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arXiv:2207.00513 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum droplets in three-dimensional Bose–Einstein condensates

Sherzod R. Otajonov

Published Apr 2, 2022• 14 citations
arXivHTMLPDF

Abstract

Abstract The properties of 3D Bose–Einstein condensate have been studied with variational and numerical methods. In the variational approach, we use the super-Gaussian trial function, and it is demonstrated that this trial function is a good approach for providing the descriptions of the quantum droplets. The analytical equations for the variational parameters are obtained. The frequency of small oscillations of quantum droplets near the equilibrium position is estimated. It is also found that periodic modulation of the coupling constants leads to resonance oscillations of the quantum droplets parameters or emission of waves depending on the amplitude of the modulations. The predictions are supported by direct numerical simulation of the governing equation.

DOI
10.1088/1361-6455/ac6365

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Abstract The properties of 3D Bose–Einstein condensate have been studied with variational and numerical methods. In the variational approach, we use the super-Gaussian trial function, and it is demonstrated that this trial function is a good approach for providing the descriptions of the quantum droplets. The analytical equations for the variational parameters are obtained. The frequency of small oscillations of quantum droplets near the equilibrium position is estimated. It is also found that periodic modulation of the coupling constants leads to resonance oscillations of the quantum droplets parameters or emission of waves depending on the amplitude of the modulations. The predictions are supported by direct numerical simulation of the governing equation.
The dynamics of a two-dimensional Bose-Einstein condensate in a presence of quantum fluctuations is studied. The properties of localized density distributions, quantum droplets (QDs), are analyzed by means of the variational approach. It is demonstrated that the super-Gaussian function gives a good approximation for profiles of fundamental QDs and droplets with nonzero vorticity. The dynamical equations for parameters of QDs are obtained. Fixed points of these equations determine the parameters of stationary QDs. The period of small oscillations of QDs near the stationary state is estimated. It is obtained that periodic modulations of the strength of quantum fluctuations can actuate different processes, including resonance oscillations of the QD parameters, an emission of waves and a splitting of QDs into smaller droplets.
Nonlinear oscillations of a 3D radial symmetric Bose-Einstein condensate under periodic variation in time of the atomic scattering length have been studied analytically and numerically. The time-dependent variational approach is used for the analysis of the characteristics of nonlinear resonances in the oscillations of the condensate. The bistability in oscillations of the BEC width is invistigated. The dependence of the BEC collapse threshold on the drive amplitude and parameters of the condensate and trap is found. Predictions of the theory are confirmed by numerical simulations of the full Gross-Pitaevski equation.
Quantum droplets have recently emerged as a novel liquid state of matter in a mixture of two-component ultracold Bose gases under the equilibrium condition between the competing attractive inter- and repulsive intraspecies forces. Quantum droplets represent a self-bound liquid state in purely nonlinear systems and as such there is not any management applied to their study up to now. We here introduce the nonlinear management via space-periodic Feshbach resonance technique to a binary Bose–Einstein condensate and investigate theoretically the structure, property and dynamics of the one-dimensional quantum droplets therein, three findings are made: small droplets with single hump (solitonlike), moderate droplets composed of two solitons, large droplets feature a modulated flat-top shape. Particularly, by means of linear-stability and direct perturbed simulations, we prove that the droplets have a much wider stable region than their counterparts without taking the nonlinear management. The variational approximation is adopted and validated with numerical results in terms of small droplets. The results predicted here may be realized in the frontier ultracold atoms experiments aided by Feshbach-resonance, opening up a new channel to the quantum droplets studies.
Static and dynamic properties of matter-wave solitons in dense Bose–Einstein condensates, where three-body interactions play a significant role, have been studied by a variational approximation (VA) and numerical simulations. For experimentally relevant parameters, matter-wave solitons may acquire a flat-top shape, which suggests employing a super-Gaussian trial function for VA. Comparison of the soliton profiles, predicted by VA and those found from numerical solution of the governing Gross–Pitaevskii equation shows good agreement, thereby validating the proposed approach.
Abstract We consider quantum droplets in dipolar Bose–Einstein condensates embedded in optical lattices (OLs) within the framework of Gross–Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole–dipole interaction (DDI) provides an additional mechanism for self-binding. We analyze the linear stability as well dynamics of quantum droplets. We find an effective potential for the width and show that the optimum width for formation of quantum droplet increases as the DDI increases. We study dynamics of the stable droplets and see that its width oscillates with time, and the amplitude of oscillation increases with the increase of DDI. In presence of OLs, width of a stable droplet changes quasi-periodically while the location of density profile changes periodically in space. The frequency of oscillation depends sensitively on the lattice parameters. We check dynamical stability of a quantum droplet through numerical simulation.
The Bose-Einstein condensate is generally described by the three-dimensional equations. However, the two-dimensional case is of great interest too. First, these results can be directly used to study the planar condensates. Second, it serves as a test version for numerical calculations and as a quasi-two-dimensional approximation for the complete three-dimensional problem. An additional advantage of the two-dimensional case is that with the help of variational approach one can obtain analytical results for the stationary state and for the expansion of the condensate. In this paper, we propose a new approach for investigation of the Bose condensate in harmonic potential based on the variational method. We consider the inverse problem of reconstructing coefficients of the two-dimensional Gross-Pitaevskii equation with harmonic potential under the initially given condensate parameters (particle number, interatomic interaction, cloud widths). In particular, we solve the stationary problem in the framework of three trial functions, i.e., the Gaussian, the super-Gaussian and the Thomas-Fermi functions. In addition to that, the virial relation valid for arbitrary trial functions is obtained. We introduce the concept of special nonlinearity parameter in order to differentiate various degrees of the interatomic interaction. We perform comparison the variational solutions for the problem with the results of numerical calculations at the various levels of nonlinearity. For example, the relative differences between the condensate widths (Q1 and Q2) obtained by the variational method and the results of the direct numerical simulation is presented. It can be seen that the Gaussian and the super-Gaussian approximate the widths of the condensate with the accuracy better than 5% at various levels of nonlinearity. We demonstrate that the suggested approach allows one to solve the inverse problem of reconstructing all the equation parameters under the initially given condensate parameters. The joint use of the variational approach to the problem together with the numerical calculations allows one efficiently link the characteristics of the Bose condensate with the parameters of the Gross-Pitaevskii equation. Moreover, the proposed analytical-numerical approach can be used for broader class of problems than we discuss in the paper, for example, for investigating the self-focusing electromagnetic waves.