No abstract

We report on the experimental realization of electric quantum walks, which mimic the effect of an electric field on a charged particle in a lattice. Starting from a textbook implementation of discrete-time quantum walks, we introduce an extra operation in each step to implement the effect of the field. The recorded dynamics of such a quantum particle exhibits features closely related to Bloch oscillations and interband tunneling. In particular, we explore the regime of strong fields, demonstrating contrasting quantum behaviors: quantum resonances versus dynamical localization depending on whether the accumulated Bloch phase is a rational or irrational fraction of 2π.

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., non-random) case, we allow any unitary operator, which commutes with translations, and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling, and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order we find a non-random drift, which depends only on the mean of the control process and not on the initial state. In diffusive scaling the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the non-random limit, i.e., when the coins chosen are all very close, or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.Comment: 34 pages, 20 figure

We study one-dimensional quantum walks in a homogenous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion, and Anderson localization, depend very sensitively on the value of the electric field, Φ, e.g., on whether Φ/(2π) is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about the behavior on finite time scales.

The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties -a vanishing group velocity and the absence of transport -with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its manybody eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.The concept of disorder induced localisation has been introduced in the seminal work by Anderson [1] who captured the mechanism responsible for the absence of diffusion of waves in disordered media. This mechanism is specifically well understood in the single-particle case, where one can show that in the presence of a suitable random potential, all eigenfunctions are exponentially localised [1]. In addition to this spectral characterisation of localisation there is a notion of dynamical localisation, which requires that the transition amplitudes between lattice sites decay exponentially [2,3].Naturally, there is a great interest in extending these results to the many-body setting [4,5]. In the case of integrable systems that can be mapped to free fermions, such as the XY chain, results on single particle localisation can be applied directly [6,7]. A far more intricate situation arises in interacting systems. Such many-body localisation [8,9] has received an enormous attention recently. In terms of condensed-matter physics, this phenomenon allows for systems to remain an insulator even at non-zero temperature [10], in principle even at infinite temperature [11]. In the foundations of statistical mechanics, such many-body localised systems provide examples of systems that fail to thermalise. When pushed out of equilibrium, signatures of the initial condition will locally be measurable even after long times, in contradiction to what one might expect from quantum statistical mechanics [4,12,13].Despite great efforts to approach the phenomenon of manybody localisation, many aspects are not fully understood and a comprehensive definition is still lacking. Similar to the case of single-particle Anderson localisation, there are two complementary approaches to capture the phenomenon. On the one hand probes involving real-time dynamics [14,15] have been discussed, showing excitations "getting stuck", or seeing suitable signatures in density-auto-correlation functions [16], le...