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Journal of Physics A Mathematical and General

The relation between the scaling of Husimi functions and the linear phase insensitive amplification of the corresponding quantum states and its implications

Dragomir M Davidović, D Lalovic

Published Jul 21, 1996• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/29/14/007

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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.
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.
The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.
We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum-classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time-dependent Schrödinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase-space Schrödinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum-classical correspondence, and quasiclassical trajectory simulations of quantum dynamics.