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Physical Review A

Theoretical investigation of phenomena in the closed Raman-driven four-level symmetrical model

Hong Yuan Ling

Published Apr 1, 1994• 13 citations
Publisher

Abstract

We construct a solution to the linear complex absorption coefficient for the closed Raman-driven four-level symmetrical model. We derive, from this solution, asymptotic expressions for the spectra around and away from the two-photon resonance in the absence as well as in the presence of the Doppler broadening. They are used for the derivations of analytical conditions and the discussion of the physical origins of various phenomena that do not appear in the traditional two-level system.

DOI
10.1103/physreva.49.2827

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We construct a solution to the linear complex absorption coefficient for the closed Raman-driven four-level symmetrical model. We derive, from this solution, asymptotic expressions for the spectra around and away from the two-photon resonance in the absence as well as in the presence of the Doppler broadening. They are used for the derivations of analytical conditions and the discussion of the physical origins of various phenomena that do not appear in the traditional two-level system.
Our investigation is focused on obtaining analytical results describing the resonance fluorescence and weak probe absorption spectra of a driver Lambda-system under the Raman resonance condition. For the saturation field limit and within the rotating wave approximation (RWA) the resonance fluorescence and weak probe absorption spectra are calculated for the case of unequal driving fields strengths and non-zero one-photon detunings, generalizing the previously known results. A novel contribution derived is the nonpositive nonlorentzian part of the spectrum, originating from the specific quantum properties of the atomic correlation functions. This contribution is asymptotically small under given conditions and, therefore, determines behavior of the resonance fluorescence spectrum wings and the absorption spectrum in the vicinities of the driving field frequencies. Beyond the RWA, the spectral structures due to the off-resonant four-photon atomic excitations are calculated. They consist of a coherent and two broadened spectral lines centered at the four-photon frequencies 2(omega) L-(omega) 1L and 2(omega) +1)/ - (omega) L4/ respectively, where (omega l and (omega) (superscript 1$L are the driving laser frequencies.
This paper deals with the absorption spectra of a weak probe in a four-level Doppler-broadened system driven by three coherent fields. The main aim is to extend earlier studies of the spontaneous emission spectrum and to present a comprehensive survey of the spectral features of this system. In addition to a derivation of exact formulae for the spectra, we give an explanation with the help of an appropriate set of dressed atomic states. We also get a deeper insight into the physical origin of gain in view of the existence of a population inversion between the levels of the lasing transition. Finally, we explore the effect of Doppler broadening on the absorption profile of the weak probe.
Two-photon absorption is theoretically analyzed within the semiclassical formalism of radiation-matter interaction. We consider an ensemble of inhomogeneously broadened three-level atoms subjected to the action of two counterpropagating fields of the same frequency. By concentrating in the limit of large detuning in one-photon transitions, we solve perturbatively the Bloch equations in a non-usual way. In this way we derive an analytical expression for the width of the two-photon resonance that makes evident sub-Doppler two--photon spectroscopy. We also derive an analytical expression for the Stark shift of the two-photon resonance.
This paper discusses the origin of splitting of two-photon spectra of three-level cascade configurations having a near resonant intermediate level. A clear distinction between splitting mechanisms in homogeneously and inhomogeneously broadened systems is demonstrated. When one of the laser fields is a weak probe the velocity averaged two-photon response is primarily caused by two resonantly excited velocity groups; the absence of these implies a split spectrum as suggested by Chebotaev and collaborators. A simple derivation of the probe response is presented. When both fields have non-negligible intensities resonant excitation is inhibited and instead all level populations pulsate. Still a split spectrum can appear. A simple graph showing the regions where this occurs is given and analyzed.
The absorption spectra of a pair of two-level atoms driven on resonance by a coherent laser field are considered analytically including the dipole-dipole interaction between the atoms. Analytical formulae for the absorption spectra are derived for four regions of the dipole-dipole interaction Omega 12 : Omega >> gamma >> mod Omega 12 mod , Omega >> mod Omega 12 mod >> gamma Omega approximately= mod Omega 12 mod >> gamma , mod Omega 12 mod >> gamma >> Omega (2 gamma being Einstein's coefficient A, and Omega the Rabi frequency). The formulae, containing both absorption-like and dispersion-like terms, reveal explicitly the resonant structure of the absorption spectra. At great intensities of the driving field the absorption is negative in a certain range of frequencies. The dipole-dipole interaction between the atoms is shown to modify considerably the form of this absorption spectrum. For sufficiently strong dipole-dipole interaction it exhibits dispersion-like behaviour in the vicinity of 2 Omega with a new region of negative absorption. The spectrum is asymmetric, albeit slightly. For strong dipole-dipole interaction and a weak driving field the spectrum consists of one absorption-like peak, located at omega - omega 0 = Omega 12 .
A class of molecular two-photon processes, occurring under transient or steady-state conditions, is analyzed exactly for certain cases using a density-matrix formalism. Our results are relevant to the recently observed phenomena of transient coherent Raman beats and steady-state two-photon absorption of oppositely directed laser beams. Lowest-order perturbation theories have shown that these processes are insensitive to the Doppler effect, and consequently the linewidth is unaffected by Doppler broadening and by elastic collisions that change molecular velocity. The solutions presented here are of the same form for the Raman and two-photon problems and reveal significant power-dependent frequency shifts and line broadening that correct the ideal Doppler-free solutions. For the transient case, we assume a molecular three-level quantum structure that can be switched in or out of optical resonance with cw laser radiation by means of a pulsed dc Stark field, but the solutions also apply when resonant optical pulses are introduced. During the resonant condition when the Stark pulse is on, the three levels are prepared in coherent superposition, and during the nonresonant condition following the Stark pulse, the laser field(s) stimulates the coherently prepared sample in a transient two-photon process. One of these processes, the Ramanbeat effect, has been observed in forward scattering, but the second one, transient two-photon emission, should be observable in backward scattering. Bloch-like equations are derived for this three-level problem that facilitate an exact pulse solution for state preparation. Following the pulse, transient decay is well-approximated by a perturbation calculation. For the steady-state case, an exact solution is also obtained and is of interest for continuous-wave spectroscopy or for transient experiments requiring an initial preparation of quantum states, prior to Stark switching. These solutions exhibit a power-dependent line broadening and a frequency shift of magnitude $\ensuremath{\sim}\frac{\ensuremath{\Delta}({\ensuremath{\alpha}}^{2}\ensuremath{-}{\ensuremath{\beta}}^{2})}{({\ensuremath{\Delta}}^{2}+\frac{1}{{T}_{2}^{2}})}$, in agreement with our earlier estimate, where $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ are the Rabi flopping frequencies for the two intermediate transitions and $\ensuremath{\Delta}$ is the off-resonance tuning behavior for one of them.