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The Journal of Chemical Physics

The theory of photon echoes for strongly coupled electron–phonon systems

James L. Skinner

Published Oct 1, 1982• 29 citations
Publisher

Abstract

We calculate the photon echo decay for a strongly coupled electron–phonon system, without making unrealistic assumptions about excitation pulse durations. We find that asymptotically, the photon echo decay is exponential, and we provide a formal expression for the decay rate 1/T2 which is independent of pulse duration. In contrast to the usual results, our expression for 1/T2 is nonperturbative in the electron–phonon interaction. We calculate the dependences of the echo amplitude on pulse power and duration, finding that the strong electron–phonon coupling modifies the simple two level system result. We also focus on the relationship between the zero phonon line shape in the homogeneous optical absorption spectrum, and the photon echo decay rate. We find that the zero phonon line shape is Lorentzian, with width 1/πT2. We show that the usual optical Bloch equation approach to analyzing time-domain experiments, which normally is only applicable if the electron–phonon coupling is weak, can be extended to the case of strong coupling. These strong coupling Bloch equations are adequate for analyzing the time dependence, but not the pulse power and duration dependences, of time-domain experiments. Our results indicate that within the Condon approximation, electron–phonon coupling cannot explain the very low temperature anomaly of recent photon echo experiments on impurities in molecular crystals.

