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Optics and Spectroscopy

Soliton of Bose–Einstein condensate in a trap with rapidly oscillating walls: I. Multiscale method and analysis of soliton motion in the limit of extremely fast wall oscillations

Nikolay A. Veretenov, N. V. Vysotina, Leonid A. Nesterov, Nikolay N. Rosanov

Published Nov 1, 2015• 3 citations
Publisher

Abstract

Motion of a soliton of Bose–Einstein condensate of atoms captured by a trap with rapidly oscillating walls has been studied. This motion can be described using both the Gross–Pitaevskii equation for a condensate wave function and an approximate equation in the form of the Newton equation for the soliton center coordinate. An analytical approach for solving the Newton equation has been developed. This approach is based on the multiscale method where the solution is sought for in the form of small-parameter expansion. This parameter is a ratio of the frequency of intrinsic slow soliton oscillations around the equilibrium position to the frequency of fast oscillations of the trap walls. In the first part of the study, an approach based on two time scales is described and the case of extremely fast wall oscillations is investigated. The calculation performed within the zero approximation shows a very good coincidence with the numerical solution of the Newton equation with respect to all parameters. A good agreement with the numerical solutions of the Gross–Pitaevskii equation is also demonstrated for calculations of the parameters such as oscillation frequency and shift of the soliton equilibrium position under the action of the wall motion. In the second part, the role of corrections to the obtained solution is analyzed for a decreasing wall-oscillation frequency and the range of applicability of the used analytical approach is discussed.

DOI
10.1134/s0030400x15110247

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Soliton of Bose–Einstein condensate in a trap with rapidly oscillating walls: II. Analysis of the soliton behavior upon a decrease in the wall oscillation frequency

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This work is a continuation of our study [1], in which a two-scale analytical approach to the investigation of a soliton oscillon in a trap with rapidly oscillating walls has been developed. In terms of this approach, the solution to the equation of motion of the soliton center is sought as a series expansion in powers of a small parameter, which is a ratio of the intrinsic frequency of slow soliton oscillations to the frequency of fast trap wall oscillations. In [1], we have examined the case ε ≪ 1, in which, to describe the motion of the soliton, it is sufficient to restrict the consideration to the zero approximation of the sought solution. However, when the frequency of wall oscillations begins to decrease, while the parameter begins to increase, it is necessary to take into account corrections to the zero approximation. In this work, we have calculated corrections of the first and second orders in to this approximation. We have shown that, with an increase in, the role played by the corrections related to fast oscillations of the trap walls increases, which results in a complex shape of the envelope of oscillations of the soliton center. It follows from our calculations that, if the difference between the amplitudes of wall oscillations is not too large, the analytical solution of the equation of motion of the soliton center will coincide very well with the numerical solution. However, with an increase in this difference, as well as with a decrease in the wall oscillation frequency, the discrepancy between the numerical and analytical solutions generally begins to increase. Regimes of irregular oscillations of the soliton center arise. With a decrease in the frequency of wall oscillations, the instability boundary shows a tendency toward a smaller difference between the wall oscillation amplitudes. In general, this leads to enlargement of the range of irregular regimes. However, at the same time, stability windows can arise in this range in which the analytical and numerical solutions correlate rather well with each other. Our comparative analysis of the analytical and numerical solutions has allowed us not only to study their properties in detail, but also to draw conclusions on the limits of applicability of the analytical approach.

