Coupling rotational and translational motion via a continuous measurement in an optomechanical sphere

We consider a measurement of the position of a spot painted on the surface of a trapped nano-optomechanical sphere. The measurement extracts information about the position of the spot and in doing so measures a combination of the orientation and position of the sphere. The quantum back-action of the measurement entangles and correlates these two degrees of freedom. Such a measurement is not available for atoms or ions, and provides a mechanism to probe the quantum mechanical properties of trapped optomechanical spheres. In performing simulations of this measurement process we also test a numerical method introduced recently by Rouchon and collaborators for solving stochastic master equations. This method guarantees the positivity of the density matrix when the Lindblad operators for all simultaneous continuous measurements are mutually commuting. We show that it is both simpler and far more efficient than previous methods.