Solitary waves in coupled nonlinear waveguides with Bragg gratings

Mak, William C. K.; Chu, P.L.; Malomed, Boris A.

doi:10.1364/josab.15.001685

Cited by 116 publications

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“…The ME spectrum gives rise to the well-known bandgap structure. Note that, in the model of two linearly coupled Bragg gratings (i.e., a system of four equations for counterpropagating waves in two cores), the coupling leads to shrinkage or closure of the spectral gap [16]. In the present case, the effect of the coupling is more complex.…”

Section: Linear Spectrummentioning

confidence: 82%

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Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices

Gubeskys

Malomed

2007

Phys. Rev. A

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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 bandgaps 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, bi-stability 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, ∆, between the OLs in the two cores is also studied. It is concluded that ∆ = π/2 only mildly deforms the picture, while ∆ = π changes it drastically, replacing the symmetry-breaking bifurcations by pseudo-bifurcations, 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 inter-species nonlinear interaction becomes stronger than the intra-species nonlinearity.

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Section: Linear Spectrummentioning

confidence: 82%

“…In optics, spontaneous symmetry breaking in two-component gap solitons was studied in models of dualcore [16] and triple-core [17] fiber Bragg gratings (solitons in triangular and planar triple-core optical waveguides without gratings were considered in Refs. [18]).…”

Section: Introduction and The Modelmentioning

confidence: 99%

Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices

Gubeskys

Malomed

2007

Phys. Rev. A

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“…8). The effects of adding the linear inter-species coupling between the components has also been studied in some detail [313,[323][324][325][326][327][328][329][330][331][332][333]. One of the salient features of adding the linear inter-species coupling is the suppression or promotion of the transition to miscibility (cf.…”

Section: Spinor/multicomponent Condensatesmentioning

confidence: 99%

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

2008

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Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

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“…Due to this linear coupling, a soliton in this system necessarily has its components in both cores. As shown by Mak, Malomed and Chu [1998a], an obviously symmetric soliton with equal components in the two cores, E F 1 = E F 2 and E B1 = E B2 , becomes unstable when its energy exceeds a certain threshold. The instability gives rise to a pitchfork bifurcation that generates a pair of stable asymmetric solitons which are mirror images of each other.…”

Section: Introduction § § § Introductionmentioning

confidence: 98%

“…Finally, we mention a dual-core nonlinear optical fiber, each core containing a Bragg grating, which is described by four equations (Mak, Malomed and Chu [1998a]),…”

Section: Introduction § § § Introductionmentioning

confidence: 99%

Optical solitons in periodic media with resonant and off-resonant nonlinearities

et al. 2001

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Solitary waves in coupled nonlinear waveguides with Bragg gratings

Cited by 116 publications

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Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices

Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices

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

Optical solitons in periodic media with resonant and off-resonant nonlinearities

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