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The Journal of Physical Chemistry B

Optical N-Wave-Mixing Spectroscopy with Strong and Temporally Well-Separated Pulses: The Doorway−Window Representation

Maxim F. Gelin, Dassia Egorova, Wolfgang Domcke

Published Mar 22, 2011• 32 citations
Publisher

Abstract

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.

DOI
10.1021/jp112055h

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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 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.
We predict that the four-wave mixing (FWM) excitation spectrum for a two-level system becomes extremely asymmetrical when steady-state excitation is replaced by pulsed excitation, using temporally nonoverlapping probe and pump pulses. We also show that when a resonant probe pulse precedes the detuned pump pulse, the FWM signal may become more intense than when the pulses overlap. These effects are attributed to free-induction decay (FID): the coherence induced in the two-level system by a short resonant probe pulse survives the pulse (FID) and replaces it in producing a strong nonlinear signal on interaction with the strong pump pulse. Moreover, by plotting the FWM signal intensity as a function of the delay between the probe and pump pulses, an efficient method is proposed for determining short transverse relaxation times or fast chemical reaction rates.
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 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.
Considering an electronic two-level system coupled to vibrational degrees of freedom and driven by short and intense non-overlapping laser pulses, we introduce the concept of nonperturbative response functions. These response functions are expressed in terms of effective electronic transition dipole moments which depend on the strength of the field-matter coupling and on the pulse durations. It is shown that the nonlinear polarization representing four-wave-mixing signals can elegantly be expressed in terms of these nonperturbative response functions to all orders in the field-matter coupling. The nonperturbative response functions generalize the interpretative power of the familiar third-order response functions to four-wave-mixing experiments with intense laser pulses.
An efficient method has been developed for the calculation of third-order time- and frequency-resolved optical signals. To obtain the general four-wave mixing signal, seven auxiliary density matrices have to be propagated in time. For the special cases of two-pulse photon-echo and transient-grating signals, two or three density matrices, respectively, are required. The method is limited to weak laser fields (it is thus valid within the third-order perturbation theory) but allows for any pulse durations and automatically accounts for pulse-overlap effects. To illustrate the method, we present the explicit derivation of the three-pulse photon-echo signal. Any other third-order optical signal can be calculated in the same manner. As an example, two- and three-pulse photon-echo and transient-grating signals for a weakly damped displaced harmonic oscillator have been calculated.