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arXiv:1811.08995 · Physical Review A

Solitons in one-dimensional systems at the BCS-BEC crossover

Tianhao Ren, I. L. Aleǐner

Published Jan 29, 2019• 6 citations
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Abstract

We developed a comprehensive semiclassical theory of solitons in one-dimensional systems at the BCS-BEC crossover to provide a semiclassical explanation of their excitation spectra. Our semiclassical results agree well with the exact solutions on both the deep BCS and the deep BEC side and explain qualitatively the smooth crossover between them. Especially, we showed that the minimum energy of the $S=1/2$ excitation is achieved exactly at the Fermi momentum ${k}_{F}=\ensuremath{\pi}n/2$, where $n{m}_{F}$ (${m}_{F}$ is the mass of the fermionic atom) is the total mass density of the system. This momentum remains unchanged along the whole crossover whether the mass is contained in the bosonic molecules as on the deep BEC side or in the fermionic atoms as on the deep BCS side. This phenomenon comes about as a result of a special feature of one-dimensional systems that the conventional quasiparticle is not stable with respect to soliton formation. It is valid not only in exactly solvable models, but also on the level of semiclassical theory. Besides, we also resolved the inconsistency of existing semiclassical theory with the exact solution of solitonlike $S=0$ excitations on the deep BCS side by a different proposal of soliton configuration.

DOI
10.1103/physreva.99.013626

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We developed a comprehensive semiclassical theory of solitons in one-dimensional systems at the BCS-BEC crossover to provide a semiclassical explanation of their excitation spectra. Our semiclassical results agree well with the exact solutions on both the deep BCS and the deep BEC side and explain qualitatively the smooth crossover between them. Especially, we showed that the minimum energy of the S=1/2S=1/2S=1/2 excitation is achieved exactly at the Fermi momentum kF=\ensuremathπn/2{k}_{F}=\ensuremath{\pi}n/2kF​=\ensuremathπn/2, where nmFn{m}_{F}nmF​ (mF{m}_{F}mF​ is the mass of the fermionic atom) is the total mass density of the system. This momentum remains unchanged along the whole crossover whether the mass is contained in the bosonic molecules as on the deep BEC side or in the fermionic atoms as on the deep BCS side. This phenomenon comes about as a result of a special feature of one-dimensional systems that the conventional quasiparticle is not stable with respect to soliton formation. It is valid not only in exactly solvable models, but also on the level of semiclassical theory. Besides, we also resolved the inconsistency of existing semiclassical theory with the exact solution of solitonlike S=0S=0S=0 excitations on the deep BCS side by a different proposal of soliton configuration.
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.
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.
We propose an eigenfunction expansion method to solve the integral equations raised by one-dimensional δ-interaction integrable models. Based on this method, the crossover from BCS superfluid of singlet pairs to BEC of molecules is exactly analyzed in both ground and excited states in terms of the momentum distributions. In the exact one-dimensional model, we derive the expressions and calculate the dependence of several physical quantities on the inverse of the dimensionless coupling parameter γ. These physical quantities include the ground state energy, chemical potential, momentum cutoff, charge velocity, excitation energy, excitation momentum, spin gap and spin velocity. Our exact results agree well with previous numerical results. In the practical quasi-one-dimensional model, we apply the method to the ground state and the results give a clear BCS–BEC crossover picture.
We investigate the behavior of dark solitons in a superfluid Fermi gas along the BCS-BEC crossover by solving the Bogoliubov--de Gennes equations and looking for real and odd solutions for the order parameter. We show that in the resonance unitary region, where the scattering length is large, the density profile of the soliton has a deep minimum, differently from what happens in the BCS regime. The superfluid gap is found to be significantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter.
Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent T-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.
We demonstrate the existence of supersonic bell soliton in the Bardeen-Cooper-Schrieffer-Bose-Einstein condensate (BCS-BEC) crossover regime. Starting from the extended Thomas-Fermi density functional theory of superfluid order parameter, a density transformation is used to map the hydrodynamic mean field equation to a Lienard type equation. As a result, bell solitons are obtained as exact solutions, which is further verified by the numerical solution of the dynamical equation. The stability of the soliton is established and its behaviour in the entire crossover domain is obtained. It is found that, akin to the case of vortices, the bell solitons yield highest contrast in the BEC regime.