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Journal of Physics A Mathematical and General

General solutions of the pseudo-diffusion equation of squeezed states

Jamil Daboul, Marcelo A. Marchiolli, Salomon Sylvain Mizrahi

Published Aug 21, 1995• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/28/16/019

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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.
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.
Here we consider the Husimi function (HF) P for the squeezed states and calculate the marginal and correlation distribution functions (MDFs and CDFs) when P is projected onto the photon-number states. According to the value of the squeezing parameter one verifies the occurence of oscillations and beats as already appointed in the literature. We verify that these phenomena are entirely contained in the correlation function. In particular, we show that since the Husimi and its MDFs satisfy partial differential equations where the squeeze parameter plays the role of time, the solutions (the squeezed functions obtained from `initial' unsqueezed functions) can be expressed by means of kernels responsible for the `propagation' of squeezing. From the calculational point of view, this method presents advantages for calculating the MDFs (compared with a direct integration over one of the two phase-space variables of P ) since one can use the symmetry properties of the differential equations.
As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the P representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.
Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.