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Physics Today

Sodium Atoms Kicked by Standing Waves Provide a New Probe of Quantum Chaos

Graham P. Collins

Published Jun 1, 1995• 9 citations
Publisher

Abstract

The question of how to quantize a system whose classical dynamics is chaotic has long been a puzzle. The hallmark of classical chaos is sensitive dependence on initial conditions: A small perturbation of the system changes its evolution in a way that grows exponentially with time. But in quantum mechanics a small perturbation of the initial state will in general be mapped into a small perturbation of the final state. In reeveral cent years the study of quantum systems whose underlying classical dynamics is chaotic has been an active area of research.

DOI
10.1063/1.2808053

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The question of how to quantize a system whose classical dynamics is chaotic has long been a puzzle. The hallmark of classical chaos is sensitive dependence on initial conditions: A small perturbation of the system changes its evolution in a way that grows exponentially with time. But in quantum mechanics a small perturbation of the initial state will in general be mapped into a small perturbation of the final state. In reeveral cent years the study of quantum systems whose underlying classical dynamics is chaotic has been an active area of research.
The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1\hbar^{-1}ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
How classical chaos emerges from quantum mechanics remains a central open question, as the unitary evolution of isolated quantum systems forbids exponential sensitivity to initial conditions. A key insight is that this quantum-classical link is provided by measurement processes. In this work, we identify gain competition in a chaotic photon gas as an operational quantum measurement that selects single motional modes from an initial superposition through stochastic, nonlinear amplification. We show that this mechanism naturally gives rise to classical chaotic behavior, most notably sensitivity to initial conditions. Our results provide a concrete physical mechanism for the quantum-classical transition in a chaotic system and demonstrate that essential aspects of quantum measurement-state projection, Born-rule-like selection, and irreversibility-can naturally emerge from intrinsic gain dynamics.