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Contemporary Physics

Exploring fractional revivals and sub-Planck structures: a walk in phase space with Wigner and Kirkwood

J. Banerji

Published May 1, 2007• 13 citations

Abstract

The nonlinear evolution of certain quantum systems such as a Gaussian wave packet in an infinite potential well, can lead to fractional revivals of the initial wave packet. Analogously, a coherent state propagating through a Kerr medium can form the so-called Schrödinger cats and cat-like states. Recently, it was shown that a superposition of Schrödinger cats can give rise to sub-Planck structures in the Wigner distribution. The purpose of this review is to introduce and apply, in the above context, a phase space distribution ‘which differs but little from Wigner's and yet, had remained obscure for decades. This is the Kirkwood – Rihaczek (KR) distribution, which has aroused considerable interest in recent years. It is thus appropriate and timely to see how it compares with the Wigner distribution in analysing the above phenomena. Our detailed analysis shows that the KR distribution is just as adequate in explaining these phenomena as the Wigner distribution. Sub-Planck structures are obtained in the KR distribution as well, and despite important differences, the two phase space pictures are comparable.

DOI: 10.1080/00107510701666869

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The nonlinear evolution of certain quantum systems such as a Gaussian wave packet in an infinite potential well, can lead to fractional revivals of the initial wave packet. Analogously, a coherent state propagating through a Kerr medium can form the so-called Schrödinger cats and cat-like states. Recently, it was shown that a superposition of Schrödinger cats can give rise to sub-Planck structures in the Wigner distribution. The purpose of this review is to introduce and apply, in the above context, a phase space distribution ‘which differs but little from Wigner's and yet, had remained obscure for decades. This is the Kirkwood – Rihaczek (KR) distribution, which has aroused considerable interest in recent years. It is thus appropriate and timely to see how it compares with the Wigner distribution in analysing the above phenomena. Our detailed analysis shows that the KR distribution is just as adequate in explaining these phenomena as the Wigner distribution. Sub-Planck structures are obtained in the KR distribution as well, and despite important differences, the two phase space pictures are comparable.

We calculate the Wigner quasi-probability distribution for position and momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite well potential, using both x- and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear at fractional multiples of the exact revival time.

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, PW(n)(x,p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, PW(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like “mini-packets” which appear at fractional multiples of the exact revival time.

We study the dynamics of superposed wave packets in a specific nonlinear Hamiltonian which models the wave packet propagation in Kerr-like media and the dynamics of Bose–Einstein condensates. We show the dependence of fractional revival times on the initial superposed wave packets using analysis based on the expectation values, Rényi entropy and Wigner function. We also show how the selective identification of fractional revivals using moments of appropriate observables depends on the number of wave packets present in the initial state.

Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining appropriate higher moments.

Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.

We find the existence of sub-Planck scale structures in the P\"oschl-Teller potential, which is an exactly solvable potential with both symmetric and asymmetric features. We analyze these structures in both cases by looking at the Wigner distribution of the state evolved from an initial coherent state up to various fractional revival times. We also investigate the sensitivity to perturbations of the P\"oschl-Teller potential and we verify that, similar to the harmonic oscillator, the presence of sub-Planck structure in phase space is responsible for a high sensitivity to phase-space displacements.