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The Journal of Chemical Physics

Single molecule photon emission statistics of driven three-level systems

Yonggang Peng, Yujun Zheng, Frank L. H. Brown

Published Mar 13, 2007• 39 citations
Publisher

Abstract

We study the statistics of photon emission from three-level single molecule systems. The generating function method [Y. Zheng and F. L. H. Brown, Phys. Rev. Lett. 90, 238305 (2003)] is used to calculate steady state absorption line shapes and Mandel's Q parameter as a function of excitation frequency, as well as the time dependence associated with approach to the steady state. The line shape calculations confirm known results derived via other methods, while the Q parameter results display complex frequency dependences not amenable to simple interpretation. This study confirms the applicability of the generating function formalism to multilevel quantum systems, including the proper modeling of quantum coherence effects.

DOI
10.1063/1.2567235

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We study the statistics of photon emission from three-level single molecule systems. The generating function method [Y. Zheng and F. L. H. Brown, Phys. Rev. Lett. 90, 238305 (2003)] is used to calculate steady state absorption line shapes and Mandel's Q parameter as a function of excitation frequency, as well as the time dependence associated with approach to the steady state. The line shape calculations confirm known results derived via other methods, while the Q parameter results display complex frequency dependences not amenable to simple interpretation. This study confirms the applicability of the generating function formalism to multilevel quantum systems, including the proper modeling of quantum coherence effects.
We extend the generating function technique for calculation of single molecule photon emission statistics (Zheng, Y.; Brown, F. L. H. Phys. Rev. Lett. 2003, 90, 238305) 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 when they arise. 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.
Based on the generating function formalism, we investigate broadband photon statistics of emission for single dimers and trimers driven by a continuous monochromatic laser field. In particular, we study the first and second moments of the emission statistics, which are the fluorescence excitation line shape and Mandel's Q parameter. Numerical results for this line shape and the Q parameter versus laser frequency in the limit of long measurement times are obtained. We show that in the limit of small Rabi frequencies and laser frequencies close to resonance with one of the one-exciton states, the results for the line shape and Q parameter reduce to those of a two-level monomer. For laser frequencies halfway the transition frequency of a two-exciton state, the photon bunching effect associated with two-photon absorption processes is observed. This super-Poissonian peak is characterized in terms of the ratio between the two-photon absorption line shape and the underlying two-level monomer line shapes. Upon increasing the Rabi frequency, the Q parameter shows a transition from super- to sub- to super-Poissonian statistics. Results of broadband photon statistics are also discussed in the context of a transition (frequency) resolved photon detection scheme, photon tracking, which provides a greater insight in the different physical processes that occur in the multi-level systems.
We derive the moment generating function for photon emissions from a single molecule driven by laser excitation. The frequencies of the fluoresced photons are explicitly considered. Calculations are performed for the case of a two-level dye molecule, showing that measured photon statistics will display a strong and nonintuitive dependence on detector bandwidth. Moreover, it is demonstrated that the antibunching phenomenon, associated with negative values of Mandel's Q parameter, results from correlations between photons with well separated frequencies.
In this paper, we summarize the photon emission statistical properties of single molecule system, using the recently developed generating function method. Through the introduce of one or double “auxiliary” variables, we can investigate the first moment and second moment of photon emission statistics driven by different external fields, such as the line shapes and Mandel's Q parameters, the photon emission probabilities, the probability distribution of time between successive emission, the waiting time and the waiting time distribution, the cross correlation, the joint probabilities, etc. Among, the first moment of photon statistics and the experimental results of optical amplification are in good agreement.