Quantum chaos in an ion trap: the delta-kicked harmonic oscillator

We propose a general method by which the overlap parameter, first proposed by Peres, of a quantum mechanical chaotic system can be determined. We show explicitly how this could be carried out for the deltakicked harmonic oscillator, a system capable of displaying chaos classically. We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic oscillator could be realized, and investigated. In classical mechanics, deterministic chaos is often most simply described as exponential sensitivity to initial conditions, meaning that initially neighbouring classical trajectories diverge extremely rapidly with time. Due to the necessity of preserving the inner product, this kind of divergence between two possible initial states cannot occur quantum mechanically. The question of what then constitutes the quantum mechanical equivalent of chaos immediately arises. An interesting proposal by Peres [1] is to examine the initial state |ψ〉 evolving under two slightly differing (classically chaotic) Hamiltonians Ĥ1 and Ĥ2. The overlap O = ∣