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Proceedings of the Physical Society

Radiation damping of degenerate states II. Two-photon transitions in two-molecule systems

Michael R. Philpott

Published Jan 1, 1967• 6 citations
Publisher

Abstract

The two-photon transitions of a system of two dissimilar molecules are studied in the dipole approximation. Special attention is paid to transitions in which both molecules participate simultaneously. By use of a reduced Hamiltonian the influence of long-range intermolecular forces is also studied. The three main transitions considered occur when: (i) no molecular level is in resonance with a primary photon, (ii) one primary photon excites a molecular level, and (iii) two primary photons excite a molecular level. It is shown that in no case can the mixing of excited states by intermolecular forces be represented as a change to an effective transition dipole moment for one of the molecules. The one-photon decay of a state in which both molecules are excited is also treated.

DOI
10.1088/0370-1328/90/1/317

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Journal of the Optical Society of America B•May 1, 1991•15 citations
The two-photon transitions of a system of two dissimilar molecules are studied in the dipole approximation. Special attention is paid to transitions in which both molecules participate simultaneously. By use of a reduced Hamiltonian the influence of long-range intermolecular forces is also studied. The three main transitions considered occur when: (i) no molecular level is in resonance with a primary photon, (ii) one primary photon excites a molecular level, and (iii) two primary photons excite a molecular level. It is shown that in no case can the mixing of excited states by intermolecular forces be represented as a change to an effective transition dipole moment for one of the molecules. The one-photon decay of a state in which both molecules are excited is also treated.
The calculation of the retarded fourth-order resonance interaction between identical molecules with one shared quantum of excitation energy is described. The interaction is calculated in the electric dipole approximation for molecular transitions with the help of time-ordered graphs and a reduced Hamiltonian which treats molecule-photon and molecule-molecule interactions on the same footing. Interactions involving spins, valence forces and permanent electric or magnetic multipoles are not considered. In the near and wave zones the leading terms of the retarded interaction are proportional to R -6 and R -2 respectively, where R is the intermolecular separation.
Rabi oscillations reflect the existence of a coherent superposition of two molecular eigenstates during the interaction of molecules with a strong, resonant electromagnetic field. This phenomenon is not restricted to two-level systems. We demonstrate its occurrence in a three-level system where the radiation induces a two-photon absorption. The intermediate level is (slightly) detuned from resonance. For a theoretical description the dressed-state picture is adopted. The two-photon excitation is performed either by using one laser, yielding a one-color two-photon process, or by using two counterpropagating, spatially overlapping laser beams, yielding a two-color two-photon process. Rabi oscillations are produced by varying the fluence, i.e., by changing either the laser power or the molecule-laser interaction time. For the one-color case the two-photon transition dipole moment for an SF6{\mathrm{SF}}_{6}SF6​ transition has been determined from the relation between laser fluence and number of Rabi oscillations. For the two-color cases the detuning with the intermediate level is varied by tuning the respective laser frequencies while keeping the sum frequency constant. A dramatic change in the degree of excitation as a function of the intermediate detuning has been observed. \textcopyright{} 1996 The American Physical Society.
One of the purposes of this paper is to develop an analytical many-level generalized rotating wave approximation (GRWA), including the effects of permanent dipoles, for the excitation of many-level molecules through the simultaneous absorption of two photons. Included are expressions for the two-photon laser–molecule coupling C, and its two components Cd and Cv corresponding to the direct permanent dipole and the virtual state excitation mechanisms, respectively, and related observables such as the time-dependent populations of the initial and final states of the excitation process and resonance profiles. This GRWA treatment also includes an energy shift parameter e, which causes shifts in the position of the resonance energy as the laser intensity increases. The effects of permanent dipoles are very different in Cv and e versus Cd. These effects have been discussed previously for Cd using analytic two-level RWA approaches. The analytical results for Cv and e obtained here are new as is their use in discussing the influence of permanent dipoles in the parts of the two-photon excitation process involving virtual states. In the absence of permanent dipoles Cd is zero whereas Cv and e are not; they equal the corresponding perturbation theory results. The GRWA and perturbative results are related by Bessel function damping functions which, for dipolar molecules, damp out the divergence of the perturbative results as the laser intensity increases. Illustrative examples are given for a two-photon excitation involving a model chromophore which has a significant virtual state contribution through both the laser–molecule coupling and the energy shift parameter.
The spectral and dynamic effects associated with a non-zero difference, d, between the permanent dipole moments of two molecular states involved in a transition, are investigated for a variety of electromagnetic field-molecule interactions. Both analytical and exact numerical techniques are used to solve the time-dependent Schrodinger equation and to obtain the temporal populations of the molecular states and the associated absorption spectra, for molecular interactions with a Gaussian pulsed, or one or two continuous wave (CW), laser(s).;Analytical rotating wave approximation results are derived for a single CW laser interacting with a two-level d {dollar}\\ne{dollar} 0 molecule with an excited state lifetime, and for a two-level d {dollar}\\ne{dollar} 0 molecule interacting with two CW lasers. A symmetry-adapted Riemann product integral method for exact calculations of pulsed laser-molecule interactions is developed and applied to specific problems. The Floquet technique for the treatment of periodic Hamiltonians is extended to accommodate the interaction of a molecule with two CW lasers.;The one-field RWA results are used to predict and interpret the effects of permanent dipoles on the temporal behaviour of the molecular states for a decaying two-level system interacting with a CW laser, and for a non-decaying two-level system interacting with a pulsed laser. It is shown that the excited state lifetime and the pulse duration can act as internal and external probes, respectively, of the temporal effects of permanent dipoles, relative to the often-studied d = 0 problem. The two-field analytical RWA solutions are used to help discuss and predict the effects of permanent dipole moments in two-laser multi-photon transitions. Several examples of multi-photon, two-colour, two-level resonance profiles, calculated in the RWA and by using exact Floquet techniques, are used to demonstrate how particular multi-photon transitions can be optimized, and to help quantify the conditions for the validity of the two-colour RWA. The effects of neighbouring energy levels on the transition of interest, for both probe and pump-probe laser-molecule interactions, are investigated using a series of three- to seven-level model molecular systems.
It has been observed experimentally [H.R. Xia, C.Y. Ye, and S.Y. Zhu, Phys. Rev. Lett. 77, 1032 (1996)] that quantum interference between two molecular transitions can lead to a suppression or enhancement of spontaneous emission. This is manifest in the fluorescent intensity as a function of the detuning of the driving field from the two-photon resonance condition. Here we present a theory that explains the observed variation of the number of peaks with the mutual polarization of the molecular transition dipole moments. Using master equation techniques we calculate analytically as well as numerically the steady-state fluorescence, and find that the number of peaks depends on the excitation process. If the molecule is driven to the upper levels by a two-photon process, the fluorescent intensity consists of two peaks regardless of the mutual polarization of the transition dipole moments. If the excitation process is composed of both a two-step, one-photon process and a one-step, two-photon process, then there are two peaks on transitions with parallel dipole moments and three peaks on transitions with antiparallel dipole moments. This latter case is in excellent agreement with the experiment.