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arXiv:cond-mat/0508215 · Physical Review A

Dynamical selection in developing fermionic pairing

Roman A. Barankov, Leonid S Levitov

Published Mar 17, 2006• 35 citations
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Abstract

We consider the development of pairing in an ideal Fermi gas in the presence of a time-dependent BCS interaction. The pairing amplitude time evolution is controlled by the interplay of linear instability of the unpaired state and nonlinear interactions which limit the growth. This results in oscillatory time dependence with predictable characteristics, selecting periodic soliton trains of a specific form, described by a Jacobi elliptic function. While the parameters of the soliton train, such as the period, amplitude, and time lag, fluctuate among different realizations, the elliptic function form remains robust. The parameter fluctuations are accounted for by the randomness of particle distribution in the initial unpaired state.

DOI
10.1103/physreva.73.033614

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We consider the development of pairing in an ideal Fermi gas in the presence of a time-dependent BCS interaction. The pairing amplitude time evolution is controlled by the interplay of linear instability of the unpaired state and nonlinear interactions which limit the growth. This results in oscillatory time dependence with predictable characteristics, selecting periodic soliton trains of a specific form, described by a Jacobi elliptic function. While the parameters of the soliton train, such as the period, amplitude, and time lag, fluctuate among different realizations, the elliptic function form remains robust. The parameter fluctuations are accounted for by the randomness of particle distribution in the initial unpaired state.
Motivated by recent efforts to achieve cold fermions pairing, we study the nonadiabatic regime of the Bardeen-Cooper-Schrieffer state formation. After the interaction is turned on, at times shorter than the quasiparticle energy relaxation time, the system oscillates between the superfluid and normal state. The collective nonlinear evolution of the BCS-Bogoliubov amplitudes u(p), v(p), along with the pairing function Delta, is shown to be an integrable dynamical problem which admits single soliton and soliton train solitons. We interpret the collective oscillations as Bloch precession of Anderson pseudospins, where each soliton causes a pseudospin 2pi Rabi rotation.
We study the time evolution of a system of fermions with pairing interactions at a finite temperature. The dynamics is triggered by an abrupt increase in the BCS coupling constant. We show that if initially the fermions are in a normal phase, the amplitude of the BCS order parameter averaged over the Boltzmann distribution of initial states exhibits damped oscillations with a relatively short decay time. The latter is determined by the temperature, the single-particle level spacing, and the ground-state value of the BCS gap for the new coupling. In contrast, the decay is essentially absent when the system was in a superfluid phase before the coupling increase.
We characterize the collective modes of a soliton train in a quasi-one-dimensional Fermi superfluid, using a mean-field formalism. In addition to the expected Goldstone and Higgs modes, we find novel long-lived gapped modes associated with oscillations of the soliton cores. The soliton train has an instability that depends strongly on the interaction strength and the spacing of solitons. It can be stabilized by filling each soliton with an unpaired fermion, thus forming a commensurate Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We find that such a state is always dynamically stable, which paves the way for realizing long-lived FFLO states in experiments via phase imprinting.
Dark solitons in superfluid Bose gases decay through the snake instability mechanism, unless they are strongly confined. Recent experiments in superfluid Fermi gases have also interpreted soliton decay via this mechanism. However, we show using both an effective field numerical simulation and a perturbative analysis that there is a qualitative difference between soliton decay in the BEC and BCS regimes. On the BEC side of the interaction domain, the characteristic snaking deformations are induced by fluctuations of the amplitude of the order parameter, while on the BCS side, fluctuations of the phase destroy the soliton core through the formation of local Josephson currents without the presence of snaking deformations. The latter mechanism is qualitatively different from the snaking instability and this difference should be experimentally detectable.
We describe the dynamics of two component non-interacting ultracold Fermions which are initially in thermal equilibrium and undergo a rapid quench to either the repulsive or attractive side of a Feshbach resonance. The short time dynamics is dominated by the exponentially growing collective modes. We study the Stoner instability and formation of ferromagnetic textures on the repulsive side, and the pairing instability towards BCS or FFLO-like states (determined by the population imbalance) on the attractive side. In each case, we evaluate the growth rate of unstable modes and predict the typical lengthscale of textures to be formed.
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.