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arXiv:1709.00862 · New Journal of Physics

Dark soliton collisions in superfluid Fermi gases

Wout Van Alphen, G Lombardi, S N Klimin, J Tempere

Published May 4, 2018• 7 citations
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Abstract

In this work dark soliton collisions in a one-dimensional superfluid Fermi gas are studied across the BEC-BCS crossover by means of a recently developed finite-temperature effective field theory (2015 Eur. Phys. J. B 88 122). The evolution of two counter-propagating solitons is simulated numerically based on the theory’s nonlinear equation of motion for the pair field. The resulting collisions are observed to introduce a spatial shift into the trajectories of the solitons. The magnitude of this shift is calculated and studied in different conditions of temperature and spin-imbalance. When moving away from the BEC-regime, the collisions are found to become inelastic, emitting the lost energy in the form of small-amplitude density oscillations. This inelasticity is quantified and its behavior analyzed and compared to the results of other works. The dispersion relation of the density oscillations is calculated and is demonstrated to show a good agreement with the spectrum of collective excitations of the superfluid.

DOI
10.1088/1367-2630/aac2a9

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In this work dark soliton collisions in a one-dimensional superfluid Fermi gas are studied across the BEC-BCS crossover by means of a recently developed finite-temperature effective field theory (2015 Eur. Phys. J. B 88 122). The evolution of two counter-propagating solitons is simulated numerically based on the theory’s nonlinear equation of motion for the pair field. The resulting collisions are observed to introduce a spatial shift into the trajectories of the solitons. The magnitude of this shift is calculated and studied in different conditions of temperature and spin-imbalance. When moving away from the BEC-regime, the collisions are found to become inelastic, emitting the lost energy in the form of small-amplitude density oscillations. This inelasticity is quantified and its behavior analyzed and compared to the results of other works. The dispersion relation of the density oscillations is calculated and is demonstrated to show a good agreement with the spectrum of collective excitations of the superfluid.
We study, both analytically and numerically, the formation and propagation of dark solitons in a superfluid Fermi gas in the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC). Starting from a superfluid order-parameter equation, we derive a Korteweg--de Vries equation for weak nonlinear excitations under quasi-one-dimensional and long-wavelength approximations. We present dark soliton solutions valid for both BCS and BEC limits and also for the crossover, and show that dark solitons in different superfluid regimes possess different features. Particularly, a dark soliton in the BCS (BEC) regime has larger (smaller) propagating velocity and smaller (larger) spatial width. Upon moving to the boundary of the condensate, it generally decelerates and generates small radiations, which display different behavior in different superfluid regimes. We study also a head-on collision between two dark solitons and demonstrate that the phase shift due to the collision changes nonmonotonically along the BCS-BEC crossover. All analytical results are checked by numerical simulations, and good agreement between them is found.
We use a finite-temperature effective field theory recently developed for superfluid Fermi gases to investigate the properties of dark solitons in these superfluids. Our approach provides an analytic solution for the dip in the order parameter and the phase profile across the soliton, which can be compared with results obtained in the framework of the Bogoliubov--de Gennes equations. We present results in the whole range of the BCS-BEC crossover, for arbitrary temperatures and taking into account Gaussian fluctuations about the saddle point. The obtained analytic solutions yield an exact energy-momentum relation for a dark soliton showing that the soliton in a Fermi gas behaves like a classical particle even at nonzero temperatures. The spatial profile of the pair field and for the parameters of state for the soliton are analytically studied.
In this paper the properties of dark solitons in superfluid Fermi gases with spin imbalance are studied by means of a recently developed effective field theory [Eur. Phys. J. B 88, 122 (2015)] suitable to describe the BEC-BCS crossover in ultracold gases in an extended range of temperatures as compared to the usual Ginzburg--Landau treatments. The spatial profiles for the total density and for the density of the excess-spin component, and the changes of their properties across the BEC-BCS crossover are examined in different conditions of temperature and imbalance. The presence of population imbalance is shown to strongly affect the structure of the soliton excitation by filling its core with unpaired atoms. This in turn influences the dynamical properties of the soliton since the additional particles in the core have to be dragged along thus altering the effective mass.
We study soliton collisions and the decay of solitons into sound in superfluid Fermi gases across the Bose–Einstein condensate to Bardeen–Cooper–Schrieffer (BEC–BCS) crossover by performing numerical simulations of the time-dependent Bogoliubov–de Gennes equations. This decay process occurs when the solitons are accelerated to the bulk pair-breaking speed by an external potential. A similar decay process may occur when solitons are accelerated by an inelastic collision with another soliton. We find that soliton collisions become increasingly inelastic as we move from the BEC to the BCS regime, and the excess energy is converted into sound. We interpret this effect as being due to the evolution of Andreev bound states localized within the soliton.
In the present article the snake instability mechanism for dark solitons in superfluid Fermi gases is studied in the context of a recently developed effective field theory [S. N. Klimin et al., Eur. Phys. J. B 88, 122 (2015)]. This theoretical treatment has proven to be suitable to study stable dark solitons in quasi-one-dimensional setups across the BEC-BCS crossover. In this paper the nodal plane of the stable soliton solution is perturbed by adding a transverse modulation. The numerical solution of the system of coupled nonlinear differential equations describing the amplitude of the perturbation leads to an estimate of the growth rate and characteristic length scale of the instability, which are calculated for a wide range of interaction regimes and compared to other theoretical predictions. The behavior of the maximum transverse size that the atomic cloud can have in order to preserve the stability is described across the BEC-BCS crossover. The analysis of the effects of spin imbalance on this critical length reveals a stabilization of the soliton with increasing imbalance and therefore provides the experimental community with a method to achieve the realization of stable solitons in real three-dimensional configurations, without reducing the system dimensionality.
Families of dark solitons exist in superfluid Fermi gases. The energy-velocity dispersion and number of depleted particles completely determine the dynamics of dark solitons on a slowly varying background density. For the unitary Fermi gas, we determine these relations from general scaling arguments and conservation of local particle number. We find solitons to oscillate sinusoidally at the trap frequency reduced by a factor of 1/31/\sqrt{3}1/3​. Numerical integration of the time-dependent Bogoliubov--de Gennes equation determines spatial profiles and soliton-dispersion relations across the BEC-BCS crossover, and proves consistent with the scaling relations at unitarity.