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Physical Review A

General method for deforming quantum dynamics into classical dynamics while keeping ħ fixed

K. R. W. Jones

Published Jul 1, 1993• 9 citations
Publisher

Abstract

Using Weinberg's generalized nonlinear quantum dynamics [Ann. Phys. (N.Y.) 194, 336 (1989)], we propose a simple way to form an interpolative dynamical system that joins the classical and quantum regimes. By altering a dimensionless control parameter \ensuremath{\lambda}\ensuremath{\in}[0,1], one can smoothly metamorphose quantum evolution into classical evolution, keeping the Hilbert-space dimension and physical constants unchanged. The method suggests an approach to studies of dynamical chaos suppression in quantized classically chaotic systems.

DOI
10.1103/physreva.48.822

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Using Weinberg's generalized nonlinear quantum dynamics [Ann. Phys. (N.Y.) 194, 336 (1989)], we propose a simple way to form an interpolative dynamical system that joins the classical and quantum regimes. By altering a dimensionless control parameter \ensuremath{\lambda}\ensuremath{\in}[0,1], one can smoothly metamorphose quantum evolution into classical evolution, keeping the Hilbert-space dimension and physical constants unchanged. The method suggests an approach to studies of dynamical chaos suppression in quantized classically chaotic systems.
We introduce a simple deformed quantization prescription that interpolates the classical and quantum sectors of Weinberg's nonlinear quantum theory. The result is a novel classical limit where ℏ\hbarℏ is kept fixed while a dimensionless mesoscopic parameter, λ∈[0,1]λ\in[0,1]λ∈[0,1], goes to zero. Unlike the standard classical limit, which holds good up to a certain timescale, ours is a precise limit incorporating true dynamical chaos, no dispersion, an absence of macroscopic superpositions and a complete recovery of the symplectic geometry of classical phase space. We develop the formalism, and discover that energy levels suffer a {\em generic perturbation\/}. Exactly, they become E(λ2ℏ)E(λ^{2}\hbar)E(λ2ℏ), where λ=1λ= 1λ=1 gives the standard prediction. Exact interpolative eigenstates can be similarly constructed. Unlike the linear case, these need no longer be orthogonal. A formal solution for the interpolative dynamics is given, and we exhibit the free particle as one exactly soluble case. Dispersion is reduced, to vanish at λ=0λ= 0λ=0. We conclude by discussing some possible empirical signatures, and explore the obstructions to a satisfactory physical interpretation.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
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This paper explores the potential for controlling chaotic dynamics within dynamical systems by leveraging the principles of quantum entanglement. The core claim is that the inherent complexity of chaotic systems can be precisely manipulated through the utilization of quantum entanglement. We propose a mechanism whereby entanglement is used to target and modify key parameters governing the chaotic behavior. The theoretical framework presented outlines a methodology for achieving this control, focusing on the intricate relationship between classical chaos and quantum correlations. This work offers a novel approach to chaos control, potentially leading to advancements in areas such as secure communication, precision sensing, and advanced control systems. The paper details the mathematical formulation, focusing on the manipulation of the system's Hamiltonian and the corresponding evolution equations. The central equation governing the system's dynamics is presented, alongside methods for quantifying the degree of entanglement and its impact on the system's trajectory. Finally, we discuss the challenges and future directions of this research area.
Using a new state-dependent, λλλ-deformable, linear functional operator, Qψλ{\cal Q}_ψ^λQψλ​, which presents a natural C∞C^{\infty}C∞ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case Qψ1{\cal Q}_ψ^{1}Qψ1​ reproduces linear quantum mechanics, whereas Qψ0{\cal Q}_ψ^{0}Qψ0​ admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.
Quantum control tasks are classified either as classical-like or as quantum requiring interference of pathways. We study the generation of interference pathways and relate them to the fidelity of the control target at a fixed time for various tasks. The model drift Hamiltonian studied is the two-dimensional Henon-Heiles (HH) potential. This system shows regular classical dynamics for low energies and chaotic dynamics for higher energies. A control operator supported by the whole momentum space and therefore connecting the entire Hilbert phase space is a random spiky potential. The other extreme is a smooth control potential. Intermediate cases are obtained by filtering the random spiky potential in momentum space. The fidelity of achieving a control task was related to the connectivity in phase space of the control operators. Typical quantum tasks such as generating a superposition of generalized coherent states rely on interfering pathways. For these cases the nonlinearity in the drift or control Hamiltonian is a necessary requirement for creating interferences. Control over rapidly diverging components of the wave function is achieved by the use of highly nonlocal control operators. Quantum control under chaotic drift was found to give a better yield than control under regular dynamics for such cases. For classical tasks we study the transformation of an initial generalized coherent state to another one. The best fidelity is obtained for regular or harmonic regions of the potential and smooth control operators. The approach to the classical limit is checked by decreasing the effective value of $\ensuremath{\hbar}$. Control under both quantum and classical tasks suffered from the decrease of $\ensuremath{\hbar}$ and the approach to classical proximity. Classical control tasks which rely heavily on maintaining a generalized coherent state throughout the evolution were found to be dysfunctional and lead to a completely uncontrolled situation once the classical chaos starts to appear.