Arbitrary-order even and odd winking states of excited Schrödinger's cats

We consider how the higher-energy excited Schrodinger's cats can be transformed into the even and odd winking states, the main feature of which consists in the periodic jumping of the probability density distribution from one to other regions of space. In particular, the winking states can be formed among closely displaced ground and arbitrary-order coherent states involved into the Schrodinger's “alive” and “dead” cat dynamics. From a pedagogical standpoint it is worth mentioning that the general analytical solution for arbitrary-order winking states can be obtained practically without any calculations. We use only the method of mathematical induction and the “often forgotten” quantum-mechanical Husimi and Taniuti symmetry transformation. We show that the Husimi and Taniuti gauge invariance establishes a one to one correspondence between the harmonic oscillator eigenstates and the higher-energy coherent states of the same order in such a way that their space and time-dependent phase factors are identical for all orders of coherent states. By way of illustration, we exemplify the main logical-operation properties of winking states up to the fifth-order.