We consider the collisional evolution towards equilibrium of a spatially homogeneous and isotropic phonon gas of a three-dimensional superfluid with a concave acoustic excitation branch, at a non-zero but arbitrarily low temperature
T. Three-phonon collisions
1ϕ↔2ϕ are forbidden by conservation of energy-momentum. Four-phonon collisions
2ϕ→2ϕ of Landau and Khalatnikov lead, after a time
∝T−7, only to a partial thermal equilibrium, a Bose law of non-zero chemical potential for the phonons, because they conserve the total number of phonons. Relaxation towards complete thermochemical equilibrium is therefore ensured by the much slower five-phonon collisions
2ϕ↔3ϕ of Khalatnikov, in a time
∝T−9. Using kinetic equations on the occupation numbers of the phonon modes and explicitly calculating the
2ϕ→3ϕ collisional amplitude with quantum hydrodynamics at low temperature, we determine the corresponding evolution of the fugacity
zϕ of the phonon gas from the non-degenerate regime
zϕ=0+ to complete equilibrium
zϕ=1−. Using the conservation of total energy, we find that the fugacity varies with a non-integer power law
∝t4/5 at short times and an exponential law at long times; the speed of change of entropy, always positive, is asymptotically proportional to the square of the speed of change of fugacity,
(d/dt)Sϕ∝[(d/dt)zϕ]2, as Landau predicted for an arbitrarily slow adiabatic transformation. Our results bring to a close the study initiated by Khalatnikov in 1950 and could be experimentally verified in a gas of cold spin-1/2 fermionic atoms in the sector of the BEC-BCS crossover where the acoustic branch has a concave start, or in superfluid liquid helium-4 at sufficiently high pressure.
We study the phononic collective modes of the pairing field and their corresponding signature in both the order-parameter and density response functions for a superfluid Fermi gas at all temperatures below
Tc in the collisionless regime. The spectra of collective modes are calculated within the Gaussian pair fluctuation approximation. We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a nonperturbative analytic continuation of the pairing field propagator. At low temperature, we recover the known exponential temperature dependence of the damping rate and velocity shift of the Anderson-Bogoliubov branch. In the vicinity of
Tc, we find analytically a weakly damped collective mode whose velocity vanishes with a critical exponent of
1/2, and whose quality factor diverges logarithmically with ${T}_{c}\ensuremath{-}T$, thereby clarifying an existing debate in the literature [Andrianov and Popov, Theor. Math. Phys. 28, 829 (1976); Ohashi and Takada, J. Phys. Soc. Jpn. 66, 2437 (1997)]. A transition between these two phononic branches is visible at intermediary temperatures, particularly in the BCS limit where the phase-phase response function displays two maxima.
We develop a microscopic theory of the dynamic structure factor to describe the Bogoliubov-Anderson-Goldstone phonon mode and its damping rate in a strongly interacting Fermi gas at finite temperature. It is based on a density functional approach---the so-called superfluid local density approximation. The accuracy of the theory is quantitatively examined by comparing the theoretical predictions with recent experimental measurements for the local dynamic structure factor of a nearly homogeneous unitary Fermi gas at low transferred momentum [S. Hoinka et al., Nat. Phys. 13, 943 (2017)], without any free parameters. We calculate the dynamic structure factor as functions of temperature and transferred momentum, and determine the temperature evolution of the phonon damping rate, by considering the dominant decay process of the phonon mode via scatterings off fermionic quasiparticles. These predictions can be confronted with future Bragg scattering experiments on a unitary Fermi gas near the superfluid transition.
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.