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Motion of a soliton of Bose–Einstein condensate of atoms captured by a trap with rapidly oscillating walls has been studied. This motion can be described using both the Gross–Pitaevskii equation for a condensate wave function and an approximate equation in the form of the Newton equation for the soliton center coordinate. An analytical approach for solving the Newton equation has been developed. This approach is based on the multiscale method where the solution is sought for in the form of small-parameter expansion. This parameter is a ratio of the frequency of intrinsic slow soliton oscillations around the equilibrium position to the frequency of fast oscillations of the trap walls. In the first part of the study, an approach based on two time scales is described and the case of extremely fast wall oscillations is investigated. The calculation performed within the zero approximation shows a very good coincidence with the numerical solution of the Newton equation with respect to all parameters. A good agreement with the numerical solutions of the Gross–Pitaevskii equation is also demonstrated for calculations of the parameters such as oscillation frequency and shift of the soliton equilibrium position under the action of the wall motion. In the second part, the role of corrections to the obtained solution is analyzed for a decreasing wall-oscillation frequency and the range of applicability of the used analytical approach is discussed.
This work is a continuation of our study [1], in which a two-scale analytical approach to the investigation of a soliton oscillon in a trap with rapidly oscillating walls has been developed. In terms of this approach, the solution to the equation of motion of the soliton center is sought as a series expansion in powers of a small parameter, which is a ratio of the intrinsic frequency of slow soliton oscillations to the frequency of fast trap wall oscillations. In [1], we have examined the case ε ≪ 1, in which, to describe the motion of the soliton, it is sufficient to restrict the consideration to the zero approximation of the sought solution. However, when the frequency of wall oscillations begins to decrease, while the parameter begins to increase, it is necessary to take into account corrections to the zero approximation. In this work, we have calculated corrections of the first and second orders in to this approximation. We have shown that, with an increase in, the role played by the corrections related to fast oscillations of the trap walls increases, which results in a complex shape of the envelope of oscillations of the soliton center. It follows from our calculations that, if the difference between the amplitudes of wall oscillations is not too large, the analytical solution of the equation of motion of the soliton center will coincide very well with the numerical solution. However, with an increase in this difference, as well as with a decrease in the wall oscillation frequency, the discrepancy between the numerical and analytical solutions generally begins to increase. Regimes of irregular oscillations of the soliton center arise. With a decrease in the frequency of wall oscillations, the instability boundary shows a tendency toward a smaller difference between the wall oscillation amplitudes. In general, this leads to enlargement of the range of irregular regimes. However, at the same time, stability windows can arise in this range in which the analytical and numerical solutions correlate rather well with each other. Our comparative analysis of the analytical and numerical solutions has allowed us not only to study their properties in detail, but also to draw conclusions on the limits of applicability of the analytical approach.
The motion of the center of a soliton in a trap with oscillating walls is studied analytically and numerically for the case in which the intrinsic frequency of small soliton oscillations in the equilibrium state considerably exceeds the frequency of wall oscillations. this problem can be solved either by applying the gross–pitaevskii equation, which most exactly describes the behavior of the soliton in the trap, or by using the approximate, “mechanical,” equation of motion of the newtonian type for the center of the soliton. an approximate analytical solution of the mechanical equation is obtained and is compared with the numerical solution of the newton equation, while the latter solution is compared with the numerical solution of the gross–pitaevskii equation. good agreement between the first two solutions is revealed. it is also shown that there is a range of parameters in which the numerical solutions of the newton and gross–pitaevskii equations are closest to each other. the frequency-sweeping effect of soliton center oscillations is revealed. an approximate analytical formula for the limiting frequency of these oscillations is obtained and the numerical analysis of this phenomenon is performed.
Dynamics of solitons in a one-dimensional trap with motionless and moving walls is studied by the example of bright solitons of an atomic Bose-Einstein condensate. Newton-type equations of the soliton's center motion are proposed and analyzed, and regimes of periodic, almost-periodic, and deterministically chaotic dynamics are presented. The predictions based on these approximate equations are in excellent agreement with that based on the Gross-Pitaevskii equation in the case of sufficiently narrow and slow solitons.
We have proposed a mechanical model that corresponds to the Newton equation for describing the dynamics of an oscillon, viz., a soliton-like cluster of the Bose–Einstein condensate (with atomic attraction) placed above an oscillating atomic mirror in a uniform gravitational field. The model describes the stochastic Fermi acceleration and periodic, quasi-periodic, and chaotic motion of the oscillon center, as well as hysteresis phenomena in the case of a slow variation of mirror oscillation frequency, which are in good agreement with the results obtained using the Gross–Pitaevskii equation.
We model the three-dimensional (3D) trapped Bose-Einstein condensate with the Gross-Pitaevskii equation (GPE) incorporating higher-order nonlinear interaction effects. Based on the F-expansion method and self-similar approach, we analytically derive the exact soliton solutions of the 3D GPE, identifying the oscillatory mode under certain experimental settings. The oscillatory kinetic parameter values we obtained coincide very well with those reported in previous work, which is based on the variational approach, showing the applicability of the theoretical treatment presented in this work.