Discrete Gross-Pitaevskii Equation Research Articles

Transient ordering in the Gross-Pitaevskii lattice after an energy quench within a nonordered phase

We numerically investigate heating-and-cooling quenches taking place entirely in the non-ordered phase of the discrete Gross-Pitaevskii equation on a three-dimensional cubic lattice. In equilibrium, this system exhibits a U(1)-ordering phase transition at an energy density which is significantly lower than the minimum one during the quench. Yet, we observe that the post-quench relaxation is accompanied by a transient U(1) ordering, namely, the correlation length of U(1) fluctuations significantly exceeds its equilibrium pre-quench value. The longer and the stronger the heating stage of the quench, the stronger the U(1) transient ordering. We identify the origin of this ordering with the emergence of a small group of slowly relaxing lattice sites accumulating a large fraction of the total energy of the system. Our findings suggest that the transient ordering may be a robust feature of a broad class of physical systems. This premise is consistent with the growing experimental evidence of the transient U(1) order in rather dissimilar settings.

Dynamics of topological defects after photoinduced melting of a charge density wave

Charge-density-wave order in a solid can be temporarily "melted" by a strong laser pulse. Here we use the discrete Gross-Pitaevskii equation on a cubic lattice to simulate the recovery of the CDW long-range phase coherence following such a pulse. Our simulations indicate that the recovery process is dramatically slowed down by the three-dimensional topological defects - CDW dislocations - created as a result of strongly nonequilibrium heating and cooling of the system. Overall, the simulated CDW recovery was found to be remarkably reminiscent of a recent pump-probe experiment in LaTe$_3$.

Discrete vortex quantum droplets

We analyze the existence and characteristics of discrete vortex quantum droplets with intrinsic topological charges S of up to 5 in two-component Bose-Einstein condensates in a deep two-dimensional (2D) optical lattice. The 2D discrete Gross-Pitaevskii equation with a Lee-Huang-Yang (LHY) correction term is used to model the condensate, which incorporates local contact interactions and LHY corrections induced by quantum fluctuations. The characteristics of vortical modes with S = 1 and S = 2 are studied in detail. The chemical potential μ and peak density ρmax of the modes are calculated as functions of the norm N. The μ−N curves of on-site-centered and off-site-centered modes with the same number of lattice sites are nearly identical. We make an analysis and reach a conclusion consistent with these simulation results. This system contains a nontrivial type of vortex mode known as the mixed-vortex mode, in which the two components have the same density pattern and different topological charges. To the best of our knowledge, we are the first to report vortex modes with isomorphic density patterns and heteromorphic vortices in the field of quantum droplets.

Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice

We propose theoretically an experimentally realizable method to demonstrate the Lyapunov instability and to extract the value of the largest Lyapunov exponent for a chaotic many-particle interacting system. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We suggest to use imperfect time-reversal of system's dynamics known as Loschmidt echo, which can be realized experimentally by reversing the sign of the Hamiltonian of the system. The routine involves tracking and then subtracting the noise of virtually any observable quantity before and after the time-reversal. We support the theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent can indeed be extracted from the Loschmidt echo routine. We also discuss possible values of experimental parameters required for implementing this proposal.

Fluctuating hydrodynamics for a discrete Gross-Pitaevskii equation: Mapping onto the Kardar-Parisi-Zhang universality class

We show that several aspects of the low-temperature hydrodynamics of a discrete Gross-Pitaevskii equation (GPE) can be understood by mapping it to a nonlinear version of fluctuating hydrodynamics. This is achieved by first writing the GPE in a hydrodynamic form of a continuity and an Euler equation. Respecting conservation laws, dissipation and noise due to the system's chaos are added, thus giving us a nonlinear stochastic field theory in general and the Kardar-Parisi-Zhang (KPZ) equation in our particular case. This mapping to KPZ is benchmarked against exact Hamiltonian numerics on discrete GPE by investigating the non-zero temperature dynamical structure factor and its scaling form and exponent. Given the ubiquity of the Gross-Pitaevskii equation (a.k.a. nonlinear Schrodinger equation), ranging from nonlinear optics to cold gases, we expect this remarkable mapping to the KPZ equation to be of paramount importance and far reaching consequences.

