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Effects of Injected Atomic Coherence on Multiwave Mixing.

Timothy Carty

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

Discussion begins with a brief account of atomic level-pumping and reasons why atomic coherence is typically not considered in cw work on optical interactions. This dissertation is divided into four parts: semiclassical treatments of one-photon electric- dipole atom-field single-mode interactions and multimode interactions, and corresponding treatments for the two-photon interaction. We present the effects of injected atomic coherence on the polarization of the medium, the slowly varying envelope wave equation, the single- and multiwave mixing coefficients, and weak field propagation in a homogeneously broadened two-level medium. Spatial and temporal phase matching of the injected coherence to a field mode is crucial throughout, since the field may not be able to remain in phase with the induced and injected polarizations. One-photon injected coherence contributes directly to the polarization at the atomic resonance frequency. The perfectly phase-matched case leads to a linear superposition of an exponentially decaying field (Beer's law) and a constant field driven by the injected coherence. The interaction of an injected coherence with a detuned field produces frequency-symmetric sidebands about the pump field polarization. The sideband spacing equals the atom-field detuning. To probe the injected coherence we inject a weak resonant field. The resulting three-wave mixing leads to multiwave mixing coefficients that are unaffected to first-order in the weak sidemodes, but the injected coherence adds inhomogeneous terms to the coupled-mode equations. For both single- and multimode interactions the injected coherence does not affect the exponential growth/decay of the sidemodes, but it supports a weak field that may propagate if properly phase matched. For two-photon media the injected coherence requires at least one field interaction in order to produce a polarization, which then appears in the single- and multiwave mixing coefficients. The exponential growth/decay rate is modified by the injected coherence. For a centrally-tuned pump the injected coherence contributes the standard multiwave mixing terms as well as additional effects. Four-wave mixing is discussed as a means of relaxing the spatial phase matching constraint on the injected coherence.

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Effects of injected atomic coherence on one-photon single-mode interactions

Timothy Carty, Murray Sargent

We investigate the semiclassical electric-dipole-allowed interaction of a unidirectional plane-wave electric field with a homogeneously broadened two-level medium prepared in a coherent superposition of energy eigenstates. Atomic level pumping and the effects of phase matching the new polarization contributions are discussed. The slowly varying polarization is derived and applied to the problem of the propagation of a weak field. The polarization of the medium acquires a direct (field-independent) contribution due to the injected coherence. We find that the slowly varying envelope propagation equation retains the standard single-mode absorption coefficient but develops additional driving terms. Phase matching in both time and space is crucial for the injected coherence to couple to a field. The phase-matched case of weak-field propagation proves to be a linear superposition of two processes: the usual Beer's law exponential decay (or growth) and a constant field supported by the medium. The injected coherence enables a weak electric field to persist in an absorbing medium.

Physical Review A•Aug 1, 1990•3 citations

Effects of injected atomic coherence on single-mode frequency locking in a cavity

Timothy Carty, Murray Sargent

The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.

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Annual Meeting Optical Society of America•Jan 1, 1986•0 citations

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This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.

•Jan 1, 1986•0 citations

Theory of multiwave mixing in two- and three-level media.

Sunghyuck An

This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.