DOI
10.1063/1.444283

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Optica Acta International Journal of Optics•Nov 1, 1981•0 citations
We calculate the photon echo decay for a strongly coupled electron–phonon system, without making unrealistic assumptions about excitation pulse durations. We find that asymptotically, the photon echo decay is exponential, and we provide a formal expression for the decay rate 1/T2 which is independent of pulse duration. In contrast to the usual results, our expression for 1/T2 is nonperturbative in the electron–phonon interaction. We calculate the dependences of the echo amplitude on pulse power and duration, finding that the strong electron–phonon coupling modifies the simple two level system result. We also focus on the relationship between the zero phonon line shape in the homogeneous optical absorption spectrum, and the photon echo decay rate. We find that the zero phonon line shape is Lorentzian, with width 1/πT2. We show that the usual optical Bloch equation approach to analyzing time-domain experiments, which normally is only applicable if the electron–phonon coupling is weak, can be extended to the case of strong coupling. These strong coupling Bloch equations are adequate for analyzing the time dependence, but not the pulse power and duration dependences, of time-domain experiments. Our results indicate that within the Condon approximation, electron–phonon coupling cannot explain the very low temperature anomaly of recent photon echo experiments on impurities in molecular crystals.
In this paper we investigate the influence of optical density or superradiance on the photon echo decay using the coupled Maxwell–Bloch equations. Our theory applies to optically thick but dilute media, where the transition dipole–dipole interaction between nearby chromophores is negligible, but the far-field interaction is not. Corrections to the (normalized) echo decay in the optically thin limit, INe(tπ)=exp [−4tπ/T2] (tπ and T2 are the interpulse separation times and the coherence relaxation times, respectively), are found by expanding the polarization, excited state population and pulse electric fields in powers of ζ=αz, where α is the inverse Beer’s absorption length, and z is the coordinate along the pulse propagation axis. Numerical solutions for optically thick samples are also presented, and when ζ≫1 the normalized echo intensity is found to obey the simple analytic form: INe(tπ)=2 exp [−tπ/T1]−1, when tπ/T1<ln 2 and INe(tπ)=0 for longer times. (T1 is the population relaxation time.) In this limit the echo decay is independent of the coherence relaxation time T2, illustrating the profound effect intermolecular coupling may have on any nonlinear optical measurement. We show that at low temperatures, when T2=2T1, the echo decay rate increases as the optical density increases. However, for higher temperatures, when T2 is sufficiently shorter than 2T1, the opposite behavior results, that is, the echo decay rate decreases as the optical density increases.
The effects of optical density (OD) on photon echo experiments is examined both theoretically and experimentally. The theoretical treatment involves coupled Maxwell-Optical Bloch equations, and the experiments are picosecond photon echo studies of the mixed molecular crystal, pentacene in naphthalene. Under high power excitation (π/2, π) in high OD samples, the echo decays are predicted and observed to be faster than decays arising solely from the intrinsic molecular T2. The increase in the echo decay rate vanishes in the limit of low power or low OD, permitting the intrinsic molecular T2 to be measured. Previously reported concentration effects are shown to actually be optical density effects. The experimentally determined intrinsic molecular T2 of pentacene in naphthalene is shown to be independent of concentration and temperature (below ∼2 K) and T2≠2T1. It is suggested that the difference between T2 and 2T1 is due to the static multiplet level structure arising from the electronic state-phonon coupling.
An approximate excited state vibrational Hamiltonian, He, which accounts for both linear and diagonal quadratic electron-phonon coupling is derived and shown to be adequate for mode frequency changes up to ~30%. The associated linear response function which is valid for finite temperature in the absence of damping is obtained. 2-point correlation function or, equivalently, the line broadening function, g(t;T), that provides a good physical description of pure electronic dephasing and vibrational damping for linearly coupled modes is postulated. A linear response function that accounts for pure electronic dephasing and damping for systems with either linearly or quadratically coupled modes is proposed. Applications of the new linear response function include the single site absorption and hole-burned spectra of the special pair band of the bacterial reaction center and the temperature dependence of the single site absorption spectrum of Al-phthalocyanine tetrasulphonate in glassy ethanol. A 4-point correlation function which is valid for finite temperature in the absence of damping using the approximate excited state vibrational Hamiltonian, He, is derived. An impulsive stimulated photon echo signal for a single-mode system with very large inhomogeneous broadening is calculated using the 4-point correlation function. An expression for ordinary photon echo with two finite pulses is derived. 4-point correlation function for linearly coupled modes is expressed in terms of g(t;T) and can be used to calculate all non-linear spectroscopic signals. Applications of the 4-point correlation function are limited to impulsive 2-pulse photon echo (IPE) signals. IPE signals are generated for different values of the Huang-Rhys factor (S) at a fixed temperature and at different temperatures for a fixed value of S in order to see how the echo profile changes as one varies S or temperature. An initial fast non-exponential decay, which reflects the contribution of the phonon sideband, followed by vibrational oscillations (quantum beats) due to linear electron-phonon coupling and a slow exponential decay, on picosecond time scales, due to the zero-phonon line are observed in the IPE signals. Application of IPE to the special pair absorption band of the bacterial reaction center is also included.
Using a stochastic model, we calculate the temperature dependence of the homogeneous linewidth and the two-pulse photon-echo decay rate of optically active impurities in glasses. The theory applies to the low-temperature regime T\ensuremath{\lesssim}10 K, where it is assumed that the dephasing arises from an elastic dipole interaction of the impurity with an array of thermally excited tunneling systems. Averaging over the tunneling parameters and using relaxation rates appropriate to vitreous silica, we find that the photon-echo decay rate varies as T1.1{T}^{1.1}T1.1 and the homogeneous linewidth as T1.2{T}^{1.2}T1.2 for 0.1\ensuremath{\le}T\ensuremath{\le}1 K, assuming a constant density of states for the tunneling systems. Both the photon-echo amplitude and the optical dipole-moment correlation function decay exponentially in time. The rates, however, are different, with a ratio of photon-echo dephasing time to optical-absorption dephasing time of 6.9 at T=1 K, assuming a constant density of states. The relation of our results to various experimental studies is discussed.
The interaction between an intense radiation field and atoms that undergo collisions in an atomic vapor is studied. With the assumption that velocity-changing collisions are state independent and weak, we obtain an analytic expression for the time-dependent density-matrix elements by solving a quantum-mechanical transport equation using a dressed-state approach. Our analytical calculation is used to predict the form of the signal in an extended-pulse photon echo. The results show good agreement with an experiment carried out by Yodh et al. [Phys. Rev. Lett. 53, 659 (1984)]. It is found that the decay rate of the echo signal decreases rapidly with increasing duration of the second pulse in the pulse excitation sequence. A comparison between theory and experiment is presented and a physical interpretation of the results is given.
We present a theory of optical two-photon rotary echo for the case when the direct couplings between the three levels due to spontaneous decay to and from the intermediate state dominate the indirect couplings (collisional, etc.) to the passive levels and the phase-interrupting (P.I.) collisions cannot be ignored. For this situation which generally obtains in the visible/U.V. region which is relevant for many two-photon transient experiments, the ‘two-photon Bloch equations’ with non-resonant intermediate state are non-trivially different from the familiar (two-level) Bloch equations with or without direct coupling. An explicit analytical calculation demonstrates two-photon rotary echo with the usual dependence on the total phase shift. The dependence of the echo peak on pulse delay time is governed by a damping constant in which ‘intensity modification’ involves only the P.I. collision constant and not the spontaneous decay constant in sharp contrast to the one-photon case. For the case of counter-propagating beams, the intensity modified echo damping constant turns out to be especially simple. We suggest that the features unique to the two-photon rotary echo can be usefully exploited for the empirical determination of the polarization decay constant.