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Physical Review A

Theory of four-wave mixing using an amplitude approach

B. Dubetsky, P. R. Berman

Published Feb 1, 1993• 13 citations
Publisher

Abstract

A theory of four-wave mixing is developed using an amplitude approach in the Schr\"odinger picture. It is shown that the four-wave mixing signal produced when three fields are incident on an ensemble of two-level atoms can be viewed as arising from constructive interference of the signals emitted at the various atomic sites. Fourteen distinct amplitudes contribute to the four-wave mixing signal. Some of these amplitudes involve final states in which one or two atoms are excited, and some involve final states in which photons are emitted into modes other than a phase-matched mode of interest. The amplitude approach has the advantage of clearly showing the atom-field correlations. It helps one to understand the underlying physical mechanisms responsible for creation of the four-wave mixing signal, as well as features in the line shape resulting from collisions and effects relating to atomic recoil on the absorption or emission of radiation. These so-called collision- and recoil-induced resonances are calculated explicitly using the amplitude approach, and an interpretation of the results in terms of the interference between different amplitudes is given. Our results, calculated for a finite interaction time between the atoms and the fields, are compared with analogous calculations in which an effective ground-state decay time is assumed. It is shown that the results differ if the atom-field interaction time is not the longest relevant time parameter in the problem.

DOI
10.1103/physreva.47.1294

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A theory of four-wave mixing is developed using an amplitude approach in the Schr"odinger picture. It is shown that the four-wave mixing signal produced when three fields are incident on an ensemble of two-level atoms can be viewed as arising from constructive interference of the signals emitted at the various atomic sites. Fourteen distinct amplitudes contribute to the four-wave mixing signal. Some of these amplitudes involve final states in which one or two atoms are excited, and some involve final states in which photons are emitted into modes other than a phase-matched mode of interest. The amplitude approach has the advantage of clearly showing the atom-field correlations. It helps one to understand the underlying physical mechanisms responsible for creation of the four-wave mixing signal, as well as features in the line shape resulting from collisions and effects relating to atomic recoil on the absorption or emission of radiation. These so-called collision- and recoil-induced resonances are calculated explicitly using the amplitude approach, and an interpretation of the results in terms of the interference between different amplitudes is given. Our results, calculated for a finite interaction time between the atoms and the fields, are compared with analogous calculations in which an effective ground-state decay time is assumed. It is shown that the results differ if the atom-field interaction time is not the longest relevant time parameter in the problem.
An interpretation of collision-induced extra resonances in multiwave mixing is constructed using a fully quantum-mechanical calculation carried out in the Schr\"odinger representation. The phase-matched signal is seen to arise as an interference between the amplitudes for scattering of the incident fields at different atomic sites. When collisions occur, the separation of the signal into a collision-induced resonance plus a background term appears naturally in this approach. The origin of the phase-matching condition, conservation of energy (and breaking of the Manley-Rowe relations), the role played by relaxation mechanisms, and the fact that the fields need not be temporally coherent to produce the extra resonances, all can be easily understood within the context of this calculation.
A theory of resonantly enhanced degenerate four-wave mixing in two-level systems including the effects of atomic and photon angular momentum is presented in the absence of pump-induced saturation of the transition. It is shown that there exist three distinct quantum-mechanical amplitudes leading to the third-order polarization density. These quantum-mechanical amplitudes are shown to be sensitive to the states of polarization of the incident fields. The quantum-mechanical transport equation in the mmm representation is used to calculate the output signal in the various regimes of laser detuning, atomic linewidth, and polarization states of the radiation field for collinear interaction.
A very large number of physical phenomena in the resonant interaction of atomic-molecular systems with the radiation fields can be studied by treating the atomic system as a two level system and by examining its response to the applied fields. Various perturbative results are well known. A number of recent experimental works require the knowledge of the nonperturbative response functions, when both the applied fields are strong. These include a variety of experiments on instabilities(1) in lasers involving the switching from a single mode to for example two mode operation. Similarly recent nonlinear mixing experiments, where not only four wave mixing but also higher mixing in saturating fields has been studied. Pump probe experiments have also been repeated with probes whose intensity could be large. The purpose of this paper is to formulate the general theory and use it to understand the experiments. The formulation we present is also well suited for studying the effect of collisions. Details of various collisional induced coherences in fourwave and six wave mixing etc. can be studied.
Correlations in space and time of the light intensity from extended sources, observed by Brown and Twiss and by Forrester, were analyzed by these authors primarily in terms of field oscillations. This paper discusses an atomic process which brings out the intensity correlations, namely, the photoionization of a pair of atoms following photon emission by another pair of independently excited atoms. The calculated probability of this process depends sinusoidally on the relative positions of the four atoms and also, when the source atoms emit different frequencies, on the time interval between the photoionizations, in agreement with the macroscopic treatment. The oscillations arise from an interference of probability amplitudes which is not affected by the random phases of the source atoms. The calculation follows a standard approach but involves some novel detail.
Using both a bare-atom and dressed-atom picture of atom-field interactions, we present a theory of the pressure-induced resonances that can arise when four radiation fields interact with a three-level quantum system. The four fields drive two coupled atomic transitions which share a common level, the levels being arranged in a ``ladder'' scheme. As the field frequencies are varied, certain resonant structures in the excitation spectrum of the upper state population appear only in the presence of collisions. The resonant structures result from processes that have been referred to in the past as collisionally aided radiative excitation (CARE) and pressure-induced extra resonances in four-wave mixing (PIER4). We show that both CARE- and PIER4-type resonances have a common origin, which can be linked to conservation of energy. In contrast to conventional PIER4 signals, line shapes calculated in this work result from a ``single atom'' rather than a collective emission. Moreover, the signals vanish unless the applied fields are relatively coherent, again in contrast to conventional PIER4.
A theory of four-wave mixing including effects of velocity-changing collisions is presented. Three fields with frequencies \ensuremath{\omega}, \ensuremath{\omega}, and \ensuremath{\omega}+\ensuremath{\delta} are incident on a vapor of ``two-level'' atoms having upper state b and lower state a. Two of the fields are counterpropagating and the third (of frequency \ensuremath{\omega}+\ensuremath{\delta}) makes a small angle with one of the others. The frequency \ensuremath{\omega} is a nearly resonant (inside the Doppler width) with the a-b transition frequency. The phase-conjugate signal emitted at frequency \ensuremath{\omega}-\ensuremath{\delta} is calculated as a function of \ensuremath{\delta}. Using a simple collision model in which collisions are phase interrupting in their effect on atomic coherence and velocity-changing in their effect on level populations, we discuss the conditions under which resonances characterized by the upper or lower radiative and collision rates can be observed. Assuming that the total (a+b) state population is conserved in the absence of collisions, it is shown that velocity-changing collisions can ``open'' the system and lead to a resonance characterized by the lower-state width (convoluted with the residual Doppler width). With increasing pressure, the width of this induced resonance structure decreases monotonically. For sufficiently high pressure, the collisional redistribution of velocity classes is complete---the system ``recloses'' and the narrow resonance disappears. The interplay of the collision-induced opening, line narrowing, and reclosing of the system is discussed, as is the relationship of these narrow resonances to the so-called pressure-induced extra resonances of Bloembergen and co-workers [Indian J. Pure Appl. Phys. 16, 151 (1978); Phys. Rev. Lett. 46, 111 (1981)].