Time-Optimal Transport of a Harmonic Oscillator: Analytic Solution

Motivated by the experimental transport of a trap with a quantum mechanical system modeled as a harmonic oscillator (h.o.) the corresponding classical problem is investigated. Protocols for the fastest possible transport of a classical h.o. in a wagon over a distance d are derived where both initially and finally the wagon is at rest and the h.o. is in its equilibrium position and also at rest. The acceleration of the wagon is assumed to be bounded. For fixed oscillator frequency \Omega it is shown that there are in general three switches in the acceleration and for special values of \Omega only one switch. In the latter case the optimal transport time is T_abs , that of a wagon without oscillator. The optimal transport time and the switch times are determined. It is shown that in some cases it is advantageous to go backwards for a while. In addition a time-dependent \Omega(t), bounded by \Omega_ and \Omega+ , is allowed. In this case the behavior depends sensitively on {\Omega}_+ and is spelled out in detail. In particular, depending on \Omega_+ , T_abs may be obtained in continuously many ways.