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Journal of Physics B Atomic and Molecular Physics

Coherence effects in ordinary Raman scattering in the presence of two-photon resonance

A. D. Wilson‐Gordon, Harry Friedmann

Published Apr 14, 1979• 4 citations
Publisher

Abstract

In the presence of coherent saturated near-resonant two-photon excitation, the ordinary Raman scattering line is replaced by a dynamic Stark shifted or Stark split doublet, and Rabi nutations appear in the time-resolved ordinary Raman spectrum. The relative intensity of the peaks of the doublet depends on the shape and on the average frequency of the exciting pulse. Striking new features arise from the interfering contributions of the upper and lower states of the two-photon transition to the Raman scattering process. The theory is applied to Raman scattering in atomic Na in the presence of the 3 2 S 1 2/(F=1, M F =1) to 4 2 D 3 2,5/2/(F=3,M F =3) two-photon resonances.

DOI
10.1088/0022-3700/12/7/020

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In the presence of coherent saturated near-resonant two-photon excitation, the ordinary Raman scattering line is replaced by a dynamic Stark shifted or Stark split doublet, and Rabi nutations appear in the time-resolved ordinary Raman spectrum. The relative intensity of the peaks of the doublet depends on the shape and on the average frequency of the exciting pulse. Striking new features arise from the interfering contributions of the upper and lower states of the two-photon transition to the Raman scattering process. The theory is applied to Raman scattering in atomic Na in the presence of the 3 2 S 1 2/(F=1, M F =1) to 4 2 D 3 2,5/2/(F=3,M F =3) two-photon resonances.
We have studied the stimulated hyper-Raman scattering process in the presence of two strong electromagnetic fields whose frequency modes are initially populated and are near resonance with the two atomic transition frequencies. Using the Green's-function formalism, the excitation spectra are calculated in the limit of high photon densities of both laser fields. Expressions are derived for the relative intensities describing the following processes when (i) both pump fields are at resonance, (ii) the detunings take opposite values corresponding to the cooperative two-photon resonance, and (iii) both laser fields are off resonance. In processes (i) and (ii), the spectra consist of a single peak describing the stimulated hyper-Raman scattering and two pairs of sidebands induced by the laser fields. A new peak appears in process (iii) in addition to those appearing in processes (i) and (ii). The induced spectra are symmetric at resonance but asymmetric in the off-resonance processes, and the extent of the asymmetry depends strongly on the numerical values of the detunings. It is shown that when the two laser fields are treated classically, the derived results are identical to those obtained when the fields are quantized only when photon-photon correlations arising from each laser field are discarded. Numerical calculations for selected values of Rabi frequencies and detunings are graphically presented and discussed.
Abstract Double resonance, resonance Raman and resonance fluorescence are discussed in terms of a superposition of adiabatic states. The adiabatic states are the instantaneous eigenstates of a two‐level Hamiltonian which includes the interaction with a near‐resonant coherent pulse and which has been transformed by a unitary transformation so that its only time‐dependence is that of the field envelope of the pulse. Resonance Raman results from spontaneous transitions to a third, lower state from the adiabatic following state which reduces to the initial (lower) state of the two‐level system in the absence of radiation. Resonance fluorescence is produced by spontaneous transition to the lower third state from the upper adiabatic state which reduces to the upper state of the two‐level system in the absence of radiation. Differential equations allowing the calculation of the probability amplitudes of the adiabatic states are given and a method of solving these equations by successive approximation is proposed. Saturation effects such as dynamic Stark shift and splitting (Autler—Townes effect) and optical nutation are interpreted in terms of perturbation of adiabatic energy levels and interference effects between adiabatic states. The possibility of observing optical nutation in a two‐level system by using time‐resolved double resonance experiments is suggested. Decay of the states of the two‐level system is taken into account leading to extended adiabaticity conditions which show that adiabatic following becomes possible even at resonance. For weak fields it is shown that adiabatic following at resonance can produce light scattering narrower than the linewidth as predicted by Heitler. The adiabatic states theory has also been extended to near‐resonance multiphoton interaction for systems that can be described by an effective two‐level Hamiltonian.
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.
High order Raman generation has received considerable attention as a possible method for generating ultrashort pulses. A large number of Raman orders can be generated when the Raman-active medium is pumped by two laser pulses that have a frequency separation equal to the Raman transition frequency. High order Raman generation has been studied in the different temporal regimes, namely: adiabatic, where the pump pulses are much longer than the coherence time of the transition; transient, where the pulse duration is comparable to the coherence time; and impulsive, where the bandwidth of the ultrashort pulse is wider than the transition frequency. To date, almost all of the work has been concerned with generating as broad a spectrum as possible, but we are interested in studying the spectra of the individual orders when pumped in the transient regime. We concentrate on looking at extra peaks that are generated when the Raman medium is pumped with linearly chirped pulses. The extra peaks are generated on the low frequency side of the Raman orders. We discuss how linear Raman scattering from two-photon dressed states can lead to the generation of these extra peaks.
The authors study the phenomenon of coherent Raman beats in 15N^{15}\mathrm{N}15NH3{\mathrm{H}}_{3}H3​ using a CO2{\mathrm{O}}_{2}O2​ laser and Stark switching. In particular, a three-level system in which the two upper levels always remain split by a few MHz is prepared by switching the transitions into resonance with a short Stark pulse whose bandwidth is large enough to coherently excite both transitions. When one of the coherently excited levels remains in resonance with the laser after the Stark pulse, this system exhibits a qualitatively new effect which has not been seen heretofore. For Stark shifts on the order of the upper-state splitting or smaller, the Raman beat is amplitude modulated at a frequency related to the optical nutation frequency. This is interpreted as being due to an interaction between the two-photon coherent Raman beat process and a single-photon optical nutation process which occurs simultaneously. Numerical calculations as well as a simple analytic model are presented to support this interpretation. By reducing the laser power, one can make the modulation of the Raman beat disappear. From the Raman signal in this regime the permanent electric dipole moment of 15N^{15}\mathrm{N}15NH3{\mathrm{H}}_{3}H3​ in an excited vibrational state is determined and the Raman beat decay rate measured. By comparing the latter result with a delayed optical nutation measurement we show that phase-changing collisions are negligible for the transition studied.
The theory of resonance Raman effect, resonantly enhanced two-photon absorption, and resonance fluorescence---valid also for strong fields---is treated in a unified and simplified way. Dressed states, Bloch equations, and perturbation theory are used to calculate line positions, and integrated intensities both for steady state and transient excitation. The case of adiabatic following is solved explicitly; it occurs when an off-resonance incident pulse is turned on and off slowly. It leads to the identification of Raman scattering, two-photon absorption, and Rayleigh scattering with an adiabatic process while fluorescence and consecutive two-photon absorption with nonadiabaticity. Time-dependent spectra are defined in a rigorous way. Our formulas agree in the various limits with those that appear in the literature.