States without a linear counterpart in Bose-Einstein condensates

D’Agosta, Roberto; Presilla, Carlo

doi:10.1103/physreva.65.043609

Cited by 73 publications

(95 citation statements)

References 23 publications

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“…The most interesting results, however, are related with the emergence of new properties which do not have a counterpart in the linear case. Stationary solutions of the GPE which do not reduce to any of the eigenfunctions for a vanishing nonlinearity have been studied in [6] using a tight binding approximation, in and two-wells systems, in [13][14][15][16][17].The mean-field model of a quasi-1D BEC trapped in a KP potential is governed by the following nonlinear Schrödinger (or Gross-Pitaevski) equation:where µ is the chemical potential and g the nonlinear coupling constant. The KP external potential is given by equispaced delta-functions: V (x) = p ∞ n=−∞ δ(x−na), having a lattice constant a.…”

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confidence: 99%

Nonlinear Krönig-Penney model

Smerzi

2004

Phys. Rev. E

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We study the nonlinear Schrödinger equation with a periodic delta-function potential. This realizes a nonlinear Krönig-Penney model, with physical applications in the context of trapped Bose-Einstein condensate alkaly gases and in the transmission of signals in optical fibers. We find analytical solutions of zero-current Bloch states. Such wave-functions have the same periodicity of the potential, and, in the linear limit, reduce to the Bloch functions of the Krönig-Penney model. We also find new classes of solutions having a periodicity different from that of the external potential. We calculate the chemical potential of such states and compare it with the linear excitation spectrum.PACS numbers: 03.75. Kk, 03.75. Lm, 05.45.Yv Nonlinearity can deeply modify the Bloch theory of non-interacting atoms trapped in periodic potentials. Loop structures, energetic and dynamical instabilities, solitons and "generalized" Bloch states (i.e. states which do not share the same periodicity of the lattice), all arise in the context of a nonlinear Schrödinger (or GrossPitaevskii) equation. Applications span, for instance, the physics of dilute Bose-Einstein condensed gas trapped in optical lattices [1][2][3][4][5][6][7][8][9] or the propagation of signals in optical fibers [10] In this paper we study analitically the nonlinear Schrödinger equation with an external Krönig-Penney (KP) potential, given by a periodic array of deltafunctions. The linear Schrödinger equation with the same potential has been solved quite early in the '30, playing a distinguished role as a model in metal's theory [11,12].It is noticeable that also several properties of the nonlinear Schrödinger equation, with the same KP external potential, can be derived analitically. The most interesting results, however, are related with the emergence of new properties which do not have a counterpart in the linear case. Stationary solutions of the GPE which do not reduce to any of the eigenfunctions for a vanishing nonlinearity have been studied in [6] using a tight binding approximation, in and two-wells systems, in [13][14][15][16][17].The mean-field model of a quasi-1D BEC trapped in a KP potential is governed by the following nonlinear Schrödinger (or Gross-Pitaevski) equation:where µ is the chemical potential and g the nonlinear coupling constant. The KP external potential is given by equispaced delta-functions: V (x) = p ∞ n=−∞ δ(x−na), having a lattice constant a. Since the external potential has a step-like shape, it is useful to rewrite the GPE in hydrodynamic form. With ψ (x) = ρ (x) exp [−iΘ (x)] (and in dimensionless units), we have:, aρ → ρ and normalization n±1 n dx ρ(x) = 1. N 0 is the number of atoms in each well, and the integration constants α, β are fixed by the boundary conditions. In particolar, α has a simple physical meaning, being the current carried by the order parameter ψ(x):(where J a 2 h → J). Bloch state. We derive a class of stationary solutions of Eq.s(2) having (i) the same periodicity of the external potential (Bloch states), an...

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confidence: 99%

Nonlinear Krönig-Penney model

Smerzi

2004

Phys. Rev. E

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“…We will give diagrams to illustrate this. Similar diagrams for a quartic potential can be found in [13], however they do not correspond to any analytic solutions known to us.…”

Section: Symmetry Breaking States (Symmetric Wells)mentioning

confidence: 75%

Statics and dynamics of Bose-Einstein condensates in double square well potentials

et al. 2006

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In this paper we treat the behavior of Bose Einstein condensates in double square well potentials, both of equal and different depths. For even depth, symmetry preserving solutions to the relevant nonlinear Schrödinger equation is known, just as in the linear limit. When the nonlinearity is strong enough, symmetry breaking solutions also exist, side by side with the symmetric one. Interestingly, solutions almost entirely localized in one of the wells are known as an extreme case. Here we outline a method for obtaining all these solutions for repulsive interactions. The bifurcation point at which, for critical nonlinearity, the asymmetric solutions branch off from the symmetry preserving ones is found analytically. We also find this bifurcation point and treat the solutions generally via a Josephson Junction model.When the confining potential is in the form of two wells of different depth, interesting new phenomena appear. This is true of both the occurrence of the bifurcation point for the static solutions, and also of the dynamics of phase and amplitude varying solutions. Again a generalization of the Josephson model proves useful. The stability of solutions is treated briefly.

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“…which predicts equilibria, X = X c [see Eq. (5)], at points with U (X c ) = 0. First, for vanishingly small X, the expansion of potential (10) yields, for both forms of P (A) defined by Eqs.…”

Section: The Analytical Approachmentioning

confidence: 99%

Symmetry breaking in competing single-well linear-nonlinear potentials

et al. 2018

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The combination of linear and nonlinear potentials, both shaped as a single well, enables competition between the confinement and expulsion induced by the former and latter potentials, respectively. We demonstrate that this setting leads to spontaneous symmetry breaking (SSB) of the ground state in the respective generalized nonlinear Schrödinger (Gross-Pitaevskii) equation, through a spontaneous off-center shift of the trapped mode. Two different SSB bifurcation scenarios are possible, depending on the shape of the nonlinearity-modulation profile, which determines the nonlinear potential. If the profile is bounded (remaining finite at |x| → ∞), at a critical value of the integral norm the spatially symmetric state loses its stability, giving rise to a pair of mutually symmetric stable asymmetric ones via a direct pitchfork bifurcation. On the other hand, if the nonlinear potential is unbounded, two unstable asymmetric modes merge into the symmetric metastable one and destabilize it, via an inverted pitchfork bifurcation. Parallel to a systematic numerical investigation, basic results are obtained in an analytical form. The settings can be realized in Bose-Einstein condensates and nonlinear optical waveguides.

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States without a linear counterpart in Bose-Einstein condensates

Cited by 73 publications

References 23 publications

Nonlinear Krönig-Penney model

Nonlinear Krönig-Penney model

Statics and dynamics of Bose-Einstein condensates in double square well potentials

Symmetry breaking in competing single-well linear-nonlinear potentials

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