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

Journal of the Physical Society of Japan

Note on the Optical Equivalence Theorem

Yutaka Kano

Published Feb 1, 1974• 0 citations
Publisher

Abstract

The optical equivalence theorem is examined for the particular class of the electromagnetic fields. It is shown that the distribution function can be taken as a non-negative function, if the density operator describing the states of the electromagnetic field is diagonal in the occupation number representation. Hence the equivalence is not only formal but also essential for this class of the electromagnetic fields, so long as the description of the optical coherence theory is concerned. As a typical example the “diagonal” representation of the density operator for blackbody radiation is studied.

DOI
10.1143/jpsj.36.526

Related papers

Some new non-diagonal representations for density operators of optical fields

Hari Prakash, Naresh Chandra, Vachaspati

Some new non-diagonal representations for density operators of radiation in terms of coherent states are given. These involve only two integrations per mode. The weight functions are expressed in terms of coherence functions and of occupation number space matrix elements of the density operators. When a compact form of the weight function in the diagonal representation does not exist, one of these new representations may become preferable. One such example is discussed.

Journal of Physics A Mathematical Nuclear and General•Nov 21, 1974•0 citations

Coherent-State Representations for the Photon Density Operator

Kevin E. Cahill

The "diagonal" $P$ representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the $P$ representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the $P$ representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the $P$ representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.

Physical Review•Jun 21, 1965•86 citations

Reformulation of the Optical Equivalence Theorem in Terms of the LAGUERRE Polynomials

Jan Peřina, Ladislav Mišta

Abstract The “diagonal” weighting function Φ(α) of the density matrix expressed in the coherent states is examined being decomposed in the base of the Laguerre polynomials. A criterion of the convergence in L2‐space is obtained. This decomposition is also studied as a generalized function. The equivalence of the FOCK and SUDARSHAN representations of the density matrix is proved using present formulae. Some theorems from the moment theory are applied and general assumptions are given under which the Φ‐representation is an ordinary non‐negative function or a generalized function whose support is composed of a finite number of points. The results are demonstrated on the examples of the thermal and coherent fields, their superposition and on the Fock state of the field. The decomposition of the Φ‐function in terms of the LAGUERRE polynomials is useful for approximative determination of the Φ‐function from a finite number of moments.

Annalen der Physik•Jan 1, 1969•10 citations

Representation of Operators in Quantum Optics

Marvin M. Miller, Eli A. Mishkin

The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.

Physical Review•Dec 25, 1967•27 citations

Coherence Properties of Blackbody Radiation. II. Correlation Tensors of the Quantized Field

C. L. Mehta, E. Wolf

Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.

Physical Review•Jun 1, 1964•63 citations

A sincere tribute to E.C.G Sudarshan's phenomenal contribution toward quantum theory of optical coherence

Arindam Kumar Chatterjee, Anik Rudra, Soham Chakraborty

The diagonal representation and optical equivalence theorem are the E. C. G. Sudarshan's mid 20th century adventures in non-classical optics. It basically deals with a quantum mechanical description of photons to explain the quantum properties of light. Inspired by Sudarshan's pioneering work we try to explain the every minute mathematical details of his paper "Equivalence of semi-classical and quantum mechanical descriptions of statistical light beams". In this article we are going to go through some of the basics in developing quantum optics, then land up in E.C.G's original work and try to present it as rigorous as possible. We show some of its important applications in various classes of physics problems.

arXiv:1812.03854•Nov 21, 2018•0 citations
The optical equivalence theorem is examined for the particular class of the electromagnetic fields. It is shown that the distribution function can be taken as a non-negative function, if the density operator describing the states of the electromagnetic field is diagonal in the occupation number representation. Hence the equivalence is not only formal but also essential for this class of the electromagnetic fields, so long as the description of the optical coherence theory is concerned. As a typical example the “diagonal” representation of the density operator for blackbody radiation is studied.
The "diagonal" PPP representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the PPP representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the PPP representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the PPP representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.
Abstract The “diagonal” weighting function Φ(α) of the density matrix expressed in the coherent states is examined being decomposed in the base of the Laguerre polynomials. A criterion of the convergence in L2‐space is obtained. This decomposition is also studied as a generalized function. The equivalence of the FOCK and SUDARSHAN representations of the density matrix is proved using present formulae. Some theorems from the moment theory are applied and general assumptions are given under which the Φ‐representation is an ordinary non‐negative function or a generalized function whose support is composed of a finite number of points. The results are demonstrated on the examples of the thermal and coherent fields, their superposition and on the Fock state of the field. The decomposition of the Φ‐function in terms of the LAGUERRE polynomials is useful for approximative determination of the Φ‐function from a finite number of moments.
The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.
Expressions are derived for the electromagnetic correlation tensors of blackbody radiation defined on the basis of the theory of the quantized field. Correlation functions of all order are considered, but second-order ones are discussed in detail; it is found that these are identical with those obtained on the basis of semi-classical theory in part I of this investigation. This result illustrates a recent theorem of E.C.G. Sudarshan relating to the equivalence between semiclassical and quantum mechanical description of statistical light beams.
The diagonal representation and optical equivalence theorem are the E. C. G. Sudarshan's mid 20th century adventures in non-classical optics. It basically deals with a quantum mechanical description of photons to explain the quantum properties of light. Inspired by Sudarshan's pioneering work we try to explain the every minute mathematical details of his paper "Equivalence of semi-classical and quantum mechanical descriptions of statistical light beams". In this article we are going to go through some of the basics in developing quantum optics, then land up in E.C.G's original work and try to present it as rigorous as possible. We show some of its important applications in various classes of physics problems.