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Physical Review A

Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density: Analytic solution and a diagrammatic approach

Darin J. Ulness, A. C. Albrecht

Published Feb 1, 1996• 43 citations
Publisher

Abstract

Degenerate four-wave mixing using broadband, non-transform-limited (incoherent) laser light having a Lorentzian spectral density is treated analytically for a Bloch two-level system. We consider the configuration in which an incoherent beam from a Lorentzian source is split into two beams having distinct k vectors, k and ${\mathit{k}}^{\ensuremath{'}}$. The twin beam along ${\mathit{k}}^{\ensuremath{'}}$ has been subjected to a controllable delay, \ensuremath{\tau}, relative to k. These twin beams, focused into the sample of interest, generate new fourth fields, the signal at ${\mathit{k}}_{\mathit{s}}$=2k-${\mathit{k}}^{\ensuremath{'}}$ being examined here as a function of \ensuremath{\tau}. A set of factorized time correlation (FTC) diagrams is introduced to organize the calculation and to yield both analytic expressions as well as physical insight. Not only is the signal not symmetric in \ensuremath{\tau} but the signal can peak at \ensuremath{\tau}\ensuremath{\ne}0 even in the absence of inhomogeneous broadening. Finally, the physical insights and the use of the FTC diagrams are extended to rationalize several other incoherent light spectroscopies. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.1081

