Stability of the Ground State of a Harmonic Oscillator in a Monochromatic Wave

Abstract

Classical and quantum dynamics of a harmonic oscillator in a monochromatic wave is studied in the exact resonance and near resonance cases. This model describes, in particular, a dynamics of a cold ion trapped in a linear ion trap and interacting with two lasers fields with close frequencies. Analytically and numerically a stability of the ``classical ground state'' (CGS) -- the vicinity of the point ($x=0, p=0$) -- is analyzed. In the quantum case, the method for studying a stability of the quantum ground state (QGS) is suggested, based on the quasienergy representation. The dynamics depends on four parameters: the detuning from the resonance, $δ=\ell-Ω/ω$, where $Ω$ and $ω$ are, respectively, the wave and the oscillator's frequencies; the positive integer (resonance) number, $\ell$; the dimensionless Planck constant, $h$, and the dimensionless wave amplitude, $ε$. For $δ=0$, the CGS and the QGS are unstable for resonance numbers $\ell=1, 2$. For small $ε$, the QGS becomes more stable with increasing $δ$ and decreasing $h$. When $ε$ increases, the influence of chaos on the stability of the QGS is analyzed for different parameters of the model, $\ell$, $δ$ and $h$.