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arXiv:1511.06864 · New Journal of Physics

Effective Floquet–Gibbs states for dissipative quantum systems

Tatsuhiko Shirai, Juzar Thingna, Takashi Mori, Sergey Denisov, Peter Hänggi A, Seiji Miyashita

Published May 10, 2016• 81 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/18/5/053008

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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.
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.
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.
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.
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.
Although the statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies epsilon_{j} can be placed in a finite frequency interval 0< or =epsilon_{j}<omega (with omega the driving frequency and variant Planck's over 2pi=1 ). Therefore, if we describe a Floquet system approximately by restricting its available state space to be spanned by a finite number N of basis states, then the number of near degeneracies ( |epsilon_{j}-epsilon_{k}|< or =delta for arbitrarily small fixed delta ) in this interval grows without limit as N is increased. As we noted in a previous paper, this leads to pathologies, including drastic changes in Floquet states, as N tends to infinity. In earlier work on Floquet systems in contact with a heat bath these difficulties were often put aside by fixing N while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, with some major differences from the usual time-independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated space spanned by the N lowest energy eigenstates of the undriven system is limited, as N increases indefinitely, only by the usual Born-Markov approximation, which neglects bath memory effects. As we seek higher accuracy by increasing N , we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive here the equations for the steady-state reduced density matrix without restriction on the size of quasienergy splittings. In general, this matrix is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where near degeneracies of quasienergies appear. In spite of the Floquet state pathologies, the explicit form of our results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as N approaches infinity.
It is proved that the energy absorption in a periodically driven classical spin system is exponentially slow in frequency, which results in a two-step relaxation called the Floquet prethermalization. This result is shown by establishing the classical limit of the quantum spin dynamics. The Floquet prethermal state is well described by the Gibbs ensemble with respect to the static effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion. On the other hand, the same effective Hamiltonian does not reproduce the local dynamics for an exponentially long time. This is due to the chaoticity of classical dynamics and in stark contrast to quantum spin systems, in which the effective Hamiltonian well reproduces the exact quantum dynamics up to an exponentially long time.