Schrödinger's cat states and their nonlinear solitonic analogues

Abstract

Nonlinear solitonic analogues of Schrödinger's cat states arise in the framework of the nonlinear Schrödinger equation (NLSE) model with confining harmonic oscillator potential. In-phase or out-of-phase displaced canonical(linear) Schrödinger's coherent states (well known as the “male” or “female” Schrödinger's cats) periodically oscillate and interfere in the crossing central point of confining harmonic oscillator potential. They are self-localized and robust linear solitary waves that do not disperse and preserve their identity after repeated collisions, and, in this sense, they are analogous to the well-known nonlinear bound states of the NLSE solitons. We discuss the main parallels and distinctions between canonical Schrödinger's cat states and their nonlinear solitonic analogues. Our primary aim is to reveal the main features of nonlinear solitonic analogues of Schrödinger's cat states restricted by the quantum-mechanical normalization condition and interpretation. The fulfillment of this condition means that, in general terms, the hypothesis of the possibility to develop nonlinear quantum mechanics can be tested directly by computational experiments with generalized NLSE models. We clarify how the strong short-range nonlinear forces lead to considerable decreasing (or increasing) of the oscillation periods between even and odd soliton-like Schrödinger's coherent states, when they closely approach each other. We demonstrate the possibility to realize the coherent superposition of practically noninteracting displaced solitonic Schrödinger's coherent states with opposite phases. Our analytical results do give a quite good qualitative and quantitative check of the numerical results known so far.