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Physical Review A

Theory of coherent two-photon resonance

Masamoto Takatsuji

Published Feb 1, 1975• 111 citations
Publisher

Abstract

A theory of coherent two-photon resonance is developed by applying a canonical transformation on the Heisenberg equation of atomic motion. The theory leads both to the optical Bloch equation and the second-order induced polarization which drives the classical Maxwell's equation. The expressions for the Rabi frequency and the induced polarization are given explicitly for two-photon absorption and Raman effect. The results are applied to coherent steady-state propagation, and two kinds of pulses are derived for different-frequency two-photon absorption, including the atomic frequency shift.

DOI
10.1103/physreva.11.619

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Physical Review A•Jul 1, 1975•56 citations
A theory of coherent two-photon resonance is developed by applying a canonical transformation on the Heisenberg equation of atomic motion. The theory leads both to the optical Bloch equation and the second-order induced polarization which drives the classical Maxwell's equation. The expressions for the Rabi frequency and the induced polarization are given explicitly for two-photon absorption and Raman effect. The results are applied to coherent steady-state propagation, and two kinds of pulses are derived for different-frequency two-photon absorption, including the atomic frequency shift.
The authors develop a theoretical analysis which describes coherent nonlinear resonant effects where two optical pulses of different frequencies undergo two-photon self-induced transparency (two-photon SIT) in an inhomogeneously broadened three-level system. The field (Maxwell) and atomic (Schrodinger; density matrix representation) equations are coupled in a self-consistent manner by virtue of the resultant macroscopic polarizations at the two frequencies. This treatment is accomplished under circumstances where atomic coherence plays a predominant role. The resulting equations yield a new two-photon area theorem which gives first the intensity dependent, two-photon absorption law analogous to Beer's law for the one-photon transition. Secondly, a 2 pi Lorentzian pulse solution is derived which propagates without loss at the modified pulse velocity.
A theory of resonance two-photon absorption, which describes this effect as a combination of two-stage and two-photon processes, is developed in a unified mathematical formalism. Expressions for the transition probabilities and kinetic equations are derived, the absorption coefficient is determined for different and identical frequencies, and the shift of the maximum of the resonance two-photon absorption line is obtained. Investigations are made of the intensity dependences of these expressions and of the kinetics of resonance two-photon transitions. The results are compared with available experimental data.
The propagation of two pulses in a three-level absorber is treated by using Maxwell's equations and an effective "two-photon Bloch equations" which are analogous to the one-photon Bloch equations, except for two-photon coupling and frequency shifts. I derive relaxation-dependent shifts in the equilibrium $2n\ensuremath{\pi}$ Lorentzian pulses. Part of these shifts depends on weighted time integrals of harmonics of the area and hence remove the propagation degeneracy of pulses having equal areas but different shapes. Absorption of small-area pulses is studied as a function of absorption length, input energies, and relaxation, and it is found that a local enhancement in the transmission occurs as a function of the absorption length.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
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.
In this paper we develop a theoretical analysis which describes a coherent transient effect of two entering pulses---one laser and one Stokes---in a Raman-active medium. The field (Maxwell) and atomic (Schr\"odinger, density-matrix representation) equations are coupled in a self-consistent manner in time and space by virtue of the resultant second-order nonlinear macroscopic polarizations at the two frequencies. This treatment is accomplished under the circumstances where atomic coherence plays a predominant role. The equations derived here may be referred to as "Raman Bloch-Maxwell equations." The resultant equations yield a new aspect of transient stimulated Raman scattering and thus coherent Raman propagation. The transient-pulse behavior may be understood as self-induced modulation of coherent amplification and absorption. The self-induced modulation for the step-function input pulse is described by a combination of Jacobian elliptic functions depending on the input intensity. Computer calculations of transient-pulse evolution reveal interesting pulse breakup phenomena with peak amplification, energy transmission, pulse advance or delay, and pulse-narrowing effects.