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Journal of Physics B Atomic and Molecular Physics

The effects of laser cross correlation in the three-level ladder system

T. A. B. Kennedy, S. Swain

Published Jun 28, 1984• 23 citations
Publisher

Abstract

The authors have shown that cross correlation between the fluctuating phases of two lasers (as introduced by Wong and Eberly (1977) and Dalton and Knight (1982-3)) produces significant effects on the shape and asymmetry of the optical double-resonance spectra. If suitable sources can be found, these features should be readily observable.
DOI
10.1088/0022-3700/17/12/002

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Cross correlation refers to the situation in which the fluctuations in the fields of two or more lasers are correlated in some way. Recently, Thomas et al. [ Opt. Lett.6, 298 ( 1981); Phys. Rev. Lett.48, 867 ( 1982)] proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and went on to exploit this property in the observation of narrow Ramsay fringes. Dalton and Knight [ J. Phy. B15, 3997 ( 1982)] discussed the theoretical aspects of cross correlation and showed that it produced interesting effects in population trapping for example. We have demonstrated that it has significant effects on the optical double-resonance (ODR) spectra [ J. Phys. B17, L389 ( 1984)]. On the other hand, the effects of non-Lorentzian laser line shapes have been shown by various workers to have significant effects on a wide range of phenomena, such as multiphoton ionization, resonance fluorescence, and ODR. In this paper we present a theory of the interactions of lasers whose fluctuations are governed by the phase-diffusion process with atomic systems that includes the effects of cross correlation and non-Lorentzian line shapes. We derive the basic equations (which in general have to be solved numerically) and obtain analytic results in the case in which the line shape is approximately Lorentzian near line center but falls off faster than a Lorentzian in the wings. We believe that this analytic model shows all the essential features of the interplay between cross correlation and non-Lorentzian line shapes. The theory is applied to the ODR experiment. In particular, we find that the asymmetry of the spectra is strongly affected by these two features, and so is the ODR spectral line shape when the saturating laser intensity is close to the threshold for producing two peaks.
Thomas et al. (1981, 1982) proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and subsequently exploited this property in the observation of very narrow Ramsey fringes. Cross correlation has been discussed theoretically by Dalton and Knight (1982, 1983) and shown to have interesting effects in population trapping. For such effects to be important, the cross correlation coefficient must be as large as possible. The authors discuss the degree of correlation between the two modes of a two-mode laser using the approach of Scully and Lamb (1967, 1968) and show it can be large. They also evaluate the linewidths of the two laser modes. They find that if the laser parameters for the two modes are equal, the two-mode linewidth is one half the value of the linewidth of the corresponding single-mode laser, well above threshold.
A three-level system driven by two coherent fields on two-photon resonance leads to the trapping of population in a superposition of initial and final states which is immune to further photoexcitation, manifesting itself in a narrow coherence minimum in the absorption spectrum. The authors analyse the effects of laser field bandwidths and cross-correlations on these phenomena. They find that the inclusion of laser bandwidths dephases atomic coherences which leads to the destruction of population trapping and of the narrow coherence minimum in the absorption spectrum. When the driving fields are not cross-correlated, the atomic coherences are similarly dephased in both lambda and ladder systems. If, however, the driving fields are critically cross-correlated, two-photon coherences in the lambda system are unaffected by laser fluctuations; trapping and the coherence minimum persist. For the ladder system, on the other hand, the cross-correlation does not restore the atomic coherences involved.
A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio 3x3x3x (23x\frac{2}{3x}3x2​), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.