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arXiv:2107.11779 · Journal of Physics B Atomic Molecular and Optical Physics

Signature of supersolidity in a driven cubic–quartic nonlinear Schrödinger equation

Argha Debnath, Jammu Tarun, Ayan Khan

Published Jan 18, 2022• 7 citations
arXivHTMLPDF

Abstract

Abstract We present analytical solution, which is periodic in nature, for a driven cubic–quartic nonlinear Schrödinger equation (DCQNLSE) is placed in a bi-chromatic optical lattice. The solution indicates the creation of density wave. Since, beyond mean-field contribution in quasi one dimensional and one dimensional geometry differs on the even exponents of the nonlinearity thus we extend our analysis toward quadratic–cubic–quartic and quadratic–cubic nonlinearities as well. Later, we study the dynamics of DCQNLSE. Our study indicates the existence of stripe phase along with considerable phase coherence. These findings allow us to comment on the possible emergence of supersolid phase in a condensate.

DOI
10.1088/1361-6455/ac4c8e

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Abstract We present analytical solution, which is periodic in nature, for a driven cubic–quartic nonlinear Schrödinger equation (DCQNLSE) is placed in a bi-chromatic optical lattice. The solution indicates the creation of density wave. Since, beyond mean-field contribution in quasi one dimensional and one dimensional geometry differs on the even exponents of the nonlinearity thus we extend our analysis toward quadratic–cubic–quartic and quadratic–cubic nonlinearities as well. Later, we study the dynamics of DCQNLSE. Our study indicates the existence of stripe phase along with considerable phase coherence. These findings allow us to comment on the possible emergence of supersolid phase in a condensate.
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In a one-dimensional shallow optical lattice, in the presence of both cubic and quintic nonlinearity, a superfluid density wave is identified in a Bose-Einstein condensate. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In a certain parameter domain, the competition between lattice potential and the interactions is shown to give rise to a stripe phase, where atoms are confined in finite domains. In a pure two-body case, apart from the known superfluid and insulating phases, a density wave insulating phase is found to exist, possessing two frequency modulations commensurate with the lattice potential.
Abstract Supersolid is an exotic state of matter, showing crystalline order with a superfluid background, observed recently in dipolar Bose–Einstein condensate in a trap. Here, we present exact Bloch wave function of the self-trapped supersolid phase, in the presence of mean-field and beyond mean-field interaction. Our general solutions of the amended nonlinear Schrödinger equation are obtained through Möbius transform, connecting a wide class of supersolid solutions to the ubiquitous cnoidal waves. The solutions yield the supersolid phase in the self-trapped quantum matter, where an array of quantum droplets exist, accompanied by a constant condensate. For the supersolid phase, the chemical potential for one class of solutions is the same as that of self-trapped quantum droplets, and is lower for the general non-perturbative solution. Due to the destabilizing effects of fluctuations on long range order in one dimension, the realization of the supersolid phase may be possible in a finite system.
Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schr\"odinger equation nor in the integrable nonlinear Schr\"odinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.
The cubic–quintic nonlinear Schrödinger equation with a periodic potential, which is expressed in terms of the Jacobian function dn, is used to model a Bose–Einstein condensate trapped in an optical lattice (in most cases, the dn potential is very close to the ordinary sinusoidal one). The quintic term in the equation represents either effects of three-body collisions (which is relevant in the case when the ordinary two-body collisions are very weak), or an effective deviation of the trapped condensate from the one-dimensionality. The cases of both repulsive and attractive two-body interactions are considered. The same model governs the planar propagation of light in waveguides with the cubic–quintic optical nonlinearity, the periodic potential being induced by transverse modulation of the refractive index. In the case of the attractive quintic term (which is always the case if it accounts for the residual non-one-dimensionality of the tightly trapped condensate), a new family of exact periodic solutions is constructed in terms of Jacobian elliptic functions. Using semi-analytical and numerical methods, a stability region for these periodic states is identified. The effect of boundaries is shown to be very weak. E Kengne dedicates this work to Taa Bah Deffo Takotue Innocent, the Chairman of the Bandjoun Community in Douala (Cameroon).
We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.