GeneralizedQ-functions

Abstract

The modulus squared of a class of wave functions 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 wave function, 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 stateρ 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 stateρ, 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 stateρ by convolution with the Wigner function for the generalized squeezed state.

Recent Progress in Optical Fiber Research

“…Distribution functions in general have different properties and are associated with various dynamical equations, so they may be described most conveniently by distribution functions having different characteristics (Hillery et al, 1984). Other distribution functions that have been considered in the past include those of Kirkwood (Kirkwood, 1933), Margenau-Hill (Johansen & Luis, 2004;Margenau & Hill, 1961; Terletsky, 1937) 2 , Husimi (Husimi, 1940), Q-functions (Husimi, 1940;Kano, 1965;Smith, 2006), Page (Page, 1952), Glauber-Sudarshan (Glauber, 1963;Sudarshan, 1963),…”

The quantum state vector in phase space and Gabor's windowed Fourier transform

“…In the signals literature, |Φ(q, p)| 2 is known as the spectrogram of the signal ψ. It can be regarded as a generalized Husimi function [47] (see below). In particular, when the window function is the simple Gaussian corresponding to the oscillator ground state (see below),…”

Optical Fibers in Phase Space: A Theoretical Framework

“…Distribution functions in general have different properties and are associated with various dynamical equations, so they may be described most conveniently by distribution functions having different characteristics (Hillery et al, 1984). Other distribution functions that have been considered in the past include those of Kirkwood (Kirkwood, 1933), Margenau-Hill (Johansen & Luis, 2004;Margenau & Hill, 1961;Terletsky, 1937) 2 , Husimi (Husimi, 1940), Q-functions (Husimi, 1940;Kano, 1965;Smith, 2006), Page (Page, 1952), Glauber-Sudarshan (Glauber, 1963;Sudarshan, 1963), Rihaczek (Rihaczek, 1968), and Choi-Williams (Choi & Williams, 1989). Another very used function is the ambiguity function (Bastiaans, 1980;Marks & Hall, 1979;Woodward, 1963).…”