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Physica D Nonlinear Phenomena

Method of quasiclassical approximation for c-number projection in coherent states basis

G. P. Berman, Alexander Iomin, G. M. Zaslavsky

Published Oct 1, 1981• 39 citations
Publisher
DOI
10.1016/0167-2789(81)90008-7

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Many field states in quantum optics may be regarded as quasiclassical in that they possess a fairly well-defined phase and amplitude in the limit in which the average number of photons is large. Examples include coherent states and squeezed states that have a coherent component substantially larger than the squeezed noise contribution. For such states, calculations involving sums over the coefficients in the Fock-state basis can be considerably simplified in the limit of large photon number. The Jaynes-Cummings model and its extensions (to, e.g., three-level, two-mode configurations) may be advantageously studied from this perspective. New results on the three-level, two-mode problems, on the influence of cavity losses, and on the various time scales of interest will be presented here. The method typically identifies a set of privileged, long-lived atomic states, which lead to a quasiclassical evolution that agrees with the prediction of semiclassical treatments over a characteristic time scale; linear combinations of the long-lived states, on the other hand, become superpositions of macroscopically distinct states in time and result in an evolution that exhibits many uniquely quantum effects.
In a wide range of quantum gravity theories, quasiclassical geometries, which are solutions to the Einstein field equations approximately, are described by "coherent states." Here we propose a Hamiltonian formalism for gravitational dynamics with respect to this coherent state basis, which generates time evolution of the spacetime with respect to a clock at infinity. Since the coherent states are not orthogonal, an initial quasiclassical geometry is dynamically driven into a superposition of different amplitudes. Our framework provides a dynamical mechanism for tunneling between geometries that is ubiquitous in a number of approaches to quantum gravity, from loop quantum gravity to the Euclidean path integral. We apply our framework to the problem of black hole evaporation, providing a hint at how unitarity may be preserved with the inclusion of quantum corrections to the semiclassical evolution of the black hole.
Quantization of classical dynamical systems with a Poisson structure on homogeneous Kähler manifolds is considered. The quantization follows the method invented by Berezin and represents the unitary transition operator as a quasiclassical path integral in the coherent-state basis. In case the coherent-state manifold appears as a (degenerate) rank-one co-adjoint orbit of the symmetry group, an explicit representation of the transition amplitude in terms of classical data can be derived for large values of the highest weight, which corresponds to the quasiclassical approximation. This representation is further shown to perfectly agree, in contrast to some earlier approaches, with the known exact results and may provide non-trivial asymptotics of physical relevance.