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Physical review. B, Condensed matter

Time-dependent single molecule spectral lines

Taras Plakhotnik

Published Feb 15, 1999• 13 citations
Publisher

Abstract

A general conceptual problem of time-dependent single molecule spectra is discussed theoretically in the framework of recently developed intensity-time-frequency correlation spectroscopy. It is shown that the new method is closely related to a ``gedanken'' three-pulse photon echo experiment done on an ensemble of identical molecules interacting with statistically identical microscopic environments. The correlation function is an integral transform (under certain conditions a Fourier transform) of the echo amplitude as a function of the delay between the first and the second pulses.

DOI
10.1103/physrevb.59.4658

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Physical Review Letters•Oct 29, 2001•82 citations
A general conceptual problem of time-dependent single molecule spectra is discussed theoretically in the framework of recently developed intensity-time-frequency correlation spectroscopy. It is shown that the new method is closely related to a ``gedanken'' three-pulse photon echo experiment done on an ensemble of identical molecules interacting with statistically identical microscopic environments. The correlation function is an integral transform (under certain conditions a Fourier transform) of the echo amplitude as a function of the delay between the first and the second pulses.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
A theoretical analysis is carried out for the recently developed three-pulse photon echo spectroscopy employing dual frequency combs (DFC) as the light sources. In this method, the molecular sample interacts with three pulse trains derived from the DFC, and the generated third-order signal is displayed as a two-dimensional (2D) spectrum that depends on the waiting time introduced by employing the asynchronous optical sampling method. Through the analysis of the heterodyne-detected signal interferogram using a local oscillator derived from one of the optical frequency combs, we show that the 2D spectrum closely matches the spectrum expected from a conventional approach with four pulses derived from a single femtosecond laser pulse, and the waiting time between the second and third field-matter interactions is given by the down-converted detection time of the interferogram. The theoretical result is applied to a two-level model system with solvation effect described by solvatochromic spectral density. The model 2D spectrum reproduces spectral features such as the loss of frequency correlation, dephasing, and spectral shift as a function of the population time. We anticipate that the present theory will be the general framework for quantitative descriptions of DFC-based nonlinear optical spectroscopy.
A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.
We develop the theory of nonlinear spectroscopy for a single molecule undergoing stochastic dynamics and interacting with a sequence of two laser pulses. We find general expressions for the photon counting statistics and the exact solution to the problem for the Kubo-Anderson process. In the limit of impulsive pulses the information on the photon statistics is contained in the molecule's dipole correlation function. The selective limit, the semiclassical approximation, and the fast modulation limit exhibit rich general behaviors of this new type of spectroscopy. We show how the design of external fields leads to insights on ultrafast dynamics of individual molecules that are different from those found for an ensemble.
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.