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The Journal of Chemical Physics

Generalized non-Markovian optical Bloch equations

Adrián A. Budini

Published Feb 1, 2007• 25 citations
Publisher

Abstract

By considering single chromophore systems whose radiative decay can be written in terms of a nonlocal Lindblad-type evolution, the authors extend the formalism of generalized optical Bloch equations [Y. Zheng and F. L. H. Brown, Phys. Rev. Lett. 90, 238305 (2003)] to non-Markovian dynamics. They demonstrate that photon statistical properties such as bunching and antibunching, as well as sub- and super-Poissonian photon statistics can be fitted in the context of non-Markovian dynamics. The nonlocal effects may arise due to the interaction with a complex structured environment. In this case, the photon statistics can be related with the parameters that define the microscopic system-environment interaction. Alternatively, the authors demonstrate that effective dynamics such as triplet blinking, where the system is coupled via incoherent transitions to an extra dark state, can also be worked out in terms of generalized non-Markovian optical Bloch equations. The corresponding memory contributions are mapped with those that arise from the microscopic approach.

DOI
10.1063/1.2434148

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By considering single chromophore systems whose radiative decay can be written in terms of a nonlocal Lindblad-type evolution, the authors extend the formalism of generalized optical Bloch equations [Y. Zheng and F. L. H. Brown, Phys. Rev. Lett. 90, 238305 (2003)] to non-Markovian dynamics. They demonstrate that photon statistical properties such as bunching and antibunching, as well as sub- and super-Poissonian photon statistics can be fitted in the context of non-Markovian dynamics. The nonlocal effects may arise due to the interaction with a complex structured environment. In this case, the photon statistics can be related with the parameters that define the microscopic system-environment interaction. Alternatively, the authors demonstrate that effective dynamics such as triplet blinking, where the system is coupled via incoherent transitions to an extra dark state, can also be worked out in terms of generalized non-Markovian optical Bloch equations. The corresponding memory contributions are mapped with those that arise from the microscopic approach.
The statistics of photon emission from a single molecule under continuous wave excitation are considered. In particular, we study stochastic model systems where photon emission rates evolve in time with non-Markovian dynamics. Our calculations are based on the recently introduced generalized optical Bloch equation (GBE) formalism, but with numerical complications beyond those seen in previous Markovian stochastic models. A spectral representation is introduced to facilitate the numerical solution of the GBE equations for these more challenging cases.
In this paper we derive an extra class of non-Markovian master equations where the system state is written as a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between all of them, resembling the structure of a classical rate equation. The system dynamics may develop strong nonlocal effects such as the dependence of the stationary properties with the system initialization. These equations are derived from alternative microscopic interactions, such as complex environments described in a generalized Born-Markov approximation and tripartite system-environment interactions, where extra unobserved degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Conditions that guarantee the completely positive condition of the solution map are found. Quantum stochastic processes that recover the system dynamics in average are formulated. We exemplify our results by analyzing the dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.
We derive the generating function for single molecule photon emission events within the context of the stochastically modulated optical Bloch equations. Statistical properties of single molecule photon counting experiments are deduced from a set of coupled differential equations only slightly more complicated than the Bloch equations themselves. This formulation allows for the study of photon bunching and antibunching within a single theoretical framework and provides a description of these behaviors that emphasizes the connection with single molecule experiments. Application is made to the spectroscopy of a chromophore coupled to a single two level system.
In this paper, we formulate a quantum jump approach associated with single fluorescent systems whose radiative decay is defined by a Lindblad-like equation with a memory kernel. We base our results in a generalized Born–Markov approximation, which allows us to get our results from a full microscopic description. In the weak-laser-excitation regime, the photon-emission process can be described through a delay function, which in a natural way provides the basis for unravelling the dynamics in terms of a stochastic density matrix. As in the Markovian case, the stochastic dynamics consists in periods where the evolution is smooth and non-unitary, interrupted at random times where the photon-detection events produce a sudden change in the system state. The phenomenon of non-Poissonian intermittent fluorescence induced by complex environments (Budini 2006 Phys. Rev. A 73 061802) is analysed in the context of this approach.
We analyze the appearance of non-Markovian effects in the dynamics of a bipartite system coupled to a reservoir, which can be described within a class of non-Markovian equations given by a generalized Lindblad structure. A master equation is derived, which we term the quantum Bloch-Boltzmann equation, describing both motional and internal states of a test particle in a quantum framework. When due to the preparation of the system or to decoherence effects one of the two degrees of freedom is amenable to a classical treatment and not resolved in the final measurement, though relevant for the interaction with the reservoir, non-Markovian behaviors such as stretched exponential or power law decay of coherences can occur.
We investigate the transient dynamics of photon statistics through two-time correlation functions for optical field coupled to a non-Markovian environment, described by the Fano-type Hamiltonian. We exactly solve the time-evolution of an initially nonclassical state which exhibits photon antibunching. We find that the transient correlations at different times ttt yield a smooth transition from antibunching to bunching photon statistics in the weak system-environment coupling regime. In the strong-coupling regime, the two-time correlations exhibit oscillations that persists both in the transient process and in the steady-state limit. The oscillatory behavior of photon statistics is a manifestation of strong non-Markovian memory dynamics where the system remains in nonequilibrium from its environment. We also find that the antibunching-to-bunching transition in the weak-coupling regime and the photon statistical oscillations in the strong-coupling regime are strongly influenced by the environment temperature.