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Journal of the Optical Society of America B

Quantum theory of spontaneous emission and excitation near a phase-conjugating mirror

Peter W. Milonni, Erik J. Bochove, Richard J. Cook

Published Oct 1, 1989• 15 citations
Publisher

Abstract

We present a quantum-electrodynamic theory of spontaneous emission in the presence of a phase-conjugating mirror (PCM) and show that the radiative relaxation rate is increased owing to the amplification of quantum noise by the PCM. For any finite value of the PCM reflectivity an atom in any initial state will relax to a steady state with a finite probability of being excited. In particular, there is a finite probability of spontaneous excitation of a ground-state atom placed near the PCM. The fluorescence spectrum of an atom near a PCM is generally described by two Lorentzian functions.

DOI
10.1364/josab.6.001932

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Relaxation of the density operator of an atom near a four-wave mixing phase conjugator (PC) is studied. The relaxation operator Tau can be expressed in the Cartesian components of the atomic dipole operator and correlation functions of the vacuum electromagnetic field. We evaluate these correlation functions in terms of the Fresnel coefficients for reflection and transmission of a polarized monochromatic plane wave. Our expression for Tau includes both linear and nonlinear interactions in the medium, and it reduces to well-known results in the limits of a pure dielectric, a mirror, and empty space. The example of a model two-state atom in combination with a model PC is worked out in detail. From the rate equations for the populations of the levels we infer that the relaxation acquires a contribution from both ordinary spontaneous decay and stimulated transitions which are induced by the PC. It is shown that an atom in its ground state has a finite probability of being excited, and that the excited state has a finite population in the long-time limit. We also work out the case of a degenerate two-level atom in combination with a realistic model of a transparent PC, and it is shown that our present results for the transition rates are consistent with earlier calculations of the fluorescence yield.

Unknow•Mar 1, 1992•0 citations
We present a quantum-electrodynamic theory of spontaneous emission in the presence of a phase-conjugating mirror (PCM) and show that the radiative relaxation rate is increased owing to the amplification of quantum noise by the PCM. For any finite value of the PCM reflectivity an atom in any initial state will relax to a steady state with a finite probability of being excited. In particular, there is a finite probability of spontaneous excitation of a ground-state atom placed near the PCM. The fluorescence spectrum of an atom near a PCM is generally described by two Lorentzian functions.
The properties of a phase conjugate mirror (PCM) interacting radiatively with other objects are investigated. The spontaneous emission rate of a two-level atom (TLA) is attributed to the dipole emission of the upper energy level into vacuum field modes incident on the front face of the PCM, the emission into vacuum modes transmitted from the rear port of the PCM, enhanced by a factor 1 + R (R = reflectivity), and the emission of a photon while the atom is in the lower level, accompanied by annihilation of two pump photons, weighted by a factor of R per emitted photon. These processes lead to an enhancement factor of 1 + R for the spontaneous decay rate and a steady state upper level occupation probability R/(2R + 2), both in agreement with solution of the Heisenberg equations of TLA operators.1 The momentum transfer is zero, and the steady state emission rate is A12R(R + 2)/(2R + 2) photons/s atom. The stimulated emission and absorption constants are enhanced in a similar way. Contrary to the case of a conventional mirror, the Lippman fringes are present only for coherent radiation states.
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.
We demonstrate, by solution of the Heisenberg equations of motion of the pseudo-spin and momentum operators of a two-level atom interacting with a phase- conjugate mirror (PCM) and a quantized radiation field, that a force is exerted by the PCM on the ground state atom. The expression found for the force is F=gRA(P1-P2)pvn̂, where P1 and P2 are the ground and excited state occupation probabilities, A=Einstein A-coefficient, pv=photon momentum at resonance, R=PCM reflectivity, n̂=unit outward normal to the PCM, and g=an aperture factor (g=9/64 for a dipole parallel to the plane of the PCM and solid angle=Ω= 2╔) . This result has a simple physical interpretation in terms of vacuum fields that are conjugated or amplified by the PCM (1,2).
The spontaneous radiation of a classical oscillator and that of a two-level atom both in the vicinity of a phase-conjugate mirror are calculated and compared. In either case the phase-conjugate mirror may be replaced by a phase-conjugate "image" of the source. We study the dependence of the decay constants and the characteristic frequencies on the phase-conjugate reflectivity and the difference between the characteristic frequency of the radiator and the pump frequency of the phase-conjugate mirror.
Relaxation of the density operator of an atom near a four-wave mixing phase conjugator (PC) is studied. The relaxation operator Tau can be expressed in the Cartesian components of the atomic dipole operator and correlation functions of the vacuum electromagnetic field. We evaluate these correlation functions in terms of the Fresnel coefficients for reflection and transmission of a polarized monochromatic plane wave. Our expression for Tau includes both linear and nonlinear interactions in the medium, and it reduces to well-known results in the limits of a pure dielectric, a mirror, and empty space. The example of a model two-state atom in combination with a model PC is worked out in detail. From the rate equations for the populations of the levels we infer that the relaxation acquires a contribution from both ordinary spontaneous decay and stimulated transitions which are induced by the PC. It is shown that an atom in its ground state has a finite probability of being excited, and that the excited state has a finite population in the long-time limit. We also work out the case of a degenerate two-level atom in combination with a realistic model of a transparent PC, and it is shown that our present results for the transition rates are consistent with earlier calculations of the fluorescence yield.