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Physical Review A

Classical equations for quantum squeezing and coherent pumping by the time-dependent quadratic Hamiltonian

A. F. R. de Toledo Piza

Published Feb 1, 1995• 11 citations
Publisher

Abstract

The time evolution, under the general time-dependent quadratic quantum Hamiltonian, of mean values of a class of observables that includes the field quadratures and their dispersions, is obtained exactly in classical Hamiltonian form. Quantum states are described in terms of density operators, so that arbitrary initial states (both pure states and statistical mixtures) are allowed.

DOI
10.1103/physreva.51.1612

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The time evolution, under the general time-dependent quadratic quantum Hamiltonian, of mean values of a class of observables that includes the field quadratures and their dispersions, is obtained exactly in classical Hamiltonian form. Quantum states are described in terms of density operators, so that arbitrary initial states (both pure states and statistical mixtures) are allowed.
The dynamic decoupling of the correlation function hierarchy under the general quadratic Hamiltonian is exploited to extend the results obtained for coherent pumping and squeezing in Ref. . Closed c-number dynamic equations are given explicitly for three- and four-operator averages, for an arbitrary (pure or mixed) initial state of the field. At this level an independent c-number constant of motion appears, related to two-photon correlation properties of the initial state. Extension to higher correlation functions is straightforward. \textcopyright{} 1996 The American Physical Society.
A general framework for a time-dependent variational approach in terms of squeezed coherent states is given with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. By using a formula given by Balian and Brezin for canonical decompositions of generalized Bogoliubov transformations, we show that this approach is entirely equivalent to that given by Jackiw and Kerman in terms of the most general Gaussian functions, which is originally motivated for the variational definition of the effective action in the quantum field theory. The resultant equations of motion have structure of classical Hamiltonian systems consisting of displacement degree of freedom and squeezing degree of freedom in the configuration space. The initial conditions for solving these equations, particularly for the squeezing degree of freedom, are discussed in connection with the minimum-uncertainty relations of the squeezed coherent state. For a simple one-dimensional quantum-mechanical Hamiltonian with potentials of quadratic terms or less, the present formulation gives an exact result for the time-evolution of the Gaussian wave packet.
The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2(Rn)L^2(\mathbb R^n)L2(Rn) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
It is by now well recognized how the quantum time evolution for time-dependent Hamiltonians that are quadratic both in position and momentum variables generates ‘‘squeezing.’’ On the other hand, ‘‘squeezed states’’ and their ‘‘high-n generalizations’’ are known to provide a simple classical mechanical description of general quantum time evolution. In this paper, these ideas are combined to show the relevance of ‘‘generalized squeezed states’’ in the semiclassical limit of the evolution of time-dependent wave packets for general time-dependent Hamiltonians, including the multidimensional case.