Spontaneous symmetry breaking of binary fields in a nonlinear double-well structure

Abstract We introduce a one-dimensional two-component system with the self-focusing cubic nonlinearity concentrated at a symmetric set of two spots. Effects of the spontaneous symmetry breaking (SSB) of localized modes were previously studied in the single-component version of this system. In this work, we study the evolution (in the configuration space of the system) and SSB scenarios for two-component modes of three generic types, as concerns the spatial symmetry of each component: symmetric–symmetric (Sm–Sm), antisymmetric–antisymmetric (AS–AS), and symmetric–antisymmetric (S–AS) ones. In the limit case of the nonlinear potential represented by two δ -functions, solutions are obtained in a semi-analytical form. They feature novel properties, in comparison with the previously studied single-component model. In particular, the SSB of antisymmetric modes is possible solely in the two-component system, and, obviously, S–AS states exist only in the two-component system too. In the general case of the symmetric pair of finite-width nonlinear potential wells, evolution scenarios are very complex. In this case, new results are reported, first, for the single-component model. These are pairs of broken-antisymmetry modes, and of twin-peak symmetric ones, which are generated by saddle-mode bifurcations separated from the transformations previously studied in the single-component setting. With regard to these findings, complex scenarios of the evolution of the two-component solution families are realized in terms of links connecting pairs of modes of three simplest types: (A) two-component ones with unbroken symmetries; (B) single-component modes featuring density peaks in both potential wells; (C) single-component modes which are trapped, essentially, in a single well.

[1]  Boris A. Malomed,et al.  Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites , 2011, 1105.0266.

[2]  Paré,et al.  Approximate model of soliton dynamics in all-optical couplers. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[3]  Marek Trippenbach,et al.  Spontaneous symmetry breaking of solitons trapped in a double-channel potential , 2007, 0704.1601.

[4]  L. Salasnich,et al.  Rabi?Josephson oscillations and self-trapped dynamics in atomic junctions with two bosonic species , 2010, 1012.2989.

[5]  Xiao-Qiang Xu,et al.  Stability and dynamical property for two-species ultracold atoms in double wells , 2008 .

[6]  A. A. Kolokolov,et al.  Stationary solutions of the wave equation in a medium with nonlinearity saturation , 1973 .

[7]  Marek Trippenbach,et al.  Two - dimensional solitons in media with the stripe - shaped nonlinearity modulation , 2010, 1006.3687.

[8]  Boris A. Malomed,et al.  Spontaneous symmetry breaking in a nonlinear double-well structure , 2008, 0810.0859.

[9]  Hassan K. Khalil,et al.  Nonlinear Systems Third Edition , 2008 .

[10]  B. Malomed,et al.  Modulational instability of a wave scattered by a nonlinear center. , 1993, Physical review. B, Condensed matter.

[11]  Boris A. Malomed,et al.  Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices , 2007, 0705.0364.

[12]  S. Ashhab,et al.  External Josephson effect in Bose-Einstein condensates with a spin degree of freedom , 2002 .

[13]  M. Tsubota,et al.  Rabi-Josephson Transitions in Spinor Bose-Einstein Condensates , 2009 .

[14]  Dmitry A. Zezyulin,et al.  Nonlinear modes for the Gross–Pitaevskii equation—a demonstrative computation approach , 2007 .

[15]  B. A. Malomed,et al.  Competition between the symmetry breaking and onset of collapse in weakly coupled atomic condensates , 2010, 1003.3835.

[16]  A. Posazhennikova,et al.  Nonequilibrium Josephson oscillations in Bose-Einstein condensates without dissipation. , 2009, Physical review letters.

[17]  P. Kevrekidis,et al.  Symmetry breaking in symmetric and asymmetric double-well potentials. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  H. Stoof,et al.  Ultracold Quantum Fields , 2009 .

[19]  B. A. Malomed,et al.  Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps , 2008, 0802.1821.

[20]  B. A. Malomed,et al.  Spontaneous symmetry breaking in photonic lattices: Theory and experiment , 2004, cond-mat/0412381.

[21]  A. Smerzi,et al.  Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein Condensates , 1997, cond-mat/9706221.

