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arXiv:1901.04294 · Laser Physics

Influence of a dark soliton on the reflection of a Bose–Einstein condensate by a square barrier

Qiao-Ling Cheng, Wen-Kai Bai, Yao-Zhong Zhang, Bo Xiong, Tao Yang

Published Nov 9, 2018• 7 citations
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Abstract

Abstract We study the quantum reflection of a two-dimensional disk-shaped Bose–Einstein condensate with dark soliton excitation by a square potential barrier. For the given geometry, the dark soliton initially located at the centre of the condensate cloud survives long enough to investigate the reflection process. We show the time evolution of the reflection probability with respect to various widths of the barrier. The asymptotic value of the reflection probability is decreased by the existence of a dark soliton, and is highly sensitive to the initial orientation of the dark soliton, which also affects the excitation properties during the process of condensate and barrier interaction.

DOI
10.1088/1555-6611/aaea78

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Abstract We study the quantum reflection of a two-dimensional disk-shaped Bose–Einstein condensate with dark soliton excitation by a square potential barrier. For the given geometry, the dark soliton initially located at the centre of the condensate cloud survives long enough to investigate the reflection process. We show the time evolution of the reflection probability with respect to various widths of the barrier. The asymptotic value of the reflection probability is decreased by the existence of a dark soliton, and is highly sensitive to the initial orientation of the dark soliton, which also affects the excitation properties during the process of condensate and barrier interaction.
We study dynamical behaviors of a Bose–Einstein condensate (BEC) containing a dark soliton reflected from potential wells and potential barriers, respectively. The orientation angle of the dark soliton and the width of the potential change play key roles on the reflection probability R s. Variation of the reflection probability with respect to the orientation angle θ of the dark soliton can be well described by a cosine function R s ∼ cos [λ(θ – π/2)], where λ is a parameter determined by the width of the potential change. There are two characteristic lengths which determine the reflection properties. The dependence of the reflection probability on the width of the potential change shows distinct characters for potential wells and potential barriers. The length of the dark soliton determines the sensitiv width of potential wells, whereas for potential barriers, the decay length of the matter wave in the region of the barrier qualifies the sensitive width of the barrier. The time evolution of the density profiles of the system during the reflection process is studied to disclose the different behaviors of matter waves in the region of the potential variation.
We study the dynamic behavior of a Bose-Einstein condensate (BEC) containing a dark soliton separately reflected from potential drops and potential barriers. It is shown that for a rapidly varying potential and in a certain regime of incident velocity, the quantum reflection probability displays the cosine of the deflection angle between the incident soliton and the reflected soliton, i.e., R(θ)∼cos⁡2θR(θ) \sim \cos 2θR(θ)∼cos2θ. For a potential drop, R(θ)R(θ)R(θ) is susceptible to the widths of potential drop up to the length of the dark soliton and the difference of the reflection rates between the orientation angle of the soliton θ=0θ=0θ=0 and θ=π/2θ=π/2θ=π/2, δRsδR_sδRs​, displays oscillating exponential decay with increasing potential widths. However, for a barrier potential, R(θ)R(θ)R(θ) is insensitive for the potential width less than the decay length of the matter wave and δRsδR_sδRs​ presents an exponential trend. This discrepancy of the reflectances in two systems is arisen from the different behaviors of matter waves in the region of potential variation.
By using multiple-scale method, we analytically study the soliton dynamics of Bose–Einstein condensate (BEC) trapped in a double square well potential. It is shown that dark soliton can be stable, which is generated in BEC trapped in a double square well. The amplitude and the width of dark soliton are related to the initial position of the soliton. Within the double square well, because of the confinement of the potential, the amplitude of the dark soliton is larger, while the width of the dark soliton is smaller. Furthermore, we also find that the amplitude and the width of the soliton are closely related to the depth of the double square well. With the increase of the depth, the amplitude of the dark soliton increases remarkably, while the width of the dark soliton decreases slowly.
We consider, both analytically and numerically, the evolution of two-dimensional (2D) nonlinear matter-wave pulses in a Bose-Einstein condensate with a disk-shaped trap and repulsive atom-atom interactions. Due to the strong confinement in the axial direction the sound speed of the system is c=(1/21/4)c0,{c=(1/2}^{1/4}{)c}_{0},c=(1/21/4)c0​, where c0{c}_{0}c0​ is the corresponding value without the trap. From the 3D order-parameter equation of the condensate, we derive a soliton-bearing Kadomtsev-Petriashvili equation with positive dispersion. When the trapping potential is weak in two transverse directions, a low-depth plane dark soliton can propagate in the condensate with a changing profile but preserving its structure down to the boundary of the condensate. We show that high-depth plane dark solitons are unstable to long-wavelength transverse disturbances. The instability appears as a longitudinal modulation of the soliton amplitude decaying into vortices. We also show how a dark lumplike 2D nonlinear excitation can be excited in the system. Furthermore, a dark lump decaying algebraically in two spatial directions can propagate rather stable in the condensate, but disappears near the boundary of the condensate where two vortices are nucleated. The vortices move in opposite directions along the boundary and when meeting merge creating a new lump. Finally, we also provide results for head-on and oblique collisions of two lumps in the system.
We first present an analytical solution of the single and double solitions of Bose—Einstein condensates trapped in a double square well potential using the multiple-scale method. Then, we show by numerical calculation that a dark soliton can be transmitted through the square well potential. With increasing depth of the square well potential, the amplitude of the dark soliton becomes larger, and the soliton propagates faster. In particular, we treat the collision behaviour of the condensates trapped in either equal or different depths of the double square well potential. If we regard the double square well potential as the output source of the solitons, the collision locations (position and time) between two dark solitons can be controlled by its depth.
We investigate the effect of interatomic interactions on the quantum-mechanical reflection of Bose-Einstein condensates from regions of rapid potential variation. The reflection process depends critically on the density and incident velocity of the condensate. For low densities and high velocities, the atom cloud has almost the same form before and after reflection. Conversely, at high densities and low velocities, the reflection process generates solitons and vortex rings that fragment the condensate. We show that this fragmentation can explain the anomalously low reflection probabilities recently measured for low-velocity condensates incident on a silicon surface.