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Journal of Physics B Atomic Molecular and Optical Physics

Two-dimensional solitons in periodically modulated double-well potentials

C. Sudheesh, Nir Bar‐Gill, Boris A. Malomed, Gershon Kurizki

Published Oct 4, 2010• 7 citations
Publisher

Abstract

We introduce a model of a two-dimensional (2D) self-attractive medium embedded into a quasi-1D symmetric double-well potential (DWP), whose depth is subject to periodic modulations (management). The model applies to matter waves in Bose–Einstein condensates (BECs), as well as to the nonlinear transmission of light (in spatial and temporal domains alike). It is known that, in the absence of the management, the DWP induces the spontaneous symmetry breaking (SSB) of 2D solitons, when their norm exceeds a critical value. Above the SSB point, symmetric solitons are unstable, while the system supports stable asymmetric ones. The DWP also admits Josephson oscillations of solitons between the two wells. We study effects of periodic modulations of the DWP's depth on the stability of symmetric, asymmetric, and oscillating solitons. Stability areas for the solitons of these three types are produced in the plane of the modulation frequency and amplitude. The shape of the stability borders is strongly affected by the proximity of the modulation frequency to the frequency of free oscillations of the soliton in the static DWP structure, the solitons being destroyed by the modulations with an arbitrarily small amplitude at the point of the exact resonance. Similar results are obtained for an extended 2D model, which combines the transverse DWP and a longitudinal periodic potential (optical lattice, in terms of BEC). The findings are compared with those reported in other recently studied management models for 1D and 2D solitons, which makes it possible to draw general conclusions about the stability limits of solitons under the resonant management.

DOI
10.1088/0953-4075/43/20/205304

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We introduce a model of a two-dimensional (2D) self-attractive medium embedded into a quasi-1D symmetric double-well potential (DWP), whose depth is subject to periodic modulations (management). The model applies to matter waves in Bose–Einstein condensates (BECs), as well as to the nonlinear transmission of light (in spatial and temporal domains alike). It is known that, in the absence of the management, the DWP induces the spontaneous symmetry breaking (SSB) of 2D solitons, when their norm exceeds a critical value. Above the SSB point, symmetric solitons are unstable, while the system supports stable asymmetric ones. The DWP also admits Josephson oscillations of solitons between the two wells. We study effects of periodic modulations of the DWP's depth on the stability of symmetric, asymmetric, and oscillating solitons. Stability areas for the solitons of these three types are produced in the plane of the modulation frequency and amplitude. The shape of the stability borders is strongly affected by the proximity of the modulation frequency to the frequency of free oscillations of the soliton in the static DWP structure, the solitons being destroyed by the modulations with an arbitrarily small amplitude at the point of the exact resonance. Similar results are obtained for an extended 2D model, which combines the transverse DWP and a longitudinal periodic potential (optical lattice, in terms of BEC). The findings are compared with those reported in other recently studied management models for 1D and 2D solitons, which makes it possible to draw general conclusions about the stability limits of solitons under the resonant management.
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, $\ensuremath{\lesssim}5$ 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.
Models of two-dimensional (2D) traps, with double-well structure in the third direction, for Bose-Einstein condensates are introduced with attractive or repulsive interactions between atoms. The models are based on systems of linearly coupled 2D Gross-Pitaevskii equations, where the coupling accounts for tunneling between the wells. Each well carries an optical lattice (OL) (stable 2D solitons cannot exist without OLs). The linear coupling splits each finite band gap in the spectrum of the single-component model into two subgaps. The main subject of the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons and localized vortices (SSB was not considered before in 2D settings). Using variational approximation (VA) and numerical methods, we demonstrate that, in a system with attraction or repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or vortices), respectively. The corresponding bifurcation destabilizes the original solution branch and gives rise to a stable branch of asymmetric solitons or vortices. The VA provides for an accurate description of the emerging branch of asymmetric solitons. In the model with attraction, all stable branches eventually terminate due to the onset of collapse. Stable asymmetric solitons in higher finite band gaps and vortices with a multiple topological charge are found too. The models also give rise to first examples of embedded solitons and embedded vortices (the states located inside Bloch bands) in two dimensions. In the linearly coupled system with opposite signs of the nonlinearity in the two cores, two distinct types of stable solitons and vortices are found, dominated by either the self-attractive component or the self-repulsive one. In the system with a mismatch between the two OLs, a pseudobifurcation is found: when the mismatch attains its largest value $(\ensuremath{\pi})$, the bifurcation does not happen, as branches of different solutions asymptotically approach each other, but fail to merge.
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.
We introduce a one-dimensional model of Bose-Einstein condensates (BECs), combining the double-well potential, which is a usual setting for the onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic modulation of the nonlinearity, which may be implemented by means of the Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity that is repulsive or attractive on the average are considered. In the former case, the main effect produced by the application of the FRM is spontaneous self-trapping of the condensate in either of the two potential wells in parameter regimes where it would remain untrapped in the absence of the management. In the weakly nonlinear regime, the frequency of intrinsic oscillations in the FRM-induced trapped state is very close to half the FRM frequency, suggesting that the effect is accounted for by a parametric resonance. In the case of the attractive nonlinearity, the FRM-induced effect is the opposite, i.e., enforced detrapping of a state which is self-trapped in its unmanaged form. In the latter case, the frequency of oscillations of the untrapped mode is close to a quarter of the driving frequency, suggesting that a higher-order parametric resonance may account for this effect.
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 $\ensuremath{\omega}$.
We propose a model of a nonlinear double-well potential (NDWP), alias a double-well pseudopotential, with the objective to study an alternative implementation of the spontaneous symmetry breaking (SSB) in Bose-Einstein condensates (BECs) and optical media, under the action of a potential with two symmetric minima. In the limit case when the NDWP structure is induced by the local nonlinearity coefficient represented by a set of two delta functions, a fully analytical solution is obtained for symmetric, antisymmetric, and asymmetric states. In this solvable model, the SSB bifurcation has a fully subcritical character. Numerical analysis, based on both direct simulations and computation of stability eigenvalues, demonstrates that, while the symmetric states are stable up to the SSB bifurcation point, both symmetric and emerging asymmetric states, as well as all antisymmetric ones, are unstable in the model with the delta functions. In the general model with a finite width of the nonlinear-potential wells, the asymmetric states quickly become stable, simultaneously with the switch of the SSB bifurcation from the subcritical to supercritical type. Antisymmetric solutions may also get stabilized in the NDWP structure of the general type, which gives rise to a bistability between them and asymmetric states. The symmetric states require a finite norm for their existence, an explanation to which is given. A full diagram for the existence and stability of the trapped states in the model is produced. Experimental observation of the predicted effects should be possible in BEC formed by several hundred atoms.