[22]  Boris A. Malomed,et al.  Solitons in nonlinear lattices , 2011 .

[23]  A. Smerzi,et al.  Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping , 1997 .

[24]  Luca Salasnich,et al.  Atomic Josephson junction with two bosonic species , 2009, 0905.2080.

[25]  Statics and dynamics of Bose-Einstein condensates in double square well potentials. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Gang-Ding Peng,et al.  Soliton switching and propagation in nonlinear fiber couplers: analytical results , 1993 .

[27]  Michael Albiez,et al.  Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction. , 2005, Physical review letters.

[28]  A Study of The Formation of Stationary Localized States Due to Nonlinear Impurities Using The Discrete Nonlinear Schrödinger Equation , 1996, cond-mat/9606125.

[29]  Yuri S. Kivshar,et al.  Coupled mode theory for Bose-Einstein condensates , 2000, QELS 2000.

[30]  Stationary solutions of the Gross-Pitaevskii equation with linear counterpart , 1998, cond-mat/9809203.

[31]  Peter S. Lomdahl,et al.  The discrete self-trapping equation , 1985 .

[32]  F. Toigo,et al.  Nonlinear quantum model for atomic Josephson junctions with one and two bosonic species , 2010, 1002.2881.

[33]  Panayotis G. Kevrekidis,et al.  Symmetry-Breaking Bifurcation in Nonlinear Schrödinger/Gross-Pitaevskii Equations , 2007, SIAM J. Math. Anal..

[34]  G. Gligoric,et al.  Interface solitons in one-dimensional locally coupled lattice systems , 2010, 1008.2178.

[35]  Luc Bergé,et al.  Wave collapse in physics: principles and applications to light and plasma waves , 1998 .

[36]  M. Oberthaler,et al.  Classical bifurcation at the transition from Rabi to Josephson dynamics. , 2010, Physical review letters.

[37]  M. Lewenstein,et al.  Josephson oscillations in binary mixtures of F=1 spinor Bose-Einstein condensates , 2009, 0902.3206.

[38]  Spatial quadratic solitons guided by narrow layers of a nonlinear material , 2011, 1105.0100.

[39]  B. Malomed,et al.  Solitons supported by localized nonlinearities in periodic media , 2011, 1103.1557.

[40]  Sandro Stringari,et al.  Theory of ultracold atomic Fermi gases , 2007, 0706.3360.

[41]  M. Weinstein,et al.  Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation , 2003, nlin/0309020.

[42]  Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects , 2011 .

[43]  Mark Edwards,et al.  Symmetry-breaking and Symmetry-restoring Dynamics of a Mixture of Bose-Einstein Condensates in a Double Well , 2008, 0811.1921.

[44]  Zheng-Wei Zhou,et al.  Bose-Einstein condensates on a ring with periodic scattering length: Spontaneous symmetry breaking and entanglement , 2008 .

[45]  K. W. Mahmud Quantum phase-space picture of Bose-Einstein condensates in a double well (17 pages) , 2005 .

[46]  W. Hai,et al.  Stability and chaotic behavior of a two-component Bose–Einstein condensate , 2006 .

[47]  B. Malomed,et al.  Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials , 2011, 1106.5209.

[48]  B. A. Malomed,et al.  Two-component nonlinear Schrodinger models with a double-well potential , 2008, 0805.0023.

[49]  Gerard J. Milburn,et al.  Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential , 1997 .

[50]  Molina Mi,et al.  Nonlinear impurities in a linear chain. , 1993 .

[51]  B A Malomed,et al.  Symmetry breaking in linearly coupled dynamical lattices. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  N. Akhmediev,et al.  Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. , 1993, Physical review letters.

[54]  Boris A. Malomed,et al.  Solitary waves in coupled nonlinear waveguides with Bragg gratings , 1998 .

[55]  A. Messiah Quantum Mechanics , 1961 .

[56]  Yuri S. Kivshar,et al.  Optical Solitons: From Fibers to Photonic Crystals , 2003 .

[57]  B. N. Brockhouse,et al.  Pseudopotentials in the theory of metals , 1966 .