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arXiv:1410.5364 · Physical Review A

Scattering theory for Floquet-Bloch states

Thomas Bilitewski, Nigel R. Cooper

Published Mar 2, 2015• 126 citations
arXivHTMLPDF

Abstract

Motivated by recent experimental implementations of artificial gauge fields for gases of cold atoms, we study the scattering properties of particles that are subjected to time-periodic Hamiltonians. Making use of Floquet theory, we focus on translationally invariant situations in which the single-particle dynamics can be described in terms of spatially extended Floquet-Bloch waves. We develop a general formalism for the scattering of these Floquet-Bloch waves. An important role is played by the conservation of Floquet quasienergy, which is defined only up to the addition of integer multiples of $\ensuremath{\hbar}\ensuremath{\omega}$ for a Hamiltonian with period $T=2\ensuremath{\pi}/\ensuremath{\omega}$. We discuss the consequences of this for the interpretation of ``elastic'' and ``inelastic'' scattering in cases of physical interest. We illustrate our general results with applications to the scattering of a single particle in a Floquet-Bloch state from a static potential and the scattering of two bosonic particles in Floquet-Bloch states through their interparticle interaction. We analyze examples of these scattering processes that are closely related to the schemes used to generate artificial gauge fields in cold-atom experiments, through optical dressing of internal states, or through time-periodic modulations of tight-binding lattices. We show that the effects of scattering cannot, in general, be understood by an effective time-independent Hamiltonian, even in the limit $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}$ of rapid modulation. We discuss the relative sizes of the elastic scattering (required to stabilize many-body phases) and of the inelastic scattering (leading to deleterious heating effects). In particular, we describe how inelastic processes that can cause significant heating in the current experimental setup can be switched off by additional confinement of transverse motion.

