Single Molecule Photon Counting Statistics for Quantum Mechanical Chromophore Dynamics

Abstract

We extend the generating function technique for calculation of single molecule photon emission statistics [Y. Zheng and F. L. H. Brown, Phys. Rev. Lett., 90,238305 (2003)] to systems governed by multi-level quantum dynamics. This opens up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, the present methodology allows for calculation of photon statistics that are spectrally resolved and subject to quantum coherence. Several model calculations illustrate the generality of the technique and highlight quantitative and qualitative differences between quantum mechanical models and related stochastic approximations. Calculations suggest that studying photon statistics as a function of photon frequency has the potential to reveal more about system dynamics than the usual broadband detection schemes.

Photon emission statistics and photon tracking in single-molecule spectroscopy of molecular aggregates: Dimers and trimers

“…The equations for the generating functions are exact within the rotating wave approximation and the limits set by the Hamiltonian. This method has already been applied successfully to single two-level chromophores 39, 40, 46-49, 56, 57 and their extension to multi-level quantum systems, 58,59 and has also been used by Sanda and Mukamel. 52 Using numerical methods, we apply the general equations to investigate the fluorescence excitation line shape (we will refer to this, for short, as the line shape) and Mandel's Q parameter as a function of laser frequency in the limit of long measurement times, i.e., much longer than any dynamical time scale present, for both dimers and trimers of interacting molecules.…”

“…Two of the exceptions are the study by Gopich and Szabo 60 concerning the distribution of the number of donor and acceptor photons from single molecule Förster resonance energy transfer measurements and the work of Bel, Zheng, and Brown, 58 in which statistical moments were calculated for multi-level quantum systems, based on an extension of the generating function method. Using extended generating functions, we determine a general scheme for calculating statistical moments of transition resolved (frequency resolved) photons emitted from a molecular aggregate.…”

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

“…It comes as no surprise that considerable effort has been expended on the theory of interpreting/modeling SMS trajectories, at various levels of sophistication. 8, Recent work by us [37][38][39][40][41][42][43][44][45] and others [46][47][48][49][50] has established generating function techniques as a somewhat general means for calculating statistical quantities of single molecule photon counting experiments, including quantum mechanical evolution of the chromophore. The only fundamental limitations to this approach are that you must consider the spontaneous emission of photons to be governed by rate processes and the directly calculated quantities are statistical moments and probabilities of the number of photons emitted over a given time interval.…”

“…This allows for an elementary treatment of the generating function equations via simple matrix operations. 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.…”

“…Furthermore, extension of the calculations presented here to general multilevel quantum systems is straightforward and follows the general considerations necessary for modeling of multilevel systems within the generating function approach. 41 This paper is organized as follows. In Sec.…”