The disorder-induced localization of few bosons interacting via a contact potential is investigated through the analysis of the level-spacing statistics familiar from random matrix theory. The model we consider is defined in a continuum and describes one-dimensional bosonic atoms exposed to the spatially correlated disorder due to an optical speckle field. %
First, we identify the speckle-field intensity required to observe, in the single-particle case, the Poisson level-spacing statistics, which is characteristic of localized quantum systems, in a computationally and experimentally feasible system size. Then, we analyze the two-body and the three-body systems, exploring a broad interaction range, from the noninteracting limit up to moderately strong interactions. Our main result is that the contact potential does not induce a shift towards the Wigner-Dyson level-spacing statistics, which would indicate the emergence of an ergodic chaotic state, indicating that localization can occur also in interacting few-body systems in a continuum. We also analyze how the ground-state energy evolves as a function of the interaction strength
The many-body localization transition for Heisenberg spin chain with a speckle disorder is studied. Such a model is equivalent to a system of spinless fermions in an optical lattice with an additional speckle field. Our numerical results show that the many-body localization transition in speckle disorder falls within the same universality class as the transition in an uncorrelated random disorder, in contrast to the quasiperiodic potential typically studied in experiments. This hints at possibilities of experimental studies of the role of rare Griffiths regions and of the interplay of ergodic …
We evaluate the localization length of the wave (or Schroedinger) equation in the presence of a disordered speckle potential. This is relevant for experiments on cold atoms in optical speckle potentials. We focus on the limit of large disorder, where the Born approximation breaks down and derive an expression valid in the "quasi-metallic" phase at large disorder. This phase becomes strongly localized and the effective mobility edge disappears.
The effect of a weak three-dimensional (3d) isotropic laser speckle disorder on various thermodynamic properties of a dilute Bose gas is considered at zero temperature. First, we summarize the derivation of the autocorrelation function of laser speckles in 1d and 2d following the seminal work of Goodman. The goal of this discussion is to show that a Gaussian approximation of this function, proposed in some recent papers, is inconsistent with the general background of laser speckle theory. Then we propose a possible experimental realization for an isotropic 3d laser speckle potential and derive…
Quantum reservoir computing is a machine-learning approach designed to exploit the dynamics of quantum systems with memory to process information. As an advantage, it presents the possibility to benefit from the quantum resources provided by the reservoir combined with a simple and fast training strategy. In this work, this technique is introduced with a quantum reservoir of spins and it is applied to find the ground-state energy of an additional quantum system. The quantum reservoir computer is trained with a linear model to predict the lowest energy of a particle in the presence of different…
We study the localization properties of non-interacting waves propagating in a speckle-like potential superposed on a one-dimensional lattice. Using a decimation/renormalization procedure, we estimate the localization length for a tight-binding Hamiltonian where site-energies are square-sinc-correlated random variables. By decreasing the width of the correlation function, the disorder patterns approaches a $δ$-correlated disorder, and the localization length becomes almost energy-independent in the strong disorder limit. We show that this regime can be reached for a size of the speckle grains …
We investigate the localization properties of atoms moving in a three-dimensional optical lattice in the presence of an uncorrelated disorder potential having the same probability distribution $P(V)$ as laser speckles. We find that the disorder-averaged (single-particle) Green's function, calculated via the coherent potential approximation, is in very good agreement with exact numerics. Using the transfer-matrix method, we compute the phase diagram in the energy-disorder plane and show that its peculiar shape can be understood from the self-consistent theory of localization. In particular, we …
Strongly correlated fermions in three- and two-dimensional optical lattices with experimentally realistic speckle disorder are investigated. We extend and apply the statistical dynamical mean-field theory, which treats local correlations non-perturbatively, to incorporate on-site and hopping-type randomness on equal footing. Localization due to disorder is detected via the probability distribution function of the local density of states. We obtain a complete paramagnetic ground state phase diagram for experimentally realistic parameters and find a strong suppression of the correlation-induced …
We discuss the stochastic mean-field theory (SMFT) method which is a new approach for describing disordered Bose systems in the thermodynamic limit including localization and dimensional effects. We explicate the method in detail and apply it to the disordered Bose-Hubbard model at finite temperature, with on-site box disorder, as well as experimentally relevant unbounded speckle disorder. We find that disorder-induced condensation and reentrant behavior at constant filling are only possible at low temperatures, beyond the reach of current experiments [Pasienski et al., arXiv:0908.1182]. Inclu…
