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Physical Review A

Cumulant reduced density matrices as measures of statistical dependence and entanglement between electronic quantum domains with application to photosynthetic light harvesting

Julie T. Skolnik, David A. Mazziotti

Published Sep 27, 2013• 29 citations
Publisher

Abstract

Recent ultrafast spectroscopy experiments have demonstrated the entanglement of chromophores in photosynthetic light harvesting. Here we apply the cumulant parts of reduced density matrices (RDMs) to measure the statistical dependence and entanglement between electronic quantum domains in light harvesting. Because the cumulant RDMs are invariant to one-electron unitary transformations, they provide a measure of electron correlation and entanglement that is independent of the orbital basis set. Specifically, we apply the cumulant to a three-chromophore subsystem of the Fenna-Matthews-Olson complex, which has been shown to exhibit similar entanglement as the full system. Time-dependent Frobenius norms of the cumulant $p$-RDMs for $2\ensuremath{\le}p\ensuremath{\le}6$ reveal correlation and entanglement in groupings of $p$ chromophores. The results show that the entanglement of pairs of chromophores is significantly more important than the entanglement in higher groupings of the chromophores. Data from the model are generally consistent with recent findings from ultrafast spectroscopy. Beyond their application to light harvesting, the Frobenius norms of the cumulant RDMs provide a useful measure of statistical dependence, correlation, and entanglement between electronic quantum domains with applications to molecules and materials.

