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Physica Scripta

Effects of quantum corrections on the modulational instability of Bose–Einstein condensates trapped in a periodic optical lattice

Olivier Tiokeng Lekeufack, Subramaniyan Sabari, Serge Bruno Yamgoué, Timoléon Crépin Kofané

Published Oct 1, 2020• 6 citations
Publisher

Abstract

Abstract Through a semiquantal procedure, we study the perturbed modulation of amplitude and phase of Gross-Piteavskii equation describing trapped Bose–Einstein condensates in an optical lattice potential. By introducing quantum correctional parameters, the problem is quantized and leads to the derivation of a novel dynamical instability criterion. Additional degrees of freedom carrying quantum properties play a central role on the refine of the instability bandwidth, and, combined to the strength of optical lattice potential, entail unstable modes into full stability. A set of computational tools exhibited various features that bear instability characteristics, to confirm analytically predicted results. The quantum fluctuations thus have a stabilizing effect on the dynamics of harmonically trapped Bose–Einstein condensates in an optical lattice potential.

DOI
10.1088/1402-4896/abbd6a

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We investigate the dynamical instability of Bose-Einstein condensates (BECs) with higher-order interactions immersed in an optical lattice with weak driving harmonic potential. For this, we compute both analytically and numerically a modified Gross-Pitaevskii equation with higher-order nonlinearity and external potentials generated by magnetic and optical fields. Using the time-dependent variational approach, we derive the ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. Through an effective potential, we obtain the modulational instability condition of BECs and discuss the effect of the higher-order interaction in the dynamics of the condensates in presence of optical potential. We perform direct numerical simulations to support our analytical results, and good agreement is found.
We study the stabilization properties of dipolar Bose–Einstein condensate in a deep one-dimensional optical lattice with an additional external parametrically modulated harmonic trap potential. Through both analytical and numerical methods, we solve a dimensionless nonlocal nonlinear discrete Gross–Pitaevskii equation with both the short-range contact interaction and the long-range dipole–dipole interaction. It is shown that, the stability of dipolar condensate in modulated deep optical lattice can be controled by coupled effects of the contact interaction, the dipolar interaction and the external modulation. The system can be stabilized when the dipolar interaction, the contact interaction, the average strength of potential and the ratio of amplitude to frequency of the modulation satisfy a critical condition. In addition, the breather state, the diffused state and the attractive-interaction-induced-trapped state are predicted. The dipolar interaction and the external modulation of the lattice play important roles in stabilizing the condensate.
We investigate the dynamical instabilities of a one-dimensional ultracold Bose-Bose mixture with long-range dipole-dipole interactions, trapped in deep optical lattices and subject to periodically varying contact interaction. The effect of beyond-mean-field corrections due to quantum fluctuations is considered. In the tight-binding regime, we employ Wannier functions to derive a discrete nonlinear Schrödinger equation that captures the dynamics of the system. Using linear stability analysis coupled to multiple scale expansion, we systematically study the modulational and parametric instabilities of the system in the miscible regime, identifying the conditions under which these instabilities emerge. The corresponding instability domains are found and examined. The roles of long-range interactions and quantum fluctuations are highlighted, demonstrating their significant impact on the stability and dynamics of the lattice system. Our analytical predictions are validated through direct numerical simulations, which confirm the instability diagrams through the effective onset of instabilities and reveal the intricate interplay between interaction strength, quantum fluctuation, and the long-range nature of the dipole-dipole forces. This work provides insights into the control and manipulation of ultracold quantum gases in optical lattices, which have potential applications in quantum simulation and condensed-matter physics.
We present a comprehensive study of modulational instability (MI) in a binary Bose-Einstein condensate with spin-orbit coupling, confined to a deep optical lattice. The system is modeled by a set of discrete Gross-Pitaevskii equations. Using linear stability analysis, we derive the explicit MI conditions for the system, elucidating the critical and distinct roles played by spin-orbit coupling, inter-species nonlinearity, and intra-species nonlinearity. Our analysis, conducted for both unstaggered and staggered fundamental modes, reveals markedly different instability landscapes for these two configurations. The analytical predictions are confirmed by extensive numerical simulations of the full nonlinear dynamics, which vividly illustrate the spatiotemporal evolution of wave amplitudes, phase coherence, and energy localization during the instability process. The numerical results, obtained via a fourth-order Runge-Kutta method, show excellent agreement with the linear stability theory and provide a complete picture of the MI-induced pattern formation.
We study the dynamical stability of Bose-Einstein condensates in an optical lattice with a time-periodic modulation potential and a constant acceleration force simultaneously. We derive the explicit expressions of quasienergies and obtain the stability diagrams in the parameter space of the interaction strength and the modulation amplitude. The ratio of the acceleration force to the modulation frequency characterizes two cases: integer and non-integer resonances. For integer resonances, the critical interaction strength gcg_{\mathrm{c}}gc​ shows an alternate behavior where the completely unstable regions correspond to the negative effective tunneling strength. Among non-integer resonances, we observe that gcg_{\mathrm{c}}gc​ peaks are centered around half-integer resonances for which the completely unstable regions disappear, accompanied with a whole displacement of gcg_{\mathrm{c}}gc​. Compared with integer and half-integer resonances, the crossovers between them show no explicit dependence of gcg_{\mathrm{c}}gc​ on the modulation amplitude.
Motivated by the recent experiment [Nature 530 (2016) 194] in which a stable droplet in a dipolar quantum gas has been created by the interaction-induced instability, we focus on the modulation instability of an optically-trapped dipolar Bose–Einstein condensate with three-body interaction. Within the mean-field level, we analytically solve the discrete cubic-quintic Gross–Pitaevskii equation with dipole–dipole interaction loaded into a deep optical lattice and show how combined effects of the three-body interaction and dipole–dipole interaction on the condition of modulational instability. Our results show that the interplay of the three-body interaction and dipole–dipole interaction can dramatically change the modulation instability condition compared with the ordinary Gross–Pitaevskii equation. We believe that the predicted results in this work can be useful for the future possible experiment of loading a Bose–Einstein condensate of 164 Dy atoms with strong magnetic dipole–dipole interaction into an optical lattice.