Superfluidity versus Anderson Localization in a Dilute Bose Gas

Paul, Tobias; Schlagheck, Peter; Lebœuf, P.; Pavloff, Nicolas

doi:10.1103/physrevlett.98.210602

Cited by 81 publications

(98 citation statements)

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“…Nevertheless, a detailed study [490] has shown that expansion of a quasi-2D cloud may lead to weak and even strong localization using currently available speckle patterns. Other works have revealed that Anderson localization may occur during transport processes in repulsive BECs [491][492][493]. In this case, however, and for condensates at equilibrium, the interaction-induced delocalization dominates disorder-induced localization, except for the case of weak interactions [494] (see also earlier work in Ref.…”

Section: Matter-waves In Disordered Potentialsmentioning

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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Section: Matter-waves In Disordered Potentialsmentioning

confidence: 99%

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

2008

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“…A central aim in this context is the realization and unambiguous identification of strong Anderson localization with Bose-Einstein condensates, which was attempted by several experimental groups [1][2][3] with recent success [4,5], and theoretically studied both from the perspective of the expansion process of the condensate [6,7] as well as from the scattering perspective [8,9]. Complementary studies were focused on localization properties of Bogoliubov quasiparticles [10,11], on dipole oscillations in the presence of disorder [12,13], as well as on the realization of Bose glass phases [14,15].…”

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“…The past years have witnessed an increasing number of theoretical and experimental research activities on the behavior of ultracold atoms in magnetic or optical disorder potentials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. A central aim in this context is the realization and unambiguous identification of strong Anderson localization with Bose-Einstein condensates, which was attempted by several experimental groups [1][2][3] with recent success [4,5], and theoretically studied both from the perspective of the expansion process of the condensate [6,7] as well as from the scattering perspective [8,9].…”

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Coherent Backscattering of Bose-Einstein Condensates in Two-Dimensional Disorder Potentials

Hartung

Wellens²,

Müller³

et al. 2008

Phys. Rev. Lett.

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We study quantum transport of an interacting Bose-Einstein condensate in a two-dimensional disorder potential. In the limit of a vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. We reproduce this result, obtained from the numerical integration of the Gross-Pitaevskii equation, by a diagrammatic theory of weak localization in the presence of nonlinearity. DOI: 10.1103/PhysRevLett.101.020603 PACS numbers: 05.60.Gg, 03.75.Kk, 67.85.ÿd, 72.15.Rn The past years have witnessed an increasing number of theoretical and experimental research activities on the behavior of ultracold atoms in magnetic or optical disorder potentials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. A central aim in this context is the realization and unambiguous identification of strong Anderson localization with Bose-Einstein condensates, which was attempted by several experimental groups [1][2][3] with recent success [4,5], and theoretically studied both from the perspective of the expansion process of the condensate [6,7] as well as from the scattering perspective [8,9]. Complementary studies were focused on localization properties of Bogoliubov quasiparticles [10,11], on dipole oscillations in the presence of disorder [12,13], as well as on the realization of Bose glass phases [14,15].The above-mentioned topics (apart from Ref.[7]) mainly refer to processes that are essentially one dimensional (1D) by nature. Qualitatively new phenomena, however, do arise in two or three spatial dimensions, due to the scenario of weak localization. The latter manifests in a slight reduction of the transmission probability of an incident wave through a disordered region as compared to the classically expected value, due to constructive interference between backscattered paths and their time-reversed counterparts. This interference phenomenon particularly leads to a cone-shaped enhancement of the backscattering current in the direction reverse to the incident beam, which was indeed observed [16] and theoretically analyzed [17] in light scattering processes from disordered media. Related weak localization effects also arise in electronic mesoscopic physics, leading to characteristic peaks in the magnetoresistance [18,19].In this Letter, we investigate the phenomenon of coherent backscattering with atomic Bose-Einstein condensates that propagate in presence of two-dimensional (2D) disorder potentials. An essential ingredient that comes into play here is the interaction between the atoms of the condensate. On the mean-field level, this is accounted for by the nonlinear term in the Gross-Pitaevskii equation describing the time evolution of the condensate wave function. Indeed, nonlinearities do also appear in scattering processes of light, e....

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“…In the case of matter waves, the localization of Bose-Einstein condensates (BEC) is, at present, a very active research topic investigated by several experimental and theoretical groups [1,2,3,4,5,6,7,8], in particular with the recent observation of the (1D) localization of matter waves in a disordered optical potentials [9,10]. Although, in these experiments, the atom-atom interactions were negligible, the basic question remains how effects of interference between multiply scattered waves, such as weak or strong localization, are affected by interactions.…”

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Speckle Instability: Coherent Effects in Nonlinear Disordered Media

Grémaud

Wellens

2010

Phys. Rev. Lett.

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We numerically investigate the properties of speckle patterns formed by nonlinear point scatterers. We show that, in the weak localization regime, dynamical instability appears, eventually leading to chaotic behavior of the system. Analysing the statistical properties of the instability thresholds for different values of the system size and disorder strength, a scaling law is emphasized. The later is also found to govern the smallest decay rate of the linear system, putting thus forward the crucial importance of interference effects. This is also underlined by the fact that coherent backscattering is still observed even in the chaotic regime.PACS numbers: 42.25.Dd 42.65.sf 42.65.-k As first described by Anderson in 1958, the impact of disorder on the transport properties of waves, depending on the dimensionality and the disorder strength, ranges from weak to strong localization. In the case of matter waves, the localization of Bose-Einstein condensates (BEC) is, at present, a very active research topic investigated by several experimental and theoretical groups [1,2,3,4,5,6,7,8], in particular with the recent observation of the (1D) localization of matter waves in a disordered optical potentials [9,10]. Although, in these experiments, the atom-atom interactions were negligible, the basic question remains how effects of interference between multiply scattered waves, such as weak or strong localization, are affected by interactions.BEC's in disordered potentials appear to be good candidates to study these questions, since, in the mean field regime, the condensate is still described by a single coherent wave function governed by a nonlinear wave equation (Gross-Pitaevski equation). Thus, the condensate in principle retains its ability to display interference effects also in presence of (not too strong) interactions. Similar nonlinear equations describe propagation of light in disordered nonlinear media [11]. In contrast, the situation is quite different in the case of electronic transport [12], where the interactions combined with finite temperature effects give rise to dephasing, which in general destroy the disorder-induced coherent effects.Even if the theoretical description of coherent effects in nonlinear disordered systems is far from complete, an important step was done recently by the development of a diagrammatic theory for coherent backscattering in presence of nonlinearity [13]. This approach relies on the assumption of a unique stationary solution of the nonlinear wave equation under consideration. This assumption is expected to be valid for very weak nonlinearities (which, however, may still considerably affect the height of the coherent backscattering cone [8,13]). On the other hand, it is known that larger nonlinearities can induce speckle instabilities, such that no stationary state is reached at long times [8,14]. A clear explanation for the physical origin of this effect, however, is still missing. Theoretical predictions of the nonlinearity threshold above which instabilities develop, were attemp...

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Superfluidity versus Anderson Localization in a Dilute Bose Gas

Cited by 81 publications

References 32 publications

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

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

Coherent Backscattering of Bose-Einstein Condensates in Two-Dimensional Disorder Potentials

Speckle Instability: Coherent Effects in Nonlinear Disordered Media

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