Entanglement and entangling power of the dynamics in light-harvesting complexes

We study the evolution of quantum entanglement during exciton energy transfer (EET) in a network model of the Fenna-Matthews-Olson (FMO) complex, a biological pigment-protein complex involved in the early steps of photosynthesis in sulfur bacteria. The influence of Markovian as well as spatially and temporally correlated (non-Markovian) noise on the generation of entanglement across distinct chromophores (site entanglement) and different excitonic eigenstates (mode entanglement) is studied for different injection mechanisms, including thermal and coherent laser excitation. Additionally, we study the entangling power of the FMO complex under natural operating conditions. While quantum information processing tends to favor maximal entanglement, near unit EET is achieved as the result of an intricate interplay between coherent and noisy processes where the initial part of the evolution displays intermediate values of both forms of entanglement.

Figure 1

Entanglement of different bi-partitions of the FMO complex dynamics subject to a local form of noise where each FMO site couples to a damped resonant mode. The effective dephasing rates are taken to be equal to the optimal values derived for a fully Markovian model, as in Ref. [7], but the degree of Markovianity of the model can be tuned by varying the parameter $f$, as described in the text. The entanglement curves are for the 6 splits of the form $(1,\dots ,i)|(i+1,\dots ,7)$, $i\in \{1,\dots ,6\}$, where $i$ corresponds to FMO site $i$. The behavior of $p_{\mathrm{sink}}$ for different values of $f$ is also shown. Large values of $f$ render the dynamics fully Markovian and we recover the results of Ref. [7], whereas one can observe longer coherence times while at the same time enhancing $p_{\mathrm{sink}}$ in the low-$f$ domain, as the one exemplified by the value $f=1$. Note the nonmonotonic behavior of the transfer rate as a function of $f$, as emphasized by the arrows.

Figure 2

Entanglement of different bipartitions of the FMO complex vs. time ($\mathrm{ps}$), in the presence of a tunable source of noise correlations described by a nonlocal bosonic bath and for different values of the ratio $f$. For small $f$, the environmental correlations yield a non-Markovian form of noise and the time scale for entanglement persistence increases while, within this model, the efficiency for transport $p_{\mathrm{sink}}$ decreases.

Figure 3

Contour plot for the transfer efficiency in the FMO complex as a function of time (in $\mathrm{ps}$) and of the parameter $f$ for a non-Markovian model with a nonlocal bosonic bath linearly interacting with the FMO.

Figure 4

Contour plot for the entangling power in the FMO complex, as a function of time (in $\mathrm{ps}$) and of the parameter $f$, for the nonlocal-bath non-Markovian model. For low-noise levels, the entangling power of the evolution can be nonmonotonic in time, indicating coherent oscillations in the system. The entangling power decreases monotonically with increasing noise level. In comparison, the entangling power of a CNOT gate is 1, and a SWAP gate is 2. This means that the FMO complex has nontrivial entangling power for a large parameter range.

Figure 5

Entanglement, quantified by the logarithmic negativity, in the FMO complex for local (dashed lines) and spatially correlated (continuous lines) dephasing noise, in presence of the thermal injection bath ($n_{\mathrm{th}}=100$, ${\Gamma }_1=1$), within the Markov approximation. The inset shows the site population behavior as a function of time.

Figure 6

Entanglement in an FMO complex (evolving by a Markovian dynamics) when irradiated with a laser pulse (see the text for more details). Dephasing noise is either local (dashed lines) or spatially correlated (continuous lines). Strong entanglement between sites $1$ and $2$ is clearly illustrated by the behavior of the top curve [bipartition $1-(2-7)$] as opposed to the bottom one [bipartition $(1-2)-(3-7)$]. Similarly, strong entanglement between sites $5$ and $6$ is best exemplified by the central plot where we compute the entanglement across bipartition $(1-5)-(6-7)$. The inset shows the site population distribution in the FMO complex for $t=75\hspace{0.5em}\mathrm{fs}$.

Figure 7

The logarithmic negativity in the FMO complex (evolving by Markovian dynamics) for local dephasing noise in the exciton basis. Initially, one excitation is in site $1$. The curves are for the 6 splits of the form $(1,\dots ,i)|(i+1,\dots ,7)$, $i\in \{1,\dots ,6\}$, where now $i$ corresponds to exciton $i$ and they are ordered with increasing exciton energies. Inset shows exciton population vs. time ($\mathrm{ps}$).