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The Journal of Chemical Physics

Theory of quantum beats in time-resolved multiphoton ionization of molecules

Yoshihiro Nomura, Yuichi Fujimura, Hirohiko Kono

Published Feb 1, 1988• 11 citations
Publisher

Abstract

An expression for the probability of time-resolved three-photon ionization via coherently excited resonant states of a molecular system is derived in the perturbative density matrix formalism with the aid of the Liouville space Feynman diagram. It is shown how the time evolution of the molecular coherence in the first resonant state can be detected as a function of the delay time between the pumping and probing lasers through the probing two-photon ionization process. The three-photon ionization process consists of the simultaneous and sequential processes which are classified by the Liouville space Feynman diagrams. The possibility of appearance of quantum beats in the pump–probe three-photon ionization is discussed by performing model calculations of the probability of ionization. The role of the direct and redistributed processes is discussed. The redistributed process is induced by intramolecular processes in the resonant state. The theoretical treatment is applied to the (1+2) three-photon ionization of pyrazine observed by Knee et al. It is demonstrated that the fast decay component (110 ps) of the pyrazine spectra originates from the intramolecular dephasing of the vibronic coherence created by the pumping pulse laser, and this component appears only when the redistributed process (the ionization process through triplet levels) is absent.

DOI
10.1063/1.454128

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An expression for the probability of time-resolved three-photon ionization via coherently excited resonant states of a molecular system is derived in the perturbative density matrix formalism with the aid of the Liouville space Feynman diagram. It is shown how the time evolution of the molecular coherence in the first resonant state can be detected as a function of the delay time between the pumping and probing lasers through the probing two-photon ionization process. The three-photon ionization process consists of the simultaneous and sequential processes which are classified by the Liouville space Feynman diagrams. The possibility of appearance of quantum beats in the pump–probe three-photon ionization is discussed by performing model calculations of the probability of ionization. The role of the direct and redistributed processes is discussed. The redistributed process is induced by intramolecular processes in the resonant state. The theoretical treatment is applied to the (1+2) three-photon ionization of pyrazine observed by Knee et al. It is demonstrated that the fast decay component (110 ps) of the pyrazine spectra originates from the intramolecular dephasing of the vibronic coherence created by the pumping pulse laser, and this component appears only when the redistributed process (the ionization process through triplet levels) is absent.
Time - dependent behaviors of molecular coherence in resonant multiphotoni onization spectra of molecules are theoretically investigated . An expression for the transition probability of time-resolved two-photon ionization processes via coherently excited resonant intermediate states is derived in the perturbative density matrix formalism. Model calculations of the probability of ionization are performed to predict appearance of quantum beats in pump-probe two-photon ionization experiments. The quantum beat signals in the time-resolved ionization spectra are compared with those in time-resolved total photon emission spectra.
A joint experimental and theoretical study of four-photon ionization of Li is performed using a pump-probe setup with the laser frequency tuned in between the excitation energies of the 2s2s2s--4p4p4p and 2s2s2s--4f4f4f three-photon transitions. The three-dimensional photoelectron momentum pattern is recorded as a function of the delay time between two otherwise identical laser pulses. The process is modeled by numerically solving the time-dependent Schr\"odinger equation. While complicated interference patterns arise in the electron spectrum when the two pulses partially overlap in time, the expected quantum-beat phenomenon between Li atoms in either the 4p4p4p or the 4f4f4f state are seen when the fast oscillations involving the 2s2s2s initial state are averaged over. The qualitative agreement between the predicted and observed beating patterns is very good.
Mechanisms of two-photon transitions of a molecular ensemble induced by stationary-phase-incoherent light beams are theoretically studied. The time evolution of the vibronic coherence created by the phase-incoherent light beams is analyzed by using the Liouville-space Feynman diagram properly taking into account vibronic dephasings. It is shown that effects of coherence between sequential transitions make a dominant contribution to the time profile of the pump-probe two-photon transition induced by the phase-incoherent light beams. The time profiles are calculated within a simple model to demonstrate differences in the time evolution of the vibronic coherence between the phase-incoherent and pulsed-coherent light spectroscopy. A theoretical consideration is also given to nonresonant two-photon fluorescence of coumarin 440 in solution excited by incoherent light beams, which shows subfemtosecond oscillation with the incident photon-field frequencies.
Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.
A full density-matrix theory for quantum beats in optical transmission-correlation or pump-probe type of experiments is developed. The model used is a three-level system with split excited-state levels. It is demonstrated on the basis of perturbation theory that quantum beats should be observable in such experiments. Theoretical and numerical results are compared with the complex details, including damped sinusoidal decays, observed in recent femtosecond relaxation studies of large organic dye molecules. It is also shown that there is no separately identifiable decay component corresponding to any homogeneous or inhomogeneous optical dephasing time in such experiments. Similar beating should also be observable in ultrafast spectroscopy using incoherent light and in Raman-type energy-level configurations where the ground state contains split levels.
We present a theory of quantum beats in photoionization from a coherent superposition of D32,5.2{D}_{\frac{3}{2},5.2}D23​,5.2​ levels in alkali-metal atoms. The semiclassical density-matrix formalism is used to describe both the excitation of the D32,52{D}_{\frac{3}{2},\frac{5}{2}}D23​,25​​ levels by a two-photon resonant pulse and the subsequent photoionization by a delayed probe pulse. Quantum beats occur in the ionization signal as a function of the delay of the probe pulse. The depth of modulation is determined for different light polarizations.