Quantum wave packet revivals

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Abstract

The numerical prediction, theoretical analysis, and experimental verification of the phenomenon of wave packet revivals in quantum systems has flourished over the last decade and a half. Quantum revivals are characterized by initially localized quantum states which have a short-term, quasi-classical time evolution, which then can spread significantly over several orbits, only to reform later in the form of a quantum revival in which the spreading reverses itself, the wave packet relocalizes, and the semi-classical periodicity is once again evident. Relocalization of the initial wave packet into a number of smaller copies of the initial packet (‘minipackets’ or ‘clones’) is also possible, giving rise to fractional revivals. Systems exhibiting such behavior are a fundamental realization of time-dependent interference phenomena for bound states with quantized energies in quantum mechanics and are therefore of wide interest in the physics and chemistry communities.

We review the theoretical machinery of quantum wave packet construction leading to the existence of revivals and fractional revivals, in systems with one (or more) quantum number(s), as well as discussing how information on the classical period and revival time is encoded in the energy eigenvalue spectrum. We discuss a number of one-dimensional model systems which exhibit revival behavior, including the infinite well, the quantum bouncer, and others, as well as several two-dimensional integrable quantum billiard systems. Finally, we briefly review the experimental evidence for wave packet revivals in atomic, molecular, and other systems, and related revival phenomena in condensed matter and optical systems.

Introduction

The study of localized, time-dependent solutions to bound state problems in quantum mechanics has been of interest to both researchers and students of the subject alike since the earliest days of the development of the field. Schrödinger [1] and others [2], [3], [4] discussed the connections between the quantum and classical descriptions of nature by exhibiting explicit wave packet solutions to many familiar problems, including the cases of the free-particle, uniform acceleration (constant electric or gravitational field), harmonic oscillator (forerunner of coherent and squeezed states), and uniform magnetic field. Many such examples then appeared in early textbooks [5], [6], [7] only a decade later, discussing both wave packet spreading and periodic time-dependence in a way which is easily accessible to students even today.

Despite Schrödinger's hope [1] that “…wave groups can be constructed which move round highly quantised Kepler ellipses and are the representations by wave mechanics of the hydrogen electron…” (without spreading, as with the constant width harmonic oscillator packet he derived), early investigators soon found [3], [8] that such dispersion was a natural feature of wave packets for the Coulomb problem.

Attempts at constructing localized semi-classical solutions (of the coherent-state type) for the Coulomb problem, following up on Schrödinger's suggestions, continued with theoretical results [9], [10], [11], [12], [13], [14], [15], [16] appearing in the literature much later. It was, however, the development of modern experimental techniques, involving the laser-induced excitation of atomic Rydberg wave packets, including the use of the ‘pump-probe’ [17] or ‘phase-modulation’ [18] techniques to produce, and then monitor the subsequent time-development of, such states which led to widespread interest in the physics of wave packets. (For reviews of the subject, see [19], [20], [21], [22], [23].) Updated proposals for the production of such states in the context of Rydberg atoms, where one could study the connections between localized quantum mechanical solutions and semi-classical notions of particle trajectories, led first to the creation of such spatially localized states [24], to experiments which observed their return to near the atomic core [25], and then the observation of the classical Kepler periodicity [26], [27] of Rydberg wave packets, over only a few cycles in the early experiments.

However, this interest also led to the prediction of qualitatively new features in the long-term time development in such bound state systems, such as quantum wave packet revivals. Parker and Stroud [28] were the first to find evidence for this behavior in numerical studies of Rydberg atoms, while Yeazell and Stroud [30], [31] and others [32], [33] soon confirmed their predictions experimentally.

The phenomenon of wave packet revivals, which has now been observed in many experimental situations, arises when a well-localized wave packet is produced and initially exhibits a short-term time evolution with almost classical periodicity (Tcl) and then spreads significantly after a number of orbits, entering a so-called collapsed phase where the probability is spread (not uniformly) around the classical trajectory. On a much longer time scale after the initial excitation, however, called the revival time (with TrevTcl), the packet relocalizes, in the form of a quantum revival, in which the spreading reverses itself and the classical periodicity is once again apparent. Even more interestingly, many experiments have since observed additional temporal structures, with smaller periodicities (fractions of Tcl), found at times equal to rational fractions of the revival time (pTrev/q). These have been elegantly interpreted [34] as the temporary formation of a number of ‘mini-packets’ or ‘clones’ of the original packet, often with 1/q of the total probability, exhibiting local periodicities Tcl/q, and have come to be known as fractional revivals. Observations of fractional revivals have been made in a number of atomic [31], [32], [33] and molecular [35] systems.

A simple picture [36] of the time-dependence of the quantum state leading to these behaviors, modeling the individual energy eigenstates and their exponential (exp(−iEnt/ℏ)) time-dependent factors as an ensemble of runners or race-cars on a circular track, is often cited. The quantum mechanical spreading arises from the differences in speed, while the classical periodicity of the system is observable over a number of revolutions (or laps). For longer times, however, the runners/race-cars spread out and no correlations (or clumpings) are obvious, while after the fastest participants have lapped their slower competitors (once or many times), obvious patterns can return, including smaller ‘packs’ of racers, clumped together, which model fractional revivals.

