Modulational instability in Bose-Einstein condensates in optical lattices

Konotop, V. V.; Salerno, Mario

doi:10.1103/physreva.65.021602

Cited by 336 publications

(363 citation statements)

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“…We note in passing that the dimensionality reduction of the GP equation can also be done self-consistently, using multiscale expansion techniques [106,107]. It is also worth mentioning that more recently a rigorous derivation of the 1D GP equation was presented in Ref.…”

Section: Lower-dimensional Gp Equationsmentioning

confidence: 99%

“…In this case, and in connection to the above discussion, a quite relevant issue is the possibility of nonlinear localization of matter-waves in the gaps of the linear spectrum, i.e., the formation of fundamental nonlinear structures in the form of gap solitons, as observed in the experiment [44]. The underlying mechanism can effectively be described by means of the so-called Bloch-wave envelope approximation near the band edge, first formulated in the context of optics [171], and then used in the BEC context as well [172,106,173] (see also Ref. [174] and the review [175]).…”

Section: Reduced Mean-field Models For Becs In Optical Latticesmentioning

confidence: 99%

“…(22) is simply the 1D optical lattice V (z) = V 0 cos 2 (kz). Similarly, in the quasi-2D case, an "egg-carton potential" V (x, y) = V 0 cos 2 (k x x) + cos 2 (k y y) is relevant for disk-shaped condensates.We note in passing that the dimensionality reduction of the GP equation can also be done self-consistently, using multiscale expansion techniques [106,107]. It is also worth mentioning that more recently a rigorous derivation of the 1D GP equation was presented in Ref.…”

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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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Section: Lower-dimensional Gp Equationsmentioning

confidence: 99%

Section: Reduced Mean-field Models For Becs In Optical Latticesmentioning

confidence: 99%

mentioning

confidence: 99%

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

2008

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“…Additionally, the MI has been examined recently in the context of optical lattices in BECs both in one-dimensional and quasi-one dimensional systems, as well as in multiple dimensions. In such settings, it has been predicted theoretically [12,13] and verified experimentally [14,15] to lead to destabilization of plane waves, and in turn to delocalization in momentum space (equivalent to localization in position space, and the formation of solitary wave structures).…”

Section: Introductionmentioning

confidence: 99%

Modulational instability of Gross-Pitaevskii-type equations in1+1dimensions

et al. 2003

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The modulational stability of the nonlinear Schrödinger (NLS) equation is examined in the case with a quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in Bose-Einstein condensates (BECs). The theoretical analysis invokes a lens-type transformation that converts the Gross-Pitaevskii into a regular NLS equation with an additional growth term. This analysis suggests the particular interest of a specific time-varying potential (∼ (t + t ⋆ ) −2 ). We examine both this potential, as well as the time independent one numerically and conclude by suggesting experiments for the production of solitonic wave-trains in BEC.

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“…A more adequate analysis of both periodic and localized modes of the continuous Gross-Pitaevskii (GP) equation with a periodic lattice potential is still missing. Some of the recent studies of the GP model [13,14,15,16] point towards interesting features of the BEC dynamics that should be observed beyond the applicability of discrete models.…”

Section: Introductionmentioning

confidence: 99%

Bose-Einstein condensates in optical lattices: Band-gap structure and solitons

et al. 2003

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We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.

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Modulational instability in Bose-Einstein condensates in optical lattices

Cited by 336 publications

References 13 publications

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

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

Modulational instability of Gross-Pitaevskii-type equations in1+1dimensions

Bose-Einstein condensates in optical lattices: Band-gap structure and solitons

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