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Physical Review A

Time- and frequency-gated spontaneous emission as a tool for studying vibrational dynamics in the excited state

Maxim F. Gelin, Andrei V. Pisliakov, Wolfgang Domcke

Published Jun 17, 2002• 42 citations
Publisher

Abstract

The theory of time- and frequency-gated (TFG) spontaneous emission (SE) spectra is elaborated. The present formulation generalizes previous derivations, clarifies the interrelations between different existing expressions, and establishes the validity of certain commonly assumed approximations. We obtain various explicit expressions for TFG SE spectra, which are suitable for performing actual calculations for nontrivial systems and which allow us to establish generic (that is, model-independent) properties of TFG spectra. The doorway-window picture of temporally and spectrally resolved spectra is further developed. It is shown that, to the leading order in the pump and probe pulses, the TFG SE signal is equivalent to the stimulated-emission contribution to the integral pump-probe spectrum in the case of nonoverlapping pulses. The theory is illustrated for the example of an electronic two-level system with a single Condon-active harmonic vibrational mode that is coupled to a thermal bath. The effect of imperfect time and frequency resolution is studied. It is pointed out that the TFG SE spectrum carries information not only on the strength of the system-bath coupling, but also on the relative magnitude of the bath correlation time.

DOI
10.1103/physreva.65.062507

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The theory of time- and frequency-gated (TFG) spontaneous emission (SE) spectra is elaborated. The present formulation generalizes previous derivations, clarifies the interrelations between different existing expressions, and establishes the validity of certain commonly assumed approximations. We obtain various explicit expressions for TFG SE spectra, which are suitable for performing actual calculations for nontrivial systems and which allow us to establish generic (that is, model-independent) properties of TFG spectra. The doorway-window picture of temporally and spectrally resolved spectra is further developed. It is shown that, to the leading order in the pump and probe pulses, the TFG SE signal is equivalent to the stimulated-emission contribution to the integral pump-probe spectrum in the case of nonoverlapping pulses. The theory is illustrated for the example of an electronic two-level system with a single Condon-active harmonic vibrational mode that is coupled to a thermal bath. The effect of imperfect time and frequency resolution is studied. It is pointed out that the TFG SE spectrum carries information not only on the strength of the system-bath coupling, but also on the relative magnitude of the bath correlation time.
The effect of overlapping pump and gate pulses on time- and frequency-gated spontaneous emission spectra is explored for a model of material dynamics that accounts for strong nonadiabatic and electron-vibrational coupling effects, vibrational relaxation, and optical dephasing, thus representing characteristic features of photoinduced excited-state dynamics in large molecules in the gas phase or in condensed phases. The behaviors of the sequential, coherent, and doorway-window contributions to the spontaneous emission spectrum are studied separately. The interrelation between the sequential and coherent contributions is demonstrated to be sensitive to the carrier frequencies of the pump and gate pulses and also to the optical dephasing rate, opening the possibility of an experimental determination of the latter. The coherent contribution is shown to dominate the spectrum at specific emission frequencies.
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.
We present a novel approach to molecular spectroscopy with light pulses of arbitrary strength and duration. The key quantity is the frequency-resolved net energy transfer dE/dω which reveals at which frequencies energy is transferred from the field to the molecule (absorption) or from the molecule to the field (stimulated emission). It is shown that dE/dω can be expressed as the Fourier transform of the cross-correlation function of the molecular polarization P(t) and the time derivative of the applied field, dℰ/dt. In this sense, it is formally equivalent to the absorption cross section under weak-field conditions which, as commonly known, can be represented as the Fourier transform of the autocorrelation function S(t). The time-dependent polarization P(t) is determined by exact integration of the time-dependent Schrödinger equation including the light-matter interaction to all orders. It is shown that under weak-field conditions the expression for dE/dω reduces to the well-known cross section formula in the time-dependent picture of spectroscopy, multiplied by the spectral intensity of the light pulse. Therefore, we consider the expression for the frequency-resolved energy transfer, which is valid for arbitrary electric fields, as the natural extension of the absorption cross section in the weak-field limit. Furthermore, dE/dω is shown to be formally equivalent to the change of the spectral intensity, ΔI(ω), of an optical pulse after transmission through a sample, the latter being derived by solving Maxwell’s equations under well-known approximations. The theory is applied to a simple one-dimensional model with two electronic states and the frequency-resolved energy transfer is investigated as a function of the field strength. For sufficiently strong fields, dE/dω exhibits transitions between essentially all vibrational levels in the ground and all states in the excited electronic manifold. The new expression distinguishes between absorption and emission and that is clearly seen in the spectra.
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).
We have extended the doorway-window representation of optical pump-probe spectroscopy with weak pulses toward N-wave-mixing spectroscopy with temporally well-separated pulses of arbitrary strength. The expressions for the signals in the strong-pulse doorway-window representation are derived in the framework of the nonperturbative theory of N-wave-mixing spectroscopy. The strong-pulse doorway-window representation is complementary to the equation-of-motion phase-matching approach. The latter fully accounts for pulse-overlap effects in signals induced by weak pulses but is computationally more expensive. The performance of the doorway-window approximation for temporally well-separated strong pulses is illustrated for an electronic two-level system with an underdamped Condon-active vibrational mode.
The specific origin of oscillations in time-resolved optical signals, in particular, for complex systems with nontrivial interstate couplings and nonseparable electron-nuclear motion, is often difficult to assign. Here, we show that coherent oscillations in two-dimensional photon-echo are capable of self-analysis; their beating maps provide a tool to tell apart ground-state bleach (GSB), stimulated emission (SE), and excited-state absorption (ESA) contributions to the oscillatory signal component. Because GSB carries information on ground-state coherence while SE and ESA reflect the excited-state coherence, the observed oscillations can be unambiguously assigned to ground-state or excited-state coherent motion. The findings prove especially advantageous for systems with dense detectable manifolds of states pertaining to each electronic state. An analogous analysis for frequency-resolved (dispersed) pump-probe spectroscopy is discussed briefly.