Symmetry breaking in symmetric and asymmetric double-well potentials

Abstract

Motivated by recent experimental studies of matter waves and optical beams in double-well potentials, we study the corresponding solutions of the nonlinear Schrödinger equation. Using a Galerkin-type approach, we obtain a detailed handle on the nonlinear solution branches of the problem, starting from the corresponding linear ones, and we predict the relevant bifurcations for both attractive and repulsive nonlinearities. The dynamics of the ensuing unstable solutions is also examined. The results illustrate the differences that arise between the steady states and the bifurcations emerging in symmetric and asymmetric double wells.

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

“…The answer, naturally, depends on the form of the potential. The above reduction has been extremely successful in tackling double well potentials in BECs [223][224][225], as well as in optical systems [226]. In this simplest case, a two-mode description is sufficient to extract the prototypical dynamics of the system with M = 2.…”

Symmetry-Breaking Effects for Polariton Condensates in Double-Well Potentials

“…(20) and for the anti-symmetric one of Eq. (19), as well as for the asymmetric branch which is theoretically predicted for the parameters of our doublewell potential to bifurcate from the anti-symmetric solution for l [ 0:7722 and [49] (where the spectrum is chiefly on the imaginary axis), here the spectrum contains predominantly decaying modes with ImðxÞ\0: For the stable symmetric ground state in Fig. 3, all modes are decaying except for the symmetry mode associated with x ¼ 0; while for the unstable anti-symmetric mode of the bottom panel the eigenfrequency associated with the growth is purely imaginary with ImðxÞ [ 0: On the other hand, for the asymmetric modes of Fig.…”

“…(21). It is anticipated that the presence of loss and gain will not (generically) modify the nature of the bifurcations in comparison to the Hamiltonian case [49]. Namely, an asymmetric solution will bifurcate from the symmetric one in the focusing nonlinearity case of s ¼ À1; due to a non-vanishing contribution of the anti-symmetric part in the solution, while on the contrary, an asymmetric mode will emanate from the anti-symmetric one in the defocusing nonlinearity setting of s ¼ 1 (due to a symmetric contribution within the solution).…”

“…Importantly, the latter technique was used for the study of a ''polariton Josephson junction'' [38], in the spirit of earlier studies on ''bosonic Josephson junctions'' [39] in the context of atomic BECs. Importantly, a large volume of theoretical studies has accompanied these developments, first in the context of atomic BECs, through investigations related to finite-mode reductions and symmetry-breaking bifurcations [40][41][42][43][44][45][46][47][48][49], quantum effects [50], and nonlinear variants of the double-well potential [51], and more recently in the context of polariton condensates, especially as concerns Josephson oscillations therein [52][53][54][55]. It should be mentioned in passing that similar (spontaneous symmetry breaking) effects have been monitored in the realm of nonlinear optics: in this context, formation of asymmetric states in dual-core fibers [56][57][58][59][60][61][62], self-guided laser beams in Kerr media [63], and optically-induced dual-core waveguiding structures in photorefractive crystals [64] have been reported.…”

Switch between the Types of the Symmetry Breaking Bifurcation in Optically Induced Photorefractive Rotational Double-Well Potential

“…In nonlinear systems, because additional nonlinearity always give rise to the effect of SSB, phase transition and SSB are also important issues and attract great attention. Among many kinds of nonlinear systems, double-well potential (DWP) or dual-core system is the most essential model employed to study the phase transition and SSB of the nonlinear states [2][3][4][5][6][7][8][9]. In DWP system, the important process relating to the phase transition and SSB is symmetric breaking bifurcation (SBB), which determines the process of symmetric states transiting to the asymmetric ones [10][11][12][13][14][15][16][17].…”