We analyze how accurately supervised machine learning techniques can predict the lowest energy levels of one-dimensional noninteracting ultracold atoms subject to the correlated disorder due to an optical speckle field. Deep neural networks with different numbers of hidden layers and neurons per layer are trained on large sets of instances of the speckle field, whose energy levels have been preventively determined via a high-order finite difference technique. The Fourier components of the speckle field are used as feature vector to represent the speckle-field instances. A comprehensive analysi…
Motivated by the recent experimental and theoretical progresses in the exploration of the effect of disorder in interacting system, we examine the effect of two types of correlated disorder, the quasi-periodic potential and speckle disorder potential, on one- and two-particle problem with exact diagonalization (ED) method. We give the phase diagram for single particle in the presence of quasi-periodic potential and also analyse the effect of strong interaction on the phase diagram for ground state in two dimensions. For the speckle disorder potential case, we examine both the effect of correla…
We study the tail of the spectrum for non-interacting bosons in a blue-detuned random speckle potential. Using an instanton approach we derive the asymptotic behavior of the density of states in d dimensions. The leading corrections resulting from fluctuations around the saddle point solution are obtained by means of the Gel'fand-Yaglom method generalized to functional determinants with zero modes. We find a good agreement with the results of numerical simulations in one dimension. The effect of weak repulsive interactions in the Lifshitz tail is also discussed.
We report on the impact of variable-scale disorder on 3D Anderson localization of a non-interacting ultracold atomic gas. A spin-polarized gas of fermionic atoms is localized by allowing it to expand in an optical speckle potential. Using a sudden quench of the localized density distribution, we verify that the density profile is representative of the underlying single-particle localized states. The geometric mean of the disordering potential correlation lengths is varied by a factor of four via adjusting the aperture of the speckle focusing lens. We observe that the root-mean-square size of t…
We have studied the phase diagram of a quasi-two-dimensional interacting Bose gas at zero temperature in the presence of random potential created by laser speckles. The superfluid fraction and the fraction of particles with zero momentum are obtained within the mean-field Gross-Pitaevskii theory and in diffusion Monte Carlo simulations. We find a transition from the superfluid to the insulating state, when the strength of the disorder grows. Estimations of the critical parameters are compared with the predictions of the percolation theory in the Thomas-Fermi approximation. Analytical expressio…
We develop a procedure to modify the correlations of a speckle potential. This procedure, that is suitable for spatial light modulator devices, allows one to increase the localization efficiency of the speckle in a narrow energy region whose position can be easily tuned. This peculiar energy-dependent localization behavior is explored by pulling the potential through a cigar-shaped Bose-Einstein condensate. We show that the percentage of dragged atoms as a function of the pulling velocity depends on the potential correlations below a threshold of the disorder strength. Above this threshold, in…
We study the Anderson localization of atomic gases exposed to simple-cubic optical lattices with a superimposed disordered speckle pattern. The two mobility edges in the first band and the corresponding critical filling factors are determined as a function of the disorder strength, ranging from vanishing disorder up to the critical disorder intensity where the two mobility edges merge and the whole band becomes localized. Our theoretical analysis is based both on continuous-space models which take into account the details of the spatial correlation of the speckle pattern, and also on a simplif…
We investigate, both experimentally and theoretically, the static geometric properties of a harmonically trapped Bose-Einstein condensate of ${}^6$Li$_2$ molecules in laser speckle potentials. Experimentally, we measure the in-situ column density profiles and the corresponding transverse cloud widths over many laser speckle realizations. We compare the measured widths with a theory that is non-perturbative with respect to the disorder and includes quantum fluctuations. Importantly, for small disorder strengths we find quantitative agreement with the perturbative approach of Huang and Meng, whi…
Anderson localization has been observed for a variety of media, including ultracold atomic gases with speckle disorder in one and three dimensions. However, observation of Anderson localization in a two-dimensional geometry for ultracold gases has been elusive. We show that a cause of this difficulty is the relatively high percolation threshold of a speckle potential in two dimensions, resulting in strong classical localization. We propose a realistic point-like disorder potential that circumvents this percolation limit with localization lengths that are experimentally observable. The percolat…
