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arXiv:2007.10678

Random walk of a massive quasiparticle in the phonon gas of an ultralow temperature superfluid

Yvan Castin

Published Jul 21, 2020• 1 citation
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Abstract

We consider a 3D homogeneous superfluid at low temperature $T$ with 2 types of excitations, gapless phonons with a linear dispersion relation at low wavenumber, and gapped quasiparticles with a quadratic dispersion relation around extrema. We calculate the scattering amplitude of a phonon on a quasiparticle to leading order in $T$ for all subsonic quasiparticle velocities, with a $S$-matrix formalism between exact asymptotic states dressed by virtual phonons. We then characterize the erratic motion of the quasiparticle in the superfluid due to its unceasing collisions with thermal phonons through mean force $F(k)$, longitudinal and transverse $k$-dependent momentum diffusion coefficients, and spatial diffusion coefficient. At the minimum location $k_0$ of the dispersion relation, where the velocity vanishes, $F(k)$ varies linearly with velocity with an isotropic friction coefficient; if $k_0=0$, the momentum diffusion is also isotropic and $F(k_0)=0$; if $k_0>0$, it is not, and $F(k_0)$ is nonzero but subleading with respect to friction by one order in $T$. The velocity time correlation function, whose integral is the spatial diffusion coefficient, decays with the mean velocity damping rate if $k_0=0$; if $k_0>0$, it has a second exponential component, with an amplitude and a damping rate lower by a factor $\propto T$ (it is the velocity direction thermalization rate). We also characterize force and momentum diffusion close to the stability domain sonic edge. Our general expressions are expected to be exact to leading order in $T$. We illustrate them in the BCS approximation, for a fermionic quasiparticle (an unpaired fermion) in a superfluid of spin 1/2 fermions, realisable with cold atoms in flat bottom traps. We also refute the statement of Lerch, Bartosch and Kopietz (2008), that there would be no fermionic quasiparticle in such a superfluid.

DOI
10.48550/arxiv.2007.10678

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We consider a 3D homogeneous superfluid at low temperature TTT with 2 types of excitations, gapless phonons with a linear dispersion relation at low wavenumber, and gapped quasiparticles with a quadratic dispersion relation around extrema. We calculate the scattering amplitude of a phonon on a quasiparticle to leading order in TTT for all subsonic quasiparticle velocities, with a SSS-matrix formalism between exact asymptotic states dressed by virtual phonons. We then characterize the erratic motion of the quasiparticle in the superfluid due to its unceasing collisions with thermal phonons through mean force F(k)F(k)F(k), longitudinal and transverse kkk-dependent momentum diffusion coefficients, and spatial diffusion coefficient. At the minimum location k0k_0k0​ of the dispersion relation, where the velocity vanishes, F(k)F(k)F(k) varies linearly with velocity with an isotropic friction coefficient; if k0=0k_0=0k0​=0, the momentum diffusion is also isotropic and F(k0)=0F(k_0)=0F(k0​)=0; if k0>0k_0>0k0​>0, it is not, and F(k0)F(k_0)F(k0​) is nonzero but subleading with respect to friction by one order in TTT. The velocity time correlation function, whose integral is the spatial diffusion coefficient, decays with the mean velocity damping rate if k0=0k_0=0k0​=0; if k0>0k_0>0k0​>0, it has a second exponential component, with an amplitude and a damping rate lower by a factor ∝T\propto T∝T (it is the velocity direction thermalization rate). We also characterize force and momentum diffusion close to the stability domain sonic edge. Our general expressions are expected to be exact to leading order in TTT. We illustrate them in the BCS approximation, for a fermionic quasiparticle (an unpaired fermion) in a superfluid of spin 1/2 fermions, realisable with cold atoms in flat bottom traps. We also refute the statement of Lerch, Bartosch and Kopietz (2008), that there would be no fermionic quasiparticle in such a superfluid.
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{T}_{c}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{T}_{c}Tc​, we find analytically a weakly damped collective mode whose velocity vanishes with a critical exponent of 1/21/21/2, 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 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.
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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−9T^{-9}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.
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Excitations in many areas of condensed matter physics have dispersion relations which are very unlike those of simple newtonian objects. An unusual energy-momentum relation should lead to unconventional dynamics. Unfortunately most excitations are trapped in lattices or are otherwise inaccessible. One place where dynamic experiments are possible is in the superfluids, since here the excitations propagate freely in the superfluid condensate which we can regard as a mechanical vacuum. We can now cool superfluid 3He to a temperature where the excitation mean free paths are long enough for dynamic experiments to be feasible, which implies temperatures of the order of 100μK. Since the dispersion relation is tied to the rest frame of the condensate, it will be distorted by condensate motion, providing us with a way of influencing the dispersion curve. For example, a container of superfluid with a velocity gradient will have a range of excitation dispersion curves depending on the local velocity of the condensate. This means that the density of states available to the excitations will vary with position. We discuss the dynamical implications of this in terms of the behaviour of a vibrating wire resonator in the liquid, which shows unusual and often counter-intuitive behaviour. Among other things, for example, at a low enough temperature the force on a moving object in the excitation gas becomes independent of the velocity.