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The Journal of Chemical Physics

Theory of resonance secondary emission in femtosecond laser excitation: On the connection with wave packet dynamics

Hirohiko Kono, Yuichi Fujimura

Published Nov 15, 1989• 22 citations
Publisher

Abstract

The resonance secondary emission (RSE) in femtosecond laser excitation is discussed in reference to the motion of the created wave packet moving on the excited state potential surface. The density matrix of emitted light for the multi-intermediate-level system is outlined, from which the emission correlation function is derived. The correlation function is put into the theoretical expression of the time-dependent ‘‘physical spectrum’’ for the Fabry–Perot interferometer (which is used in order to consider temporal and energetic resolution inherent in detection). The compact and practical expressions obtained connect the time- and frequency-resolved spectrum with the time evolution of the wave packet. Numerical results for a displaced harmonic oscillator model indicate that the time- and frequency-resolved spectrum can reveal how the wave packet created by a fs laser pulse travels on the excited potential surface if the response time 1/Γd of the photodetector satisfies the relation that Ω<Γd <∼ the Stokes shift (where Ω is the vibrational frequency). It is shown that the excited state wave function can be split into two terms, the one that adiabatically follows the temporal change in incident light (the adiabatic term) and the one that represents the effect of spectral broadening of light (the Fourier broadening term). It is only the Fourier broadening term that survives after the termination of incident light and reflects the motion of the created wave packet on the excited potential surface. In off-resonance excitation, the adiabatic term produces Raman-like emission and the Fourier broadening one produces fluorescence-like emission. In resonance excitation, these two terms are indistinguishable from each other with respect to emission frequency: for the duration of incident light, the adiabatic term offsets the Fourier broadening one, leading to a slow buildup of intensity in the time- and frequency-resolved spectrum (which is slower than the initial rise of the incident pulse profile).

