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Physical Review A

Multilevel quantum beats: An analytical approach

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

Published Dec 1, 1996• 93 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.54.5299

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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.
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 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.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.
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.
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.
We examine the revival structure of Rydberg wave packets. These wave packets exhibit initial classical periodic motion followed by a sequence of collapse, fractional/full revivals, and fractional/full superrevivals. The effects of quantum defects on wave packets in alkali-metal atoms and a squeezed-state description of the initial wave packets are also described. We then examine the revival structure of Rydberg wave packets in the presence of an external electric field, i.e., the revival structure of Stark wave packets. These wave packets have energies that depend on two quantum numbers and exhibit new types of interference behavior.