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Journal of the Physical Society of Japan

Frequency Dependence of Quantum Localization in a Periodically Driven System

Manabu Machida, Keiji Saito, Seiji Miyashita

Published Jan 1, 2003• 1 citation
Publisher

Abstract

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.

DOI
10.1143/jpsjs.72sc.109

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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 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.
The dynamics of the harmonic oscillator, with a fluctuating, time-dependent frequency, is studied in the quantum framework. For periodic fluctuations of period \ensuremath{\tau}, it is found that the mean energy diverges exponentially in time, provided \ensuremath{\tau} falls into certain ``bands,'' centered around ``quasiresonant'' values, for which the rate of increase is maximum. In agreement with the correspondence principle, this rate is shown to coincide exactly with the one obtained for the classic oscillator, in the same conditions [Loris Ferrari, Phys. Rev. B 56, 593 (1997)]. The notion of adiabaticity has been revisited in the case of exponentially diverging mean energy. It is found that the Heisenberg form of the Hamiltonian operator can be expressed as a new effective Hamiltonian, which is still harmonic, but has exponentially diverging coefficients. It is argued that the same results hold in general (that is, without any quasiresonance condition) if the frequency fluctuation in time is disordered. A brief discussion is presented about nonharmonic interactions leading to dissipative effects. Applications to real physical systems are outlined, with special reference to the electrons in a uniform and constant magnetic field, to which a small fluctuating component is added.
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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.
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