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Physical Review Letters

Generic Structure of Multilevel Quantum Beats

C. Leichtle, Ilya Sh. Averbukh, Wolfgang P. Schleich

Published Nov 4, 1996• 118 citations
Publisher

Abstract

Many transient signals from quantum systems result from beats among a large number of levels whose energies depend nonlinearly on the quantum number. Typical examples range from time-resolved laser femtochemistry to quantum optics of single atoms in cavities. Starting from rather general assumptions on the nature of the system, we derive approximate closed-form expressions, which describe such signals in the semiclassical limit. Our approach brings out in a most natural way the phenomenon of fractional revivals and full revivals and explains the oscillatory structures observed in recent experiments on atomic wave packets [Phys. Rev. Lett. 72, 3783 (1994)].

DOI
10.1103/physrevlett.77.3999

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We investigate the dynamics of an arbitrary atomic system (n-level atom or many n-level atoms) interacting with a resonant quantized mode of an electromagnetic field in a lossless cavity. Initially, the field is taken in a coherent state with a large average photon number. If the initial atomic state is chosen as an eigenstate of the atomic Hamiltonian in the classical field limit (semiclassical eigenstates), then the evolution of the system wave function is approximately factorized into the field and atomic parts. Explicit analytical expressions are given for the factorized wave functions. The time evolution of an arbitrary initial atomic state can be fully described in terms of factorized states. As examples, we discuss trapping states, and collapses and revivals of the atomic energy.

Physical Review A•Jun 1, 1994•50 citations
Many transient signals from quantum systems result from beats among a large number of levels whose energies depend nonlinearly on the quantum number. Typical examples range from time-resolved laser femtochemistry to quantum optics of single atoms in cavities. Starting from rather general assumptions on the nature of the system, we derive approximate closed-form expressions, which describe such signals in the semiclassical limit. Our approach brings out in a most natural way the phenomenon of fractional revivals and full revivals and explains the oscillatory structures observed in recent experiments on atomic wave packets [Phys. Rev. Lett. 72, 3783 (1994)].
We study the temporal behavior of generic transient signals originating from multilevel quantum systems. Such signals typically arise in the physics of wave packets in atoms, molecules, cavity QED, and ion traps and consist of a sum of a large number of harmonics whose frequencies depend nonlinearly on the sequential number of the harmonic. In particular, we focus on the semiclassical limit. Here, quantum beats between individual terms in the underlying sum lead to characteristic features of the signal in different time regimes, such as collapse, fractional revivals, and full revivals. We present a universal recipe for describing analytically all of the details of these features. Our approach is based on a specific representation of the sum of harmonics, which is most convenient in each of these time regions of interest. This brings out in a most natural way the phenomenon of fractional revivals and full revivals and explains their fine structures observed in recent experiments. \textcopyright{} 1996 The American Physical Society.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.
It is shown that the phenomenon of fractional revivals, which was observed in recent laser experiments on atomic wave packets, determines the long-time behavior of the Jaynes-Cummings model for a two-level atom interacting with a quantized cavity field. This provides an alternative mechanism for the generation of coherent superpositions of macroscopically distinguishable states of the field (so-called ``optical Schr\"odinger cats'') via resonant processes. The emergence of these states is associated with the appearance of atomic inversion revivals following each other 2, 3, 4, . . . times faster than the usual ones.
We examine the dynamical evolution of wave packets in a large class of quantum systems where two quantum numbers determine the energy spectrum and consequently the dynamical behavior. Using a generic Hamiltonian, we study the formation of coherent structures. The ratio of two time scales as well as the symmetry of the initial wave packet decisively determines the revival and fractional revivals of the system for which we give analytical results. The theory is applicable to a wide class of systems from diatomic molecules to ions in two-dimensional traps and two interacting Bose condensates.
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 investigate the dynamics of an arbitrary atomic system (n-level atom or many n-level atoms) interacting with a resonant quantized mode of an electromagnetic field in a lossless cavity. Initially, the field is taken in a coherent state with a large average photon number. If the initial atomic state is chosen as an eigenstate of the atomic Hamiltonian in the classical field limit (semiclassical eigenstates), then the evolution of the system wave function is approximately factorized into the field and atomic parts. Explicit analytical expressions are given for the factorized wave functions. The time evolution of an arbitrary initial atomic state can be fully described in terms of factorized states. As examples, we discuss trapping states, and collapses and revivals of the atomic energy.