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Optics and Spectroscopy

A dynamic approach to the sudden jump model in single molecule spectroscopy

Игорь Сергеевич Осадько, E. V. Khots

Published Jun 1, 2003• 1 citation
Publisher

Abstract

Various theories that use the sudden jump model to explain the time dependence of the optical band of a single molecule in a fluctuating environment are analyzed. It is shown that the dynamic theory of the optical band, based on the Hamiltonian of the system, should necessarily include the initial conditions for the quantum system. We show how these initial conditions should be chosen for the dynamic theory to properly describe the quantum jumps of the optical band of a single molecule. Unlike stochastic theories based on the sudden jump model, the dynamic theory predicts several absorptances of a single molecule and directly connects the absorptance fluctuations with this fact. The theory is used to account for the time dependence of the broadening of experimentally measured lines of individual molecules.

DOI
10.1134/1.1586739

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Various theories that use the sudden jump model to explain the time dependence of the optical band of a single molecule in a fluctuating environment are analyzed. It is shown that the dynamic theory of the optical band, based on the Hamiltonian of the system, should necessarily include the initial conditions for the quantum system. We show how these initial conditions should be chosen for the dynamic theory to properly describe the quantum jumps of the optical band of a single molecule. Unlike stochastic theories based on the sudden jump model, the dynamic theory predicts several absorptances of a single molecule and directly connects the absorptance fluctuations with this fact. The theory is used to account for the time dependence of the broadening of experimentally measured lines of individual molecules.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
The relation between the absorption coefficient of a single molecule calculated within the framework of a dynamical theory and the fluctuations in the absorption coefficient measured in experiments on scanning the laser frequency is analyzed in relation to the acquisition time of fluorescence photons. The analysis is performed for the case where the optical band of an impurity center consists of two well-resolved lines. A method of processing the fluctuating fluorescence is proposed that makes it possible to obtain the averaged absorption coefficients, which can be compared with the coefficients calculated theoretically.
Introduction The quantum theory of atoms and molecules had its origin in the famous 1913 paper of Bohr [186] in which he suggested that the interaction of radiation and atoms occurred with the transition of an atom from one stationary internal state to another, accompanied by the emission or absorption of radiation of a frequency determined by the energy difference between these states. These transitions came to be known as ‘quantum jumps’. Since all experiments until the late 1970s used to be carried out with large ensembles of atoms and molecules, these jumps were masked and could not be directly observed; they were only inferred from spectroscopic data. In fact, with the advent of quantum mechanics they eventually came to be regarded as artefacts of Bohr's simple-minded semi-classical model. But with the availability of coherent light sources and single ions prepared in ion traps [187], [188], and optically cooled [189], [190], the issue has been reopened with the experimental demonstration of quantum jumps in single ions [191], [192], [193]. These experiments have also opened up the contentious issue of wave function collapse or reduction on measurement, as they can be regarded as making collapse visible on the oscilloscope screen [191]. Evidence of the discrete nature of quantum transitions in a single quantum system had been accumulating from the observation of photon anti-bunching in single-atom resonance fluorescence [194], the tunneling of single electrons in metal–oxide-–semiconductor junctions [195] and spin-flips of individual electrons in a Penning trap [196].
In this paper, we develop a quantum-jump approach for describing the photon emission process of single fluorophore systems coupled to complex classically fluctuating reservoirs. The formalism relies on an open quantum system approach where the dynamics of the system and the reservoir fluctuations are described through a density matrix whose evolution is defined by a Lindblad rate equation. For each realization of the photon-measurement processes it is possible to define a conditional system state (stochastic density matrix) whose evolution depends on both the photon detection events and the fluctuations between the configurational states of the reservoir. In contrast to standard fluorescent systems the photon-to-photon emission process is not a renewal one, being defined by a (stochastic) waiting time distribution that in each recording event parametrically depends on the conditional state. The formalism allows calculating experimental observables such as the full hierarchy of joint probabilities associated with the time intervals between consecutive photon recording events. These results provide a powerful basis for characterizing different situations arising in single-molecule spectroscopy, such as spectral fluctuations, lifetime fluctuations and light-assisted processes.
Abstract In recent years different microscopic and spectroscopic techniques were successfully employed to detect the fluorescence signals of single dye molecules in the condensed phase. In low temperature solids the fluorescence excitation line of a single dopant molecule is a high resolution optical probe of its local nanoenvironment. As inhomogeneous contributions and ensemble averaging are absent, many effects hidden when looking at large populations of molecules become observable. Two important research areas in single molecule spectroscopy deal with the investigation of quantum optical effects and the spectral jumping behaviour of single molecule excitation lines caused by fluctuations of two‐level systems inherent to disordered solids. Rather than delivering a complete presentation, we will focus on some selected examples which emphasize the unique features of spectroscopy with a single absorber.