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arXiv:2011.11556

Towards a kinetic theory of a dark soliton gas in one-dimensional superfluids

Constanza Anahi Elsa Rivera Pereira, João D. Rodrigues, José Tito Mendonça, Hugo Terças

Published Nov 23, 2020• 0 citations
arXivHTMLPDF

Abstract

Soliton hydrodynamics is an appealing tool to describe strong turbulence in low-dimensional systems. Strong turbulence in quasi-one dimensional spuerfluids, such as Bose-Einstein condensates, involves the dynamics of dark solitons and, therefore, the description of a statistical ensemble of dark-solitons, i.e. soliton gases, is necessary. In this work, we propose a phase-space (kinetic) description of dark-soliton gases, introducing a kinetic equation that is formally similar to the Vlasov equation in plasma physics. We show that the proposed kinetic theory can capture the dynamical features of soliton gases and show that it sustains an acoustic mode, a fact that we corroborate with the help of direct numerical simulations. Our findings motivate the investigation of the microscopic structure of out-of-equilibrium and turbulent regimes in low-dimensional superfluids.

DOI
10.48550/arxiv.2011.11556

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Soliton hydrodynamics is an appealing tool to describe strong turbulence in low-dimensional systems. Strong turbulence in quasi-one dimensional spuerfluids, such as Bose-Einstein condensates, involves the dynamics of dark solitons and, therefore, the description of a statistical ensemble of dark-solitons, i.e. soliton gases, is necessary. In this work, we propose a phase-space (kinetic) description of dark-soliton gases, introducing a kinetic equation that is formally similar to the Vlasov equation in plasma physics. We show that the proposed kinetic theory can capture the dynamical features of soliton gases and show that it sustains an acoustic mode, a fact that we corroborate with the help of direct numerical simulations. Our findings motivate the investigation of the microscopic structure of out-of-equilibrium and turbulent regimes in low-dimensional superfluids.
Abstract Soliton hydrodynamics is an appealing tool to describe strong turbulence in low-dimensional systems. Strong turbulence in quasi-one-dimensional superfluids, such as Bose-Einstein condensates, involves the dynamics of dark solitons and, therefore, the description of a statistical ensemble of dark solitons, i.e. , soliton gases, is necessary. In this work, we propose a phase-space (kinetic) description of dark-soliton gases, introducing a kinetic equation that is formally similar to the Vlasov equation in plasma physics. We show that the proposed kinetic theory can capture the dynamical features of soliton gases and show that it sustains an acoustic mode, a fact that we corroborate with the help of direct numerical simulations. Our findings motivate the investigation of the microscopic structure of out-of-equilibrium and turbulent regimes in low-dimensional superfluids.
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 theoretically propose a new method of making quantum turbulence from many dark solitons in atomic Bose-Einstein condensates. We solve numerically the two-dimensional Gross-Pitaevskii equation. We set initially many solitons so that they can form a square grid. A dark soliton is known to be stable in one-dimensional systems, but unstable in two- or three-dimensional systems and decay to vortices. Our simulation shows that these solitons decay to a lot of vortices which move around in the system and eventually lead to two-dimensional quantum turbulence. The probability distribution function of the superfluid velocity obeys a Gaussian distribution in the low-velocity region and a power-law distribution in the high-velocity region. The decay of the total number of vortices obeys a power-law for a relatively long period. This scenario may be experimentally realized through interference of Bose-Einstein condensates in a trap potential.
Summary form only given. Bose-Einstein condensates (BEC) of weakly interacting gases allow for the new regime of nonlinear atom optics. Of special interest are macroscopically excited states like dark solitons or vortices. Solitons, as particle like excitations of matter wave fields, provide a link from condensate physics to fluid mechanics, nonlinear optics and fundamental particle physics. Dark solitons in matter waves are characterized by a local density minimum and, a steep phase gradient of the wavefunction at the position of the minimum. We report on experimental and theoretical investigation of dark solitons in BECs of /sup 87/Rb which are produced by the method of phase imprinting. A highly anisotropic confining potential allows us to be close to the (quasi) 1D situation where dark solitons are expected to be dynamically stable. By monitoring the evolution of the density profile we observe the evolution of density minima travelling at a smaller velocity than the speed of sound. By comparison to analytical and numerical solutions of the 3D Gross-Pitaevskii equation for our experimental conditions we identify these density minima to be moving dark solitons. We have studied in detail the creation and the dynamics of dark solitons as a function of evolution time and imprinted phase.
Dilute atomic Bose-Einstein condensates support intriguing macroscopic excitations in the form of quantized vortices and dark solitons. In this thesis we present extensive quantitative studies of the dynamics of these phenomena in the limit of zero temperature, performed by numerical simulation of the Gross- Pitaevskii equation. We show that vortices and dark solitons are inherently unstable to acceleration through the condensate, leading to the emission of sound waves. Indeed, for a single vortex/soliton, the power emitted is proportional to the square of the local acceleration. However, due to the finite size of the condensate, the vortex/soliton reinteracts with the emitted sound. This coupling has an important effect on the ensuing dynamics, and we illustrate how it can be engineered to induce net decay, stabilise, or even drive energy into the vortex/soliton. Under the harmonic confinement typically employed to trap the condensates, the acceleration-induced decay is stabilised by reabsorption of the emitted sound. However, by modifying the trap geometry, e.g. by adding an inner dimple in which the soliton resides or an optical lattice potential, it is possible to break this equilibrium and so induce a net decay of the vortex/soliton in a controllable manner. The decay rate can be considerable and should be observable under current experimental conditions. The dynamical stability of quantized vortices is also relevant to the field of turbulence in superfluid Helium, where the motion of the vortices is induced by the surrounding distribution of vortices rather than density. We extend these results to this field, and additionally consider the interactions between two and three vortices, which are also found to involve sound emission. By exciting the sound field of the condensate it is possible to drive parametrically energy into a dark soliton. In a real dissipative environment, this can be used to stabilise the soliton decay. Finally, we illustrate the links between dark solitons and vortices: a dark soliton embedded in a three-dimensional system is prone to decay into vortex rings, while a vortex in a quasi-one-dimensional geometry cannot be supported and exists as a hybrid between a vortex and dark soliton, known as a solitonic vortex.
We develop a systematic analytical approach to consider the dynamics of linear and nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. We show that, for a condensate strongly confined in two transverse directions, the ground state of the system involves the high-order eigenmodes of the transverse confining potential in the transverse directions and effective high-order Thomas-Fermi wave functions in the axial direction. The linear excitations of the system have a Bogoliubov-type spectrum with the excitation frequency varying slowly along the axial direction. We find that, in a weak nonlinear approximation, the amplitude of a nonlinear excitation is governed by a variable coefficient Korteweg--de Vries equation with additional terms contributed from the transverse structure and the inhomogeneity in the axial direction of the condensate, which results in varying amplitude, width, and velocity for dark solitons. Because of the inhomogeneity the dark solitons undergo deformation and emit radiations when traveling along the axial direction. We finally demonstrate that a dark soliton will disintegrate into several ones plus a residual wave train when passing over a steplike potential.