Dynamics of a nonlocal discrete Gross-Pitaevskii equation with defects

We study the dynamics of dipolar gas in deep lattices described by a nonlocal nonlinear discrete Gross-Pitaevskii equation. The stabilities and the propagation properties of traveling plane waves in the system with defects are discussed in detail. For a clean lattice, both energetic and dynamical stabilities of the traveling plane waves are studied. It is shown that the system with attractive local interaction can preserve the stabilities, i.e., the dipoles can stabilize the gas because of repulsive nonlocal dipole-dipole interactions. For a lattice with defects, within a two-mode approximation, the propagation properties of traveling plane waves in the system map onto a nonrigid pendulum Hamiltonian with quasimomentum-dependent nonlinearity (induced by the nonlocal interactions). Competition between defects, quasimomentum of the gas, and nonlocal interactions determines the propagation properties of the traveling plane waves. Critical conditions for crossing from a superfluid regime with propagation preserved to a normal regime with defect-induced damping are obtained analytically and confirmed numerically. In particular, the critical conditions for supporting the superfluidity strongly depend on the defect type and the quasimomentum of the plane waves. The nonlocal interaction can significantly enhance the superfluidity of the system.

Bright solitons in the one-dimensional discrete Gross-Pitaevskii equation with dipole-dipole interactions

A model of the Bose-Einstein condensate of dipolar atoms, confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction, is introduced, taking into regard the dipole-dipole (DD) and contact interactions. The model is based on the discrete nonlinear Schr\odinger equation with an additional nonlocal term accounting for the DD interactions. The existence and stability of fundamental unstaggered solitons are studied for attractive and repulsive signs of both the local and nonlocal interactions. The DD forces strongly affect the shape and stability of on-site and intersite discrete solitons. The corresponding existence and stability regions in the parametric space are summarized in the form of diagrams, which feature a multiple stability exchange between the on-site and intersite families; in the limit of the dominating DD attraction, the on-site solitons are stable, while their intersite counterparts are not. We also demonstrate that the DD interactions reduce the Peierls-Nabarro barrier and enhance the mobility of the discrete solitons.

Self-trapping of Bose-Einstein condensates in optical lattices: The effect of the lattice dimension

In the present paper, we analytically and numerically investigate the dynamics of Bose-Einstein condensates (BECs) loaded into deep optical lattices of one dimension (1D), 2D, and 3D. We focus on the self-trapped state and the effect of the lattice dimension. Under the tight-binding approximation, we obtain an analytical criterion for the self-trapped state of BEC using the time-dependent variational method. The phase diagram for self-trapping, soliton, breather, or diffusion of a BEC cloud is obtained accordingly and verified by directly solving a discrete Gross-Pitaevskii equation numerically. In particular, we find that the criterion and the phase diagrams are modified dramatically by the dimension of the optical lattices.

Nonequilibrium Gross-Pitaevskii dynamics of boson lattice models

Motivated by recent experiments on trapped ultra-cold bosonic atoms in an optical lattice potential, we consider the non-equilibrium dynamic properties of such bosonic systems for a number of experimentally relevant situations. When the number of bosons per lattice site is large, there is a wide parameter regime where the effective boson interactions are strong, but the ground state remains a superfluid (and not a Mott insulator): we describe the conditions under which the dynamics in this regime can be described by a discrete Gross-Pitaevskii equation. We describe the evolution of the phase coherence after the system is initially prepared in a Mott insulating state, and then allowed to evolve after a sudden change in parameters places it in a regime with a superfluid ground state. We also consider initial conditions with a "pi phase" imprint on a superfluid ground state (i.e. the initial phases of neighboring wells differ by pi), and discuss the subsequent appearance of density wave order and "Schrodinger cat" states.