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Long-time asymptotic states of periodically driven open quantum systems

達彦 白井

Published Mar 24, 2016• 0 citations
Publisher

Abstract

In this thesis, we study long-time asymptotic states of periodically driven quantum systems in a dissipative environment.In order to describe the subclass of the systems, we introduce the Floquet-Gibbs state, i.e., a state whose density matrix is diagonal in the basis of the Floquet states with diagonal elements given by a Boltzmann distribution over the quasienergies.We obtain sufficient conditions for the realization of the Floquet-Gibbs state in a system with an infinitesimal system-bath coupling strength, and find that these conditions severely restrict a class of suitable systems attaining this form.With the aid of a truncated Floquet Hamiltonian in the Floquet-Magnus expansion and without the rotating wave approximation, we lift the condition of the infinitesimal coupling strength and extend the idea of the Floquet-Gibbs state to a broader subclass of open quantum system with a finite dissipation effect.We also study cooperative phenomena of a periodically driven cavity system surrounded by a dissipative environment.We found a novel-type of symmetry breaking phenomenon, which originates from a synergistic effect of the microscopic dynamical effects of the driving field and interaction effects.We show that a part of this phenomenon can be understood from the concept of the Floquet-Gibbs state.It is great pleasure to express my gratitude to all the people who help, guide, and encourage me during my time as a graduate student.Without their help I definitely could not write up this thesis.First of all, I would like to thank my supervisor Professor Seiji Miyashita and Dr. Takashi Mori.Their energetic attitudes towards physics and personality continuously stimulate me.

