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Annual Meeting Optical Society of America

Functional integral representation of the dynamics of squeezed states

Gregg M. Gallatin

Published Jan 1, 1989• 0 citations
Publisher

Abstract

Functional integrals provide a powerful and useful formalism for deriving both perturbative and nonperturbative results in quantum theory. Here we consider their application to the problem of formulating the dynamics of coherent and squeezed states. We derive the functional integral representation of the coherent state density matrix by combining the closed time path representation of the density matrix in terms of functional integrals in configuration space with the projection of coherent states onto configuration space states. In this formalism the atomic variables can, without loss of generality, be integrated out exactly yielding a single functional integral which describes the combined quantum dynamics of both the atomic and the coherent states.

DOI
10.1364/oam.1989.thii5

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Functional integrals provide a powerful and useful formalism for deriving both perturbative and nonperturbative results in quantum theory. Here we consider their application to the problem of formulating the dynamics of coherent and squeezed states. We derive the functional integral representation of the coherent state density matrix by combining the closed time path representation of the density matrix in terms of functional integrals in configuration space with the projection of coherent states onto configuration space states. In this formalism the atomic variables can, without loss of generality, be integrated out exactly yielding a single functional integral which describes the combined quantum dynamics of both the atomic and the coherent states.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
To introduce the formalism of the functional integral in quantum mechanics, this chapter starts from the definition of the temporal evolution operator. It is important to emphasize that in the classical limit is not only the classic trajectory that have the grater contribution, but all the paths close to this also contribute. The chapter shows how integral formulation is perfectly consistent with Schrödinger's time-dependent differential equation. It retrieves this equation from definition of transition amplitude as a path integral. The chapter discusses functional derivative generalized to the case of the ordinary functional derivative. A very useful tool to solve problems related to functional integrals is the Wick rotation or Euclidean rotation. In quantum mechanics there are few problems we can solve exactly, and very often, when possible, we use perturbation theory to find an approximate solution. The chapter follows this approach with formalism of functional integration and construct a perturbation theory from which can be obtained a physical interpretation.
This paper presents new examples of the functional-integral description of nonrelativistic quantum processes. The main point is the development and application of a functional-integral version which gives an adequate representation of the coherent dynamics of short impulses of light in media. An exact correspondence is received between the physical picture and functional-integral trajectories as the solutions of Euler-Lagrange equations with given initial and final conditions. The observed examples demonstrate the powerfulness of the path-integral method in the functional treatment and will promote the non-operator representation of quantum mechanics and quantum statistics.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.
Using the squeezed state formalism the coherent state representation of quantum fluctuations in an expanding universe is derived. It is shown that this provides an interesting alternative to the Wigner function as a phase space representation of quantum fluctuations. The quantum to classical transition of fluctuations is simply implemented by decohering the density matrix in this representation. The entropy of the decohered vacua is derived and is shown to agree with previous results in the high squeezing limit. It is shown that the decoherence process can significantly change the predictions derived using pure states. In particular, scale-invariant power spectra are not obtained for any decoherence process, though it can be obtained by decohering in the coherent state representation.