We propose a control mechanism for the stabilization of symmetry-broken modes in Bose-Einstein condensates and optical beams trapped in double-well potentials in the presence of symmetric or asymmetric noise. The control, based on periodic $\ensuremath{\pi}$ shifts of the phase in one well, is demonstrated for solitons and in the two-mode approximation. The latter setting is considered in both the mean-field and quantum (Bose-Hubbard) regimes.
We propose a control mechanism for the stabilization of symmetry-broken modes in Bose-Einstein condensates and optical beams trapped in double-well potentials in the presence of symmetric or asymmetric noise. The control, based on periodic \ensuremathπ shifts of the phase in one well, is demonstrated for solitons and in the two-mode approximation. The latter setting is considered in both the mean-field and quantum (Bose-Hubbard) regimes.
In this paper we study the solutions to a nonlinear Schrödinger equation with harmonic and periodic potentials, motivated from the recent interest in these models as mean-field descriptions of Bose–Einstein condensates. We use a two-mode Galerkin approximation to study the dynamics of the full model. The phase plane and stability analysis of the reduced model yield very good agreement with the findings of the full partial differential equation. A particularly interesting finding of the stability analysis is a spontaneous symmetry breaking through a branching bifurcation, resulting in the stabilization of asymmetric states and the destabilization of symmetric or anti-symmetric ones. We also highlight the important differences between the cases of symmetric potentials and those of weakly asymmetric potentials.
We study the coherent nonlinear tunneling dynamics of a binary mixture of Bose-Einstein condensates in a double-well potential. We demonstrate the existence of a type of mode associated with the ``swapping'' of the two species in the two wells of the potential. In contrast to the symmetry-breaking macroscopic quantum self-trapping (MQST) solutions, the swapping modes correspond to the tunneling dynamics that preserves the symmetry of the double-well potential. As a consequence of two distinct types of broken-symmetry MQST phases where the two species localize in different potential wells or coexist in the same well, the corresponding symmetry-restoring swapping modes result in dynamics where the two species either avoid or chase each other. In view of the possibility to control the interaction between the species, the binary mixture offers a very robust system to observe these novel effects as well as the phenomena of Josephson oscillations and $\ensuremath{\pi}$ modes.
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.
We study spontaneous symmetry breaking in a system of two parallel quasi-one-dimensional traps (cores), equipped with optical lattices (OLs) and filled with a Bose-Einstein condensate (BEC). The cores are linearly coupled by tunneling (the model may also be interpreted in terms of spatial solitons in parallel planar optical waveguides with a periodic modulation of the refractive index). Analysis of the corresponding system of linearly coupled Gross-Pitaevskii equations (GPEs) reveals that spectral band gaps of the single GPE split into subgaps. Symmetry breaking in two-component BEC solitons is studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both traps; the mixed situation, with repulsion in one trap and attraction in the other (RA), is considered too. In all the cases, stable asymmetric solitons are found, bifurcating from symmetric or antisymmetric ones (and destabilizing them), in the AA and RR systems, respectively. In either case, bistability is predicted, with a nonbifurcating stable branch, either antisymmetric or symmetric, coexisting with asymmetric ones. Solitons destabilized by the bifurcation tend to rearrange themselves into their stable asymmetric counterparts. In addition to the fundamental solitons, branches of twisted (odd) solitons in the AA system, and twisted bound states of fundamental solitons in both AA and RR systems, are found too. The impact of a phase mismatch, $\ensuremath{\Delta}$, between the OLs in the two cores is also studied. It is concluded that $\ensuremath{\Delta}=\ensuremath{\pi}∕2$ only mildly deforms the picture, while $\ensuremath{\Delta}=\ensuremath{\pi}$ changes it drastically, replacing the symmetry-breaking bifurcations by pseudobifurcations, with the branch of asymmetric solutions asymptotically approaching its symmetric or antisymmetric counterpart (in the AA and RR system, respectively), rather than splitting off from it. Also considered is a related model, for a binary BEC in a single-core trap with the OL, assuming that the two species (representing different spin states of the same atom) are coupled by linear interconversion. In that case, the symmetry-breaking bifurcations in the AA and RR models switch their character, if the interspecies nonlinear interaction becomes stronger than the intraspecies nonlinearity.
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.
We investigate the phenomena of symmetry breaking and phase transition in the ground state of Bose–Einstein condensates (BECs) trapped in a double square well and in an optical lattice well, respectively. By using standing-wave expansion method, we present symmetric and asymmetric ground state solutions of nonlinear Schrödinger equation (NLSE) with a symmetric double square well potential for attractive nonlinearity. In particular, we study the ground state wave function's properties by changing the depth of potential and atomic interactions (here we restrict ourselves to the attractive regime). By using the Fourier grid Hamiltonian method, we also reveal a phase transition of BECs trapped in one-dimensional optical lattice potential.