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Chinese Physics Letters

Variational Approach to Study ${\mathscr{P}}{\mathscr{T}}$-Symmetric Solitons in a Bose–Einstein Condensate with Non-locality of Interactions

Wei Qi, Haifeng Li, Zhaoxin Liang

Published Apr 1, 2019• 7 citations
Publisher

Abstract

Considering the non-locality of interactions in a Bose–Einstein condensate, the existence and stability of solitons subject to a - symmetric potential are discussed. In the framework of the variational approach, we investigate how the non-locality of interactions affects the self-localization and stability of a condensate with attractive two-body interactions. The results reveal that the non-locality of interactions dramatically influences the shape, width, and chemical potential of the condensate. Analytically variational computation also predicts that there exists a critical negative non-local interaction strength ( p c < 0) with each fixed two-body interaction ( g 0 < 0), and there exists no bright soliton solution for p 0 < p c . Furthermore, we study the effect of the non-locality interactions on the stability of the solitons using the Vakhitov–Kolokolov stability criterion. It is shown that for a positive non-local interaction ( p 0 > 0), there always exist stable bright solitons in some appropriate parameter regimes .

DOI
10.1088/0256-307x/36/4/040501

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Variational Approach to Study PT{\mathscr{P}}{\mathscr{T}}PT-Symmetric Solitons in a Bose–Einstein Condensate with Non-locality of Interactions
Considering the non-locality of interactions in a Bose–Einstein condensate, the existence and stability of solitons subject to a - symmetric potential are discussed. In the framework of the variational approach, we investigate how the non-locality of interactions affects the self-localization and stability of a condensate with attractive two-body interactions. The results reveal that the non-locality of interactions dramatically influences the shape, width, and chemical potential of the condensate. Analytically variational computation also predicts that there exists a critical negative non-local interaction strength ( p c < 0) with each fixed two-body interaction ( g 0 < 0), and there exists no bright soliton solution for p 0 < p c . Furthermore, we study the effect of the non-locality interactions on the stability of the solitons using the Vakhitov–Kolokolov stability criterion. It is shown that for a positive non-local interaction ( p 0 > 0), there always exist stable bright solitons in some appropriate parameter regimes .
Abstract Considering the quantum fluctuation effects, the existence and stability of solitons in a Bose-Einstein condensate subjected in a P T -symmetric potential are discussed. Using the variational approach, we investigate how the quantum fluctuation affects the self-localization and stability of the condensate with attractive two-body interactions. The results show that the quantum fluctuation dramatically influences the shape, width, and chemical potential of the condensate. Analytical variational computation also predicts there exists a positive critical quantum fluctuation strength q c with each fixed attractive two-body interaction g 0 , if the quantum fluctuation strength q 0 is bigger than q c , there is no bright soliton solution existence. We also study the effects of the quantum fluctuations on the stability of solitons using the Vakhitov-Kolokolov (VK) stability criterion. A robust stable bright soliton will always exist when the quantum fluctuation strength q 0 belongs to the parameter regimes q c ≥ q 0 > 0.
We consider a Bose-Einstein condensate (BEC) with non-local inter-particle interactions. The local Gross-Pitaevskii (GP) equation is valid for the gas parameter [Formula: see text], but for [Formula: see text], the BEC is described by a modified GP equation (MGPE). We study the exact solutions of the MGPE describing bright and dark solitons. It turns out that the width of these non-local solitons has qualitatively similar behaviour as the modified healing length due to the non-local interactions of the MGPE. We also study the effect of the non-locality and gas parameter (ν) on the stability of the solitons using the Vakhitov-Kolokolov (VK) stability criterion. We show that these soliton solutions are stable according to the VK criterion. Further, the stability of these soliton solutions gets enhanced due to the non-locality of interactions.
Recent research works on ultra cold quantum gases demonstrated that dipolar Bose–Einstein condensates (BECs) exhibit rich spatiotemporal dynamic where both local and nonlocal interactions are considered. We explore theoretically the possibility of controlling the formation and dynamics of soliton molecules in binary dipolar condensates with spin-orbit coupling (SOC). We exploit the variational technique to derive the new equations of motion for the widths and amplitudes, the effective potential and the oscillation frequency of the molecules. Our study confirms the existence of stable localized bound states in an optical potential. We find that the integrity of the molecules is influenced by the physical parameters, notably the local and nonlocal interactions with the SOC. These parameters are carefully chosen by the Vakhitov–Kolokolov (VK) criterion to ensure the stability of the molecules. We present the results of numerical experiments and confirm the analytical predictions. Moreover, we show the soliton–soliton interaction in each molecule when the local interactions are strong.
Dynamics and properties of nonlinear matter waves in a trapped BEC subject to a PT-symmetric linear potential, with the trap in the form of a super-Gaussian potential, are investigated via a variational approach accounting for the complex nature of the soliton. In the process, we address how the shape of the imaginary part of the potential, that is, a gain-loss mechanism, affects the self-localization and the stability of the condensate. Variational results are found to be in good agreement with full numerical simulations for predicting the shape, width, and chemical potential of the condensate until the PT breaking point. Variational computation also predicts the existence of solitary solution only above a threshold in the particle number as the gain-loss is increased, in agreement with numerical simulations.
An extended variation approach to describing the dynamic evolution of self-attractive Bose—Einstein condensates is developed. We consider bright matter-wave solitons in the presence of a parabolic magnetic potential and a time-space periodic optical lattice. The dynamics of condensates is shown to be well approximated by four coupled nonlinear differential equations. A noteworthy feature is that the extended variation approach gives a critical strength ratio to support multiple stable lattice sites for the condensate. We further examine the existence of the solitons and their stabilities at the multiple stable lattice sites. In this case, the analytical predictions of Bose—Einstein condensates variational dynamics are found to be in good agreement with numerical simulations. We then find a stable region for successful manipulating matter-wave solitons without collapse, which are dragged from an initial stationary to a prescribed position by a moving periodic optical lattice.
An extended variation approach to describing the dynamic evolution of self-attractive Bose-Einstein condensates is developed.We consider bright matter-wave solitons in the presence of a parabolic magnetic potential and a timespace periodic optical lattice.The dynamics of condensates is shown to be well approximated by four coupled nonlinear differential equations.A noteworthy feature is that the extended variation approach gives a critical strength ratio to support multiple stable lattice sites for the condensate.We further examine the existence of the solitons and their stabilities at the multiple stable lattice sites.In this case,the analytical predictions of Bose-Einstein condensates variational dynamics are found to be in good agreement with numerical simulations.We then find a stable region for successful manipulating matter-wave solitons without collapse,which are dragged from an initial stationary to a prescribed position by a moving periodic optical lattice.