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Physical Review A

Stability limits for two-dimensional matter-wave solitons in a time-modulated quasi-one-dimensional optical lattice

Thawatchai Mayteevarunyoo, Boris A. Malomed, Monai Krairiksh

Published Nov 14, 2007• 20 citations
Publisher

Abstract

In a basic physical model where two-dimensional (2D) matter-wave solitons may be stable, namely, the Gross-Pitaevskii equation with the self-attractive nonlinearity and quasi-one-dimensional (1D) optical-lattice (OL) potential, we test robustness of the solitons against periodic time modulation of the OL strength. Stability diagrams for the 2D solitons are presented in the plane of the modulation depth and frequency. Basic features of the diagrams are explained with the help of the variational approximation for the stationary counterpart of the model. In the Bose-Einstein condensate of $^{7}\text{L}\text{i}$ atoms, the stable 2D solitons may contain the number of atoms in the range of ${10}^{4}--{10}^{5}$, relevant values of the OL strength and modulation frequency being, respectively, $\ensuremath{\lesssim}5$ recoil energies and $\ensuremath{\lesssim}10\text{ }\text{kHZ}$. Head-on collisions between stable 2D solitons moving in the unconfined direction are studied in detail too, for velocities up to $\ensuremath{\sim}5\text{ }\text{cm}∕\text{s}$. A border between quasi-elastic collisions and merger of the solitons into a single localized state is identified. In some cases, the soliton produced by the merger is stable against collapse, which was not observed before in the static OL potential either.

DOI
10.1103/physreva.76.053612

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In a basic physical model where two-dimensional (2D) matter-wave solitons may be stable, namely, the Gross-Pitaevskii equation with the self-attractive nonlinearity and quasi-one-dimensional (1D) optical-lattice (OL) potential, we test robustness of the solitons against periodic time modulation of the OL strength. Stability diagrams for the 2D solitons are presented in the plane of the modulation depth and frequency. Basic features of the diagrams are explained with the help of the variational approximation for the stationary counterpart of the model. In the Bose-Einstein condensate of 7Li^{7}\text{L}\text{i}7Li atoms, the stable 2D solitons may contain the number of atoms in the range of 104−−105{10}^{4}--{10}^{5}104−−105, relevant values of the OL strength and modulation frequency being, respectively, \ensuremath≲5\ensuremath{\lesssim}5\ensuremath≲5 recoil energies and \ensuremath≲10 kHZ\ensuremath{\lesssim}10\text{ }\text{kHZ}\ensuremath≲10 kHZ. Head-on collisions between stable 2D solitons moving in the unconfined direction are studied in detail too, for velocities up to \ensuremath∼5 cm∕s\ensuremath{\sim}5\text{ }\text{cm}∕\text{s}\ensuremath∼5 cm∕s. A border between quasi-elastic collisions and merger of the solitons into a single localized state is identified. In some cases, the soliton produced by the merger is stable against collapse, which was not observed before in the static OL potential either.
By means of the variational approximation (VA) and systematic simulations, we study dynamics and stability boundaries for solitons in a two-dimensional (2D) self-attracting Bose-Einstein condensate (BEC), trapped in an optical lattice (OL) whose amplitude is subjected to periodic time modulation (the modulation frequency $\ensuremath{\omega}$ may be in the range of several kHz). Regions of stability of the solitons against the collapse and decay are identified in the space of the model's parameters. A noteworthy result is that the stability limit may reach the largest (100%) modulation depth, and the collapse threshold may exceed its classical value in the static lattice (which corresponds to the norm of Townes soliton). The minimum norm Nmin{N}_{\text{min}}Nmin​ necessary for the stability of the solitons is identified too. It features a strong dependence on $\ensuremath{\omega}$ at low frequencies, due to a resonant decay of the soliton. Predictions of the VA are reasonably close to results of the simulations. In particular, the VA helps understand salient resonant features in the shape of the stability boundaries observed with the variation of $\ensuremath{\omega}$.
Using the variational approximation and direct simulations in real and imaginary time, we find stable two-dimensional (2D) and 3D solitons in the self-attractive Gross-Pitaevskii equation (GPE) with a potential which is uniform in one direction (z)(z)(z) and periodic in the others (however, the quasi-1D potentials cannot stabilize 3D solitons). The family of solitons includes single- and multiple-peaked ones. The results apply to Bose-Einstein condensates (BEC's) in optical lattices (OL's) and to spatial or spatiotemporal solitons in layered optical media. This is the first prediction of mobile 2D and 3D solitons in BEC's, as they keep mobility along zzz. Head-on collisions of in-phase solitons lead to their fusion into a collapsing pulse. Slow collisions between two multiple-peaked solitons whose main peaks are separated by an intermediate channel end up with their fusion into one single-peaked soliton in the middle channel, $\ensuremath{\simeq}1∕3$ of the original number of atoms being shed off. Stable localized states in the self-repulsive GPE with the low-dimensional OL combined with a parabolic trap are found too. Two such pulses in one channel perform recurrent elastic collisions, periodically featuring sharp interference patterns in the strong-overlap state.
The two-dimensional (2D) matter-wave soliton families supported by an external potential are systematically studied, in a vicinity of the junction between stable and unstable branches of the families. In this case the norm of the solution attains a minimum, facilitating the creation of such excitation. We study the dynamics and stability boundaries for fundamental solitons in a 2D self-attracting Bose-Einstein condensate (BEC), trapped in an quasiperiodic optical lattice (OL), with the amplitude subject to periodic time modulation.
By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.
Dynamics and stability of solitons in two-dimensional (2D) Bose-Einstein condensates (BEC), with one-dimensional (1D) conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media (with attractive atomic systems), the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media (repulsive BEC case) with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.
We propose an experimentally relevant scheme to create stable solitons in a three-dimensional Bose-Einstein condensate confined by a one-dimensional optical lattice (OL), using spatially uniform temporal modulation of the scattering length (through ac magnetic field tuned close to the Feshbach resonance). Another physical implication of the model is a possibility to create stable 3D “light bullets” in an optical medium with a longitudinal alternating self-focusing/defocusing structure, and periodic modulation of the refractive index in a transverse direction. We develop a variational approximation to identify a stability region in the parametric space, and verify the existence of stable breathing solitons in direct simulations. Both methods reveal that stable solitons may be supported if the average value of the nonlinear coefficient (whose sign corresponds to attraction between atoms) and the OLs strength exceed well-defined minimum values. A moderately strong OL supports multi-peaked solitons, with essential interaction between the peaks.