Quantum chaos in the Born-Oppenheimer approximation

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.PACS numbers: 05.45.+b Current research in quantum chaos is mainly focused on the classical-quantum correspondence in the semiclassical regime of classically chaotic dynamical systems [1 -3]. But yet another main avenue of quantum chaos research may derive from the observation that for some quantum systems of theoretical and practical importance the system divides in a natural way into two interacting subsystems, one of which is treated quantum mechanically, whereas the other is treated in the classical approximation.The mixed quantum-classical description of a dynamical system is justified whenever the quantum effects of one subsystem are negligible compared to the other or when the quantization of the whole system poses a severe challenge and the classical treatment of one part has to serve as a necessary guide in the further investigation. In fact, there are many examples in the physics literature where such a mixed description was already successfully applied. We mention the case of a two-level system interacting with the electromagnetic field of a laser cavity [4,5], the micromaser [6], nuclear collective motion [7], and the exciton transfer in a nonlinear molecular dimer perturbed by an intermolecular vibration [8]. It was demonstrated recently [9] that the number of photons in the electromagnetic maser field can be changed essentially in units of 1 from one photon (quantum limit) to a large number of photons (classical limit). Thus, the importance of quantum corrections in the "classical" subsystem (the cavity field) can be controlled to an astonishing degree. The molecular physics example of the nonlinear dimers is instructive because it provides an example for the justification of the division into a quantum and a classical subsystem. Describing the strong intramolecular interactions in the molecules constituting the dimer in the framework of the nonlinear discrete self-trapping equation (see, e.g. , [10,11]) one can use the weakness of the intermolecular forces between the molecules (and correspondingly small frequencies) to use a classical approximation for the intermolecular vibrations perturbing the transfer dynamics. In this case the density matrix dynamics for the exciton transfer of the quantum subsystem is characterized by a homoclinic structure on the Bloch sphere from which by perturbation with the classical oscillator a stochastic layer and chaos develops [8]. A further example is provided by a quantum system with a classical boundary moving in an anharmonic potential [12]. All the models discussed above are characterized by a mixed quantum-classical description. One of the main purposes of this Letter is to show that such systems can display quantum chaos and arise quite naturally within the framework of the Born...

Vibrating Quantum Billiards on Riemannian Manifolds

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