Photon Statistics for Single-Molecule Nonlinear Spectroscopy

Abstract

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...

Single-molecule photon emission statistics for systems with explicit time dependence: Generating function approach

“…37,41,45 Although the framework of generating function calculations remains unchanged under pulsed excitation or more general timedependent perturbations, the resulting equations must be solved numerically; this has not previously been explored in any generality. ͑Barkai and co-workers 34,52,53 considered time-dependent excitations within the generating function and related frameworks, but these works were limited to idealized square pulses and/or slowly varying perturbations.͒ The present paper demonstrates that the generating function framework is well suited for numerical calculations involving general nonconstant excitation profiles. Several model calculations are presented related to the dynamics of twoand three-level chromophore systems.…”

Reconstruction of the environmental correlation function from single-emitter photon statistics: A non-Markovian approach

“…whereσ i denotes the Pauli matrices. Within the spectral diffusion approach the two-level system density operator ρ(t|η t ) is a function of a real process η t , and obeys a stochastic Liouville equation [23][24][25] d dtρ…”

Statistics of trajectories in two-state master equations

“…As an example, we consider the simple case of a master equation with two states, + and −, with transition rates k + (from − to +) and k − (from + to −). In spite of its simplicity, this case is of interest for many physical and biological problems [11,12,13,14,15]. We will show that eq.…”

“…We study as example of our method general two-state systems that, despite their simplicity, have many non-trivial applications in problems related to single-molecule spectroscopy (see, e.g. [11,12]) and biophysics (see, e.g., [13,14,15]). Specifically, we calculate the probability of observing a given number of transitions, N , in a time, T , and the distribution of time spent in one of the two states in a time T .…”

“…− P(−, T, t + , N )+P eq + P(+, T, t + , N )]. (11) Using the above expressions for the various terms and performing the integral, we find two different expressions, one for N odd: (12) and one for N even:…”