DOI
10.15083/00073288

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In this thesis, we study long-time asymptotic states of periodically driven quantum systems in a dissipative environment.In order to describe the subclass of the systems, we introduce the Floquet-Gibbs state, i.e., a state whose density matrix is diagonal in the basis of the Floquet states with diagonal elements given by a Boltzmann distribution over the quasienergies.We obtain sufficient conditions for the realization of the Floquet-Gibbs state in a system with an infinitesimal system-bath coupling strength, and find that these conditions severely restrict a class of suitable systems attaining this form.With the aid of a truncated Floquet Hamiltonian in the Floquet-Magnus expansion and without the rotating wave approximation, we lift the condition of the infinitesimal coupling strength and extend the idea of the Floquet-Gibbs state to a broader subclass of open quantum system with a finite dissipation effect.We also study cooperative phenomena of a periodically driven cavity system surrounded by a dissipative environment.We found a novel-type of symmetry breaking phenomenon, which originates from a synergistic effect of the microscopic dynamical effects of the driving field and interaction effects.We show that a part of this phenomenon can be understood from the concept of the Floquet-Gibbs state.It is great pleasure to express my gratitude to all the people who help, guide, and encourage me during my time as a graduate student.Without their help I definitely could not write up this thesis.First of all, I would like to thank my supervisor Professor Seiji Miyashita and Dr. Takashi Mori.Their energetic attitudes towards physics and personality continuously stimulate me.
A periodically driven quantum system, when coupled to a thermal bath, relaxes to a non-equilibrium asymptotic state. To describe these states in the case of infinitesimal coupling, the notion of Floquet-Gibbs state was recently introduced. A Floquet-Gibbs state is characterized by a density matrix diagonal in the Floquet basis of the decoupled coherent system. The diagonal elements obey the Boltzmann distribution parametrized by the quasienergies of the Floquet states. It was pointed out that the Floquet-Gibbs states are realized under strict conditions that might be experimentally infeasible. Here, by using the Magnus expansion and the concept of effective Floquet Hamiltonian, we go beyond the conventionally used rotating-wave approximation and demonstrate that the condition of infinitesimal coupling can be lifted and the idea of Floquet-Gibbs states can be extended to a broader class of open quantum systems possessing finite dissipation.
A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using the so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet–Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diagonal elements obeying a Gibbs distribution, being parametrized by the corresponding Floquet quasi-energies. Addressing the non-adiabatic driving regime, upon using the Magnus expansion, we employ the concept of a corresponding effective Floquet Hamiltonian. In doing so we go beyond the conventionally used RWA and demonstrate that the idea of Floquet–Gibbs states can be extended to the realistic case of a weak, although finite system-bath coupling, herein termed effective Floquet–Gibbs states.
We investigate a long-time asymptotic state of periodically driven open quantum systems analytically. The model we consider in this paper is a free fermionic system coupled to an energy and particle reservoir. We clarify some generic properties of the system which are independent of the details of the reservoir in the high-frequency regime of the external driving. When the frequency of the external driving is much larger than the energy cutoff of the system-reservoir coupling, the low-energy properties of the system are equivalent to those of the Gibbs distribution of a Floquet effective Hamiltonian. Furthermore, we investigate the effect of finite dissipation on the system, and elucidate that we cannot suppress excitations by merely increasing the system-reservoir coupling because the excitations are created through the reservoir by the external driving when the frequency is smaller than the energy cutoff of the system-reservoir coupling.
This thesis covers my work in the field of closed, repeatedly driven, Hamiltonian systems. These systems do not exchange particles with the surrounding environment and their time-evolution is described by Hamilton's equations of motion (in the classical framework) or the Schroedinger equation (in the quantum framework). Their interaction with the environment is encoded into the time-dependence of the system's Hamiltonian. Chapter 1 is an "Overview" in which the status of the field, my contributions and future prospective are outlined. Chapters 2 to 4 provide the theoretical background which is used in Chapters 5 to 7 to derive some original results. These results show that in Hamiltonian systems, after many driving events, universal properties emerge. In particular, using the framework of the linear Boltzmann equation, I have studied the dynamics of a mobile, light impurity in a gas of heavy particles. The impurity's kinetic energy increases and, in the long time limit, approaches a non-thermal asymptotic distribution. The significance of this work is to show explicitly the emergence of a non-thermal distribution in a closed, driven system. Moreover, using the work-fluctuation theorems, I have studied the character of the energy distribution of a generic isolated system driven according a generic protocol. Both thermal and non-thermal distributions can be realized for the same system by changing the characteristics of the driving protocol. These two different regimes are separated by a dynamical phase transition. Finally, I have used the Floquet Theory and the Magnus Expansion to analyze the behavior of a generic interacting system which is driven periodically in time. For fast driving the system is unable to absorb energy and remains localized in the low energy part of the Hilbert space while for slow driving the system absorbs energy and, in the long time limit, it is delocalized in the entire Hilbert space. These two qualitatively different behaviors are separated by a many-body localization transition which is related to the break down of the Magnus expansion at the critical value of the driving frequency.
The coherent dynamics of many body quantum system is nowadays an experimental reality: by means of the cold atoms in optical lattices, many Hamiltonians and time-dependent perturbations can be engineered. In this Thesis we discuss what happens in these systems when a periodic perturbation is applied. Thanks to Floquet theory, we can see that -- if the Floquet spectrum obeys certain continuity conditions possible in the thermodynamic limit-- dephasing among Floquet quasi-energies makes local observables relax to a periodic steady regime described by an effective density matrix: the Floquet diagonal ensemble (FDE). By means of numerical examples on the Quantum Ising Chain and the Lipkin model, we discuss the properties of the FDE focusing on the difference among ergodic and regular quantum dynamics and on how this reflects on the thermal properties (T=∞T=\inftyT=∞) of the asymptotic condition. We verify thermalization in the classically ergodic Lipkin model and we demonstrate that this effect is induced by the Floquet states being delocalized and obeying Eigenstate Thermalization Hypothesis.We discuss also, in the Ising chain case, the work probability distribution, whose asymptotic condition is not described by the form (Generalized Gibbs Ensemble) that FDE acquires for local obserbvables because of integrability. Dephasing makes some correlations invisible in the local observables, but they are still present in the system. We consider also the linear response limit: when the amplitude of the perturbation is vanishingly small, the Floquet diagonal ensemble is not sufficient to describe the asymptotic condition given by LRT. For every small but finite amplitude, there are quasi-degeneracies in the Floquet spectrum giving rise to pre-relaxation to the condition predicted by Linear Response; these phenomena are strictly related to energy absorption and boundedness of the spectrum.
The asymptotic state of a periodically driven many-body quantum system in contact with an environment is investigated. The combined action of the driving and the environment steers the system towards a state being characterized by a time-periodic density operator. To compute this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, find the stroboscopic density operator as its eigen-operator and demonstrate how particle interactions affect properties of the density operator. We illustrate the idea with a periodically rocked open Bose-Hubbard dimer and discuss the relations between the interaction-induced bifurcation transitions in a mean-field dynamics and changes in the characteristics of the quantum many-body state. We argue that Floquet maps can provide insight into the system relaxation towards its asymptotic state and may help to identify the stroboscopic time-independent generator mimicking the action of the original time-dependent one.