DOI
10.1103/physreva.91.033601

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Motivated by recent experimental implementations of artificial gauge fields for gases of cold atoms, we study the scattering properties of particles that are subjected to time-periodic Hamiltonians. Making use of Floquet theory, we focus on translationally invariant situations in which the single-particle dynamics can be described in terms of spatially extended Floquet-Bloch waves. We develop a general formalism for the scattering of these Floquet-Bloch waves. An important role is played by the conservation of Floquet quasienergy, which is defined only up to the addition of integer multiples of \ensuremathℏ\ensuremathω\ensuremath{\hbar}\ensuremath{\omega}\ensuremathℏ\ensuremathω for a Hamiltonian with period T=2\ensuremathπ/\ensuremathωT=2\ensuremath{\pi}/\ensuremath{\omega}T=2\ensuremathπ/\ensuremathω. We discuss the consequences of this for the interpretation of elastic'' and inelastic'' scattering in cases of physical interest. We illustrate our general results with applications to the scattering of a single particle in a Floquet-Bloch state from a static potential and the scattering of two bosonic particles in Floquet-Bloch states through their interparticle interaction. We analyze examples of these scattering processes that are closely related to the schemes used to generate artificial gauge fields in cold-atom experiments, through optical dressing of internal states, or through time-periodic modulations of tight-binding lattices. We show that the effects of scattering cannot, in general, be understood by an effective time-independent Hamiltonian, even in the limit \ensuremathω\ensuremath→\ensuremath∞\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathω\ensuremath→\ensuremath∞ of rapid modulation. We discuss the relative sizes of the elastic scattering (required to stabilize many-body phases) and of the inelastic scattering (leading to deleterious heating effects). In particular, we describe how inelastic processes that can cause significant heating in the current experimental setup can be switched off by additional confinement of transverse motion.
This thesis considers novel phenomena arising in the few and many-body physics of ultracold atomic gases. The specific models considered are motivated by recent experimental developments. A main focus will be the theoretical description of systems used in the simulation of artificial gauge fields via time-modulated tuning of system parameters and via the coupling of internal atomic states by Raman lasers. A second aspect will be the study of systems with synthetic dimensions and the effects of the unconventional infinite range interactions that arise in this description. It explores time-dependent effects in the single and few particle setting and the collective many-body phases arising in these models. Using the framework of Floquet theory it studies the interplay of time-dependence and particle-interactions both in continuum and lattice systems. In particular, it provides a general explanation for how heating effects can arise in interacting periodically time-dependent quantum systems, identifying an underlying mechanism for heating via two-particle collisions and the relevant scaling of the associated rates with system parameters. Furthermore, the general framework is applied to specific experimental set-ups. This yields heating rates and population dynamics in agreement with the experimental data. Finally, it also proposes improvements to the experiment that allow to limit the heating rates which is required to access the strongly interacting regime. In the case of synthetic dimensions the focus lies on the intriguing interplay of artificial gauge fields and strong particle interactions of atoms confined in optical lattices. In these systems a suitable coupling of internal spin-states of the atoms allows to simulate an additional finite dimension. In contrast to the typically encountered situation of short-range interactions the particle interactions are infinite-range in this synthetic dimension. In the strong coupling limit an effective description for this system is derived and its ground state phase diagram is studied using numerical tools.
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 ${T}_{0}=2\ensuremath{\pi}/\mathrm{\ensuremath{\Omega}}$ are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a noninteracting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modeled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet quasiparticles and their scattering processes. As the energy is conserved only modulo $\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$, 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 [G. Jotzu et al., Nature (London) 515, 237 (2014)].
In this work we establish a general scattering theory for two-dimensional quantum systems with periodically modulated interactions. We show that scattering quantities, such as the scattering cross section, effective scattering length, and effective-range parameter, can be defined in a manner analogous to conventional scattering theory, even in the presence of modulation. Unlike systems with static interactions, the incident wave in our setup can be scattered inelastically into different Floquet channels, absorbing or emitting quanta from the modulation field. A family of Floquet resonances is identified, at which universal recurrence relations are derived for the Floquet scattering amplitudes. These relations are equivalent to a discrete Schr\"odinger equation describing a pointlike source trapped by an effective potential. The resonance mechanism is explained as the coincidence of the point source's energy with the bound states of the effective potential. We discuss the scattering properties both at and near the resonances, finding that the resonance width and the ratio of the inelastic to elastic cross sections can be effectively controlled through the modulation of the interaction. Our results provide a pathway to precisely manipulate many-body interactions in periodically driven quantum systems.
Controlling and manipulating complex many-body quantum systems will be a key ingredient \nfor the development of next-generation technologies. While the realisation of a \nuniversal quantum machine is still out of reach, in recent years experimental systems of \nultracold atoms have already evolved into a vivid field of research for quantum simulation. \nCrucially, such systems even allow for the successful quantum engineering of targeted \nmany-body systems by means of coherent periodic driving. The essential properties of \nthese Floquet systems encompass two main aspects: fast driving facilitates the simulation \nof effective static systems, and interactions lead to unique heating effects as energy is only \nconserved modulo the driving frequency. Within this thesis we theoretically study both \nof these aspects in respective model systems. \nIn part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott \ninsulator in a designed one-dimensional Floquet system. Here, periodic driving in combination \nwith breaking all mirror symmetries of the system can induce directed motion of \nparticles. In the limit of small excitation densities, the effectively non-interacting quantum \nratchet determines the motion of holes and doublons in the Mott insulator and can in fact \nbe used to manipulate the dynamics of such. This little quantum machine can also be \nused to drive particles against an external force, where transport is possible but requires \nthe fulfilment of a commensurability condition for long times. \nIn part II, we discuss the role of interactions for periodically driven systems by means \nof a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, \nwe develop this semiclassical formalism based on a clear separation of time scales. The \nresult is a description of the dynamics and the scattering of Floquet quasiparticles in \nsuch systems. Here, the property of discrete energy violation is naturally encoded in our \nformalism predicting the heating of interacting Floquet systems to infinite temperatures in \nthe long-time limit. As a first application of this approach, we investigate a cold atom setup \nrealising the Haldane model by means of periodic shaking. While homogeneous systems \nheat up globally, a confining potential evokes thermoelectric transport effects resulting \nfrom spatially dependent heating characteristics. Moreover, we show that the interplay \nof intrinsic heating, macroscopic diffusion and non-trivial topological properties of the \nHaldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous \nparticle transport as the result of developing temperature gradients. \nIn part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing \na theoretical framework for a possible simulation of a topological edge state in a \none-dimensional optical lattice. In this case, the one-dimensional Dirac equation with \nspatially varying mass is important, which captures the topological properties of a corresponding \nsystem of the BDI symmetry class. We analytically discuss such system and \ninvestigate the role of mean-field interaction effects. We also identify the emergence of \ndynamical instabilities in a realisation with bosonic atoms.
We study a Floquet realization of the Harper-Hofstadter model that has recently been implemented in a gas of cold bosonic atoms. We study in detail the scattering processes in this system in the weakly interacting regime due to the interplay of particle interactions and the explicit time dependence of the Floquet states that lead to band transitions and heating. We focus on the experimentally used parameters and explicitly model the transverse confining direction. Based on transition rates computed within the Floquet-Fermi golden rule we obtain band population dynamics which are in agreement with the dynamics observed in experiment. Finally, we discuss whether and how photon-assisted collisions that may be the source of heating and band population dynamics might be suppressed in the experimental setup by appropriate design of the transverse confining potential. The suppression of such processes will become increasingly important as experiments progress into simulating strongly interacting systems in the presence of artificial gauge fields.
The present thesis is devoted to quantum simulation of strongly interacting systems of ultra-cold atoms in optical lattices. It is a theoretical work which focuses on the possibility to employ strong time-periodic forcing for the coherent control of these system. This form of quantum engineering is called Floquet engineering. Experimentally, time-periodic forcing has been successfully applied to realize a variety of physical models and phenomena, especially in the regime of weak interactions. We describe two novel proposals for interesting phenomena in the regime of strong interactions that rely on lattice shaking: melting of a Mott-insulator into an excited-state superfluid via coherent coupling of Bloch bands and the creation of 1D lattice anyons. Furthermore, the role of multiphoton excitations in a driven lattice is analyzed since these processes can lead to unwanted heating and thereby impeding of successful Floquet engineering in the experiment. The introductory Chapter 1 gives an overview over the field of quantum simulations with ultra-cold atoms in optical lattices and describes the experimental progress that has been made in the recent years. In Chapter 2, Floquet theory is reviewed, which provides an excellent framework to deal with time-periodic Hamiltonians and which is the basis of the analysis presented in the following chapters. Chapter 3 deals with the proposal of coherently coupling Bloch bands of an optical lattice via resonant lattice shaking. In particular, the orbital-driven phase transition from a Mott insulating to a superfluid ground state is described in detail. In Chapter 4, a proposal of realizing 1D lattice anyons from strongly interacting bosons in a shaken and tilted lattice is worked out. Furthermore, Friedel oscillations are proposed to provide a measurable real-space signature for the anyonization. Finally, in Chapter 5 multiphoton excitations to higher Bloch bands are analyzed for the cases of a shaken and an amplitude-modulated lattice. The strength and the location of resonances, which are associated with heating, are described theoretically and numerically.