DOI
10.1063/1.457414

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Physica Scripta•Sep 1, 1983•5 citations
The resonance secondary emission (RSE) in femtosecond laser excitation is discussed in reference to the motion of the created wave packet moving on the excited state potential surface. The density matrix of emitted light for the multi-intermediate-level system is outlined, from which the emission correlation function is derived. The correlation function is put into the theoretical expression of the time-dependent ‘‘physical spectrum’’ for the Fabry–Perot interferometer (which is used in order to consider temporal and energetic resolution inherent in detection). The compact and practical expressions obtained connect the time- and frequency-resolved spectrum with the time evolution of the wave packet. Numerical results for a displaced harmonic oscillator model indicate that the time- and frequency-resolved spectrum can reveal how the wave packet created by a fs laser pulse travels on the excited potential surface if the response time 1/Γd of the photodetector satisfies the relation that Ω<Γd <∼ the Stokes shift (where Ω is the vibrational frequency). It is shown that the excited state wave function can be split into two terms, the one that adiabatically follows the temporal change in incident light (the adiabatic term) and the one that represents the effect of spectral broadening of light (the Fourier broadening term). It is only the Fourier broadening term that survives after the termination of incident light and reflects the motion of the created wave packet on the excited potential surface. In off-resonance excitation, the adiabatic term produces Raman-like emission and the Fourier broadening one produces fluorescence-like emission. In resonance excitation, these two terms are indistinguishable from each other with respect to emission frequency: for the duration of incident light, the adiabatic term offsets the Fourier broadening one, leading to a slow buildup of intensity in the time- and frequency-resolved spectrum (which is slower than the initial rise of the incident pulse profile).
The perturbative density matrix formalism is used to study the time-dependent resonance Raman scattering (RRS) and resonance hot luminescence (RHL) components of the light-induced spontaneous emission process that has been called resonance secondary radiation (RSR). A three-level system coupled in a stochastic way to a bath is used to study the RSR emission driven by a coherent laser pulse with a variable Gaussian width in time. The spectral filter in a light detection system is explicitly included in the detection step of the problem. Observable time and frequency resolved RSR spectra are then calculated numerically. The spectral and temporal properties of the time-dependent RSR spectrum are very sensitive to the width of the driving pulse when the pulse width is of the same order of magnitude as the important system dephasing constants. In this domain the RRS contribution to the RSR spectrum weakens relative to the RHL as the light pulse is shortened. In the limit that the laser pulse becomes ‘‘delta function-like’’ the RSR spectrum consists essentially of RHL. When appropriate, the numerical results are compared to the stationary RSR problem as well as to the case where the RSR emission is driven by an exponential pulse. The analytical solutions to both of these cases are presented.
The effects of the finite correlation time of the external modulation on the secondary-emission spectra are clarified emphasizing the deviation from the motional narrowing limit or the impact approximation in the case of large off-resonance excitation. The dependences of the excitation spectrum of the secondary emission on the modulation parameters and on the inhomogeneous broadening are studied systematically. It is found that the excitation spectrum of the luminescence shows a faster decrease than that of the Raman scattering as the off-resonance frequency increases. The present theory is relevant for explaining the recent experiment by Watanabe, Kinoshita, and Kushida [Chem. Phys. Lett. 126, 197 (1986)] on the \ensuremath{\beta}-carotene in solutions. A general theory of the time-resolved spectra of the secondary emission is formulated to incorporate the effect of the finite correlation time. The luminescence component shows a slow temporal decay with the longitudinal relaxation rate. It is found for the first time that the difference is unexpectedly large between the transient behavior of the luminescence intensity and that of the intermediate-state population for the slow-modulation regime and that the difference indicates the effect of finite correlation time of the frequency modulation. On the other hand, the transient behavior of the Raman scattering contains a fast component that follows the envelope of the excitation-pulse intensity and the same slow component as the luminescence. The ratio of the slow component to the total intensity depends sensitively on the modulation parameters and the off-resonance frequency. The modulation parameters of the system in question can be estimated from these dependences in combination with the spectral characteristics of the emission under stationary excitation.
The stationary second-order spectrum of the resonance-fluorescence light from a vibronic system is studied in a nonperturbative manner. In the model used the potential energy surfaces for the vibrational motion in the electronic ground state and in the excited state are assumed to be displaced and distorted relative to each other. The calculations are performed for the case when the frequency of the driving laser field is tuned to the vibrationless electronic transition. In particular, in the high-driving-field limit the vibrationless transition exhibits a triplet structure, the sidebands being narrower and higher in comparison with the well-known two-level case. The Raman lines are shifted and split into doublets, the linewidths being determined by the electronic as well as the vibrational relaxation rates.
A nonperturbative treatment of the dynamical behavior of a vibronic system in a resonant, coherent, driving light field is developed. The potential energy surfaces for the vibrational motion in the electronic ground and excited state are assumed to be displaced and distorted relative to one another. A system of three integrodifferential equations is derived from the infinite set of density-matrix equations. Both approximate analytical and numerical solutions are presented for the intensity and the intensity-correlation function of the scattered light. The results are valid for arbitrary ratios of the Rabi frequency to the relaxation rates. A system with fast vibrational relaxation behaves as a two-level system. For systems with slow vibrational relaxation, the applicability of perturbation theory requires pump fields that are weaker than those of a two-level system. The Rabi oscillations occurring in a strong pump field are damped less than those of a two-level system, and the intensity substantially decreases. The long-time behavior (determined by the vibrational relaxation time) is characterized by the increase of the intensity toward its stationary value.
Summary form only given. The femtosecond dynamics and the coherence properties of resonant secondary emission (RSE) from quantum wells is a subject of considerable current interest. We use a novel approach in which the sample is resonantly excited by a pair of phase-locked pulses, and the spontaneously emitted radiation is detected by femtosecond upconversion. This novel experimental technique allows us, for the first time, to investigate the temporal evolution and the coherence properties of RSE simultaneously.
The problem of resonant interaction between an electromagnetic radiation field and a vibronic system, such as a molecule or a localized electronphonon system, is considered. The vibronic coupling is assumed to result from displaced and distorted potential energy surfaces for the vibrational motion in the electronic quantum states involved in the resonant interaction with the radiation field. Furthermore, the vibrational energy relaxation is taken into account without assumptions about Markovian-type relaxation. In order to study correlation properties of higher than second order in the electric field strength of the light emitted a method of calculation of the expectation value of an arbitrary product combination of effective time evolution operators of the vibronic system is developed. The calculations are performed by utilizing Feynman's disentangling and operator ordering techniques and yield closed solutions in the low-temperature limit. The theory is used for studying the intensity and the normally ordered intensity correlation function of the resonance fluorescence light from a two-atom molecule excited by a relatively weak laser field, which undergoes phase fluctuations. It is shown that the vibronic coupling can give rise to an anticorrelation effect which is substantially stronger than in the case of resonance fluorescence from an atom. For the case of small vibrational damping a general formula which directly connects the asymptotic long-time behaviour of the normalized intensity correlation function with the vibrational relaxation in the molecule is deduced. It should be convenient for the determination of very small vibrational decay rates.