Dynamics of Entanglement and Uncertainty relation in Coupled Harmonic Oscillator System : Exact Results

The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schr\"{o}dinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time-dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions we derive the analytical expressions for von Neumann and R\'{e}nyi entropies. Making use of Wigner distribution function defined in phase space, we derive the time-dependence of position-momentum uncertainty relations. In order to show the dynamics of entanglement and uncertainty relation graphically we introduce two toy models and one realistic quenched model. While the dynamics can be conjectured by simple consideration in the toy models, the dynamics in the realistic quenched model is somewhat different from that in the toy models. In particular, the dynamics of entanglement exhibits similar pattern to dynamics of uncertainty parameter in the realistic quenched model.