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arXiv:2208.02994 · Physical Review A

Tunneling and revival of Anderson localization in a Bose-Einstein condensate

Sriganapathy Raghav, Barun Halder, Pradosh Basu, Utpal Roy

Published Dec 12, 2022• 5 citations
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Abstract

We provide an analytical model to fabricate an exponential localization of a Bose-Einstein condensate under a bichromatic optical lattice. Such localization is famously known as Anderson localization. The degree of localization is investigated using the participation ratio to recognize the laser parameter domain for Anderson localization. The exponential nature of the localization is proved, and we also identify the localization length. The tunneling of an Anderson-localized condensate with time is observed, and the revival phenomenon of Anderson localization is reported. Slowing down of Anderson localization is noticed for higher laser intensity. We also study the dynamical and structural stability of the condensate during Anderson localization, which suggests the preferred values of laser power and the time instance to encounter a minimal mean difference in the presence of noise.

DOI
10.1103/physreva.106.063304

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We provide an analytical model to fabricate an exponential localization of a Bose-Einstein condensate under a bichromatic optical lattice. Such localization is famously known as Anderson localization. The degree of localization is investigated using the participation ratio to recognize the laser parameter domain for Anderson localization. The exponential nature of the localization is proved, and we also identify the localization length. The tunneling of an Anderson-localized condensate with time is observed, and the revival phenomenon of Anderson localization is reported. Slowing down of Anderson localization is noticed for higher laser intensity. We also study the dynamical and structural stability of the condensate during Anderson localization, which suggests the preferred values of laser power and the time instance to encounter a minimal mean difference in the presence of noise.
Abstract Mean field and beyond mean field model calculations of Bose–Einstein condensates trapped in optical lattices have shown that initially homogeneous condensates can evolve into self-trapped, strongly localized states in the presence of weak boundary dissipation, a phenomenon called self-localization . A dynamical phase transition from extended to localized states can be observed when the effective nonlinearity exceeds a critical threshold Λ eff c . We investigate this phase transition to self-localization in the mean field approximation of the discrete nonlinear Schrödinger equation. We quantitatively characterize the properties of the discrete breathers , i.e. the nonlinear localized solutions, at the phase transition. This leads us to propose and numerically verify an analytical lower bound Λ eff L for the critical nonlinearity based on the idea of self-induced Anderson localization.
Anderson localization of waves in disordered media was originally predicted[1] fifty years ago, in the context of transport of electrons in crystals[2]. The phenomenon is much more general[3] and has been observed in a variety of systems, including light waves[4, 5]. However, Anderson localization has not been observed directly for matter waves. Owing to the high degree of control over most of the system parameters (in particular the interaction strength), ultracold atoms offer opportunities for the study of disorder-induced localization[6]. Here we use a non-interacting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice - a system that features a crossover between extended and exponentially localized states, as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated through investigations of the transport properties and spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. This controllable system may be used to investigate the interplay of disorder and interaction (ref. 7 and references therein), and to explore exotic quantum phases[8, 9].
We provide an exact analytical model for the dynamics of a 1D Bose–Einstein condensate loaded in a bichromatic optical lattice. Although a host of exact solutions result from this novel method, we mainly concentrate on the solitonic excitations. The trapping potential and its depth of lattice frustration can be varied by tuning the power and the wavelengths of the two overlaying laser beams. Both attractive and repulsive regimes are thoroughly investigated. In the attractive domain, we obtain bright soliton, which reveals interesting variations with the depth of lattice frustration. Localization of the matter wave density is demonstrated as one of the applications in this regime. In the repulsive domain, dark soliton is obtained when the potential resembles an optical lattice. With appropriate tuning of the potential parameter, the dark soliton becomes modulated with an oscillatory background and gradually transforms to bright solitary trains.
The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.
We numerically explore the long-time expansion of a one-dimensional Bose-Einstein condensate in a disorder potential employing the Gross-Pitaevskii equation. The goal is to search for unique signatures of Anderson localization in the presence of particle-particle interactions. Using typical experimental parameters, we show that the time scale for which the nonequilibrium dynamics of the interacting system begins to diverge from the noninteracting system exceeds the observation times up to now accessible in the experiment. We find evidence that the long-time evolution of the wave packet is characterized by (sub)diffusive spreading and a growing effective localization length, suggesting that interactions destroy Anderson localization.
By direct numerical simulation of the time-dependent Gross-Pitaevskii equation, we study different aspects of the localization of a noninteracting ideal Bose-Einstein condensate (BEC) in a one-dimensional bichromatic quasiperiodic optical-lattice potential. Such a quasiperiodic potential, used in a recent experiment on the localization of a BEC [Roati et al., Nature (London) 453, 895 (2008)], can be formed by the superposition of two standing-wave polarized laser beams with different wavelengths. We investigate the effect of the variation of optical amplitudes and wavelengths on the localization of a noninteracting BEC. We also simulate the nonlinear dynamics when a harmonically trapped BEC is suddenly released into a quasiperiodic potential, as done experimentally in a laser speckle potential [Billy et al., Nature (London) 453, 891 (2008)]. We finally study the destruction of the localization in an interacting BEC due to the repulsion generated by a positive scattering length between the bosonic atoms.