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 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.

Periodically driven quantum systems can be used to realize quantum pumps, ratchets, artificial gauge fields and novel topological states of matter. Starting from the Keldysh approach, we develop a formalism, the Floquet-Boltzmann equation, to describe the dynamics and the scattering of quasiparticles in such systems. The theory builds on a separation of time-scales. Rapid, periodic oscillations occurring on a time scale T0 = 2π/Ω, are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a non-interacting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modelled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet-quasiparticles and their scattering processes. As the energy is conserved only modulohΩ, the interacting system heats up in the long-time limit. As a first application of this approach, we compute the heating rate for a cold-atom system, where a periodical shaking of the lattice was used to realize the Haldane model [1].Periodically modulated quantum systems can effectively be described by a static Hamiltonian. This theoretical concept has recently evolved into a major experimental tool used by many groups to generate new states of matter.Early experiments [2,3] used, for example, that one can effectively change the strength as well as sign of the hopping of atoms in an optical lattice, allowing to realize new types of band structures. Periodic driving has also be used to realize directed transport in quantum ratchets [4]. More recently, the realization of emergent Gauge fields and topological band structures has been at the focus of many studies. Examples of such experiments include the generation of Gauge fields and superfluids with finite momentum [5,6], the generation of topological quantum walks [7] and of effective electric fields in a discrete quantum simulator [8], the realization of (Floquet-) topological insulators with photons [9], the generation of spin-orbit coupling [10], the direct measurement of Chern numbers and Berry phases in the Hofstadter Hamiltonian [11,12] and quantized charge pumps [26].In a periodically driven system, the Hamiltonian has only a discrete time-translational symmetry, H(t + T 0 ) = H(t). As a consequence, the total energy is not conserved but quantized changes of energy are possible, ∆E = nhΩwith Ω = 2π/T 0 and n ∈ Z. For non-interacting systems the absence of energy conservation has mostly no effect. The situation is, however, different when interactions in a many-particle system are considered. For a generic closed system, one can expect that in the longtime limit, t → ∞, the system approaches the state with the highest entropy consistent with the conservation laws. In the absence of some cooling mechanism, e.g., by an external bath [27] or by emitting radiation, one can therefore expect that generic interacting Floquet systems heat up to infinite temperatures [28] (an exception are many-body localized systems [29]). This important (and well-known) aspect has received relativel...

To describe a mobile defect in polyacetylene chains, Su, Schrieffer and Heeger formulated a model assuming two degenerate energy configurations that are characterized by two different topological phases. An immediate consequence was the emergence of a soliton-type edge state located at the boundary between two regions of different configurations. Besides giving first insights in the electrical properties of polyacetylene materials, interest in this effect also stems from its close connection to states with fractional charge from relativistic field theory. Here, using a one-dimensional optical lattice for cold rubidium atoms with a spatially chirped amplitude, we experimentally realize an interface between two spatial regions of different topological order in an atomic physics system. We directly observe atoms confined in the edge state at the intersection by optical real-space imaging and characterize the state as well as the size of the associated energy gap. Our findings hold prospects for the spectroscopy of surface states in topological matter and for the quantum simulation of interacting Dirac systems.

We present an in-situ method to measure the birefringence of a single vacuum window by means of microwave spectroscopy on an ensemble of cold atoms. Stress-induced birefringence can cause an ellipticity in the polarization of an initially linearly-polarized laser beam. The amount of ellipticity can be reconstructed by measuring the differential vector light shift of an atomic hyperfine transition. Measuring the ellipticity as a function of the linear polarization angle allows us to infer the amount of birefringence ∆n at the level of 10 −8 and identify the orientation of the optical axes. The key benefit of this method is the ability to separately characterize each vacuum window, allowing the birefringence to be precisely compensated in existing vacuum apparatuses.Many experiments in quantum optics rely on an accurate control of the polarization of the laser beams [1][2][3][4][5][6]. Optical access of laser beams to ultrahigh vacuum apparatus is offered by vacuum windows, which are, in general, affected by stress-induced birefringence occurring after mounting and bake-out. While the typical values of the induced birefringence ∆n are in the order of 10 −6 , values significantly below this magnitude require special attention in mounting the vacuum viewports to avoid deformations [7]. It is, thus, important for precision applications to be able to characterize the amount of birefringence of each individual window. Knowing the amount of birefringence and the orientation of the principal axes makes it possible to avoid polarization distortions either by aligning the incoming linear polarization onto one of the optical axis, or by fully compensating the birefringence by means of optical (e.g., Soleil-Babinet compensator) or mechanical techniques [8]. However, characterizing the polarization distortion outside of the vacuum with conventional polarimeters is not sufficient to reconstruct separately the birefringence of the two vacuum viewports, which the laser beam must transit. One solution which has been proposed to obviate this problem requires employing wedged vacuum windows and picking off the beam back-reflected from the inside facet [9]. However, this is not directly applicable to standard viewports or vacuum cells.In this note, we demonstrate an in-situ method to reconstruct the stress-induced birefringence ∆n of a vacuum window and the orientation angle θ 0 of the optical axes. Our scheme makes use of the atoms themselves as a sensitive probe to detect any ellipticity caused by mechanical stresses acting on the vacuum window, as illustrated in figure 1(a). While varying the angle θ of the incident linear polarization, we measure the light shift δ of a hyperfine transition by means of microwave spectroscopy. We will show that the recorded signal behaves aswhere the proportionality constant is fully determined by the atomic properties, k is the probe laser wavevector, L the thickness of the vacuum window, and S 0 denotes A half-wave plate and rotatable polarizer allow the linearpolarization of a probe laser beam ...