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Statistics of the electromagnetic field radiated in a two-photon process

Yu. M. Golubev

Published Jan 1, 1979• 2 citations
OpenAlex

Abstract

The statistical properties of the emission of a two-photon single-mode laser are theoretically analyzed in terms of quantum electrodynamics, using the kinetic equation for the electromagnetic field. It is found that in contrast to suggestions made by others, there is nothing unique in the properties of two-photon emission. Thus, the relative fluctuations of the numbers of photons in this radiation cannot be smaller than in coherent emission, and therefore two-photon emission in the form considered cannot be used to produce noiseless sources.

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Physical Review A•Jun 1, 1982•221 citations
The statistical properties of the emission of a two-photon single-mode laser are theoretically analyzed in terms of quantum electrodynamics, using the kinetic equation for the electromagnetic field. It is found that in contrast to suggestions made by others, there is nothing unique in the properties of two-photon emission. Thus, the relative fluctuations of the numbers of photons in this radiation cannot be smaller than in coherent emission, and therefore two-photon emission in the form considered cannot be used to produce noiseless sources.
A theory of a quantum amplifier based on two-photon transitions is presented. The model consists of an assemblage of atoms interacting with a single mode of the light field via two-photon absorption and emission. The treatment is quantum mechanical, and the emphasis is on the photon-statistical aspects of the problem. Difference-differential equations for the density matrix of the field mode are derived. The equations are solved for a special case. Differential equations for the expectation vaues of the number of photons and of the field amplitudes, and for their higher statistical moments, are also obtained. Approximate solutions for the average number of photons are found, and the effect of the amplification process on the statistical fluctuations of the number of photons and the field amplitudes is discussed. The results of this study suggest that such an amplifier does not have the coherence properties of a laser amplifier because it tends to increase the statistical fluctuations of the field.
It is predicted that the beam from a two-photon laser, if it can be made to operate in a coherent mode, will have properties quite different from the classical properties of the standard single-photon laser. Whereas the coherent beam of the single-photon laser is a dipole mode of the electromagnetic field, that of the two-photon laser is expected to be a monopole or quadrupole mode. It is predicted that the coherence properties of the two-photon laser will show up rather dramatically in non-linear processes.
Abstract One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
In this paper the difference between two-photon dipole-forbidden and cascade transitions of an inverted system of atoms is studied. It is shown that the fluctuations of the electromagnetic field (EMF) density in the case of two-photon dipole-forbidden transitions are larger than the ones in the case of cascade emission. In the process of the two-photon dipole-forbidden cooperative emission the photons are created in pairs and there appears coherence between pairs of photons (biphotons). In this situation the distribution of energy between the photons in one pair may be random. For two-photon dipole-forbidden transitions the time-behavior of the square density fluctuations of EMF is similar to those of the density of EMF in the case of cascade emission. When the distance between two atoms excited relative to dipole-forbidden transitions is less than the radiation wavelength, then the kinetics of such atoms is analogous to the cascade emission of a single atom.
Photon statistics in some fully quantized models of the interaction of a two-level atom with a single-mode radiation field have been studied using the operator equations of motion. Expressions for the photon number distribution and the mean photon number are presented for various initial conditions. It is found that the mean photon number may show decays and revivals of coherence similar to those of the atomic inversion in the coherent-state Jaynes-Cummings model. Application of these models to the study of multiphoton laser, absorption, and emission processes is also discussed.