Quantum Stairs and Multi-Rabi Chaos in a Driven Anharmonic Oscillator

Abstract

A single magnetized electron driven by the EM wave in the vicinity of the cyclotron frequency can exhibit large hysteretic resonance caused by a tiny relativistic change of its mass [1]. Consistent with the theory [1] this effect has recently been observed in experiment [2]. The theory of this most fundamental multi-stable interaction of light with matter can be developed using a simple model of a quantum anharmonic oscillator driven by a periodic force. Making a common assumption that quantum transitions occur only between neighboring slightly-nonequidistant eigenstates of the oscillator, one can describe the dynamics of the system by infinite number of coupled kinetic equations for the density matrix elements at each eigenstate. We found that the reaction of the system (expressed in the terms of expectation energy of excitation) dramatically depends on the speed of sweeping frequency of the driving force near the cyclotron resonance. If the driving frequency is swept downward infinitesimally slow and no dissipation is present, the system’s response shows strongly pronounced train of "quantum stairs" at the raising slope of the function "energy vs. driving frequency" (Fig. 1) starting at the main (cyclotron) frequency Ωr which is a resonant frequency of the unperturbed (i.e. harmonic) oscillator. The height of each of these stairs is ħΩr and they are equidistantly spaced by ΔΩsp = Ωn−Ωn−1 such that ΔΩsp/Ωr = ħΩr/moc2 = krre/α, where kr = Ωr/c, re = e2/moc2 is a classical electron radius, and α = e2/ħc = 1/137 is a fine structure constant; e.g., at λr = 2mm, ΔΩsp = 180.76 Hz. The stair of each consequent order n can be interpreted as an adiabaticly slow Landau-Zenner transition between (n-1)th and n-th excited level respectively. However, when the frequency sweeping is sufficiently fast, these transitions become oscillatory with the oscillations at each one of them being due to a Rabi frequency pertinent to that individual transition. Since all of them are coupled and since due to the anharmonicity all the Rabi oscillations form an infinite set of incommensurate frequencies, these oscillations evolve into strongly chaotic motion (Fig. 2). These quantum effects are universal and should exist in any anharmonic oscillator as long as its anharmonicity is much stronger than dissipation, i.e. when ΔΩspτ ≫ 1, where τ is relaxation time of the system (for a single cyclotron electron with its energy dissipation attributed to the synchrotron radiation, ΔΩspτ = 3/2α = 205.5). This work is supported by AFOSR.