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Physical Review A

Path-integral approach to problems in quantum optics

Mark S. Hillery, Muhammad Suhail Zubairy

Published Jul 1, 1982• 72 citations
Publisher

Abstract

A formalism for applying path integrals to certain problems in nonlinear optics is considered. The properties of a coherent-state propagator are discussed and a path-integral representation for the propagator is presented. This representation is then employed in evaluating the propagator for general single-mode and multimode Hamiltonians which are at most quadratic in the creation and destruction operators of the field. Some examples involving parametric processes are given.

DOI
10.1103/physreva.26.451

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A formalism for applying path integrals to certain problems in nonlinear optics is considered. The properties of a coherent-state propagator are discussed and a path-integral representation for the propagator is presented. This representation is then employed in evaluating the propagator for general single-mode and multimode Hamiltonians which are at most quadratic in the creation and destruction operators of the field. Some examples involving parametric processes are given.
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The quantum theory of the degenerate parametric amplifier is usually treated in the parametric approximation where the pump field is treated classically. In this paper we present a fully quantized theory of this nonlinear optical device using a path-integral approach. A perturbation series, the first term of which corresponds to the parametric approximation, is employed to evaluate explicitly the coherent-state propagator. The question of the validity of the parametric approximation is considered and the conditions under which this approximation is justified are elucidated. Finally, certain correlation functions for the signal-mode operators are calculated that are needed to study squeezed states. It is shown that the quantum nature of the pump field tends to decrease the squeezing.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
We develop a path integral approach for analyzing the stationary light propagation in general dielectric media. The Hermitian form of the stationary Maxwell equations is transformed into a quantum mechanical problem of a spin 1 particle with spin-orbit coupling and position dependent mass. After appropriate ordering several path integral representations of a solution are constructed. First we keep the propagation of the polarization degrees of freedom in an operator form integrated over paths in a coordinate space. The use of spin 1 coherent states allows representing this part as a path integral over such states. Finally a path integral in a transversal momentum space with explicit transversality enforced at every time slice is also given. As an example the geometrical optics limit is discussed and the ray equation is recovered together with the Rytov rotation of the polarization vector.
The numerical evaluation of coherent-state path integrals for quantum dynamical problems is discussed for one-dimensional examples. To propagate an initial state, we use the normal and antinormal ordered coherent-state path integrals combined with a split-operator technique dividing the Hamiltonian into harmonic and anharmonic parts. For numerical purposes integrations must be approximated by quadrature formulae. This leads to a matrix multiplication scheme which is systematically tested for the double-well and Morse potentials. The method is accurate for propagation times much longer than the natural time scale of the system, and it allows for short as well as long time steps without loss of stability.