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arXiv:2012.13162 · Physical review. E

Variational approximation for two-dimensional quantum droplets

Sherzod R. Otajonov, E. N. Tsoy, F. Kh. Abdullaev

Published Dec 30, 2020• 36 citations
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Abstract

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.

DOI
10.1103/physreve.102.062217

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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.
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.
Modulational instability of a uniform two-dimensional binary Bose-Einstein condensate (BEC) in the presence of quantum fluctuations is studied. The analysis is based on the coupled Gross-Pitaevskii equations. It is shown that quantum fluctuations can induce instability when the BEC density is below a threshold. The dependence of the growth rate of modulations on the BEC parameters is found. It is observed that an asymmetry of the interaction parameters and/or initial densities of the components typically decreases the growth rate. Further development of the instability results in the BEC breaking up into a set of quantum droplets. These droplets merge dynamically with each other, so the total number of droplets decreases rapidly. The rate of this decrease is evaluated numerically for different initial parameters.
In this work, we study the stationary structures and the breathing mode behavior of a two-dimensional self-bound binary Bose droplet. We employ an analytical approach using a variational ansatz with a super-Gaussian trial order parameter and compare it with the numerical solutions of the extended Gross-Pitaevskii equation. We find that the super-Gaussian is superior to the often used Gaussian ansatz in describing the stationary and dynamical properties of the system. For sufficiently large nonrotating droplets the breathing mode is energetically favorable compared to the self-evaporating process. However, for small self-bound systems our results differ based on the ansatz. Inducing angular momentum by imprinting multiply quantized vortices at the droplet center, this preference for the breathing mode persists independent of the norm.
We consider quantum droplets in dipolar Bose-Einstein condensates embedded in optical lattices within the framework of Gross-Pitaevskii equations. In dipolar BECs, the long-range and isotropic dipole-dipole interaction 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 dipole-dipole interaction 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 dipole-dipole interaction. In presence of optical lattices, 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.
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.
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.