A different metaphor involves the (deterministic) shuffling of an initially highly ordered deck of playing cards. One shuffling method involves splitting the deck into two equal halves, and then alternately placing the bottom card from each half into a pile, reforming and reordering the deck. After only a few such shuffles, the original order is seemingly completely lost and the cards appear to have randomized. After only a few more turns, however, clear patterns of ordered subsets of suits and ranks appear, increasingly so until after only eight such shuffles the deck has returned to its original highly ordered state. Only the simplest of mathematical concepts is required to describes these analogs of fractional and full revivals.

The Dynamics of wave packets of highly-excited states of atoms and molecules, including a discussion of wave packet revivals and fractional revivals, and descriptions of the experimental observations of these phenomena, were discussed in the excellent 1991 review by Averbukh and Perelman [37], while a nicely accessible general discussion by Bluhm and Kostelecký [38] has also appeared. There have been developments in the field since then, and many of the basic quantum mechanical concepts behind revival behavior have also begun to appear in the pedagogical literature, so it seems appropriate to provide a review of some of the fundamental ideas behind the short- and long-term behavior of quantum wave packets, describing both the classical periodicity and revival behavior of wave packets in many model systems, and their experimental realizations.

The theoretical machinery required to understand many aspects of revival behavior is sufficiently accessible, and potentially of enough general interest, that our review of the subject will contain many tutorial aspects, such as

  • (i)

    the use of familiar model systems (such as the infinite well and others) as illustrative of the fundamental concepts,

  • (ii)

    an emphasis on examples in which exact revival behavior is found (to be used as benchmarks for more realistic systems),

  • (iii)

    a focus on semi-classical methods, including the WKB approximation, which are appropriate for many wave packet systems constructed from large n energy eigenstates,

  • (iv)

    general discussions of how information on both the classical periodicity and quantum revival times are encoded in the energy eigenvalue spectrum,

  • (v)

    references not only to the original research literature, but to many of the pedagogical papers appearing on the subject.

In these areas, and others, we hope to extend the reviews in [37], [38] in useful ways.

We start in Section 2 with a general introduction to the theoretical tools required for understanding revival behavior, including the autocorrelation function. We then turn to discussions of many model systems (in Section 3) which illustrate various aspects of the quantum mechanical time development of localized wave packets, including background material on unbound systems (free-particle and uniform acceleration cases) and for ones exhibiting only periodic behavior (the harmonic oscillator). The infinite well is then discussed in great detail, as are other familiar one-dimensional (1D) problems. Two-dimensional (2D) quantum systems, especially quantum billiard geometries, are studied in Section 4. We then briefly review experimental evidence for revival behavior in Section 5 in atomic (Coulomb) and molecular (vibrational, rotational) systems, as well as in situations where the quantum revivals are due to the quantized nature of the electromagnetic field in two-state atom-field systems, or in the excitation spectrum of Bose–Einstein condensates, and finally we discuss related revival phenomena in a variety of optical systems.

Section snippets

Autocorrelation function

The study of the time-development of wave packet solutions of the Schrödinger equation has a long history and often makes use of the concept of the overlap (〈ψt|ψ0〉) of the time-dependent quantum state (|ψt〉) with the initial state (|ψ0〉). For example, early work by Mandelstam and Tamm [39] on the time-development of isolated quantum systems led to the inequality|〈ψt0〉|2cos2ΔHtvalidfor0⩽t⩽πℏ2ΔH,where ΔH=〈H2〉−〈H〉2 is the uncertainty in the free-particle energy of the wave packet. These ideas

Model systems

The time-dependence of localized quantum wave packets, including possible quantum revival behavior, has been discussed for a large number of pedagogically familiar, and physically relevant, 1D model systems. We briefly review several such cases, and then focus attention on the infinite well as a benchmark case where exact quantum revival behavior is found.

Two- and three-dimensional quantum systems

A number of integrable 2D infinite well or quantum billiard geometries lend themselves to the study of quantum revival behavior in systems with several quantum numbers, and we focus here on three polygonal billiard footprints, namely the square (N=4), equilateral triangle (N=3), and circular (N→∞) infinite wells. Discussions of time-dependent wave packet solutions of the first and third cases go back to at least de Broglie [4] and also provide useful examples of the connections of the

Experimental realizations of wavepacket revivals

The existence of revival and fractional revival behavior in quantum bound states, first found numerically in simulations of Rydberg atoms [28], has led to a number of experimental tests in atomic, molecular, and other systems. We briefly review some of the experimental evidence for quantum wave packet revivals, while noting that excellent reviews of wave packet physics [19], [20], [21], [22], [23], [37] have appeared elsewhere. In addition, we discuss other experimental realizations of quantum

Discussion and conclusions

The connections between the energy eigenvalue spectrum of a quantum bound state system and the classical periodicities of the system have been a standard subject in quantum theory since the first discussions of the correspondence principle by Bohr.

Some semiclassical techniques, such as periodic orbit theory, can connect the quantum energy spectrum with classical closed or periodic orbits, but often do so in a way which does not exhibit the time-dependence of quantum wave packets. Truly dynamic

Acknowledgements

We thank M. Doncheski for fruitful and enjoyable collaborations on many projects. We are very grateful to I. Averbukh, M. Belloni, R. Bluhm, H. Fielding, A. Kostelecký, I. Marzoli, W. Schleich, C. Stroud, W. van der Zande, and D. Villeneuve for helpful comments and communications. Some of the original work of the author cited here was supported, in part, by the National Science Foundation under Grant DUE-9950702.

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