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arXiv:quant-ph/9910083 · Journal of Physics A Mathematical and General

Marginal and correlation distribution functions in the squeezed-states representation

Marcelo A. Marchiolli, Salomon Sylvain Mizrahi, V. V. Dodonov

Published Nov 30, 1999• 3 citations
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Abstract

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.

DOI
10.1088/0305-4470/32/49/311

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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.
We find a new κ-parameter squeezed coherent state | p , q ⟩ κ representation, which possesses well-behaved features, i.e., its Wigner function's marginal distribution in the `` q -direction'' and in the `` p -direction'' is the Gaussian form exp {−κ( q '− q ) 2 }, and exp {( p '− p ) 2 /κ}, respectively. Based on this, the Husimi function of | p , q ⟩ κ is also obtained, which is a Gaussian broaden version of the Wigner function. The | p , q ⟩ κ state provides a good representative space for studying various properties of the Husimi operator.
By use of the squeezed-number-state basis, we study several features of the squeezed thermal states. Higher-order squeezing properties are found to be entirely determined by the second-order ones. The photon-number distribution is expressed in closed form in terms of a Legendre polynomial. We point out that at a critical value of the squeeze parameter, pairwise oscillations of the photon-number distribution set in. Finally, a compact analytical expression of the exact lth-order correlation function is derived and examined.
A new operator representation, called squeezed coherent state representation, is introduced since Husimi operator has the form of squeezed coherent state. We fisrt introduce its specific integral expression. When κ = 1, this representation is reduced to the usual P function. As an example, we calculate the squeezed coherent state representation for thermal field to illustrate a difference between P function and the squeezed coherent state representation. Especially, in order to better apply this representation to quantum optics, we reveal the integral transformations between the squeezed coherent state representation, respectively, and the following three functions: Wigner function, Q function, and Husimi function.
Using the description of the linear phase insensitive amplification of a quantum state available in the literature we prove that to every state , produced from the initial state by such amplification, there corresponds a phase space distribution , where is the Husimi function of the initial state. The scaling parameter satisfies the relation and decreases with time while, as we have shown earlier, the scaled function is again a Husimi distribution. We prove, using these facts, that if the definition of the phase insensitive amplification is physically correct then the distribution of the phase of a state is unambiguously given as the corresponding marginal distribution obtained from its Husimi distribution represented in polar coordinates.