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Progress of Theoretical Physics Supplement

Frequency Dependence of Quantum Localization in a Periodically Driven System

Manabu Machida, Keiji Saito, Seiji Miyashita

Published Jan 1, 2002• 0 citations
Publisher

Abstract

We study the quantum localization of a GOE random matrix model driven by a periodic external field. After the diffusion is suppressed, the average energy oscillates around some finite value Esat. We study how Esat depends on the frequency of the external field. In order to investigate this phenomenon, we also study the frequency dependence of the relevant number of Floquet states Nmin, and discuss its dependence in the view of the Anderson localization in the region where the frequency is not large.

DOI
10.1143/ptps.145.290

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We study the quantum localization of a GOE random matrix model driven by a periodic external field. After the diffusion is suppressed, the average energy oscillates around some finite value Esat. We study how Esat depends on the frequency of the external field. In order to investigate this phenomenon, we also study the frequency dependence of the relevant number of Floquet states Nmin, and discuss its dependence in the view of the Anderson localization in the region where the frequency is not large.
We study the quantum localization phenomena for a random matrix model belonging to the Gaussian orthogonal ensemble (GOE). An oscillating external field is applied on the system. After the transient time evolution, energy is saturated to various values depending on the frequencies. We investigate the frequency dependence of the saturated energy. This dependence cannot be explained by a naive picture of successive independent Landau-Zener transitions at avoided level crossing points. The effect of quantum interference is essential. We define the number of Floquet states which have large overlap with the initial state, and calculate its frequency dependence. The number of Floquet states shows approximately linear dependence on the frequency, when the frequency is small. Comparing the localization length in Floquet states and that in energy states from the viewpoint of the Anderson localization, we conclude that the Landau-Zener picture works for the local transition processes between levels.
Under a periodically driven external field or a temporally oscillating boundary, quantum few-body systems whose corresponding classical systems show chaos are modeled by the random matrices with an oscillating parameter. In such systems, time evolution of the system from the ground state shows the quantum diffusion. However, due to periodic oscillation of the parameter, quantum interference effect prevents the energy of the system from diverging and the energy saturates after some transient time. We show an example where the energy eventually fluctuates around a constant saturated energy. Since the value of this saturated energy depends on the frequency of the driving, we also investigate the dependence of the saturated energy on the frequency in the small frequency region.
Many-body localization transition in a periodically driven quantum system is investigated using a solution of a matching Bethe lattice problem for Floquet states of a quantum random energy model with a generalization to more realistic settings. It turns out that an external periodic field can both suppress and enhance localization depending on field amplitude and frequency which leads to three distinguishable regimes of field enhanced, controlled and suppressed delocalization. The results can be verified experimentally in systems of cold atoms and/or interacting spin defects in semiconductors.
The transport and localization of quantum droplets in time-dependent periodic disorder potentials are systematically studied. Our numerical results indicate that such exquisite periodically driven structures may support quantum phases that are principally different compared to static systems. We observe an intriguing anomalous diffusion, with a diffusion exponent depending on the strength of quantum fluctuations and on noise parameters (amplitude and frequency). We show that the interplay of quantum fluctuations and noise tends to suppress localization especially in a large droplet, while the transport is subdiffusive in a small droplet only for a very weak noise. For reducing modulation strength and at fixed dephasing time, the system may exhibit Anderson localization of the Floquet states in the small droplet. We also elucidate the role of thermal effects in the transport properties of quantum droplets. The obtained anomalous diffusion is justified in the frame of a heuristic model.
A periodically driven two-state dynamics, being analysed within the Floquet formalism, exhibits localization of the amplitude dynamics in an infinite frequency range, extending from the bare tunnel splitting up to infinity. In contrast, the suppression of tunnelling in a driven symmetric double well is restricted to a limited frequency regime, extending from the bare tunnel splitting up to the first resonance frequency with higher-lying states. With the amplitude dynamics of a periodically driven two-level system not being restricted to describe coherent tunnelling transport only, the localization phenomenon within the infinite frequency range does allow for novel applications for systems in strong laser fields.
The dynamics of periodically driven quantum systems coupled to a thermal environment is investigated. The interaction of the system with the external coherent driving field is taken into account exactly by making use of the Floquet picture. Treating the coupling to the environment within the Born-Markov approximation one finds a Pauli-type master equation for the diagonal elements of the reduced density matrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, time-periodic density matrix which describes the long-time behavior of the system. Taking the example of a periodically driven particle in a box, the stationary solution is determined numerically for a wide range of driving amplitudes and temperatures. It is found that the quasistationary distribution differs substantially from a Boltzmann-type distribution at the temperature of the environment. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain number of Floquet states. This number of Floquet states turns out to be nearly independent of the temperature. The plateau region is sharply separated from an exponential tail of the stationary distribution which expresses a canonical Boltzmann-type distribution over the mean energies of the Floquet states. These results are explained in terms of the structure of the matrix of transition rates for the dissipative quantum system. Investigating the corresponding classical, nonlinear Hamiltonian system, one finds that in the semiclassical range essential features of the quasistationary distribution can be understood from the structure of the underlying classical phase space.