Influence of Spectral Diffusion on Single-Molecule Photon Statistics Y. He and E. Barkai Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46556, USA (Received 15 December 2003; published 5 August 2004) We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave field. Using a generating function formalism an exact analytical formula for Mandel’s Q parameter is obtained. The solution, which is valid for weak and strong excitation fields, exhibits transitions between (i) quantum sub-Poissonian and classical super-Poissonian behaviors, and (ii) fast to slow modulation limits. DOI: 10.1103/PhysRevLett.93.068302 PACS numbers: 82.37.–j, 05.10.Gg, 33.80.–b, 42.50.Ar Physical, chemical, and biological systems are inves- tigated in many laboratories using single molecule spec- troscopy [1]. The investigation of the distribution of the number of photons emitted from a single molecule source is the topic of extensive theoretical research, e.g., [2–5]. An important mechanism responsible for fluctuations in photon statistics from single molecule sources is spectral diffusion. In many cases the absorption frequency of the molecule randomly changes due to different types of interactions between the molecule and its environment (e.g., [1,6 –8] and references therein). For example, in low temperature glasses flipping two level systems embedded in the glassy environment induce spectral jumps in the absorption frequency of the single molecule under inves- tigation [6]. In this way the molecule may come in and out of resonance with the continuous wave laser field with which it is interacting. A second mechanism responsible for fluctuations of photon counts is the quantum behavior of the spontaneous emission process [9,10]. In his fundamental work Mandel [11] showed that a single atom in the process of resonance fluorescence, in the absence of spectral diffusion, exhibits sub-Poissonian photon statistics [12]. The photon statis- tics is characterized by Mandel’s Q parameter Q N 2 N 2 N 1; (1) where N is the number of emitted photons within a certain time interval. The case Q< 0 is called sub- Poissonian behavior, while Q> 0 is called super- Poissonian behavior. Briefly sub-Poissonian statistics is related to antibunching of photons emitted from a single source and to Rabi oscillations of the excited state population which favors an emission process with some periodicity in time. Sub-Poissonian statistics and anti- bunching were detected in several single molecule ex- periments, for example, [13,14]. In this Letter we obtain an exact analytical expression for the Q parameter in the long time limit, for a single molecule undergoing a stochastic spectral diffusion pro- cess. For that aim we use the powerful generating func- tion method of Zheng and Brown [15,16]. If the spectral diffusion process is slow enough super-Poissonian behav- ior of Q is expected (see details in the text). Our analyti- cal expressions classify the transitions between sub- and super-Poissonian statistics. They give the conditions on the spectral diffusion time scale and magnitude of spec- tral jumps for sub-Poissonian nonclassical behavior to be observed. Our result is valid for weak and strong excita- tion (i.e., arbitrary Rabi frequency). It yields the lower bound on Q. The solution shows how in experiment we may choose the Rabi frequency so that the quantum nature of the photon emission process becomes larger, namely, how to minimize Q in the sub-Poissonian re- gime. This is important for the efficient detection of quantum effects in single molecule spectroscopy, since choosing too small or too large values of the Rabi frequency results in very small and hence undetectable values of Q. Finally our result is used to test the semi- classical linear response theory (i.e., weak Rabi fre- quency) of Barkai, Jung, and Silbey [7,8]. The semiclassical theory yields Q> 0, while the main focus of this Letter is on the quantum regime Q< 0. Our starting point is the Zheng-Brown [15] general- ized optical Bloch equations describing a chromophore with single excited and ground state _ Us 2 Us L tV s; _ V s L tUs 2 V s W s; _ W s V s 2 1 sW s 2 1 sYs; _ Ys 2 1 sW s 2 1 sYs: (2) These equations are exact within the rotating wave ap- proximation and optical Bloch equation formalism. In Eq. (2) is the spontaneous emission rate of the elec- tronic transition and is the Rabi frequency. The time evolving detuning is L t ! L ! 0 !t, where ! L is the laser frequency, ! 0 is the molecule’s bare frequency, and !t is the stochastic spectral diffusion PHYSICAL REVIEW LETTERS week ending 6 AUGUST 2004 VOLUME 93, NUMBER 6 068302-1 0031-9007= 04=93(6)=068302(4)$22.50 2004 The American Physical Society 068302-1