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Degenerate four-wave mixing using broadband, non-transform-limited (incoherent) laser light having a Lorentzian spectral density is treated analytically for a Bloch two-level system. We consider the configuration in which an incoherent beam from a Lorentzian source is split into two beams having distinct k vectors, k and k\ensuremath′{\mathit{k}}^{\ensuremath{'}}k\ensuremath′. The twin beam along k\ensuremath′{\mathit{k}}^{\ensuremath{'}}k\ensuremath′ has been subjected to a controllable delay, \ensuremath{\tau}, relative to k. These twin beams, focused into the sample of interest, generate new fourth fields, the signal at ks{\mathit{k}}_{\mathit{s}}ks​=2k-k\ensuremath′{\mathit{k}}^{\ensuremath{'}}k\ensuremath′ being examined here as a function of \ensuremath{\tau}. A set of factorized time correlation (FTC) diagrams is introduced to organize the calculation and to yield both analytic expressions as well as physical insight. Not only is the signal not symmetric in \ensuremath{\tau} but the signal can peak at \ensuremath{\tau}\ensuremath{\ne}0 even in the absence of inhomogeneous broadening. Finally, the physical insights and the use of the FTC diagrams are extended to rationalize several other incoherent light spectroscopies. \textcopyright{} 1996 The American Physical Society.
A new method of picosecond and femtosecond transient spectroscopy yielding information about ultrafast relaxation processes without requiring ultrashort light pulses is presented. It is based on the present theoretical analysis of the resonant degenerate four-wave-mixing process excited by two temporally incoherent light beams with wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}$ which are originated from a single beam at frequency $\ensuremath{\omega}$ but have mutual time delay $\ensuremath{\tau}$. Under the assumption that the incoherent light field has Gaussian random complex amplitude and that the resonant material consists of the usual two-level atoms, the statistically averaged intensity of the output light field with ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3}=2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ at $\ensuremath{\omega}$ is calculated as a function of $\ensuremath{\tau}$. Even with the light having a much longer duration than both T1{T}_{1}T1​ (the longitudinal relaxation time) and T2{T}_{2}T2​ (the transverse relaxation time), the correlation trace, i.e., output intensity versus $\ensuremath{\tau}$, represents a decay profile determined mainly by T2{T}_{2}T2​ for both homogeneously and inhomogeneously broadened transitions, as long as the light correlation time ${\ensuremath{\tau}}_{c}$ is much shorter than the relaxation times. The correlation trace does not always represent a single-exponential decay but is sometimes slightly deformed by the T1{T}_{1}T1​ effect. However, it does not cause a significant error in the determination of T2{T}_{2}T2​. Moreover, as $\frac{{T}_{2}}{{T}_{1}}\ensuremath{\rightarrow}0$, the trace becomes a single-exponential decay curve determined only by T2{T}_{2}T2​. The feature of the results obtained by the present method is similar to that obtained by the conventional coherent transient spectroscopy with short pulses, such as the photon echo. The time resolution in the present method, however, is limited only by ${\ensuremath{\tau}}_{c}$ much shorter than the light duration. By regarding the incoherent light as a series of random ultrashort pulses, the present four-wave-mixing process is also interpreted as the ensemble of numerous transient four-wave-mixing processes caused by various combinations of these pulses.
Signal intensity as a function of mutual delay time of three incoherent pump beams in the degenerate four-wave mixing (DFWM) process has been calculated with the second-order correlation time of the fields being much shorter than both the dephasing time T 2 and the inverse of the inhomogeneous broadening width δω. The results show that when broadening is purely homogeneous and extremely inhomogeneous, the curve of the DFWM signal intensity as a function of the delay time τ shows no peak shift, while when the inhomogeneous broadening is finite, the peak shift is related to the dephasing time T 2 and the inhomogeneous width δω, and can thus be used to estimate T 2.
A multilevel model, including homogeneous and inhomogeneous broadening, is proposed for the study of time-delay four-wave mixing with incoherent light (TDFWM-IL) in absorption bands consisting of a series of vibrational transitions. A theory of TDFWM-IL is developed for this model, and a general formula for the delay-time dependence of the signal intensity Is(τ), considering both the homogeneous and inhomogeneous broadening of the transitions, is deduced. Special analyses are given to the homogeneous and inhomogeneous limits. The differences between Is(τ) derived from multilevel theory and from two-level theory, are pointed out. When the homogeneous broadening of individual transitions is much smaller than the frequency interval of adjacent transitions, Is(τ) in the multilevel theory is modulated periodically and the envelope is close to that in the two-level theory for both the homogeneous and inhomogeneous limits. When the individual transitions are homogeneously broadened enough so that an unresolved continuous band is formed, an effective two-level model can still be used approximately for the studies of TDFWM-IL in this continuous band, with both homogeneous and inhomogeneous broadening. However, the dephasing time [Formula: see text] in the formula of Is(τ) in this effective two-level model is not the real dephasing time of an individual transition of the band, but the effective one which reflects the behavior of the whole band.
The theory of an interferometric six-wave mixing quasiecho signal from a modeled molecular liquid using broadband (``noisy'') light is reported. The line-shaping function is treated by a standard mixture of a dynamic (homogeneous) dephasing process and a static (inhomogeneous) Bohr frequency distribution. Simulations are presented in either limit of the overall process to demonstrate the rephasing aspects of this nonlinear spectroscopy. The general aspects of the two-dimensional signal are inferred from factorized time correlator (FTC) diagrams. These diagrams pictorially represent the correlation functions of the random noise (associated with the broadband light) that arise upon stochastic averaging at the signal level. The entire set of FTC diagrams is seen to be isomorphic with those that describe a nonecho interferometric noisy light spectroscopy at fifth order, and the third-order electronically resonant four-wave mixing. Similarities between these three very different processes are discussed in terms of this one-to-one correspondence.
Time-resolved three-beam degenerate four-wave mixing with incoherent light for observation of population relaxation, originally proposed by Morita . [ J. Opt. Soc. Am. B4, 1269 ( 1987)] is examined both theoretically and experimentally. Theoretical calculation shows that a large part of the decaying signal in the time-resolved profile originates from the ground-state relaxation if the excited-state relaxation is faster than the ground-state relaxation. It is also shown that the use of a nonzero delay time between the k2 and k1 beams, in addition to that between the k3 and k1 beams (ks = k1 − k2 + k3, where k1–3 and ks are the wave vectors for the incident and the scattered beams, respectively), provides information on population relaxation, especially when the population-relaxation time is far longer than the correlation time of the incoherent light. These theoretical considerations are verified experimentally with the use of a triphenylmethane dye in solution.