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arXiv:2107.06931 · Physical Review A

Disorder in order: Localization without randomness in a cold-atom system

Félix Rose, Richard Schmidt

Published Jan 28, 2022• 5 citations
arXivHTMLPDF

Abstract

We present a mapping between the Edwards model of disorder describing the motion of a single particle subject to randomly positioned static scatterers and the Bose polaron problem of a light quantum impurity interacting with a Bose-Einstein condensate (BEC) of heavy atoms. The mapping offers an experimental setting to investigate the physics of Anderson localization where, by exploiting the quantum nature of the BEC, the time evolution of the quantum impurity emulates the disorder-averaged dynamics of the Edwards model. Valid in any space dimension, the mapping can be extended to include interacting particles, arbitrary disorder, or confinement and can be generalized to study many-body localization. Moreover, the corresponding exactly solvable disorder model offers means to benchmark variational approaches used to study polaron physics. Here we illustrate the mapping by focusing on the case of an impurity interacting with a one-dimensional BEC through a contact interaction. While a simple wave function based on the expansion in the number of bath excitations misses the localization physics entirely, a coherent state Ansatz combined with a canonical transformation captures the physics of disorder and Anderson localization.

DOI
10.1103/physreva.105.013324

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We present a mapping between the Edwards model of disorder describing the motion of a single particle subject to randomly positioned static scatterers and the Bose polaron problem of a light quantum impurity interacting with a Bose-Einstein condensate (BEC) of heavy atoms. The mapping offers an experimental setting to investigate the physics of Anderson localization where, by exploiting the quantum nature of the BEC, the time evolution of the quantum impurity emulates the disorder-averaged dynamics of the Edwards model. Valid in any space dimension, the mapping can be extended to include interacting particles, arbitrary disorder, or confinement and can be generalized to study many-body localization. Moreover, the corresponding exactly solvable disorder model offers means to benchmark variational approaches used to study polaron physics. Here we illustrate the mapping by focusing on the case of an impurity interacting with a one-dimensional BEC through a contact interaction. While a simple wave function based on the expansion in the number of bath excitations misses the localization physics entirely, a coherent state Ansatz combined with a canonical transformation captures the physics of disorder and Anderson localization.
We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling ggg and increasing number of scattering centers at constant density. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant ggg.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.
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].
Anderson localization – the halt of quantum transport in a strongly disordered potential – has been observed for quantum gases in optical potentials. An important stage of experiments is the creation and characterization of the optical (speckle) potential that typically features nontrivial correlations complicating the interpretation of measurements. We realize an alternative route towards Anderson localization of cold atoms using a mixture of two strongly mass-imbalanced atomic species without any externally imposed disordered potential. On short time scales, the heavy component can be considered immobile and creates a quasi-static scattering potential for the light component that experiences Anderson localization under appropriate conditions. Experimental results allow for a quantitative description based on the model of randomly distributed resonant point scatterers. This demonstrates an interesting and fruitful link between the phenomenon of (single-particle) Anderson localization and the so-called many-body localization that takes place in interacting quantum systems composed of many particles, which is the case of our experimental system.
Bose-Einstein condensates (BECs) have proven to be remarkable systems with which to study some of the foundational models of condensed matter physics. The observation of a critical velocity for the breakdown of superfluidity in a BEC and the superfluid to Mott insulator transition observed in a BEC trapped by an optical lattice are but two examples of the, by now, dozens of exciting results in this field, which combines theoretical tools from condensed matter physics with state-of-the-art experimental techniques from ultra-cold atomic physics. However, any real condensed matter system has to contend with the effects of disorder, a phenomena notably absent in the inherently clean BEC systems. We have developed and implemented a way to add well characterized disorder in a controlled way to the otherwise clean BEC system using the light field from a laser speckle pattern. Using this system, we have investigated the effects of disorder or a single Gaussian defect, on the collective dipole motion of a BEC of 7Li in an optical trap. In addition, we perform transport experiments on a weakly interacting BEC expanding in a disordered one-dimensional atom wave-guide. We have observed that in such a system, the wave nature of matter can lead to spectacular and counterintuitive phenomena. Specifically, we verify that this system exhibits Anderson localization, a phenomena fundamentally resulting from the interference of multiply scattered matter waves. In such a state, the localized gas behaves as an insulator in a regime where it is classically expected to be conducting.\nWe also present results of experiments regarding a repulsive BEC scattering from a semi-permeable, single defect potential. We investigate the transport properties of such a system with special emphasis on the velocity and defect strength dependent dissipation of the collective dipole motion of the BEC. Finally, we present the results of our experiments on the scattering properties of bright matter wave solitons. We have observed fragmentation of the soliton in a disordered potential as well as both splitting and recombination of a soliton after interacting with a single repulsive defect potential.
We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.