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arXiv:chao-dyn/9311003 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correspondence in quasiperiodic and chaotic maps: Quantization via the von Neumann equation

Joshua Wilkie, Paul W. Brumer

Published Mar 1, 1994• 15 citations
arXivHTMLPDF

Abstract

A generalized approach to the quantization of a large class of maps on a torus, i.e., quantization via the von Neumann equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well-behaved mixed quantum states for tori for which the corresponding Schr\"odinger equation has no solutions, as well as an extended spectrum for tori where the Schr\"odinger equation can be solved. Quantum-classical correspondence is demonstrated for the class of mappings considered, with the Wigner-Weyl density p(p,q,t) going to the correct classical limit. An application to the cat map yields, in a direct manner, nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the correspondence limit.

DOI
10.1103/physreve.49.1968

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•Jan 1, 2014•0 citations
A generalized approach to the quantization of a large class of maps on a torus, i.e., quantization via the von Neumann equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well-behaved mixed quantum states for tori for which the corresponding Schr"odinger equation has no solutions, as well as an extended spectrum for tori where the Schr"odinger equation can be solved. Quantum-classical correspondence is demonstrated for the class of mappings considered, with the Wigner-Weyl density p(p,q,t) going to the correct classical limit. An application to the cat map yields, in a direct manner, nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the correspondence limit.
The classical-quantum correspondence is developed in its most natural setting, i.e., that of phase space distributions. Specifically, the Wigner–Weyl representation of the two fundamental quantum densities ‖n≳<n‖ and ‖n≳<m‖, which are eigenstates of the quantum Liouville operator, are shown, for integrable systems, to yield stationary and nonstationary classical Liouville eigenstates in the ℏ→0 limit. The former are uniform distributions on semiclassically quantized tori, whereas the latter are nonuniform distributions on similar tori at intermediate actions. Applications to semiclassical mechanics in action-angle variable, the Heisenberg correspondence for matrix elements and the computation of vibrational intensities are provided.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
The quantum model system is considered for which the classic analog is the known Arnold’s саг map. Due to periodicity conditions for the phase space, quantum states are represented by vectors of finite dimension N, and operators by NxN matrices. The integer parameter N characterizes a relative value of quantum effects; classic limit corresponds to N—>∞. Operator тар is suggested which governs discrete time evolution in Heisenberg representation for operators of finite shifts for position and momentum. Explicit form of evolution operator is stated in Schrodinger representation. Solution for non—stationary problem is presented and discussed for initial conditions taken as localized state, two delta—spikes, Gaussian wave packet. Quantum dynamics in terms of Husimi distribution and Wigner function, quasi—cnergy spectrum and eigenvector structure are discussed on а basis of dynamics of Heisenberg operators.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.
This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.