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Physical Review Letters

Quantum chaos in the Born-Oppenheimer approximation

R. Blümel, Bernd Esser

Published Jun 6, 1994• 51 citations
Publisher

Abstract

We show that exponential instabilities and quantum chaos occur in a system with a mixed classical-quantum description. This type of chaos is of general importance and may occur in any quantum system which divides in a natural way into a fast (quantum) and a slow (classical) subsystem.
DOI
10.1103/physrevlett.72.3658

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Abstract The theoretical description of radiation − matter interactions in terms of master equations can be approximated by a semi-classical theory when quantum fluctuations are small. This approach, the exact definition of which will emerge later, has been enormously influential in understanding instabilities and chaos in quantum optics. Although much of the theory (and experiment) has been done only in the last few years, there were already indications in the early experiments on ruby lasers of ‘spiky’ behaviour. Instabilities are often due to competition between different modes which are coupled together non-linearly. Consequently we will first describe qualitatively the modes that are important when we deal with optical systems in Fabry-Perot or ring cavities.
The relation between the dynamical properties of a coupled quasiparticle-oscillator system in the mixed quantum-classical and fully quantized descriptions is investigated. The system is considered as a model for applying a stepwise quantization. Features of the nonlinear dynamics in the mixed description such as the presence of a separatrix structure or regular and chaotic motion are shown to be reflected in the evolu- tion of the quantum state vector of the fully quantized system. In particular, it is demonstrated how wave packets propagate along the separatrix structure of the mixed description, and that chaotic dynamics leads to a strongly entangled quantum state vector. Special emphasis is given to viewing the system from a dyn- amical Born-Oppenheimer approximation defining integrable reference oscillators, and elucidating the role of the nonadiabatic couplings which complement this approximation into a rigorous quantization scheme.
A novel concept for quantum chaos in spin systems is proposed which differs distinctively from concepts currently in use. It is argued (a) that the definition of quantum chaos cannot be based on the correspondence principle and (b) that quantum chaos is not equivalent to statistical behavior. Quantum chaos in spin systems implies the existence of a new type of spectrum consisting of excitations which do not form regular patterns with a multiparameter continuum structure in the thermodynamic limit, thus removing the constraint imposed by van Hove singularities that time-dependent correlation functions for pure quantum states cannot decay more rapidly than as powers of t. The irregular spectrum is caused by strong level repulsion resulting from the lack of a sufficient number of conservation laws in nonintegrable quantum many-body systems. The similarities and differences between quantum and classical chaos and, more generally, between quantum (non)integrability and classical (non)integrability are discussed in detail.