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Forward-Backward Squeezing Propagator

Jamil Daboul

Published Jan 1, 1996• 1 citation
OpenAlex

Abstract

I show that a usual propagator cannot be defined for the pseudo-diffusion equation of the Q-functions. Instead, a forward-backward propagator is defined, which motivated a generalization of Cahill-Glauber interpolating operator. Our generalized operator {\\bf Q}(p, q;\\sigma_p^{-1},\\sigma_q) depends on two squeezing parameters \\sigma_p and \\sigma_q, and is shown to obey a generalized pseudo-diffusion equation or a diffusion equation, depending on the curve (\\sigma_p(\\mu),\\sigma_q(\\mu)) along which one moves in the (\\sigma_p, \\sigma_q) plane. An algorithm is also given for squeezing Q functions directly, using one-dimensional diffusion propagators.

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I show that a usual propagator cannot be defined for the pseudo-diffusion equation of the Q-functions. Instead, a forward-backward propagator is defined, which motivated a generalization of Cahill-Glauber interpolating operator. Our generalized operator {\bf Q}(p, q;\sigma_p^{-1},\sigma_q) depends on two squeezing parameters \sigma_p and \sigma_q, and is shown to obey a generalized pseudo-diffusion equation or a diffusion equation, depending on the curve (\sigma_p(\mu),\sigma_q(\mu)) along which one moves in the (\sigma_p, \sigma_q) plane. An algorithm is also given for squeezing Q functions directly, using one-dimensional diffusion propagators.
We show that the projection operator mod pq;yei phi )(yei phi ;pq mod , where mod pq;yei phi ) is a squeezed state, obeys a partial differential equation in which the squeeze parameter y plays the role of time. It follows that related functions, such as the probability distribution functions and the Wigner function are solutions of this equation. This equation will be called a pseudodiffusion equation, because it resembles a diffusion equation in Minkowski space. We give general solutions of the pseudo-diffusion equation, first by the method of separation of variables and then by the Fourier transform method, and discuss the limitations of the latter method. The Fourier method is used to introduce squeezing into the number states, the thermal light and the Wigner function.
For the model of Drummond and Walls [J. Phys. A 13, 725 (1980)] describing dispersive optical bistability, the s-parametrized quasidistributions W(\ensuremath{\alpha},s) introduced by Cahill and Glauber [Phys. Rev. A 117, 1882 (1969)] are investigated. The equation of motion for W(\ensuremath{\alpha},s) generally has derivative terms up to third order. For the Q function (s=-1) and the P function (s=1) the third-order terms vanish. The equation of motion is then a pseudo-Fokker-Planck equation, i.e., a Fokker-Planck equation where the diffusion matrix is not positive definite or semidefinite. It is shown that the equation of motion with the inclusion of third-order-derivative terms can be solved by the matrix continued-fraction method. In particular, results for the Wigner function (s=0) are presented and compared with the results of the Q function (s=-1). Furthermore, the results are compared also with those obtained from the equation of motion for the Wigner function where the third-order-derivative terms have been neglected.
The modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter λ specifies orderings in a mapping from the operator |ψ⟩⟨σ| to the corresponding phase space wavefunction, where σ is a given fiducial vector. The choice λ = 0 specifies the Weyl mapping and the Q -function so obtained is the usual one when |σ⟩ is the vacuum state. More generally, any choice of λ in the range (−1, 1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q -functions are non-negative by construction. They are shown to be proportional to the expectation of the system state with respect to a generalized displaced squeezed state which depends on λ and position ( p , q ) in phase space. Thus, when a system has been prepared in the state , a generalized Q -function is proportional to the probability of finding it in the generalized squeezed state. Any such Q -function can also be written as the smoothing of the Wigner function for the system state by convolution with the Wigner function for the generalized squeezed state.