes, and the variance is the sum of a ballistic term and of a delayed diffusive term, whose analytical expressions are given in the thermodynamic limit. The concave case is more exotic. It is analysed at time scales

T^{-9}

, allowing one to restrict to 2 phonons

\rightarrow

2 phonons small-angle Landau-Khalatnikov processes. The total number of phonons is conserved and the phonon mean occupation numbers at equilibrium can exhibit a negative chemical potential, assumed isotropic. The phase change variance is then the sum of a ballistic term, of a diffusive term, of exotic subsubleading terms and of a constant term. The analytic expression of some of the corresponding coefficients is obtained, as well as the diverging leading behavior of the other ones when the phonon chemical potential tends to 0. When this chemical potential is 0, the variance sub-ballistic part becomes superdiffusive, with an exponent 5/3 and an exactly-known coefficient. For a nonzero infinitesimal phonon chemical potential, a law is found, interpolating between superdiffusive and diffusive phase spreading. Also new results are obtained on the phonon Landau-Khalatnikov damping rate, in particular at negative phonon chemical potential.

Liquid helium and spin-1/2 cold-atom Fermi gases both exhibit in their superfluid phase two distinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending Landau and Khalatnikov's theory initially developed for helium [ZhETF 19, 637 (1949)], we obtain universal expressions for three- and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low temperature and compare them to those of a pure phononic origin in high-pressure liquid helium and in strongly interacting Fermi gases, paving the way to experimental observations.

It is generally accepted that the phonon gas of a superfluid always enters a weak coupling regime at sufficiently low temperatures, whatever the strength of the interactions between the underlying particles (constitutive of the superfluid). Thus, in this limit, we should always be able to calculate the damping rate of thermal phonons by applying Fermi's golden rule to the Hamiltonian

H_3

of cubic phonon-phonon coupling taken from quantum hydrodynamics, at least in the case of a convex acoustic branch and in the collisionless regime (where the eigenfrequency of the considered phonons remains much greater than the gas thermalization rate). Using the many-body Green's function method, we predict that, unexpectedly, this is not true in two dimensions, contrary to the three-dimensional case. We confirm this prediction with classical phonon-field simulations and a non-perturbative theory in

H_3

, where the fourth order is regularized by hand, giving a complex energy to the virtual phonons of the four-phonon collisional processes. For a weakly interacting fluid and a phonon mode in the long-wavelength limit, we predict a damping rate about three times lower than that of the golden rule.

We calculate the sound velocity and the damping rate of the collective excitations of a 2D fermionic superfluid in a non-perturbative manner. Specifically, we focus on the Anderson–Bogoliubov excitations in the BEC-BCS crossover regime, as these modes have a sound-like dispersion at low momenta. The calculation is performed within the path-integral formalism and the Gaussian pair fluctuation approximation. From the action functional, we obtain the propagator of the collective excitations and determine their dispersion relation by locating the poles of this propagator. We find that there is only one kind of collective excitation, which is stable at T = 0 and has a sound velocity of v F / 2 for all binding energies, i.e., throughout the BEC-BCS crossover. As the temperature is raised, the sound velocity decreases and the damping rate shows a non-monotonous behavior: after an initial increase, close to the critical temperature T C the damping rate decreases again. In general, higher binding energies provide higher damping rates. Finally, we calculate the response functions and propose that they can be used as another way to determine the sound velocity.

The stability of phonons in Bose systems is considered. It is noted that the phonons in helium II are stable against decay into two or more phonons, in contrast to the low-density Bose gas where the lifetime for decay of one phonon into two is finite and has been calculated by several authors. The consequences of phonon stability for allowable mechanisms of phonon-phonon interaction are considered. It is concluded that virtual three-phonon processes may still occur although energy-conserving three-phonon processes cannot take place. The inclusion of such virtual processes makes it plausible that the interactions between excitations can be weak, even though the linewidths measured in neutron scattering are larger than the line shifts. The quantum hydrodynamics of Landau and Khalatnikov including nonlinear terms is considered as a calculational model. In this theory the stability of the phonons is indeterminate and must be legislated. If the phonons are assumed unstable against decay, the phonon lifetime in the low-density and long-wavelength limit can be calculated, and is in agreement with the result for the atomic theory of the low-density Bose gas. This lends considerable plausibility to the quantitative validity of the Landau-Khalatnikov calculation of phonon-phonon scattering in helium II via an almost energy-conserving intermediate one-phonon state. The paper concludes by comparing the results of an idealized neutron scattering experiment as calculated from the Feynman-Cohen point of view, and from the point of view of the theory of the low-density Bose gas.