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Physical Review A

Classical chaos and its quantum measures in Rydberg states of multielectron atoms

Misha Ivanov, D. Bitouk, Kazimierz Rza̧żewski, Svetlana Kotochigova

Published Jul 1, 1995• 13 citations
Publisher

Abstract

We investigate classical and quantum one-dimensional models of two-electron atoms from the point of view of classical and quantum chaos. We propose a measure of electron-electron correlation as a quantum signature of classical chaos. The method is then applied to the three-dimensional multielectron wave functions of a Ca atom. The relation to quantum scars is stressed.

DOI
10.1103/physreva.52.149

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We investigate classical and quantum one-dimensional models of two-electron atoms from the point of view of classical and quantum chaos. We propose a measure of electron-electron correlation as a quantum signature of classical chaos. The method is then applied to the three-dimensional multielectron wave functions of a Ca atom. The relation to quantum scars is stressed.
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A discussion regarding the energy levels spectrum of quantum systems whose classical analogous has states of chaotic motion is presented. The chaotic dynamics of the classical underlying system has its manifestation in the wave functions (in the form of “scars”) and in the energy levels (in the form of “statistical repulsion” of the energy levels). The above mentioned signatures are named “quantum chaos”. A typical study of quantum chaos requires finding accurate energy eigenvalues of highly excited states, to calculate the nearest neighbors spacing between levels, to perform the “unfolding” of the spectrum in order to separate the fluctuations, and finally to find the probability distribution of the unfolded spectrum. This is exemplified by the hydrogen atom under uniform magnetic field and a quadrupole electric field.
We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.
How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.
Quantum chaotic scattering, R. Blumel microwave ionization of Rydberg atoms, P. Koch and L. Moorman geometry and quantum mechanics, L.N. Chang and Y.G. Liang where is the chaos in two-electron atoms?, R. Blumel and W.P. Reinhardt quantum classical correspondence and quantum chaos, W-M. Chang et al phase space structure and fractal dissociation boundary of a kicked molecular system, J.M Yuan et al from classical to quantum chaos, G. Zaslavsky quantum chaos in nuclear physics, F. Sakata and M. Marumori.