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The concept of the symplectic covariance property of the Wigner distribution function and the symplectic invariance of the Wigner–Rényi entropies has been leveraged to estimate the interaction time of the moving quantum state in the presence of an absolutely integrable time-dependent potential. For this study, the considered scattering centre is represented initially by the Gaussian barrier. Two modifications of this potential energy are considered: a sudden change from barrier to barrier and from barrier to well. The scattering state is prepared in the form of a Schrödinger cat, and the Moyal equation governs its further time evolution. The whole analysis of the considered scattering problem is conducted in the above-barrier regime using the phase-space representation of quantum theory. The presented concept shifts the focus from the dynamics of the quantum state to a symplectically invariant state functions in the form of the Wigner–Rényi entropies of order one and one-half. These quantities serve as indicators of the beginning and end of the interaction of the non-Gaussian state with the sufficiently fast decaying potential representing the scattering centre. The presented approach is significant because it provides a new way to estimate the interaction time of moving quantum states with the dynamical scattering centre. Moreover, it allows for studying scattering processes in various physical systems, including atoms, molecules, and condensed matter systems. [ABSTRACT FROM AUTHOR]

The phase-space approach based on the Wigner distribution function is used to study the quantum dynamics of the three families of the Schrödinger cat states identified as the even, odd, and Yurke–Stoler states. The considered states are formed by the superposition of two Gaussian wave packets localized on opposite sides of a smooth barrier in a dispersive medium and moving towards each other. The process generated by this dynamics is analyzed regarding the influence of the barrier parameters on the nonclassical properties of these states in the phase space below and above the barrier regime. The performed analysis employs entropic measure resulting from the Wigner–Rényi entropy for the fixed Rényi index. The universal relation of this entropy for the Rényi index equal one half with the nonclassicality parameter understood as a measure of the negative part of the Wigner distribution function is proved. This relation is confirmed in the series of numerical simulations for the considered states. Furthermore, the obtained results allowed the determination of the lower bound of the Wigner–Rényi entropy for the Rényi index greater than or equal to one half. [ABSTRACT FROM AUTHOR]

The phase-space approach based on the Wigner distribution function is used to study the quantum dynamics of the three families of the Schrödinger cat states identified as the even, odd, and Yurke–Stoler states. The considered states are formed by the superposition of two Gaussian wave packets localized on opposite sides of a smooth barrier in a dispersive medium and moving towards each other. The process generated by this dynamics is analyzed regarding the influence of the barrier parameters on the nonclassical properties of these states in the phase space below and above the barrier regime. The performed analysis employs entropic measure resulting from the Wigner–Rényi entropy for the fixed Rényi index. The universal relation of this entropy for the Rényi index equal one half with the nonclassicality parameter understood as a measure of the negative part of the Wigner distribution function is proved. This relation is confirmed in the series of numerical simulations for the considered states. Furthermore, the obtained results allowed the determination of the lower bound of the Wigner–Rényi entropy for the Rényi index greater than or equal to one half. [ABSTRACT FROM AUTHOR]