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Heisenberg approach to photon emission near a phase conjugator. Technical report

Henk F. Arnoldus, Tomas F. George

Published Mar 1, 1991• 0 citations
OpenAlex

Abstract

An expression for the emitted fluorescence radiation by an atom near a phase conjugator is derived from a plane-wave expansion of the electric field operator and Heisenberg's equation of motion for the annihilation operator. The result is compared to a solution which was found previously, based on the classical Maxwell equations, where it is shown that both theories yield the same expression for the field in the radiation zone, in the limit of a transparent medium, which confirms the correctness of either approach to the problem of optical phase conjugation of atomic radiation.

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Heisenberg approach to photon emission near a phase conjugator

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Physical Review A•Jul 1, 1992•0 citations
An expression for the emitted fluorescence radiation by an atom near a phase conjugator is derived from a plane-wave expansion of the electric field operator and Heisenberg's equation of motion for the annihilation operator. The result is compared to a solution which was found previously, based on the classical Maxwell equations, where it is shown that both theories yield the same expression for the field in the radiation zone, in the limit of a transparent medium, which confirms the correctness of either approach to the problem of optical phase conjugation of atomic radiation.
An expression for the emitted fluorescence radiation by an atom near a phase conjugator is derived from a plane-wave expansion of the electric-field operator and Heisenberg's equation of motion for the annihilation operator. The result is compared to a solution that was found previously, based on the classical Maxwell equations. It is shown that both theories yield the same expression for the field in the radiation zone, in the limit of a transparent medium. This confirms the correctness of either approach to the problem of optical phase conjugation of atomic radiation.
Fluorescent emission by an atom near a phase conjugator (PC) based on four-wave mixing is studied from first principles. The Maxwell-Heisenberg equations are solved for the radiation field, and with an asymptotic expansion an expression is derived for the field in the far zone. The total emitted power that can be measured by a detector in this region is evaluated, and it is found that this power acquires three distinct contributions. First, there are photons that are emitted by the atom directly towards the detector, and without any interaction with the medium. Second, there are photons that first travel towards the surface of the PC, and they have a certain probability of being reflected in the specular direction and towards the detector. The third kind of radiation consists of phase-conjugated photons, which are emitted independently of the previous ones. It is shown that the first two processes are a result of simple atomic spontaneous decay, but that the emission of a phase-conjugated fluorescent photon involves a three-photon process. The latter process has a probability proportional to the population of the atomic ground state. It is pointed out that an atom in its ground state polarizes the nonlinear medium of the PC, which subsequently can emit spontaneously two photons. An absorption-emission-absorption process by the atom then produces a fluorescent photon, together with a spontaneous excitation of the atom.
Emission of resonance fluorescence by an atom near the surface of a four-wave mixing phase conjugator is considered. The dipole radiation field, regarded as a Heisenberg-operator field, is decomposed into plane waves with the aid of Weyl's representation of the Green's function for the wave equation. Each plane-wave component which is incident on the surface of the nonlinear medium, is reflected as its phase-conjugate image. Summation of all reflected plane waves then yields the phase conjugate replica of the incident dipole radiation. This field adds to the radiation which is emitted by the atom into the direction away from the medium. The condition under which squeezing occurs in the emitted resonance fluorescence is investigated.
Abstract : Temporal photon correlations and photon count statistics of fluorescence radiation, emitted by an atom in the vicinity of a phase conjugator, are considered. The fluorescent intensity and the two-photon correlation function are expressed in terms of photon emission operators. An explicit evaluation shows their dependence on the phase-conjugate reflectivity of the surface. The variance of the photon count distribution for long counting times is expressed in terms of the Q-factor, which is evaluated as a function of the reflectivity. It appears that for a reflectivity larger than unity, the photon correlation displays antibunching and the statistics is sub-poissonian.
The vacuum field’s role in the radiative properties of two-level atoms located externally to a phase-conjugate mirror (PCM) is investigated. The basic result of Gaeta and Boyd [ Phys. Rev. Lett.60, 2618 ( 1988)], that the PCM emits an average of R (where R is reflectivity) noise photons per mode of the radiation field, is modified. Stimulated emission of the combined PCM–atom system takes place when the atom is in its ground state; this emission is caused by a scattering–conjugation process of virtual photons. The expression for the power emitted by a system of N (N > 1) two-level atoms also contains terms that are the expectation values of products of two atomic raising or lowering operators in addition to the usual superradiant terms. The case N = 2 is solved exactly in the limit of a vanishing separation-to-wavelength ratio. The expectation values of individual and pair-product operators are derived from the Heisenberg equations of motion, and the solutions are classified according to the value of an integral of motion. When R < 1, the time evolution is characterized by four real-time constants, of which two are a complex-conjugate pair for R > 1, indicating relaxation by damped oscillations.