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Physical Review A

Localization of two-level systems

Lei Wang, Jiushu Shao

Published Feb 1, 1994• 25 citations
Publisher

Abstract

The quantum dynamics of the two-level system under a periodic external potential is mapped to the classical one of a charged particle moving in the harmonic-oscillator potential plus a magnetic field in a plane. The behavior of tunneling and localization is fully described by the radial trajectory of the particle. It is shown that localization may happen only if the radial trajectory is periodic. The possible period is an integral multiple of that of the external force. Three models are studied.

DOI
10.1103/physreva.49.r637

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The quantum dynamics of the two-level system under a periodic external potential is mapped to the classical one of a charged particle moving in the harmonic-oscillator potential plus a magnetic field in a plane. The behavior of tunneling and localization is fully described by the radial trajectory of the particle. It is shown that localization may happen only if the radial trajectory is periodic. The possible period is an integral multiple of that of the external force. Three models are studied.
This paper reviews our work on dynamical localization of the two-level system (TLS). By mapping the quantum dynamics of a driven TLS to the classical one of a particle experiencing a two-dimensional harmonic force as well as a periodic magnetic field, deep insights into the mechanism of localization have been gained. Using this scheme we are able to obtain the condition for localization of a TLS subject to the linearly polarized field. The localization induced by delta-function pulses and circularly polarized fields has been analytically investigated. It is found that the circularly-polarized-field induced localization can be fully described by the geometry of an ellipse within a unit circle on a complex plane.
Some aspects of quantum tunneling of a particle in a double-well potential periodically driven by an external force are studied within the two-level approximation. A closed expression for the temporal evolution of the occupation probability is obtained in the limit of large-amplitude oscillation by the transfer-matrix formalism. The mechanism of coherent destruction of tunneling found by Grossmann et al. [Phys. Rev. Lett. 67, 516 (1991)] is made clear from a viewpoint of interference at periodic level crossings.
Coherent tunneling can be suppressed or enhanced by exciting a vibrational mode coupled to the tunneling coordinate. The localization in a two-level system (TLS) driven by a periodic force is obtained as a semiclassical limit for such a mode. We derive and test a semiclassical quantization rule that relates the quasienergies of a classically driven TLS with the energy splittings in the model of a TLS coupled to a quantum oscillator. The dissipative stabilization of a quasilocalized state and the possibility of localization by a quasiperiodic force are analyzed from this perspective.
We study properties of a particle moving in a double-well potential in the two-level approximation placed in an additional external time-dependent field. Using previously established property (J. Phys. A 41 , 244023 (2008)) that any two-level system possesses a pseudo-supersymmetry we introduce the notion of pseudo-supersymmetric field. It is shown that these fields, even if their time dependence is not periodical, may produce the effect of localization of the particle in one of the wells of the double-well potential.