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arXiv:1807.10476

Phase spreading and temporal coherence of a pair-condensed Fermi gas at low temperature

Yvan Castin

Published Jul 27, 2018• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1807.10476

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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.
The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times ttt is the sum of a ballistic term ∝t2\propto t^2∝t2 and a diffusive term ∝t\propto t∝t with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of t2t^2t2 scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).
It is generally assumed that a condensate of paired fermions at equilibrium is characterized by a macroscopic wavefunction with a well-defined, immutable phase. In reality, all systems have a finite size and are prepared at non-zero temperature; the condensate has then a finite coherence time, even when the system is isolated. This fundamental effect, crucial for applications using macroscopic coherence, was scarcely studied. Here, we link the coherence time to the condensate phase dynamics, and show using a microscopic theory that the time derivative of the condensate phase operator θ^0\hat{\theta}_0θ^0​ is proportional to a chemical potential operator which includes both the fermionic pair-breaking and the bosonic pair-motion excitation branches. For a given realization of the number of particle NNN and of the energy EEE, the phase evolves at long times as −2μmc(E,N)t/ℏ-2\mu_{mc}(E,N) t/\hbar−2μmc​(E,N)t/ℏ where μmc(E,N)\mu_{mc}(E,N)μmc​(E,N) is the microcanonical chemical potential; fluctuations of NNN and EEE from one realization to the other then lead to a ballistic spreading of the phase and to a Gaussian decay of the temporal coherence function with a characteristic time ∝N1/2\propto N^{1/2}∝N1/2.On the contrary, in the absence of energy and number fluctuations, the decay of the temporal coherence function is exponential with a characteristic time scaling as NNN due to the diffusive motion of θ^0\hat{\theta}_0θ^0​ in the environnement created by the excited modes. We give an explict expression of this characteristic time at low temperature in the case where the bosonic branch is convex and the phonons undergo 2↔12 \leftrightarrow 12↔1 Beliaev-Landau process. Finally, we propose methods to measure each contribution to the coherence time using ultracold atoms.
Due to atomic interactions and dispersion in the total atom number, the order parameter of a pair-condensed Fermi gas experiences a collapse in a time that we derive microscopically. As in the bosonic case, this blurring time depends on the derivative of the gas chemical potential with respect to the atom number and on the variance of that atom number. The result is obtained first using linearized time-dependent Bogoliubov--de Gennes equations, then in the random-phase approximation, and then it is generalized to beyond the mean field. In this framework, we construct and compare two phase operators for the paired fermionic field: The first one, issuing from our study of the dynamics, is the infinitesimal generator of adiabatic translations in the total number of pairs. The second one is the phase operator of the amplitude of the field of pairs on the condensate mode. We explain that these two operators differ due to the dependence of the condensate wave function on the atom number.
Un condensat de paires dans un gaz tridimensionnel de fermions de spin 1/2 isolé, homogène, non polarisé, de taille finie et de température extrêmement basse, mais non nulle, subit au cours du temps un changement de phase θ ˆ ( t ) − θ ˆ ( 0 ) avec une composante aléatoire, ne serait-ce que par couplage aux phonons thermiques du gaz. À l'aide de la seconde relation de Josephson quantique reliant d θ ˆ / d t aux opérateurs nombres d'occupation des modes de phonons et d'équations cinétiques linéarisées donnant l'évolution des fluctuations des nombres d'occupation, nous accédons au comportement de la variance Var [ θ ˆ ( t ) − θ ˆ ( 0 ) ] de ce déphasage aux temps longs devant le temps de collision des phonons. Le cas où la branche de phonons est de départ convexe ressemble à celui du gaz de bosons : les processus collisionnels dominants sont ceux à trois phonons de Beliaev–Landau, si bien que la variance est la somme d'un terme balistique C t 2 et d'un terme diffusif avec retard 2 D ( t − t 0 ) , dont nous donnons les expressions analytiques à la limite thermodynamique. Le cas concave est beaucoup plus exotique. Nous l'analysons aux échelles de temps courtes devant T − 9 , ce qui permet de garder comme seuls processus collisionnels ceux 2 phonons → 2 phonons de Landau–Khalatnikov aux petits angles. Le nombre total de phonons est conservé et les nombres moyens d'occupation des phonons à l'équilibre peuvent admettre un potentiel chimique μ ϕ < 0 supposé isotrope. La variance du déphasage est alors la somme d'un terme balistique C t 2 , d'un terme diffusif 2 D t , de termes sous-sous-dominants exotiques 2 A ( π t ) 1 / 2 + 2 B ln ⁡ ( t 1 / 2 ) et d'un terme constant. Nous obtenons analytiquement l'expression des coefficients C , A et B , ainsi que le comportement dominant divergent de D et du terme constant lorsque μ
While the mechanisms governing the equilibrium properties of quantum gases are today relatively well understood, their non-equilibrium counterparts remain largely unknown. In this thesis, we theoretically explore the non-equilibrium physics of low-dimensional, interacting Bose gases in two representative configurations. In the first one, we consider an isolated Bose superfluid in two dimensions following a quench of the coupling constant. Using a quantum hydrodynamic description combined with a Keldysh field formalism, we derive quantum kinetic equations for the low-energy phononic excitations of the system. This enables us to describe the system's evolution from its short-time (prethermal) coherent dynamics to its long-time thermalization. In particular, we derive and discuss the Landau-Beliaev damping rates both in the standard perturbative approach and within a self-consistent extension of the theory. The results are also discussed in the light of recent experiments. Our second study explores a one-dimensional, periodically-driven quantum system, a Bose gas subjected to periodically-kicked interactions. In the limit of infinitely short kicks, this system was previously shown to exhibit a fast exponential spreading of the wave function. By revisiting this problem for kicks of arbitrary duration, we demonstrate that, in this case, the spreading is not exponential but sub-diffusive at long time. Our findings indicate that the breakdown of exponential spreading occurs at very short times even for extremely short kicks, suggesting that sub-diffusion is more the rule than the exception in that system.