Amplification of Schrödinger-cat state: distinguishability and interference in phase space

Amplification and cloning of the Schrödinger-cat state (even coherent-state superposition) in the optical nondegenerate parametric amplifier are analysed. In the long-time limit, distinguishability and interference in the marginal probability distributions of the Schrödinger-cat state cannot be preserved simultaneously during amplification. The mutual ratio between distinguishability and interference can be controlled by the squeezed vacuum state injected in the idler mode. Thus only one of either quantum interference of indistinguishable states or distinguished states without interference can be obtained on the output of the amplifier. The purity of the initial state in the signal mode disappears during the amplification for both cases, even if the interference is preserved. Thus the interference in the marginal probability distribution is connected with indistinguishability in the phase space rather than with purity of the state. On the other hand, the nonclassical oscillations in the photon-number distribution vanish simultaneously, as the state is more mixed, irrespective of the existence of the interference.