Anderson Localization of a Bose-Einstein Condensate in a 3D Random Potential

Skipetrov, S. E.; Minguzzi, A.; Tiggelen, B. A. van; Shapiro, Boris

doi:10.1103/physrevlett.100.165301

Cited by 87 publications

(124 citation statements)

References 29 publications

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“…The reasonable quantitative agreement between our measurements and the theory of 1D Anderson localization in a speckle potential demonstrates the high degree of control in our set-up. We thus anticipate that it can be used as a quantum simulator for investigating Anderson localization in higher dimensions 31,32 , first to look for the mobility edge of the Anderson transition, and then to measure important features at the Anderson transition that are not known theoretically, such as critical exponents. It will also become possible to investigate the effect of controlled interactions on Anderson localization.…”

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confidence: 99%

Direct observation of Anderson localization of matter waves in a controlled disorder

Billy

Josse

Zuo

et al. 2008

Nature

1,508

1,743

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In 1958, P.W. Anderson predicted the exponential localization 1 of electronic wave functions in disordered crystals and the resulting absence of diffusion. It has been realized later that Anderson localization (AL) is ubiquitous in wave physics 2 as it originates from the interference between multiple scattering paths, and this has prompted an intense activity. Experimentally, localization has been reported in light waves 3,4,5,6,7 , microwaves 8,9 , sound waves 10 , and electron 11 gases but to our knowledge there is no direct observation of exponential spatial localization of matter-waves (electrons or others). Here, we report the observation of exponential localization of a Bose-Einstein condensate (BEC) released into a one-dimensional waveguide in the presence of a controlled disorder created by laser speckle 12 . We operate in a regime allowing AL: i) weak disorder such that localization results from many quantum reflections of small amplitude; ii) atomic density small enough that interactions are negligible. We image directly the atomic density profiles vs time, and find that weak disorder can lead to the stopping of the expansion and to the formation of a stationary exponentially localized wave function, a direct signature of AL. Fitting the exponential wings, we extract the localization length, and compare it to theoretical calculations. Moreover we show that, in our one-dimensional speckle potentials whose noise spectrum has a high spatial frequency cut-off, exponential localization occurs only when the de Broglie wavelengths of the atoms in the expanding BEC are larger than an effective mobility edge corresponding to that cut-off. In the opposite case, we find that the density profiles decay algebraically, as predicted in ref 13. The method presented here can be extended to localization of atomic quantum gases in higher dimensions, and with controlled interactions.

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confidence: 99%

Direct observation of Anderson localization of matter waves in a controlled disorder

Billy

Josse

Zuo

et al. 2008

Nature

1,508

1,743

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“…The BEC ground state in a harmonic trap extends over the Thomas-Fermi radius. How such a condensate wave-packet expands in a disordered potential has been studied for different dimensionalities [14,15,16]. Because of repulsive interaction, the entire condensate is an extended object that requires a field-like description, in sharp contrast with a bright soliton that features particle-like properties.…”

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confidence: 99%

Anderson Localization of Solitons

et al. 2009

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At low temperature, a quasi-one-dimensional ensemble of atoms with attractive interaction forms a bright soliton. When exposed to a weak and smooth external potential, the shape of the soliton is hardly modified, but its center-of-mass motion is affected. We show that in a spatially correlated disordered potential, the quantum motion of a bright soliton displays Anderson localization. The localization length can be much larger than the soliton size and could be observed experimentally.PACS numbers: 03.75. Lm,72.15.Rn,05.30.Jp At zero temperature, cold atoms interacting attractively in a one dimensional (1D) system tend to cluster together, forming a bright soliton. Explicit solutions of the many-body problem can be found in some cases, for example for contact interactions [1]. Using external potentials, it has been experimentally shown how to put solitons in motion [2]. What happens to a soliton exposed to a disordered potential? If the potential is strong, it will destroy the soliton. If it is sufficiently weak and smooth not to perturb the soliton shape, one expects the soliton to undergo multiple scattering, diffusive motion and possibly Anderson localization [3]. Indeed, propagation of waves in a disordered potential is profoundly affected by Anderson localization. Multiple scattering on random defects yields exponentially localized density profiles and a suppression of the usual diffusive transport associated with incoherent wave scattering [4]. In 1D Anderson localization is a ubiquitous phenomenon [5], which has been recently observed for cold atomic matter waves [6]. It is important to understand how it is modified when interactions between particles are taken into account.We consider a Bose-Einstein condensate in a quasi-1D geometry. Within mean-field theory, it is described by the Gross-Pitaevskii energy functionalin units of E 0 = 4mω 2 ⊥ a 2 , l 0 = /2|a|mω ⊥ , and t 0 = /4a 2 mω 2 ⊥ for energy, length and time, respectively. Here, ω ⊥ denotes the transverse harmonic confinement frequency, a the atomic s-wave scattering length, and µ the chemical potential. The cases of repulsive and attractive atomic interaction are covered by g = ±1.The dynamics is extremely different in both cases, so that we first discuss the case of attractive interaction, g = −1. The ground state of (1) is the bright soliton [7] φ 0 (z − q) = N 2ξnormalized to the total number of particles N . The chemical potential is µ = −N 2 /8 and the soliton width is ξ = 2/N. This ground-state solution has an arbitrary center-of-mass (CM) position q and an arbitrary global phase θ that spontaneously break the translational and the U (1) gauge symmetry of the energy functional (1), respectively. These degrees of freedom appear as zeroenergy modes of Bogoliubov theory, and their quantum dynamics requires special attention [8,9]. The energy functional (1) is no longer translation invariant when a potential term dzV (z)|φ| 2 is added. If V (z) is sufficiently weak and smooth, the soliton shape remains unchanged to lowest order in V , and on...

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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].…”

mentioning

confidence: 99%

“…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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confidence: 99%

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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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Anderson Localization of a Bose-Einstein Condensate in a 3D Random Potential

Cited by 87 publications

References 29 publications

Direct observation of Anderson localization of matter waves in a controlled disorder

Direct observation of Anderson localization of matter waves in a controlled disorder

Anderson Localization of Solitons

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

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