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Physical Review A

Fluorescence near a phase-conjugating mirror: Amplification of quantum noise

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

Published Oct 1, 1989• 8 citations
Publisher

Abstract

We consider the spontaneous emission by an atom near a phase-conjugating mirror (PCM). The effect of the PCM on the decay rate may be understood equivalently in terms of vacuum-field modes reflected and conjugated by the PCM, or in terms of vacuum modes transmitted and amplified without conjugation by the PCM.
DOI
10.1103/physreva.40.4100

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Physical Review A•Apr 1, 1995•9 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.
The interaction of a radiating dipole with a phase conjugate mirror (PCM) is enhanced due to wavefront reversal by the PCM as well as by possible gain. Moreover, decay at two frequencies is predicted when ω o = ω m and 4(α r + αnr) d/c ≪ 1, where ω o = free oscillation frequency of the dipole, ω m = pump frequency of PCM (four-wave mixing), α r , and α nr are the radiative and nonradiative decay rates, respectively, and d is the effective distance to the PCM. Decay at multiple frequencies is predicted when 4(α r + α nr ) d/c ≳ 1. Locking of decay frequencies to the pump frequency is predicted at a critical pump intensity and at a second critical point the dipole gains energy from the PCM.
We investigate theoretically the quantum-noise properties of optical phase conjugation via four-wave mixing of nonclassical signal fields. We show that while the field generated at a phase-conjugate mirror (PCM) can be in a nonclassical state, the generated field is not necessarily useful within the context of phase conjugation. For the case in which a quadrature-squeezed signal field is incident at a PCM, we determine that the signal-to-noise ratio of one of the quadrature components of the generated field can approach the signal-to-noise ratio of the corresponding quadrature component of the signal field as the amount of squeezing of the vacuum field injected at the rear of the PCM is increased. For the case in which the signal field is amplitude squeezed, we determine that the signal-to-noise ratio of the conjugate field is always much smaller than that of the signal field. These results demonstrate that the phase-conjugation process can preserve the desirable quantum-noise properties of quadrature-squeezed fields but not those of amplitude-squeezed fields.