Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Photon-number distributions for fields with Gaussian Wigner functions

S. Chaturvedi, Venkatraman Srinivasan

Published Nov 1, 1989• 62 citations
Publisher

Abstract

We compute the generating function for the photon-number distribution for single-mode fields described by a Gaussian Wigner distribution considered previously by Agarwal and Adam [Phys. Rev. A 38, 750 (1988)] and obtain simpler formulas for the photon-number distribution than those given by them. We also give analytical results for the factorial cumulants of the photon-number distribution and consider some limiting cases.

DOI
10.1103/physreva.40.6095

Related papers

Photon distributions for nonclassical fields with coherent components

G. S. Agarwal, G. Adam

Nonclassical fields with coherent components generated in a large number of optical processes including losses can be characterized by a Gaussian Wigner function centered around the mean value of the field. The photon-number distributions for such fields are calculated and the numerical results are presented for a range of parameters. As a special case we give the number distributions for photons produced in down-conversion problems. We also give the analytical expression for the photoelectron distributions.

Physical Review A•Jun 1, 1989•74 citations

Photon distribution for one-mode mixed light with a generic Gaussian Wigner function

V. V. Dodonov, Olga V. Man’ko, Vladimir I. Man’ko

For one-mode light described by the Wigner function of a generic Gaussian form with five real parameters, the photon distribution function is obtained explicitly in terms of the Hermite polynomial of two variables. The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. The first and the second statistical moments of the photon distribution function are calculated. The generating function for the photon distribution is discussed. Various special cases, including the shifted thermal states, correlated and squeezed states, coherent states, etc., are considered.

arXiv:hep-th/9308004•Physical Review A•Apr 1, 1994•109 citations

Photon-number distributions for quantum fields generated in nonlinear optical processes

G. S. Agarwal, G. Adam

We show that in a large class of nonlinear optical processes which are used to generate the squeezed states of the radiation field, the quantum state of the generated radiation is given by a Gaussian Wigner function. This is so even when losses in the medium are included. The photon-number distributions for such fields are evaluated. The number distributions exhibit oscillatory character for the range of parameters for which the field is squeezed.

Physical Review A•Jul 1, 1988•60 citations

Wigner function for number and phase

John A. Vaccaro, David Thomas Pegg

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.

Physical Review A•May 1, 1990•91 citations

Statistical Dynamics of Quantum Oscillators and Parametric Amplification in a Single Mode

Maurice T. Raiford

The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used $P$ distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the $P$ distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the $P$ distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.

Physical Review A•Oct 1, 1970•61 citations

Multidimensional Hermite polynomials and photon distribution for polymode mixed light

V. V. Dodonov, Olga V. Man’ko, Vladimir I. Man’ko

The photon distribution function of an N-mode mixed state of light described by the Wigner function of generic Gaussian form is calculated explicitly in terms of the Hermite polynomials of 2N variables with equal pairs of indices. Simple formulas for the mean values and dispersions of photon numbers are found. The N-mode photon distribution functions for squeezed photon number states and for squeezed coherent states are expressed in terms of the Hermite polynomials of 2N and N variables, respectively.

arXiv:hep-th/9308033•Physical Review A•Jul 1, 1994•101 citations
We compute the generating function for the photon-number distribution for single-mode fields described by a Gaussian Wigner distribution considered previously by Agarwal and Adam [Phys. Rev. A 38, 750 (1988)] and obtain simpler formulas for the photon-number distribution than those given by them. We also give analytical results for the factorial cumulants of the photon-number distribution and consider some limiting cases.
Nonclassical fields with coherent components generated in a large number of optical processes including losses can be characterized by a Gaussian Wigner function centered around the mean value of the field. The photon-number distributions for such fields are calculated and the numerical results are presented for a range of parameters. As a special case we give the number distributions for photons produced in down-conversion problems. We also give the analytical expression for the photoelectron distributions.
For one-mode light described by the Wigner function of a generic Gaussian form with five real parameters, the photon distribution function is obtained explicitly in terms of the Hermite polynomial of two variables. The effective formulas reducing the two-dimensional Hermite polynomials to the classical (one-dimensional) orthogonal polynomials are given. The first and the second statistical moments of the photon distribution function are calculated. The generating function for the photon distribution is discussed. Various special cases, including the shifted thermal states, correlated and squeezed states, coherent states, etc., are considered.
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.
The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used PPP distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the PPP distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the PPP distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.
The photon distribution function of an N-mode mixed state of light described by the Wigner function of generic Gaussian form is calculated explicitly in terms of the Hermite polynomials of 2N variables with equal pairs of indices. Simple formulas for the mean values and dispersions of photon numbers are found. The N-mode photon distribution functions for squeezed photon number states and for squeezed coherent states are expressed in terms of the Hermite polynomials of 2N and N variables, respectively.