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Journal of Optics B Quantum and Semiclassical Optics

Phase distribution from the Wigner function for superpositions of squeezed displaced Fock states

G. M. Abd Al-Kader

Published Jun 1, 2003• 5 citations
Publisher

Abstract

The Wigner quasiprobability function for the superposition of squeezed displaced Fock states is reviewed. The phase distribution obtained by integrating the Wigner function over the radial variable is studied and compared with the Pegg and Barnett phase distribution. The Wigner phase distributions for some values of parameters are illustrated.

DOI
10.1088/1464-4266/5/3/352

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The Wigner quasiprobability function for the superposition of squeezed displaced Fock states is reviewed. The phase distribution obtained by integrating the Wigner function over the radial variable is studied and compared with the Pegg and Barnett phase distribution. The Wigner phase distributions for some values of parameters are illustrated.
The Wigner quasi-probability function for the superposition of squeezed displaced Fock states (SDFS's) is reviewed. The interaction of these states with a two-level atom in cavity with the presence of additional Kerr medium is studied. Exact general matrix elements of the time-dependent operators of a Jaynes-Cummings model (JCM), in the presence of a Kerr medium, with these states are derived. We have obtained the phase distribution by two different ways: one is by Pegg-Barnett formalism, the second is by integration of the Wigner function over the radial variable. Results of these two approaches are compared. The Wigner phase distributions for some values of parameters are illustrated. The behaviors of the distributions have been shown as a function of the squeeze parameter in JCM.
The Wigner quasi-probability function for the superposition of squeezed displaced Fock states (SDFS’s) is reviewed. The interaction of these states with a two-level atom in cavity with the presence of additional Kerr medium is studied. Exact general matrix elements of the time-dependent operators of a Jaynes-Cummings model (JCM), in the presence of a Kerr medium, with these states are derived. We have obtained the phase distribution by two dierent ways: one is by Pegg-Barnett formalism, the second is by integration of the Wigner function over the radial variable. Results of these two approaches are compared. The Wigner phase distributions for some values of parameters are illustrated. The behaviors of the distributions have been shown as a function of the squeeze parameter in JCM.
The phase distribution obtained within the Pegg–Barnett Hermitian phase formalism is compared to the phase distributions obtained from the s -parametrized quasiprobability distributions integrated over the "radial" variable for some real states of the field. Exact analytical formulas for the s -parametrized phase distributions of coherent states, squeezed states, and displaced number states are obtained. A general formula relating the s -parametrized phase distributions to the Pegg–Barnett distribution is derived. Numerical examples illustrating the similarities and differences are presented in a graphical form.
Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P (W) ( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P (W) ( theta ) in the limit of highly excited states.
Phase properties of the displaced number states are studied. Exact analytical formulae describing phase distributions based on different phase approaches are obtained and illustrated graphically. It is shown that the Pegg-Barnett phase distribution P( theta ) and the phase distribution P w ( theta ) associated with the Wigner function are very close to each other, while the phase distribution P Q ( theta ) associated with the Q function carries less phase information. The results have clear interpretation in terms of the area of overlap in phase space.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.