•Jan 1, 1988•0 citations
Discussion begins with a brief account of atomic level-pumping and reasons why atomic coherence is typically not considered in cw work on optical interactions. This dissertation is divided into four parts: semiclassical treatments of one-photon electric- dipole atom-field single-mode interactions and multimode interactions, and corresponding treatments for the two-photon interaction. We present the effects of injected atomic coherence on the polarization of the medium, the slowly varying envelope wave equation, the single- and multiwave mixing coefficients, and weak field propagation in a homogeneously broadened two-level medium. Spatial and temporal phase matching of the injected coherence to a field mode is crucial throughout, since the field may not be able to remain in phase with the induced and injected polarizations. One-photon injected coherence contributes directly to the polarization at the atomic resonance frequency. The perfectly phase-matched case leads to a linear superposition of an exponentially decaying field (Beer's law) and a constant field driven by the injected coherence. The interaction of an injected coherence with a detuned field produces frequency-symmetric sidebands about the pump field polarization. The sideband spacing equals the atom-field detuning. To probe the injected coherence we inject a weak resonant field. The resulting three-wave mixing leads to multiwave mixing coefficients that are unaffected to first-order in the weak sidemodes, but the injected coherence adds inhomogeneous terms to the coupled-mode equations. For both single- and multimode interactions the injected coherence does not affect the exponential growth/decay of the sidemodes, but it supports a weak field that may propagate if properly phase matched. For two-photon media the injected coherence requires at least one field interaction in order to produce a polarization, which then appears in the single- and multiwave mixing coefficients. The exponential growth/decay rate is modified by the injected coherence. For a centrally-tuned pump the injected coherence contributes the standard multiwave mixing terms as well as additional effects. Four-wave mixing is discussed as a means of relaxing the spatial phase matching constraint on the injected coherence.
We investigate the semiclassical electric-dipole-allowed interaction of a unidirectional plane-wave electric field with a homogeneously broadened two-level medium prepared in a coherent superposition of energy eigenstates. Atomic level pumping and the effects of phase matching the new polarization contributions are discussed. The slowly varying polarization is derived and applied to the problem of the propagation of a weak field. The polarization of the medium acquires a direct (field-independent) contribution due to the injected coherence. We find that the slowly varying envelope propagation equation retains the standard single-mode absorption coefficient but develops additional driving terms. Phase matching in both time and space is crucial for the injected coherence to couple to a field. The phase-matched case of weak-field propagation proves to be a linear superposition of two processes: the usual Beer's law exponential decay (or growth) and a constant field supported by the medium. The injected coherence enables a weak electric field to persist in an absorbing medium.
The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.
In this first paper of a series on the quantum theory of nondegenerate multiwave mixing applicable to traveling-wave interaction geometries, we describe our problem formulation. We consider the explicit dynamics of a subset of the field quantization modes interacting with a system of stationary two-level atoms contained in a volume much smaller than the field quantization volume. Because we make the realistic assumption of leaving the remaining infinite set of field modes as a common thermal-field reservoir, the resulting Langevin equations contain extra decay terms due to collective spontaneous emission or super-radiance. We show that all but one of these super-radiance terms are negligible in the following two limits: (a) when the number of atoms in a diffraction volume is small; (b) when the atoms are pumped far off resonance. There is, however, an anomalous decay term which does not appear in a classical model of super-radiance based upon coherently phased atomic dipoles. The magnitude of this anomalous term is neither dependent upon the number of atoms nor on the pump-frequency detuning and may not be negligible at a low pump intensity. Neglecting the super-radiance terms, we then present a general Fourier-expansion solution technique for obtaining the atomic polarization in the presence of any number of field modes. The expansion is shown to be convergent in a commonly occurring situation in which all the strong pump modes are frequency degenerate and the remaining nondegenerate modes are all weak compared to the atomic saturation intensity. In subsequent papers of this series, we will present methods to treat, with some rigor, the spatial propagation of an interacting multimode quantum field and apply these methods to traveling-wave squeezed-state generation experiments.
Multiphoton transitions are of great interest in laser spectroscopy because they permit the study of new physical phenomena and because of their potential applications. Malcuit et al.1 studied two-photon transitions in atomic sodium and showed that for appropriate phase-matching four-wave mixing can overcome the usual spontaneous decay processes. We have recently developed a quantum theory2 to treat such multiwave mixing processes in two-photon media. Because of the greater complexity of two-photon transitions, many effects arise that are absent in the one-photon case, some of which have important consequences in the generation of squeezed states. For the two-photon two-level model, the field modes have frequencies approximately one-half the frequency difference between the levels. We find that our coefficients differ from the corresponding one-photon theory in that dynamic Stark shifts have a major effect and that the pump-induced coherence between the two levels has a new term in the conjugate coupling terms, which leads to improved squeezing.
This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.
This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.