DOI
10.1103/physreva.88.032517

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Recent ultrafast spectroscopy experiments have demonstrated the entanglement of chromophores in photosynthetic light harvesting. Here we apply the cumulant parts of reduced density matrices (RDMs) to measure the statistical dependence and entanglement between electronic quantum domains in light harvesting. Because the cumulant RDMs are invariant to one-electron unitary transformations, they provide a measure of electron correlation and entanglement that is independent of the orbital basis set. Specifically, we apply the cumulant to a three-chromophore subsystem of the Fenna-Matthews-Olson complex, which has been shown to exhibit similar entanglement as the full system. Time-dependent Frobenius norms of the cumulant ppp-RDMs for 2\ensuremath≤p\ensuremath≤62\ensuremath{\le}p\ensuremath{\le}62\ensuremath≤p\ensuremath≤6 reveal correlation and entanglement in groupings of ppp chromophores. The results show that the entanglement of pairs of chromophores is significantly more important than the entanglement in higher groupings of the chromophores. Data from the model are generally consistent with recent findings from ultrafast spectroscopy. Beyond their application to light harvesting, the Frobenius norms of the cumulant RDMs provide a useful measure of statistical dependence, correlation, and entanglement between electronic quantum domains with applications to molecules and materials.
Abstract The Fenna-Matthews-Olson light-harvesting complexes have become a center of attention due to their high excitation energy transfer efficiency, which may surpass 98%. Experimental results reveal the presence of quantum entanglement in the protein structure, which seems to be associated to such a remarkably high efficiency. Notably, Redfield and Lindblad master equations have been already used to model the chromophore excitation in the protein complex as well as its interactions with the environment. The entanglement arising during the process has been quantified through several measures. This work provides a short review of methods used to model the above described system. It is also discussed the introduction of two nonconventional measures addresed to quantify the emergent entanglement.
Green sulfur bacteria is a photosynthetic organism whose light-harvesting complex accommodates a pigment-protein complex called Fenna-Matthews-Olson (FMO). The FMO complex sustains quantum coherence and quantum correlations between the electronic states of spatially separated pigment molecules as energy moves with nearly a 100% quantum efficiency to the reaction center. We present a method based on the quantum uncertainty associated to local measurements to quantify discord-like quantum correlations between two subsystems where one is a qubit and the other is a qudit. We implement the method by calculating local quantum uncertainty (LQU), concurrence, and coherence between subsystems of pure and mixed states represented by the eigenstates and by the thermal equilibrium state determined by the FMO Hamiltonian. Three partitions of the seven chromophores network define the subsystems: one chromophore with six chromophores, pairs of chromophores, and one chromophore with two chromophores. Implementation of the LQU approach allows us to characterize quantum correlations that had not been studied before, identify the most quantum correlated subsets of chromophores, and determine that, in the strongest associations of chromophores, the LQU is a monotonically increasing function of the coherence.
Ultrafast spectroscopy experiments show that photosynthetic systems can preserve quantum beats in the process of electronic energy transfer between pigments, even at room temperature. But what does this discovery imply for biology - and for quantum mechanics? This dissertation examines photosynthesis through the tools of quantum information science. We evaluate to what extent photosynthesis can be thought of as a type of quantum technology, and consider how we can apply experimental tools widely used for quantum technologies (state tomography and coherent control) to photosynthesis. Throughout, we use the Fenna-Matthews-Olson (FMO) complex of greens sulfur bacteria as a model photosynthetic pigment-protein complex.In Part I, we evaluate two mechanisms for the possible biological relevance of quantum coherent motion. First, we consider the extent to which dynamics in light-harvesting systems exhibit the quantum speedups characteristic of quantum algorithms and quantum walks. Second, we demonstrate a ratchet effect in electronic energy transfer enabled by partially coherent dynamics. To do so, we build a new model of energy transfer between weakly coupled light-harvesting complexes to understand how electronic coherence arises under natural conditions.In Part II, we present proposals for experimental probes of light harvesting dynamics with ultrafast spectroscopy. We show how the signal from pump-probe spectroscopy can be formally inverted to determine the excited state electronic density matrix. Finally, we examine the feasiblity of coherent control experiments on light-harvesting systems, and provide two realistic targets feasible with present-day technology.
Recent evidence of electronic coherence during energy transfer in photosynthetic antenna complexes has reinvigorated the discussion about whether coherence and/or entanglement have any practical functionality for these molecular systems. Here we investigate quantitative relationships between the quantum yield of a light-harvesting complex and the distribution of entanglement among its components. Our study focuses on the entanglement yield or average entanglement surviving a time scale comparable to the average excitation trapping time. We consider the Fenna–Matthews–Olson (FMO) protein of green sulfur bacteria as a prototype system and show that there is an inverse relationship between the quantum efficiency and the average entanglement between distant donor sites. Our results suggest that long-lasting electronic coherence among distant donors might help in the modulation of the light-harvesting function.
We review recent theoretical calculations of quantum entanglement in photosynthetic light harvesting complexes. These works establish, for the first time, a manifestation of this characteristically quantum mechanical phenomenon in biologically functional structures. We begin by summarizing calculations on model biomolecular systems that aim to reveal non-trivial characteristics of quantum entanglement in non-equilibrium biological environments. We then discuss and compare several calculations performed recently of excitonic dynamics in the Fenna-Matthews-Olson light harvesting complex and of the electronic entanglement present in this widely studied pigment-protein structure. We point out the commonalities between the derived results and also identify and explain the differences. We also discuss recent work that examines entanglement in the structurally more intricate light harvesting complex II (LHCII). During this overview, we take the opportunity to clarify several subtle issues relating to entanglement in such biomolecular systems, including the role of entanglement in biological function, the complexity of dynamical modeling that is required to capture the salient features of entanglement in such biomolecular systems, and the relationship between entanglement and other quantum mechanical features that are observed and predicted in light harvesting complexes. Finally, we suggest possible extensions of the current work and also review the options for experimental confirmation of the predicted entanglement phenomena in light harvesting complexes.
Transport phenomena in photosynthetic systems have attracted a great deal of attention due to their potential role in devising novel photovoltaic materials. In particular, energy transport in light-harvesting complexes is considered quite efficient due to the balance between coherent quantum evolution and decoherence, a phenomenon coined Environment-Assisted Quantum Transport (ENAQT). Although this effect has been extensively studied, its behavior is typically described in terms of the decoherence’s strength, namely weak, moderate or strong. Here, we study the ENAQT in terms of quantum correlations that go beyond entanglement. Using a subsystem of the Fenna–Matthews–Olson complex, we find that discord-like correlations maximize when the subsystem’s transport efficiency increases, while the entanglement between sites vanishes. Our results suggest that quantum discord is a manifestation of the ENAQT and highlight the importance of beyond-entanglement correlations in photosynthetic energy transport processes.