We develop the theory of non linear 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 semi-classical 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 ultra-fast dynamics of individual molecule's which are different than those found for an ensemble.PACS numbers: 82. 05.10.Gg, 42.50.Ar Nonlinear optical interaction of a sequence of laser pulses with matter provides a powerful tool for the investigation of dynamics of ensembles of molecules in a wide variety of chemical, physical and biological systems [1]. Recently, van Dijk et al [2] reported of the first experimental study of an ultra-fast pump-probe single molecule system. Unlike the previous approaches to non-linear spectroscopy where only the ensemble average response to the external fields is resolved, the new approach yields direct information on single molecule dynamics, gained through the analysis of photon counting statistics. Although the original experiment [2] was conducted on a molecule undergoing a relatively simple relaxation process, the huge potential of combining nonlinear spectroscopy with single molecule spectroscopy, inspires many unanswered theoretical questions: What are the fundamental physical limitations of the investigation of fast dynamics when spectral selectivity (defined below) is limited? How does the information contained in these experiments differ from the information contained in simpler continuous wave experiments? What is the finger print of coherence in these type of experiments, and precisely how its influence on photon statistics is suppressed due to dephasing processes? Finally, how to design the external control fields so that new information on dynamics of molecules is gained. While the answer to these questions, depends on the particular dynamics of the molecule under investigation, we present a theory based on the Kubo-Anderson model, which yields general insights on the problem.Consider a sequence of two laser pulses interacting with a single quantum system as a molecule, an atom, or a nano-crystal. The pulses are assumed to be very short compared with the radiative life-time of the emitter, so that the probability of photon emission during a pulse is negligible, hence a pair of pulses yields two photons at most. Repeating the experiment many times one may obtain the probabilities P 0 , P 1 and P 2 of emitting 0, 1 and 2 photons. In this manuscript we investigate the relation between P 0 , P 1 and P 2 and the dynamics of the underlying system interacting with the external fields.In turn this type of photon statistics reveals important in...

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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 establish the relation between the Wigner-Weisskopf theory for the description of an unstable system and the theory of coupling to an environment. According to the Wigner-Weisskopf general approach, even within the pole approximation the evolution of a total system subspace is not an exact semigroup for multichannel decay, unless the projectors into eigenstates of the reduced evolution generator W (z) are orthogonal. With multichannel decay, the projectors must be evaluated at different pole locations zα = z β , and since the orthogonality relation does not generally hold at different values of z, the semigroup evolution is a poor approximation for the multi-channel decay, even for very weak coupling. Nevertheless, if the theory is generalized to take into account interactions with an environment, one can ensure orthogonality of the W (z) projectors regardless the number of the poles. Such a possibility occurs when W (z), and hence its eigenvectors, are independent of z, which corresponds to the Markovian limit of the coupling to the continuum spectrum.

We consider the two-level system approximation of a single emitter driven by a continuous laser pump and simultaneously coupled to the electromagnetic vacuum and to a thermal reservoir beyond the Markovian approximation. We discuss the connection between a rigorous microscopic theory and the phenomenological spectral diffusion approach, used to model the interaction of the emitter with the thermal bath, and obtained analytic expressions relating the thermal correlation function to the single emitter photon statistics.