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

Quantum and Semiclassical Optics Journal of the European Optical Society Part B

SU(2) distribution functions and measurement of the fluorescence of a two-level atom

Howard M. Wiseman

Published Aug 1, 1995• 16 citations
Publisher

Abstract

It is well known that two quasi-probability distributions for an optical mode, the Wigner (W) and Husimi (Q) functions, can be measured by homodyne and heterodyne detection, respectively. These harmonic oscillator distribution functions can be generalized to apply to other algebras, in particular, the su(2) algebra of a two-level atom. In this paper I investigate the relationship between these quasi-probability functions on the Bloch sphere, and the probability distributions for the integrated homodyne and heterodyne photocurrents from the fluorescence of the atom. While there is no particularly close relation between the su(2) Wigner function and homodyne detection, the su(2) Husimi function is simply equal to the heterodyne distribution function multiplied by a weighting factor. I also show that the two-level atom heterodyne distribution function can be expressed in a form identical to the corresponding optical mode distribution function. This correspondence defines the natural analogue of optical coherent states for a two-level atom. In deriving these results the theory of effects (positive-operator-valued measures) is essential.

DOI
10.1088/1355-5111/7/4/013

Related papers

Operational and Harmonic-Analytic Aspects of Quasi-Probability Distributions

Ryo Harada

Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.

arXiv:1101.5314•Open Systems & Information Dynamics•Sep 1, 2011•0 citations

Quantum Theory of Optical Homodyne and Heterodyne Detection

Matthew J. Collett, R. Loudon, Crispin W. Gardiner

The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.

Journal of Modern Optics•Jun 1, 1987•171 citations

Characteristic functions and quantum measurements of optical observables

Yinchieh Lai, H. A. Haus

Three characteristic functions are reviewed along with the quantum theory of linear systems and used to derive the P-representation, Wigner distribution and quasi-probability density of coherent states and squeezed states. The quasiprobability density is interpreted as the outcome of measurements utilising a beamsplitter with homodyne detection. The quasiprobability density concept is generalised and its interpretation in terms of a measurement is presented. The measurement utilises a beam splitter with one of its inputs in a squeezed state followed by two homodyne detectors.

Quantum Optics Journal of the European Optical Society Part B•Dec 1, 1989•84 citations

Quasi-probability distribution functions for optical-polarization

Ravi Shankar Singh, Sunil Pratap Singh, Gyaneshwar K. Gupta

Cahill-Glauber C(s)-correspondence is employed to construct Quasi-Probability Distribution Functions (QPDFs) for optical-polarization in phase space following equivalent description of polarization in Classical Optics. The proposed scheme provides pragmatic insights as compared to obscure SU (2) quasi-distributions on Poincare sphere. QPDF (Wigner function) of bi-modal quadrature coherent states is evaluated and numerically investigated to demonstrate the application.

arXiv:1211.0436•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Dec 29, 2010•1 citation

Wigner distribution function for finite signals

Kurt Bernardo Wolf, Natig M. Atakishiyev, S. M. Chumakov

We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 1, 1997•3 citations

Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms

Jonathan P. Dowling, G. S. Agarwal, Wolfgang P. Schleich

The general theory of quantum angular momentum is used to derive the unique Wigner distribution function for arbitrary angular-momentum states. We give the explicit distribution for atomic angular-momentum Dicke states, coherent states, and squeezed states that correspond to a collection of N two-level atoms. These Wigner functions W(\ensuremath{\theta},cphi) are represented as a pseudoprobability distribution in spherical phase space with spherical coordinates \ensuremath{\theta} and cphi on the surface of a sphere of radius \ensuremath{\Elzxh} \ensuremath{\surd}j(j+1) where j is the total angular-momentum eigenvalue.

Physical Review A•May 1, 1994•247 citations
It is well known that two quasi-probability distributions for an optical mode, the Wigner (W) and Husimi (Q) functions, can be measured by homodyne and heterodyne detection, respectively. These harmonic oscillator distribution functions can be generalized to apply to other algebras, in particular, the su(2) algebra of a two-level atom. In this paper I investigate the relationship between these quasi-probability functions on the Bloch sphere, and the probability distributions for the integrated homodyne and heterodyne photocurrents from the fluorescence of the atom. While there is no particularly close relation between the su(2) Wigner function and homodyne detection, the su(2) Husimi function is simply equal to the heterodyne distribution function multiplied by a weighting factor. I also show that the two-level atom heterodyne distribution function can be expressed in a form identical to the corresponding optical mode distribution function. This correspondence defines the natural analogue of optical coherent states for a two-level atom. In deriving these results the theory of effects (positive-operator-valued measures) is essential.
Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.
The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.
Three characteristic functions are reviewed along with the quantum theory of linear systems and used to derive the P-representation, Wigner distribution and quasi-probability density of coherent states and squeezed states. The quasiprobability density is interpreted as the outcome of measurements utilising a beamsplitter with homodyne detection. The quasiprobability density concept is generalised and its interpretation in terms of a measurement is presented. The measurement utilises a beam splitter with one of its inputs in a squeezed state followed by two homodyne detectors.
We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.
The general theory of quantum angular momentum is used to derive the unique Wigner distribution function for arbitrary angular-momentum states. We give the explicit distribution for atomic angular-momentum Dicke states, coherent states, and squeezed states that correspond to a collection of N two-level atoms. These Wigner functions W(\ensuremath{\theta},cphi) are represented as a pseudoprobability distribution in spherical phase space with spherical coordinates \ensuremath{\theta} and cphi on the surface of a sphere of radius \ensuremath{\Elzxh} \ensuremath{\surd}j(j+1) where j is the total angular-momentum eigenvalue.