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Physical Review A

Four-wave mixing with time-delayed correlated fields of arbitrary bandwidths and pump intensities

Gautam Vemuri, G. S. Agarwal, Rajarshi Roy, Michael H. Anderson, J. Cooper, Stuart James Smith

Published Nov 1, 1991• 15 citations
Publisher

Abstract

We consider the generation of four-wave-mixing signals by an ensemble of two-level atoms interacting with broadband, correlated, pump and probe fields in the forward geometry. We show how the calculation of the four-wave-mixing signal reduces to the solution of six coupled Langevin equations that are numerically integrated using Monte Carlo methods. We account for the pump-induced saturation effects although the probe is weak. The finite correlation time of the pump and probe fields is also included. For weak and \ensuremath{\delta}-correlated pump fields, we reproduce the results of Morita and Yajima [Phys. Rev. A 30, 2525 (1984)]. We find that even for short correlation times of the fields, the four-wave-mixing signal, as a function of the delay between the pump and the probe fields, starts reviving with increase in the pump intensity. This revival as a function of the pump-probe delay time becomes more pronounced with further increase in the pump intensity.

DOI
10.1103/physreva.44.6009

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Journal of the Optical Society of America B•Jun 1, 1999•33 citations
We consider the generation of four-wave-mixing signals by an ensemble of two-level atoms interacting with broadband, correlated, pump and probe fields in the forward geometry. We show how the calculation of the four-wave-mixing signal reduces to the solution of six coupled Langevin equations that are numerically integrated using Monte Carlo methods. We account for the pump-induced saturation effects although the probe is weak. The finite correlation time of the pump and probe fields is also included. For weak and \ensuremath{\delta}-correlated pump fields, we reproduce the results of Morita and Yajima [Phys. Rev. A 30, 2525 (1984)]. We find that even for short correlation times of the fields, the four-wave-mixing signal, as a function of the delay between the pump and the probe fields, starts reviving with increase in the pump intensity. This revival as a function of the pump-probe delay time becomes more pronounced with further increase in the pump intensity.
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 consider the dependence of the four-wave-mixing signal from an ensemble of homogeneously broadened, two-level atoms on the statistics of the pump field. Two models for the pump field are considered: the chaotic field and the phase-diffusing field. A Monte Carlo simulation procedure is used to integrate the equations describing the four-wave-mixing process numerically. This technique allows consideration of arbitrary bandwidths for the pump and incorporates pump-induced saturation effects. Results are presented for the strength of the four-wave-mixing signal as a function of the pump intensity for both resonant and off-resonant cases. It is shown that the four-wave-mixing signal is sensitive to the statistics of the pump, particularly for strong fields.
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.
We examine the effect of propagation on the resonance Rabi sideband of the four-wave mixing (FWM) spectrum, obtained when short temporally displaced pump and probe pulses interact with an optically thick medium of two-level atoms. We find that the dependence of the time-integrated FWM signal on the pump-probe delay is considerably altered by propagation. In particular, the logarithm of the FWM signal, for the case where the probe precedes the pump, deviates from linearity and may even increase over a range of values. An explanation is given in terms of the overlap of the pump envelope with the coherent response of the atomic system to the probe, both of which are modified on propagation.
An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of two-level atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams–Lind approach [Opt. Lett.2, 94 and 3, 205 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams–Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams.