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

Theoretical Aspects of Mixtures of Thermal and Coherent Radiation

Gerard Lachs

Published May 24, 1965• 185 citations
Publisher

Abstract

Some of the statistical properties of a mode of propagation for an electromagnetic field which is a super-position of thermal and coherent radiation are derived. It is found that the electric field and the magnetic field have Gaussian probability densities. The variance of the mixed field is the same as the variance of the thermal part of the field alone, while the average of the mixed field is the same as the average of the coherent field alone. It is pointed out that this result differs from the classical theory in that the zero-point field appears only once in the variance for the mixed field while it appears once in the variance of each of the constituent fields. This result is extended to a more general class of fields, and for that class it is shown that except for the zero-point field, the quantum properties of superposition are the same as the classical properties of superposition of noisy fields. The probability distribution for the number of photons in a mode of mixed radiation is also derived. The results shows that there are fluctuations in the number of photons that arise, because of interference effects.

DOI
10.1103/physrev.138.b1012

Related papers

ON THE QUANTUM MECHANICAL ASPECTS OF THE WAVESHAPE OF ELECTROMAGNETIC RADIATION.

Gerard Lachs

Previous work of this contract on the waveshape aspects of light from a classical point of view led to an interest to investigate the quantum mechanical aspects of the waveshape of light. This work contains some results of this investigation. One of the results obtained is an upper bound on the channel capacity where the only noise in the channel is due to the zero point field. The P-representation of a general state in terms of the coherent states is utilized to obtain the probability densities of the in-phase component, quadrature component, envelope, and the number of photons of the field. This is done for thermal radiation and for radiation which is a mixture of thermal radiation and coherent radiation with the result that the thermal radiation behaves as pure Gaussian noise while the coherent radiation behaves as a classical signal. The quantum mechanical formalism is utilized to obtain the rules of superposition for a large class of fields. The results show that the signal part can be treated classically and the noise part is nearly the same as would be found from the classical superposition of noise.

•Sep 8, 1964•0 citations

Electromagnetic field fluctuations and uncertainty relation for general quantum superpositions of coherent states

A. V. Kozlovskiĭ

Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.

Journal of Modern Optics•Jun 28, 2016•3 citations

Superposition Theory for Photon Fields

J. C. Webber

An extension of previous results in the dynamics of coherent states is presented. The multimode quantized electromagnetic field state vector generated by superimposed sources, from a class that generates coherent states while acting alone, is obtained as a function of time. Equations are also given that describe the quantum statistical superposition of photon fields. The modes of the radiation field are coupled.

Physical Review•Jul 25, 1969•0 citations

The quantum statistical properties of superposition coherent states with thermal noise

Kaicheng Zhu, Huiqin Tang, Cuiliang Li +3 more

We investigate a free monochromatic electromagnetic field which is in the even or odd coherent state with thermal noise. The main tool in our analysis is the characteristic function. We establish an alystic formula for an arbitrary correlation function. We point out the non-classical oscillations of the photon-number distribution and find its generating function as well as the degree of purity of the field. We also examine the squeezing properties and antibunching feature. Besides, the usual quasiprobability densities are derived.

Journal of Modern Optics•Feb 1, 1996•6 citations

Squeezed states with thermal noise. I. Photon-number statistics

Paulina Marian, Tudor A. Marian

We investigate a free monochromatic electromagnetic field which is the superposition of a squeezed thermal radiation and a coherent one. The main tool in our analysis is the characteristic function that has a Gaussian form. We establish an analytic formula for an arbitrary correlation function, as well as its strong-squeezing limit. Besides the usual quasiprobability densities, the coherent-state, number-state, coordinate, and momentum representations of the density operator are derived. We point out the nonclassical oscillations of the photon-number distribution and find its generating function. Collaterally, displaced thermal states and squeezed thermal states are revisited as nontrivial particular cases. We examine finally the squeezing properties of the field using the distribution functions of the quadratures.

