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, $\delta =\ell -\Omega /\omega$, where $\Omega$ and $\omega$ 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, $\epsilon$. For $\delta =0$, the CGS and the QGS are unstable for resonance numbers $\ell =1,2$. For small $\epsilon$, the QGS becomes more stable with increasing $\delta$ and decreasing $h$. When $\epsilon$ increases, the influence of chaos on the stability of the QGS is analyzed for different parameters of the model, $\ell$, $\delta$ and $h$.

RevTeX, 38 pages, 24 figures

References in corpus (5)

• Quantum chaos in an ion trap: the delta-kicked harmonic oscillator
• Quantum weak chaos in a degenerate system
• Dynamical Stability and Quantum Chaos of Ions in a Linear Trap
• Localization of Quantum States at the Cyclotron Resonance
• Symmetry of Quantum Phase Space in a Degenerate Hamiltonian System

Cited by in corpus (1)

• Time evolution of interacting bosons through squeezing Hamiltonians