The Floquet-Boltzmann equation

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 $T_0=2 \pi/\Omega$, 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 modulo $\hbar \Omega$, 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.Comment: 12 pages + 3 pages of appendix, 13 figure

Directed motion of doublons and holes in periodically driven Mott insulators

Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for example, holes and doublons move in opposite direction. To investigate whether the periodic driving can be used to move particles "uphill", i.e., against an external force, we study the influence of a linear potential $- g x$. For long times, transport is only possible when the driving frequency $\omega$ and the external force $g$ are commensurate, $n_0 g = m_0 \omega$, with $\frac{n_0}{2},m_0 \in \mathbb{Z}$.Comment: 11 pages, 9 figure

Periodically driven many-body quantum systems : Quantum Ratchets, Topological States and the Floquet-Boltzmann Equation

Controlling and manipulating complex many-body quantum systems will be a key ingredient for the development of next-generation technologies. While the realisation of a universal quantum machine is still out of reach, in recent years experimental systems of ultracold atoms have already evolved into a vivid field of research for quantum simulation. Crucially, such systems even allow for the successful quantum engineering of targeted many-body systems by means of coherent periodic driving. The essential properties of these Floquet systems encompass two main aspects: fast driving facilitates the simulation of effective static systems, and interactions lead to unique heating effects as energy is only conserved modulo the driving frequency. Within this thesis we theoretically study both of these aspects in respective model systems. In part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott insulator in a designed one-dimensional Floquet system. Here, periodic driving in combination with breaking all mirror symmetries of the system can induce directed motion of particles. In the limit of small excitation densities, the effectively non-interacting quantum ratchet determines the motion of holes and doublons in the Mott insulator and can in fact be used to manipulate the dynamics of such. This little quantum machine can also be used to drive particles against an external force, where transport is possible but requires the fulfilment of a commensurability condition for long times. In part II, we discuss the role of interactions for periodically driven systems by means of a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, we develop this semiclassical formalism based on a clear separation of time scales. The result is a description of the dynamics and the scattering of Floquet quasiparticles in such systems. Here, the property of discrete energy violation is naturally encoded in our formalism predicting the heating of interacting Floquet systems to infinite temperatures in the long-time limit. As a first application of this approach, we investigate a cold atom setup realising the Haldane model by means of periodic shaking. While homogeneous systems heat up globally, a confining potential evokes thermoelectric transport effects resulting from spatially dependent heating characteristics. Moreover, we show that the interplay of intrinsic heating, macroscopic diffusion and non-trivial topological properties of the Haldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous particle transport as the result of developing temperature gradients. In part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing a theoretical framework for a possible simulation of a topological edge state in a one-dimensional optical lattice. In this case, the one-dimensional Dirac equation with spatially varying mass is important, which captures the topological properties of a corresponding system of the BDI symmetry class. We analytically discuss such system and investigate the role of mean-field interaction effects. We also identify the emergence of dynamical instabilities in a realisation with bosonic atoms

Real-space imaging of a topological protected edge state with ultracold atoms in an amplitude-chirped optical lattice

Topological states of matter, as quantum Hall systems or topological insulators, cannot be distinguished from ordinary matter by local measurements in the bulk of the material. Instead, global measurements are required, revealing topological invariants as the Chern number. At the heart of topological materials are topologically protected edge states that occur at the intersection between regions of different topological order. Ultracold atomic gases in optical lattices are promising new platforms for topological states of matter, though the observation of edge states has so far been restricted in these systems to the state space imposed by the internal atomic structure. Here we report on the observation of an edge state between two topological distinct phases of an atomic physics system in real space using optical microscopy. An interface between two spatial regions of different topological order is realized in a one-dimensional optical lattice of spatially chirped amplitude. To reach this, a magnetic field gradient causes a spatial variation of the Raman detuning in an atomic rubidium three- level system and a corresponding spatial variation of the coupling between momentum eigenstates. This novel experimental technique realizes a cold atom system described by a Dirac equation with an inhomogeneous mass term closely related to the SSH-model. The observed edge state is characterized by measuring the overlap to various initial states, revealing that this topological state has singlet nature in contrast to the other system eigenstates, which occur pairwise. We also determine the size of the energy gap to the adjacent eigenstate doublet. Our findings hold prospects for the spectroscopy of surface states in topological matter and for the quantum simulation of interacting Dirac systems

Electric quantum walks with individual atoms

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 vs. dynamical localization depending on whether the accumulated Bloch phase is a rational or irrational fraction of 2\pi.Comment: 5 pages, 4 figure

Task-based assessment of neck CT protocols using patient-mimicking phantoms—effects of protocol parameters on dose and diagnostic performance

Objectives: To assess how modifying multiple protocol parameters affects the dose and diagnostic performance of a neck CT protocol using patient-mimicking phantoms and task-based methods. Methods: Six patient-mimicking neck phantoms containing hypodense lesions of 1 cm diameter and 30 HU contrast and one non-lesion phantom were examined with 36 CT protocols. All possible combinations of the following parameters were investigated: 100- and 120-kVp tube voltage; tube current modulation (TCM) noise levels of SD 7.5, 10, and 14; pitches of 0.637, 0.813, and 1.388; filtered back projection (FBP); and iterative reconstruction (AIDR 3D). Dose-length products (DLPs) and lesion detectability (assessed by 14 radiologists) were compared with the clinical standard protocol (120 kVp, TCM SD 7.5, 0.813 pitch, AIDR 3D). Results: The DLP of the standard protocol was 25 mGy•cm; the area under the curve (AUC) was 0.839 (95%CI: 0.790-0.888). Combined effects of tube voltage reduction to 100 kVp and TCM noise level increase to SD 10 optimized protocol performance by improving dose (7.3 mGy•cm) and detectability (AUC 0.884, 95%CI: 0.844-0.924). Diagnostic performance was significantly affected by the TCM noise level at 120 kVp (AUC 0.821 at TCM SD 7.5 vs. 0.776 at TCM SD 14, p = 0.003), but not at 100-kVp tube voltage (AUC 0.839 at TCM SD 7.5 vs. 0.819 at TCM SD 14, p = 0.354), the reconstruction method at 100 kVp (AUC 0.854 for AIDR 3D vs. 0.806 for FBP, p < 0.001), but not at 120-kVp tube voltage (AUC 0.795 for AIDR 3D vs. 0.793 for FBP, p = 0.822), and the tube voltage for AIDR 3D reconstruction (p < 0.001), but not for FBP (p = 0.226). Conclusions: Combined effects of 100-kVp tube voltage, TCM noise level of SD 10, a pitch of 0.813, and AIDR 3D resulted in an optimal neck protocol in terms of dose and diagnostic performance. Protocol parameters were subject to complex interactions, which created opportunities for protocol improvement. Key points: • A task-based approach using patient-mimicking phantoms was employed to optimize a CT system for neck imaging through systematic testing of protocol parameters. • Combined effects of 100-kVp tube voltage, TCM noise level of SD 10, a pitch of 0.813, and AIDR 3D reconstruction resulted in an optimal protocol in terms of dose and diagnostic performance. • Interactions of protocol parameters affect diagnostic performance and should be considered when optimizing CT techniques