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Journal of Physics B Atomic Molecular and Optical Physics

Non-Markovian stochastic Schrödinger description of transport in quantum networks

Inés de Vega

Published Nov 28, 2011• 14 citations
Publisher

Abstract

We investigate the role of non-Markovian effects in the efficiency of excitation transfer in a ring-hub arrangement of interacting two-level systems coupled to an environment. The structure is similar to light-harvesting antennae connected to a reaction centre as it is found in natural photosynthetic systems. To this order, we use stochastic Schrödinger equations within the so-called post-Markov approximation, which under certain constraints allows us to describe the system beyond the Markov and the weak-coupling approximations commonly used in the literature.

DOI
10.1088/0953-4075/44/24/245501

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We investigate the role of non-Markovian effects in the efficiency of excitation transfer in a ring-hub arrangement of interacting two-level systems coupled to an environment. The structure is similar to light-harvesting antennae connected to a reaction centre as it is found in natural photosynthetic systems. To this order, we use stochastic Schrödinger equations within the so-called post-Markov approximation, which under certain constraints allows us to describe the system beyond the Markov and the weak-coupling approximations commonly used in the literature.
We analyze several aspects of the transport dynamics in the LH1-RC core of purple bacteria, which consists basically in a ring of antenna molecules that transport the energy into a target molecule, the reaction center, placed in the center of the ring. We show that the periodicity of the system plays an important role to explain the relevance of the initial state in the transport efficiency. This picture is modified, and the transport enhanced for any initial state, when considering that molecules have different energies, and when including their interaction with the environment. We study this last situation by using stochastic Schr{ö}dinger equations, both for Markovian and non-Markovian type of interactions.
A variety of open quantum networks are currently under intense examination to model energy transport in photosynthetic systems. Here, we study the coherent transfer of a quantum excitation over a network incoherently coupled with a structured and small environment that effectively models the photosynthetic reaction center. Our goal is to distill a few basic, possibly universal, mechanisms or effects that are featured in simple energy-transfer models. In particular, we identify three different phenomena: the congestion effect, the asymptotic unitarity, and the staircase effects. We begin with few-site models, in which these effects can be fully understood, and then proceed to study more complex networks similar to those employed to model energy transfer in light-harvesting complexes.
We investigate the quantum coherent dynamics of an externally driven effective two-level system (TLS) subjected to a slow Ohmic environment characteristic of biomolecular protein–solvent reservoirs in photosynthetic light-harvesting complexes. By means of the numerically exact quasi-adiabatic propagator path integral (QUAPI) method we are able to include non-Markovian features of the environment and show the dependence of the quantum coherence on the characteristic bath cut-off frequency ω c as well as on the driving frequency ω l and the field amplitude A . Our calculations extend from the weak-coupling regime to the incoherent strong-coupling regime. In the latter case, we find evidence for a resonant behaviour, beyond the expected behaviour, when the reorganization energy E r coincides with the driving frequency. Moreover, we investigate how the coherent destruction of tunnelling within the TLS is influenced by the non-Markovian environment.
We analytically derive transfer probabilities and efficiencies for an artificial light-harvesting photosynthetic system, which consists of a ring coupled to a central acceptor. For an incident photon pair, we find near-perfect single excitation transfer efficiency with negligible double excitation transfer in the weak-coupling regime. In the strong-coupling regime, single excitation transfer efficiency was greater than 9090%90, while the double excitation efficiency was approximately 5050%50. We have found that the three main factors which determine high transfer efficiencies are large acceptor probabilities, long acceptor decay times, and strong photon-ring coupling. A possible implementation of the theoretical framework to bio-inspired solar energy devices is also discussed.
It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schr\"odinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schr\"odinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.
When dealing with system-reservoir interactions in an open quantum system, such as a photosynthetic light-harvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations, and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of non-Markovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequently-used perturbative but non-Markovian approximations to a photosynthetic dimer model and compare their results with that of the numerically-exact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and non-Markovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that non-Markonivity can, counter-intuitively, increase with temperature, and with the coupling to the environment.