Abstract
The paper combines two topics belonging to the general theme of the spontaneous symmetry breaking (SSB) in systems including two basic competing ingredients: the self-focusing cubic nonlinearity and a double-well-potential (DWP) structure. Such systems find diverse physical realizations, chiefly in optical waveguides, made of a nonlinear material and featuring a transverse DWP structure, and in models of atomic BEC with attractive inter-atomic interactions, loaded into a pair of symmetric potential wells coupled by tunneling across the separating barrier. With the increase of the nonlinearity strength, the SSB occurs at a critical value of the strength. The first part of the paper offers a brief overview of the topic. The second part presents a model which is designed as the simplest one capable to produce the SSB phenomenology in the one-dimensional geometry. The model is based on the DWP built as an infinitely deep potential box, which is split into two wells by a delta-functional barrier at the central point. Approximate analytical predictions for the SSB are produced for two limit cases: strong (deep) or weak (shallow) splitting of the potential box by the central barrier. Critical values of the strength of the nonlinearity at the SSB point, represented by the norm of the stationary wave field, are found in both cases (the critical strength is small in the former case, and large in the latter one). For the intermediate case, a less accurate variational approximation (VA) is developed.
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References
D. Landau, E.M. Lifshitz, Quantum Mechanics (Nauka Publishers, Moscow, 1974)
S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 80, 1215 (2008); H.T.C. Stoof, K.B. Gubbels, D.B.M. Dickrsheid, Ultracold Quantum Fields (Springer, Dordrecht, 2009)
Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)
E.B. Davies, Symmetry breaking in a non-linear Schrödinger equation. Commun. Math. Phys. 64, 191–210 (1979)
J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, The discrete self-trapping equation. Physica D 16, 318–338 (1985)
A.W. Snyder, D.J. Mitchell, L. Poladian, D.R. Rowland, Y. Chen, Physics of nonlinear fiber couplers. J. Opt. Soc. Am. B 8, 2101–2118 (1991)
G. Iooss, D.D. Joseph, Elementary Stability Bifurcation Theory (Springer, New York, 1980)
E.M. Wright, G.I. Stegeman, S. Wabnitz, Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers. Phys. Rev. A 40, 4455–4466 (1989)
C. Paré, M. Fłorjańczyk, Approximate model of soliton dynamics in all-optical couplers. Phys. Rev. A 41, 6287–6295 (1990)
A.I. Maimistov, Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides. Kvant. Elektron. 18, 758–761 [Sov. J. Quantum Electron. 21, 687–690 (1991)]
N. Akhmediev, A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett. 70, 2395–2398 (1993)
B.A. Malomed, I. Skinner, P.L. Chu, G.D. Peng, Symmetric and asymmetric solitons in twin-core nonlinear optical fibers. Phys. Rev. E 53, 4084 (1996)
G.L. Alfimov, P.G. Kevrekidis, V.V. Konotop, M. Salerno, Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential. Phys. Rev. E 66, 046608 (2002)
G.J. Milburn, J. Corney, E.M. Wright, D.F. Walls, Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential. Phys. Rev. A 55, 4318–4324 (1997)
A. Smerzi, S. Fantoni, S. Giovanazzi, S.R. Shenoy, Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates. Phys. Rev. Lett. 79, 4950–4953 (1997)
V.M. Pérez-GarcÃa, H. Michinel, H. Herrero, Bose-Einstein solitons in highly asymmetric traps. Phys. Rev. A 57, 3837–3842 (1998)
M. Matuszewski, B.A. Malomed, M. Trippenbach, Spontaneous symmetry breaking of solitons trapped in a double-channel potential. Phys. Rev. A 75, 063621 (2007)
G. Schön, A.D. Zaikin, Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra small tunnel junctions. Phys. Rep. 198, 237–412 (1990)
A.V. Ustinov, Solitons in Josephson junctions. Physica D 123, 315–329 (1998)
S. Raghavan, A. Smerzi, S. Fantoni, S.R. Shenoy, Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, \(\pi \) oscillations, and macroscopic quantum self-trapping. Phys. Rev. A 59, 620–633 (1999)
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction. Phys. Rev. Lett. 95, 010402 (2005)
P.G. Kevrekidis, Z. Chen, B.A. Malomed, D.J. Frantzeskakis, M.I. Weinstein, Spontaneous symmetry breaking in photonic lattices: theory and experiment. Phys. Lett. A 340, 275–280 (2005)
T. Heil, I. Fischer,W. Elsässer, J. Mulet, C.R. Mirasso, Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers. Phys. Rev. Lett. 86, 795–798 (2000)
P. Hamel, S. Haddadi, F. Raineri, P. Monnier, G. Beaudoin, I. Sagnes, A. Levenson, A.M. Yacomotti, Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers. Nat. Photonics 9, 311–315 (2015)
A. Sigler, B.A. Malomed, Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations. Physica D 212, 305–316 (2005)
M. Liu, D.A. Powell, I.V. Shadrivov, M. Lapine, Y.S. Kivshar, Spontaneous chiral symmetry breaking in metamaterials. Nat. Commun. 5, 4441 (2014)
B.A. Malomed (ed.), Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations (Springer, Berlin, 2013)
B.A. Malomed, Symmetry breaking in laser cavities. Nat. Photonics 9, 287–289 (2015)
B.A. Malomed, Variational methods in nonlinear fiber optics and related fields. Prog. Opt. 43, 71–193 (2002)
V.I. Karpman, V.V. Solov’ev, A perturbation approach to the 2-soliton systems. Physica D 3, 487–502 (1981)
J.P. Gordon, Interaction forces among solitons in optical fibers. Opt. Lett. 8, 596–598 (1983)
F.M. Mitschke, L.F. Mollenauer, Experimental observation of interaction forces between solitons in optical fibers. Opt. Lett. 12, 355–357 (1987)
Y.S. Kivshar, B.A. Malomed, Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763–915 (1989)
B.A. Malomed, Potential of interaction between two- and three-dimensional solitons. Phys. Rev. E 58, 7928–7933 (1998)
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Malomed, B.A. (2016). Spontaneous Symmetry Breaking in Nonlinear Systems: An Overview and a Simple Model. In: Tlidi, M., Clerc, M. (eds) Nonlinear Dynamics: Materials, Theory and Experiments. Springer Proceedings in Physics, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-319-24871-4_7
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