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Journal of Physics B Atomic Molecular and Optical Physics

Autocorrelation function and quantum recurrence of wavepackets

Michael Nauenberg

Published Aug 14, 1990• 93 citations
Publisher

Abstract

Quantum wavepackets follow classical orbits and spread initially with time except for the special case when the energy levels are equally spaced, e.g. the harmonic oscillator. However, for integrable systems bound localised wavepackets can exhibit at long times quantum recurrences which do not have a classical correspondence. For the Coulomb potential the recurrence time is approximately n/3 times the Kepler period 2 pi n 3 , where n is the mean value of the principal quantum numbers of the wavepacket.

DOI
10.1088/0953-4075/23/15/001

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Quantum wavepackets follow classical orbits and spread initially with time except for the special case when the energy levels are equally spaced, e.g. the harmonic oscillator. However, for integrable systems bound localised wavepackets can exhibit at long times quantum recurrences which do not have a classical correspondence. For the Coulomb potential the recurrence time is approximately n/3 times the Kepler period 2 pi n 3 , where n is the mean value of the principal quantum numbers of the wavepacket.
A simple and elegant semiclassical theory for propagating wavepackets is presented. The development of the theory avoids the explicit use of the Green-function formalism. It involves expanding the potential to a quadratic time-dependent form and the use of multiple reference trajectories. The classical analogue in phase space predicts the accuracy of the results. The theory is applied to the calculation of the autocorrelation function in a Coulomb field in one dimension. Remarkable agreement with an exact quantum calculation is obtained in many circumstances in spite of the Coulomb singularity. All the detailed quantum behaviour, including the long-time recurrences and the spectrum, is reproduced.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
Within the framework of simple perturbation theory, recurrence time of quantum fidelity is related to the period of the classical motion. This indicates the possibility of recurrence in nearly integrable systems. We have studied such possibility in detail with the kicked rotor as an example. In accordance with the correspondence principle, recurrence is observed when the underlying classical dynamics is well approximated by the harmonic oscillator. Quantum revival of fidelity is noted in the interior of resonances, while classical-quantum correspondence of fidelity is seen to be very short for states initially in the rotational Kolmogorov-Arnold-Moser region.
Based on coherent states of the reversed harmonic oscillator, a wave function is proposed with the following three properties: (1) The mean values of position, r0, and velocity, v0, are parallel to the x axis; (2) The wave function is extended by a curve parameter w ≥ 0 such that, as a function of w, the mean values rw and vw describe a rectilinear orbit along the x axis with the mean initial values r0 and v0; (3) The mean energy is constant with respect to w by the leading order of the perturbation parameter ∈. For comparison, the classical one-dimensional equation of motion is integrated. Exact analytical equivalence between classical and the ∈ = 0 quantum limit is found, both for positive and negative mean energy. The curve parameter w is in 1-1 correspondence with time t ≥ 0.
The autocorrelation function, A(t), measures the overlap (in Hilbert space) of a time-dependent quantum mechanical wave function, psi(x,t), with its initial value, psi(x,0). It finds extensive use in the theoretical analysis and experimental measurement of such phenomena as quantum wave packet revivals. We evaluate explicit expressions for the autocorrelation function for time-dependent Gaussian solutions of the Schrodinger equation corresponding to the cases of a free particle, a particle undergoing uniform acceleration, a particle in a harmonic oscillator potential, and a system corresponding to an unstable equilibrium (the so-called `inverted' oscillator.) We emphasize the importance of momentum-space methods where such calculations are often more straightforwardly realized, as well as stressing their role in providing complementary information to results obtained using position-space wavefunctions.
In terms of the action-angle Wigner function, we analyse the effects of energy spectrum anharmonicity on the evolutions of wavepackets in long time scales. The formulation shows a close connection between classical periodic motions of phase space points and the quantum (full and partial) revivals of wavepackets. This enables us to find the conditions of (full and partial) revivals in an intuitive way. For one-dimensional cases, we obtain analytic solutions for times of equal partial revivals, i.e. an initial wavepacket splits equally into several small parts in long time scales. Numerical results for the JC model confirm the theoretical predictions.