Physical Review A•May 1, 1993•165 citations

Quantum statistics of nonclassical radiation in dissipative forward four-wave mixing. I. Method of generalized superposition of coherent and chaotic fields

Vlasta Peřinová, J. Křepelka

The Fokker-Planck equation approach to forward four-wave mixing developing from the initial coherent states is adopted to obtain a set of recursive relations for the expectation values of the normally ordered field operators. This set of equations is closed, in accordance with the model of the generalized superposition of coherent and chaotic fields, to provide differential equations for unknown quantities in this model. A numerical solution of these equations enables one to determine the photon number distribution, its factorial moments, squeezing variances, and entropy of radiation under discussion. Fully quantum-mechanical features of radiation such as its sub-Poisson behavior and squeezing of vacuum fluctuations are preserved in this approach. The validity of this approximation is discussed and numerical results are provided.

Physical Review A•Apr 1, 1989•2 citations
Some of the statistical properties of a mode of propagation for an electromagnetic field which is a super-position of thermal and coherent radiation are derived. It is found that the electric field and the magnetic field have Gaussian probability densities. The variance of the mixed field is the same as the variance of the thermal part of the field alone, while the average of the mixed field is the same as the average of the coherent field alone. It is pointed out that this result differs from the classical theory in that the zero-point field appears only once in the variance for the mixed field while it appears once in the variance of each of the constituent fields. This result is extended to a more general class of fields, and for that class it is shown that except for the zero-point field, the quantum properties of superposition are the same as the classical properties of superposition of noisy fields. The probability distribution for the number of photons in a mode of mixed radiation is also derived. The results shows that there are fluctuations in the number of photons that arise, because of interference effects.
Previous work of this contract on the waveshape aspects of light from a classical point of view led to an interest to investigate the quantum mechanical aspects of the waveshape of light. This work contains some results of this investigation. One of the results obtained is an upper bound on the channel capacity where the only noise in the channel is due to the zero point field. The P-representation of a general state in terms of the coherent states is utilized to obtain the probability densities of the in-phase component, quadrature component, envelope, and the number of photons of the field. This is done for thermal radiation and for radiation which is a mixture of thermal radiation and coherent radiation with the result that the thermal radiation behaves as pure Gaussian noise while the coherent radiation behaves as a classical signal. The quantum mechanical formalism is utilized to obtain the rules of superposition for a large class of fields. The results show that the signal part can be treated classically and the noise part is nearly the same as would be found from the classical superposition of noise.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
We investigate a free monochromatic electromagnetic field which is in the even or odd coherent state with thermal noise. The main tool in our analysis is the characteristic function. We establish an alystic formula for an arbitrary correlation function. We point out the non-classical oscillations of the photon-number distribution and find its generating function as well as the degree of purity of the field. We also examine the squeezing properties and antibunching feature. Besides, the usual quasiprobability densities are derived.
We investigate a free monochromatic electromagnetic field which is the superposition of a squeezed thermal radiation and a coherent one. The main tool in our analysis is the characteristic function that has a Gaussian form. We establish an analytic formula for an arbitrary correlation function, as well as its strong-squeezing limit. Besides the usual quasiprobability densities, the coherent-state, number-state, coordinate, and momentum representations of the density operator are derived. We point out the nonclassical oscillations of the photon-number distribution and find its generating function. Collaterally, displaced thermal states and squeezed thermal states are revisited as nontrivial particular cases. We examine finally the squeezing properties of the field using the distribution functions of the quadratures.
The Fokker-Planck equation approach to forward four-wave mixing developing from the initial coherent states is adopted to obtain a set of recursive relations for the expectation values of the normally ordered field operators. This set of equations is closed, in accordance with the model of the generalized superposition of coherent and chaotic fields, to provide differential equations for unknown quantities in this model. A numerical solution of these equations enables one to determine the photon number distribution, its factorial moments, squeezing variances, and entropy of radiation under discussion. Fully quantum-mechanical features of radiation such as its sub-Poisson behavior and squeezing of vacuum fluctuations are preserved in this approach. The validity of this approximation is discussed and numerical results are provided.