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Physical Review A

Photon statistics for a two-level system interacting with a sequence of two laser pulses

F. Shikerman, Yiwen He, Eli Barkai

Published Jun 13, 2008• 3 citations
Publisher

Abstract

We investigate photon statistics emerging from the interaction of a two-level system with two laser pulses separated by a delay interval. It is shown that ${P}_{n}(t)$, the probability of emitting $n$ photons up to time $t$, may be represented as a summation over quantum trajectories, which allows simple identification of the coherent effect arising from the existence of the delay. We obtain a relation between photon statistics emerging from the interaction with a single pulse and photon statistics emerging from the interaction with two separated pulses with finite delay. In addition, the probabilities of emitting zero, one, and two photons, the expectation number of emissions, $⟨N⟩$, and Mandel's $Q$ parameter are calculated explicitly. The case of two short and strong pulses, the so-called pump and probe setup, yielding up to two photons at most, is considered in detail. We examine different limiting behaviors of the photon statistics and obtain the upper and lower bounds for ${P}_{0}$, ${P}_{1}$, and ${P}_{2}$ in the limit of strong fields. Implications for single-molecule nonlinear spectroscopy and the theory of obtaining pairs of photons on demand are discussed briefly.

DOI
10.1103/physreva.77.063819

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We investigate photon statistics emerging from the interaction of a two-level system with two laser pulses separated by a delay interval. It is shown that Pn(t){P}_{n}(t)Pn​(t), the probability of emitting nnn photons up to time ttt, may be represented as a summation over quantum trajectories, which allows simple identification of the coherent effect arising from the existence of the delay. We obtain a relation between photon statistics emerging from the interaction with a single pulse and photon statistics emerging from the interaction with two separated pulses with finite delay. In addition, the probabilities of emitting zero, one, and two photons, the expectation number of emissions, ⟨N⟩⟨N⟩⟨N⟩, and Mandel's QQQ parameter are calculated explicitly. The case of two short and strong pulses, the so-called pump and probe setup, yielding up to two photons at most, is considered in detail. We examine different limiting behaviors of the photon statistics and obtain the upper and lower bounds for P0{P}_{0}P0​, P1{P}_{1}P1​, and P2{P}_{2}P2​ in the limit of strong fields. Implications for single-molecule nonlinear spectroscopy and the theory of obtaining pairs of photons on demand are discussed briefly.
There are many ways of calculating photon statistics in quantum optics in general and single molecule spectroscopy in particular such as the generating function method, the quantum jump approach or time ordering methods. In this paper starting with the optical Bloch equation, within the paths interpretation of Zoller, Marte and Walls we obtain the photon statistics from a sequence of laser pulses expressed by means of quantum trajectories. We find general expressions for Pn(t) - the probability of emitting n photons up to time t, discuss several consequences and show that the interpretation of the quantum trajectories (i) emphasizes contribution to the photon statistics of the coherence paths accumulated in the delay interval between the pulses and (ii) allows simple classification of the terms negligible under certain physical constraints . Applying this method to the concrete example of two square laser pulses we find the probabilities of emitting 0,1 and 2 photons, examine several limiting cases and investigate the upper and lower bounds of P0(t), P1(t) and P2(t) for a sequence of two strong pulses in the limit of long measurement times. Implication to single molecule non-linear spectroscopy and theory of pairs of photons on demand are discussed briefly.
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 develop the theory of nonlinear 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 semiclassical 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 ultrafast dynamics of individual molecules that are different from those found for an ensemble.
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.
We investigate theory of single-photon control from a two-level single-molecule source irradiated by laser pulses of various shapes and pulse durations in terms of quantum trajectories which link stochastic dynamics of the radiating source with quantum measurement theory. Using Monte Carlo wave function simulation, we analyze the detailed dissipative dynamics of the single-molecule source and the photon statistics as revealed by repeated Gedanken photon measurement on the single radiating source. We show that much of the photon statistics from the two-level single-molecule single-photon sources, including few-photon emission probability, waiting time distribution, and two-time correlation function of the fluorescent light, can be understood qualitatively from the simple picture of Rabi nutation and pi pulse in terms of pulse areas.