From the Weyl quantization of a particle on the circle to number–phase Wigner functions

Abstract

a b s t r a c tA generalized Weyl quantization formalism for a particle on the circle is shown to supply an effective method for defining the number-phase Wigner function in quantum optics. A Wigner function for the stateρ and the kernel K for a particle on the circle is defined and its properties are analysed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner functions are considered.

On some properties of number-phase Wigner function

“…with f given by (1.8). As it has been pointed out by many authors [2][3][4][5][6] the formula (1.11) gives the expectation value for the quantum phase-function f (φ) equal to the one calculated within the Pegg-Barnett approach to quantum phase [7][8][9]. Consequently, our choice for a number-phase Wigner function (1.4) seems to be appropriate when the Pegg-Barnett formulation is considered.…”

“…In recent works [1][2][3] we have developed a theory of quantum phase which is based on some enlarging of the Fock space to the Hilbert space of the square integrable functions on the circle, L 2 (S 1 ). Then the well known machinery of the Naimark projection is employed to find the respective objects in the original Fock space.…”

“…Then the well known machinery of the Naimark projection is employed to find the respective objects in the original Fock space. Such an approach to quantum phase have been considered by many authors ( for references see [1,2]). In our case we were able to define also a number-phase quasiprobability distribution which we called the number-phase Wigner function [2,3].…”

“…Such an approach to quantum phase have been considered by many authors ( for references see [1,2]). In our case we were able to define also a number-phase quasiprobability distribution which we called the number-phase Wigner function [2,3]. In short our construction can be introduced as follows: Firstly, define the self-adjoint operator Ω(φ, n) Ω(φ, n) := π {|n n|φ φ| + |φ φ|n n|} (1.1) where |n is a normalised eigenvector of the number operator n i. e. Then the number-phase Wigner function is defined as…”

The Phase Space Model of Nonrelativistic Quantum Mechanics

“…Thus, we usually assume the classical phase space to be also the quantum one. We are also aware of exceptions, like sets whose classical phase space is a cylinder A detailed analysis of this situation can be found in [ 44 , 45 , 46 , 47 ].…”