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arXiv:1506.06725 · Physical Review A

Dynamic steady state of periodically driven quantum systems

Valeriy I. Yudin, А. В. Тайченачев, M. Yu. Basalaev

Published Jan 13, 2016• 39 citations
arXivHTMLPDF

Abstract

Using the density matrix formalism, we prove the existence of the periodic steady state for an arbitrary periodically driven system described by linear dynamic equations. This state has the same period as the modulated external influence, and it is realized as an asymptotic solution $(t\ensuremath{\rightarrow}+\ensuremath{\infty})$ due to relaxation processes. The presented derivation simultaneously contains a simple and effective computational algorithm (without using either the Floquet or Fourier formalisms), which automatically guarantees a full account of all frequency components. As a particular example, for three-level $\mathrm{\ensuremath{\Lambda}}$ system we calculate the line shape and field-induced shift of the dark resonance formed by the field with a periodically modulated phase. Also we have analytically solved a basic theoretical problem of the direct frequency comb spectroscopy, when the two-level system is driven by the periodic sequence of rectangular pulses. In this case, the radical dependence of the spectroscopy line shape on pulse area is found. Moreover, the existence of quasiforbidden spectroscopic zones, in which the Ramsey fringes are significantly reduced, is predicted. Our results have a wide area of applications in laser physics, spectroscopy, atomic clocks, and magnetometry. Also they can be useful for any area of quantum physics where periodically driven systems are considered.

