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Annual Meeting Optical Society of America

Exact semiclassical theory of four-wave mixing

Yehiam Prior, A. M. Levine, N. Chencinski, W. M. Schreiber, Andrei N. Weiszmann

Published Jan 1, 1986• 0 citations
Publisher

Abstract

Nonlinear perturbative susceptibilities have been used successfully to analyze experiments in nonlinear optics for two decades. By its very nature, a perturbative approach is limited to a small value of the expansion parameter, and nonlinear susceptibilities are not useful for strong saturating fields. Presently, a theory is presented for the treatment of FWM which is applicable to arbitrary strength of the input fields, both on and off resonance. The field harmonic time dependences are removed by a transformation to the proper rotating frame, and the resulting eigenvalue problem is solved numerically. Several new effects are predicted. For strong fields resonance frequencies are shifted and lines are split (similar to the ac Stark effect). When such a split line is acted on by another strong field (a phenomenon not studied so far), a new effect of stirring is observed, which results in the broadening and merging of the two separate components. A new extra resonance appears which is induced by the strong field and is present even in the absence of any collisions or other dephasing processes or of laser phase fluctuation. This new extra resonance is a new member in the family of extra resonances which were discussed in various previous publications.

DOI
10.1364/oam.1986.tud6

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Nonlinear perturbative susceptibilities have been used successfully to analyze experiments in nonlinear optics for two decades. By its very nature, a perturbative approach is limited to a small value of the expansion parameter, and nonlinear susceptibilities are not useful for strong saturating fields. Presently, a theory is presented for the treatment of FWM which is applicable to arbitrary strength of the input fields, both on and off resonance. The field harmonic time dependences are removed by a transformation to the proper rotating frame, and the resulting eigenvalue problem is solved numerically. Several new effects are predicted. For strong fields resonance frequencies are shifted and lines are split (similar to the ac Stark effect). When such a split line is acted on by another strong field (a phenomenon not studied so far), a new effect of stirring is observed, which results in the broadening and merging of the two separate components. A new extra resonance appears which is induced by the strong field and is present even in the absence of any collisions or other dephasing processes or of laser phase fluctuation. This new extra resonance is a new member in the family of extra resonances which were discussed in various previous publications.
A nonperturbative semiclassical theory is presented for the treatment of four-wave mixing (FWM) in a four-level system. This treatment is based on a transformation to a generalized rotating frame followed by a numerical diagonalization of the resulting time-independent Hamiltonian. For strong fields, all resonant frequencies are shifted from the atomic (molecular) frequencies. Lines are split up in a way similar to the ac Stark effect for a two-level system, and for even stronger fields, the effect of stirring is observed for the first time in four-wave mixing. In addition, a new extra resonance appears which is induced by the saturating field. This field-induced resonance is the new member in the family of extra resonances (pressure induced, or stochastic fluctuation induced), and it should be observed whenever saturating fields interact with a multilevel atomic or molecular system.
A nonperturbative density-matrix formalism is used for the analysis of strong-field four-wave mixing, allowing the treatment of relaxation processes, including lifetime as well as dephasing. The combined role of proper dephasing and level saturation in the generation of extra resonances is studied. It is shown that, whereas in the absence of proper dephasing there is no pressure-induced resonance, there is always a nonzero contribution to the field-induced resonance. For increasing field intensity, a characteristic double saturation is predicted as a function of the field strength. The dependence on the various parameters is investigated, and the conditions under which these effects should be observable are identified.
A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\ensuremath{\chi}}^{(n)}$ which arise due to inelastic collisions. Various applications of these ${\ensuremath{\chi}}^{(n)}$'s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\ensuremath{\chi}}^{(n)}$'s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\ensuremath{\chi}}^{(n)}$'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\ensuremath{\chi}}^{(n)}$'s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\ensuremath{\chi}}^{(1)}$ agrees with the work of Cohen-Tannoudji and co-workers.
A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.
Extra resonances between initially unpopulated excited states (IUES) or equally populated ground states (EPGS) have been studied extensively in the context of frequency-domain nonlinear spectroscopies such as Nearly-Degenerate Four-Wave Mixing (NDFWM)1. In the absence of strong laser fields2 or effects such as pure dephasing3, stochastic fluctuations in laser fields4, and laser transients5, the NDFWM extra resonances are either very weak (in the case of EPGS) or nonexistent (in the case of IUES). It is perhaps under-appreciated that "Fourier-transform" time- and frequency-domain FWM techniques do not necessarily provide the same information. For instance, under certain circumstances6, the NDFWM and transient grating (TG) techniques form such a pair, which would imply that TG extra resonances might disappear under the same experimental conditions in which they disappear in NDFWM. This assumption is belied by the TG data shown in Fig. 1. These data were taken by exciting and probing the D1 line of Na atoms in a cell in which there was no buffer gas and the Na pressure was low enough to preclude any significant number Na-Na collisions on the experimental time scale. Even in the absence of pressure-induced dephasing, these data exhibit strong extra resonances at both the 1.77-GHz ground-state hyperfine-splitting frequency and the 189-MHz excited-state hyperfine-splitting frequency. These data have prompted us to undertake a theoretical study of the nature of extra resonances in time-domain nonlinear spectroscopies, the preliminary results of which are discussed here. We show that time-domain extra resonances increase in strength with decreasing dephasing, and that IUES resonances exist even in the complete absence of pure dephasing. Finally, we show that IUES and EPGS extra resonances have a common origin in the time domain.
A unified approach is presented for the treatment of stochastic fluctuations in four-wave-mixing processes, including uncorrelated and fully or partially cross-correlated input laser fields. The theory allows for phase as well as amplitude fluctuations and covers Markovian and non-Markovian (non-Lorentzian) input line shapes. The analogy between laser field fluctuations and molecular dephasing processes is discussed, and several types of fluctuation-induced resonances are considered. Uncorrelated input fields give rise to resonances similar to those described previously by Agarwal, while correlated input fields cause extra resonances of the type presented by us in earlier publications. Partial cross-correlations are treated here for the first time, with predictions for new effects: Oscillations in the intensity of the generated four-wave-mixing signal as a function of the degree of cross-correlation. The conditions for the experimental observation of these effects are discussed.