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arXiv:nlin/0207038 · Journal of the Physical Society of Japan

Frequency Dependence of Quantum Localization in a Periodically Driven System

Manabu Machida, Keiji Saito, Seiji Miyashita

Published Oct 15, 2002• 7 citations
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Abstract

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.

DOI
10.1143/jpsj.71.2427

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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.
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.
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.
We characterize the evolution operator of a quantum system with generic spectral properties (in agreement with random-matrix theory), with a time-dependent Hamiltonian. The results are applied to the independent-electron model for the dissipation of energy: an example of a physical application would be the absorption of low-frequency electromagnetic radiation by small metallic particles. We discuss the statistical properties of the various regimes of the model, which depend on whether the perturbation is large enough to mix levels, and on whether the frequency is low enough for the quantum adiabatic theorem to apply. We also show that the eigenstates of the evolution operator are Anderson localized in the adiabatic basis, and present results concerning the localization length. This localization causes a saturation of the energy absorption if the Hamiltonian is periodic in time. The addition of a small amount of noise to the Hamiltonian destroys this nonclassical saturation effect.
We study localization and charge dynamics in a monochromatically driven one-dimensional Anderson insulator focusing on the low-frequency, strong-driving regime. We study this problem using a mapping of the Floquet Hamiltonian to a hopping problem with correlated disorder in one higher harmonic-space dimension. We show that (i) resonances in this model correspond to adiabatic Landau-Zener (LZ) transitions that occur due to level crossings between lattice sites over the course of dynamics; (ii) the proliferation of these resonances leads to dynamics that appear diffusive over a single drive cycle, but the system always remains localized; (iii) actual charge transport occurs over many drive cycles due to slow dephasing between these LZ orbits and is logarithmic in time, with a crucial role being played by far-off Mott-like resonances; and (iv) applying a spatially varying random phase to the drive tends to decrease localization, suggestive of weak-localization physics. We derive the conditions for the strong-driving regime, determining the parametric dependencies of the size of Floquet eigenstates, and time scales associated with the dynamics, and corroborate the findings using both numerical scaling collapses and analytical arguments.
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.
We study a two-level system described by a Hamiltonian which exhibits a time-periodic variation of its diagonal matrix elements in the small coupling regime. We investigate the transition probability, the Bloch–Siegert shift, and the Rabi frequency of the multiquantum transition as functions of the amplitude and frequency of the oscillations. These quantities all exhibit a very distinct behavior depending upon whether or not the amplitude of the oscillations allows for the crossing of the levels. This behavior is monotonic for small amplitudes while it presents oscillations as a function of the frequency for larger amplitudes. These oscillations are unambiguously attributed to the Landau–Zener–Stueckelberg interference effects between the crossing point and the turning point. A discussion of the relevance of these oscillations to possible experimental observations, notably in electron-transfer reactions, is presented.