We study the Anderson localization of atomic gases exposed to three-dimensional optical speckles by analyzing the statistics of the energy-level spacings. This method allows us to consider realistic models of the speckle patterns, taking into account the strongly anisotropic correlations which are realized in concrete experimental configurations. We first compute the mobility edge $E_c$ of a speckle pattern created using a single laser beam. We find that $E_c$ drifts when we vary the anisotropy of the speckle grains, going from higher values when the speckles are squeezed along the beam propag…
I discuss whether localization in the two dimensional continuum can be stable in the presence of short range interactions. I conclude that, for an impurity model of disorder, if the system is prepared below a critical temperature $T < T_c$, then perturbation theory about the localized phase converges almost everywhere. As a result, the system is at least asymptotically localized, and perhaps even truly many body localized, depending on how certain rare regions behave. Meanwhile, for $T > T_c$, perturbation theory fails to converge, which I interpret as interaction mediated delocalization. I ca…
The effect of disorder in the intensity of the driving laser on the dynamics of a disordered three-cavity system of four-level atoms is investigated. This system can be described by a Bose-Hubbard Hamiltonian for dark-state polaritons. We examine the evolution of the first- and second-order correlation functions, the photon and atomic excitation numbers and the basis state occupation probabilities. We use the full Hamiltonian and the approximate Bose-Hubbard Hamiltonian with uniform and speckle disorder, as well as with different dipole couplings. We find that the results for the two Hamiltoni…
The full statistical distribution of the superfluid fraction characterizing one-dimensional Bose gases in random potentials is discussed. Rare configurations with extreme fluctuations of the disorder potential can fragment the condensate and reduce the superfluid fraction to zero. The resulting bimodal probability distribution for the superfluid fraction is calculated numerically in the quasi-1D mean-field regime of ultracold atoms in laser speckle potentials. Using extreme-value statistics, an analytical scaling of the zero-superfluid probability as function of disorder strength, disorder cor…
The ability to load ultracold atoms at a well-defined energy in a disordered potential is a crucial tool to study quantum transport, and in particular Anderson localization. In this paper, we present a new method for achieving that goal by rf transfer of atoms of an atomic Bose-Einstein condensate from a disorder insensitive state to a disorder sensitive state. It is based on a bichromatic laser speckle pattern, produced by two lasers whose frequencies are chosen so that their light-shifts cancel each other in the first state and add-up in the second state. Moreover, the spontaneous scattering…
Recent experiments on non-interacting ultra-cold atoms in correlated disorder have yielded conflicting results regarding the so-called mobility edge, i.e. the energy threshold separating Anderson localized from diffusive states. At the same time, there are theoretical indications that the experimental data overestimate the position of this critical energy, sometimes by a large amount. The non-trivial effect of anisotropy in the spatial correlations of experimental speckle potentials have been put forward as a possible cause for such discrepancy. Using extensive numerical simulations we show th…
We demonstrate that many-body localization of two-dimensional weakly interacting bosons in disorder remains stable in the thermodynamic limit at sufficiently low temperatures. Highly energetic particles destroy the localized state only above a critical temperature, which increases with the strength of the disorder. If the particle distribution is truncated at high energies, as it does for cold atom systems, the localization can be stable at any temperature.
Anderson localization is a single particle localization phenomena in disordered media that is accompanied by an absence of diffusion. Spin-orbit coupling (SOC) describes an interaction between a particle's spin and its momentum that directly affects its energy dispersion, for example creating dispersion relations with gaps and multiple local minima. We show theoretically that combining one-dimensional spin-orbit coupling with a transverse Zeeman field suppresses the effects of disorder, thereby increasing the localization length and conductivity. This increase results from a suppression of bac…
Anderson localization is a universal phenomenon affecting non-interacting quantum particles in disorder. In three spatial dimensions it becomes particularly interesting to study because of the presence of a quantum phase transition from localized to extended states, predicted by P.W. Anderson in his seminal work, taking place at a critical energy, the so-called mobility edge. The possible relation of the Anderson transition to the metal-insulator transitions observed in materials has originated a flurry of theoretical studies during the past 50 years, and it is now possible to predict very acc…
We study the time-dependent occupation of an impurity state hybridized with a continuum of extended or localized states. Of particular interest is the return probability, which gives the long-time limit of the average impurity occupation. In the extended case, the return probability is zero unless there are bound states of the impurity and continuum. We present exact expressions for the return probability of an impurity state coupled to a lattice, and show that the existence of bound states depends on the dimension of the lattice. In a disordered lattice with localized eigenstates, the finite …