DOI
10.1103/physreva.93.013820

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Using the density matrix formalism, we prove the existence of the periodic steady state for an arbitrary periodically driven system described by linear dynamic equations. This state has the same period as the modulated external influence, and it is realized as an asymptotic solution (t\ensuremath→+\ensuremath∞)(t\ensuremath{\rightarrow}+\ensuremath{\infty})(t\ensuremath→+\ensuremath∞) due to relaxation processes. The presented derivation simultaneously contains a simple and effective computational algorithm (without using either the Floquet or Fourier formalisms), which automatically guarantees a full account of all frequency components. As a particular example, for three-level \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ system we calculate the line shape and field-induced shift of the dark resonance formed by the field with a periodically modulated phase. Also we have analytically solved a basic theoretical problem of the direct frequency comb spectroscopy, when the two-level system is driven by the periodic sequence of rectangular pulses. In this case, the radical dependence of the spectroscopy line shape on pulse area is found. Moreover, the existence of quasiforbidden spectroscopic zones, in which the Ramsey fringes are significantly reduced, is predicted. Our results have a wide area of applications in laser physics, spectroscopy, atomic clocks, and magnetometry. Also they can be useful for any area of quantum physics where periodically driven systems are considered.
We present a general method to calculate the periodic steady state of a driven-dissipative system coupled to a transmission line (and more generally, to a reservoir) under periodic modulation of its parameters. Using Floquet's theorem, we formulate the differential equation for the system's density operator which has to be solved for a single period of modulation. On this basis we also provide systematic expansions in both the adiabatic and high-frequency regime. Applying our method to three different systems---two- and three-level models as well as the driven nonlinear cavity---we propose periodic modulation protocols of parameters leading to a temporary suppression of effective dissipation rates, and study the arising nonadiabatic features in the response of these systems.
Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micromotion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micromotion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time evolution of the system.
Driven many-body quantum systems where some parameter in the Hamiltonian is varied quasiperiodically in time may exhibit nonequilibrium steady states that are qualitatively different from their periodically driven counterparts. Here we consider a prototypical integrable spin system, the spin-1/21/21/2 transverse field Ising model in one dimension, in a pulsed magnetic field. The time dependence of the field is taken to be quasiperiodic by choosing the pulses to be of two types that alternate according to a Fibonacci sequence. We show that a steady state emerges after an exponentially long time when local properties (or equivalently, reduced density matrices of subsystems with size much smaller than the full system) are considered. We use the temporal evolution of certain coarse-grained quantities in momentum space to understand this nonequilibrium steady state in more detail and show that unlike the previously known cases, this steady state is neither described by a periodic generalized Gibbs ensemble nor by an infinite temperature ensemble. Finally, we study a toy problem with a single two-level system driven by a Fibonacci sequence; this problem shows how sensitive the nature of the final steady state is to the different parameters.
We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light-induced processes.The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time-dependent reference Hamiltonians.In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered.Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates.More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically.As examples, we apply the generalized theorem to (a) predict the dynamics of a two-level system driven by a frequency-modulated resonant oscillation, a pathological situation beyond the applicability of traditional adiabatic theorems, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can display only coherences that survive in the steady state.
Abstract Externally driven quantum systems, operating under sensitive conditions of resonance, constitute an important class of physical platforms in quantum technologies, superconducting resonant architectures, and Josephson-junction-based environments. In this class of systems, an externally imposed electromagnetic driving frequency may influence oscillatory dynamics, coherence, spectral structure, and the frequency-dependent energy distribution. This work proposes an exploratory theoretical framework for interpreting resonance-dependent energy and spectral distributions in externally driven quantum systems, treating the driving frequency not only as an external perturbation but also as an active configuration parameter of the system's response. In the proposed formulation, resonance-sensitive behavior is described through a normalized resonance-response function. The formulation distinguishes between the non-driven reference spectrum and the externally driven effective spectrum, where resonance-dependent level shifts appear as deviations from the non-driven case. Two quantities, denoted by Λ and Δ, are introduced to describe the energy distribution associated with the externally driven response. Within the proposed interpretation, Λ represents the averaged potential and externally driven contribution, while Δ represents a cycle-averaged energy of the driven response. These quantities are not intended to replace the standard Hamiltonian description of quantum systems, but to provide an additional clarifying way to analyze how externally imposed conditions in resonance may reorganize effective energetic and spectral behavior. The proposed approach is relevant to periodically driven quantum systems, Floquet-type spectral descriptions (interpreting frequency-induced modifications), superconducting resonant environments, and Josephson-junction-based systems, while remaining complementary to standard time-dependent quantum and superconducting-circuit formulations. To support the theoretical interpretation, numerical simulations are presented for three representative microwave resonance areas at 6.1, 6.9, and 7.7 GHz. The simulations employ a normalized resonance-response function and include resonance-response profiles, spectral analysis, effective energy spectra, phase-space trajectories, averaged energy-distribution components, and Gaussian probability-density maps. Additional local- and global-sweep methodology relevant to phase-space simulations further illustrate how frequency deviation from resonance and externally imposed frequency variation may reshape the distribution of phase-space activity. The simulated trajectories represent the phase-space evolution of position and momentum coordinates in a driven-oscillator model, rather than literal quantum particle paths. Similarly, the Josephson effect motivated the use of microwave frequency sweeps, since driven Josephson systems exhibit phase-sensitive and resonance-dependent behavior. Therefore, the frequency-sweep simulations should be interpreted as a preparatory resonance-response analysis, rather than as a full Josephson model. The results suggest that resonance-dependent spectral and phase-space distribution may provide useful qualitative insight for frequency-controlled quantum systems, superconducting resonant architectures, and resonance-sensitive or control platforms. This study is intended as an exploratory theoretical and simulation-based contribution, presenting how externally controlled resonance conditions influence phase-space dynamics in driven quantum systems.
In this paper, we investigate periodically driven open quantum systems within the framework of Floquet-Lindblad master equations. Specifically, we discuss Lindblad master equations in the presence of a coherent, time-periodic driving and establish their general spectral features. We also clarify the notions of transient and nondecaying solutions from this spectral perspective, and then prove that any physical system described by a Floquet-Lindblad equation must have at least one physical nonequilibrium steady state (NESS), corresponding to an eigenoperator of the Floquet-Lindblad evolution superoperator UF{\mathcal{U}}_{F}UF​ with unit eigenvalue. Since the Floquet-Lindblad formalism encapsulates the entire information regarding the NESS, it in principle enables us to obtain nonlinear effects to all orders at once. The Floquet-Lindblad formalism thus provides a powerful tool for studying driven-dissipative solid-state systems, which we illustrate by deriving the nonlinear optical response of a simple two-band model of an insulating solid and comparing it with prior results established through Keldysh techniques.