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Physical Review A

Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields

R.T. Bratfalean, P. Ewart

Published Sep 1, 1997• 21 citations
Publisher

Abstract

The spectral line shape of the four-wave mixing (FWM) signal, induced by broad bandwidth laser fields, is derived in the case of a nonresonant interaction. The laser fields are assumed to have a Lorentzian spectrum arising from Markovian stochastic fluctuations described by the phase-diffusion, chaotic field, and Gaussian-amplitude models. The usual ``phase conjugating'' geometry for FWM is considered where the nonresonant interaction is determined only by the statistics of, and the correlations between, the pump and probe fields. For each one of the four possible correlation states between the three input fields and for each of the three fluctuating field models, the line shape of the FWM signal is calculated and shown to be particularly sensitive to the field statistics.

DOI
10.1103/physreva.56.2267

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The spectral line shape of the four-wave mixing (FWM) signal, induced by broad bandwidth laser fields, is derived in the case of a nonresonant interaction. The laser fields are assumed to have a Lorentzian spectrum arising from Markovian stochastic fluctuations described by the phase-diffusion, chaotic field, and Gaussian-amplitude models. The usual ``phase conjugating'' geometry for FWM is considered where the nonresonant interaction is determined only by the statistics of, and the correlations between, the pump and probe fields. For each one of the four possible correlation states between the three input fields and for each of the three fluctuating field models, the line shape of the FWM signal is calculated and shown to be particularly sensitive to the field statistics.
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.
The effects of finite laser bandwidth on resonant degenerate four-wave mixing (DFWM) are calculated with use of a model in which the intense, counterpropagating pump beams are characterized by a chaotic field, the probe beam is weak and monochromatic, and the medium consists of a gas of two-level atoms. We present a steady-state solution in the limit where the pump-laser bandwidth exceeds all other atomic relaxation rates. Although the mean intensity due to the fluctuating fields is spatially independent (no steady-state standing-wave pattern is established), the analytic results indicate that, for intensities above the saturation intensity Isat{I}_{\mathrm{sat}}Isat​, spatially periodic saturation effects are important. Increasing bandwidth is shown to lead to an increase in the effective saturation intensity resulting in lower phase-conjugate reflectivity for IIsat{I}_{\mathrm{sat}}Isat​ than for coherent pump fields, in contrast to the results for narrow-bandwidth chaotic fields. The resonant DFWM line shape is also calculated and compared to the coherent result. We comment on the application of the model to other four-wave-mixing processes employing broad-bandwidth lasers.
Four-wave mixing in homogeneously broadened, two-level atoms, driven by time-delayed, correlated phase-diffusing optical fields is investigated. The spectrum of the driving field has an arbitrary bandwidth, i.e., it can have a Lorentzian or a Gaussian profile or an intermediate profile (Voigt). A time delay between the strong pump and weak probe field provides a partial correlation between the fields, and this time delay makes the composite field seen by the atoms non-Markovian. It is shown that the calculation of the four-wave-mixing signal reduces to the solution of six coupled Langevin equations with multiplicative noise; Monte Carlo techniques are utilized to study the response of the atoms to non-Markovian fields. A technique for numerically simulating phase-diffusing fields with colored frequency fluctuations is shown. The four-wave-mixing profiles are found to be significantly different from those for which the driving field is chaotic. The profiles show oscillatory behavior as evidence of the Rabi frequency, a feature not seen in previous works. For some parameter values, a revival of the signal is seen for an increasing time delay, analogous to the revivals seen in chaotic field signals [Phys. Rev. A 44, 6009 (1991)].
We explore theoretically the signal generated by a fluctuating radiation field via a four-wave mixing process in an atomic vapor undergoing Markovian dephasing collisions. We consider specifically a broadband laser field well detuned from the atomic resonance and calculate the signal generated using counterpropagating pump fields as a function of the time delay of the incident probe field. We assume that the pulse duration is long compared to the correlation time of the incident fields. By considering the various time-ordered pathways and treating the incident-field amplitudes as random Gaussian variables, we find that the signal generated is dominated by two specific pathways that depend only on the incident-field bandwidth and the decay rate of the ground to excited state coherence. An interesting aspect of the far-impact-detuning case is that the signal is dominated by resonances involving spatially separated atoms. Our formalism provides a basis for describing the effects of mode correlations on four-wave-mixing signals generated by stochastic fields.
We develop a theory of near-resonant four-wave mixing induced by broad-bandwidth chaotic fields in a medium composed of two-level atoms. By solving the equations of motion for the elements of the atomic density matrix using an appropriate decorrelation approximation, we derive an analytic expression for the frequency spectrum of the signal wave for the case of the bandwidth of the fluctuations of the pump field exceeding the other relaxation rates in the problem. The probe wave is considered weak and of arbitrary bandwidth. The theory is valid for pump intensities up to and exceeding the bandwidth-dependent saturation value. Finally, we show how the frequency spectrum is modified if account is taken of atomic motion. The theoretical results are of direct relevance to practical experiments involving broad-bandwidth pulsed lasers and employing an atomic vapor as the nonlinear medium.
Lorentzian models for laser line shapes lead to qualitatively incorrect results for off-resonance excitation of atoms. This paper is the first attempt to present a theory of the nonperturbative interaction of an atom with a chaotic field (representing multimode laser radiation having strong amplitude fluctuations) with a line shape falling off faster than a Lorentzian. To this end we suggest a stochastic Markovian model for a non-Lorentzian chaotic field. To solve the multiplicative stochastic differential equations describing the atom-field interaction we propose a "marginal characteristic function approach." This not only reproduces our earlier results in a more elegant way and establishes the relationship between approaches used by other authors in a different context, but also provides the simplest possible basis for our present discussion of ac Stark splitting in double optical resonance. While for a chaotic field with a Lorentzian line shape the asymmetry of the two-peaked off-resonance spectrum is reversed for all values of the detuning compared with the monochromatic case, our present model predicts a reversed peak asymmetry only for detunings smaller than a few laser bandwidths in agreement with experiment. The on-resonance spectrum is dominated by the amplitude fluctuations and is only weakly affected by changes of the spectral line shape of the laser.