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Multipartite quantum entanglement evolution in photosynthetic complexes
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10.1063/1.4742333
/content/aip/journal/jcp/137/7/10.1063/1.4742333
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/7/10.1063/1.4742333

Figures

Image of FIG. 1.
FIG. 1.

Evolution of pairwise concurrences in the FMO complex when site one is initially excited at = 77 K. All 21 pairwise concurrences computed by the convex roof – these are equal to for each subsystem of two sites, computed across the single bipartition of the pair are also shown. For entanglements 1|2 and 1|3 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Image of FIG. 2.
FIG. 2.

Entanglement evolution in the FMO complex when site six is initially excited at = 77 K. All 21 pairwise entanglements computed by the convex roof are also shown. For entanglements 5|6 and 4|5 we also plot the exact concurrence – the agreement is good enough that the difference between the convex roof and the exact calculation is not visible.

Image of FIG. 3.
FIG. 3.

Monogamy bound and convex roof computation of entanglements 1|34, 2|34, 12|3, and 12|4. Particularly in the first 200 fs the convex roof closely matches the monogamy bound.

Image of FIG. 4.
FIG. 4.

Entanglement evolution of FMO complex when site 1 is initially excited at cryogenic temperature = 77 K. The triplet site entanglement among site 1, 2, and 3 and also the pairwise site entanglement between any two of site 1, 2, and 3 are plotted. The left panel shows the dynamics of the entanglement for the system alone while the right considers the effect of the environment.

Image of FIG. 5.
FIG. 5.

Time evolution of entanglement for multiple sites at = 300 K. In the upper panel the entanglement measures across the indicated bipartitions among sites 1, 2, and 3 are shown when site 1 is initially excited. For the lower panel, site 6 is initially excited.

Image of FIG. 6.
FIG. 6.

Measures of entanglement and monogamy bounds in a four qubit system when site 1 is initially excited at temperature = 77 K. The measures computed for bipartitions 4|123, 7|123, and 7|456 by the convex roof and together with the monogamy bound are shown here. We see a larger variation in performance of the convex roof optimization here, with a smaller difference between the upper (convex roof) and lower (monogamy) bounds for 7|456 and 7|123 than for 4|123.

Image of FIG. 7.
FIG. 7.

Time evolution of various entanglement measures for subsystem 1234 for site 1 initially excited at = 77 K. The concurrence for subsystem 34 across bipartition 3|4 and the measures for subsystems 123 and 124 across bipartitions 3|12 and 4|12, respectively, are also shown. The measure across bipartition 4|123 is also shown.

Image of FIG. 8.
FIG. 8.

Entanglement measures for the four chromophore subsystem 1234 when site 1 is initially excited at temperature = 77 K. The measure across bipartition 12|34 was computed via the convex roof procedure and is shown here, together with the concurrences for pairs of chromophores 13, 14, 23, and 24. We note that in this case, we see that the entanglement 12|34 evolves similarly to both the 1|3 and 2|3 concurrences.

Image of FIG. 9.
FIG. 9.

Time evolution of concurrences and measures for the FMO complex when site 6 is initially excited at temperature = 77 K. The measures are shown are for subsystem 4567 across bipartitions 4|567, 5|467, 6|457, and 7|456. We also show the concurrences among the pairs of sites that determine the concurrence bounds for the measures across bipartitions 4|567, 5|467, 6|457, and 7|456, and the concurrence bounds themselves. The left panel shows the isolated system evolution and the right panel shows the open system dynamics with environment.

Image of FIG. 10.
FIG. 10.

Evolution of entanglement measures for the subsystems of chromophores 12345, 12346, and 12347 in the FMO complex at cryogenic temperature = 77 K for site 1 is initially excited. The measures are computed across bipartitions 5|1234, 6|1234, and 7|1234 and the corresponding monogamy bounds are also shown. Site 1, 2, 3, and 4 are sites evolved in the population pathway under this initial condition, and this data indicates that the entanglement of this subset (1234) of chromophores with the other three chromophores is small.

Image of FIG. 11.
FIG. 11.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at cryogenic temperature 77 K. The measures are shown for subsystem 14567 across bipartition 1|4567 and subsystem 24567 across bipartition 2|4567.

Image of FIG. 12.
FIG. 12.

Time evolution of entanglement measures in the FMO complex for site 6 initially excited at 77 K. (cf. Figure 7 ). The measures are computed for subsystems 3567 across bipartition 3|567, subsystem 4567 across bipartition 4|567 and subsystem 34567 across bipartition 3|4567.

Image of FIG. 13.
FIG. 13.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 1|23456, 4|123567, 5|123467, 7|123456 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 1 with the rest, 1|23456.

Image of FIG. 14.
FIG. 14.

Entanglement measures for the full FMO system at 77 K with site one initially excited. The measures are shown for the partitions 2|13456, 3|124567, 6|123457 (solid lines), together with the corresponding monogamy bounds (dotted lines). These results illustrate the performance of the convex roof optimization and also show that the largest of these measures is that which gives the entanglement of chromophore 2 with the rest, 1|23456.

Image of FIG. 15.
FIG. 15.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These four plots show measures computed across all 21 bipartitions of the seven chromophore system into a pair of chromophores and the remaining quintuplet. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

Image of FIG. 16.
FIG. 16.

Entanglement measures for the full FMO system at 77 K with site one initially excited. These six plots show measures computed across all 35 bipartitions of the seven chromophore system into three of chromophores and the remaining four. Any measure that includes either chromophore 1 or chromophore 2 (but not both) on one side of the bipartition exhibits oscillations and the value of the measure is large. Any measure that has both chromophore 1 and 2 on the same side of the bipartition takes lower values and exhibits rapid growth in the first 100 fs, but never exceeds 0.5 in value.

Image of FIG. 17.
FIG. 17.

A comparison of the effects of adding in the two-exciton subspace for different values of |α2|. The concurrence between sites one and two is plotted for the density matrix in Eq. (21) , with |α2| = 0.5, 0.1, 0.01.

Image of FIG. 18.
FIG. 18.

A comparison of the effects of adding in the two-exciton subspace γ|α|2|0000011⟩⟨0000011| for different values of γ, with |α|2 = 0.5. The concurrence between sites one and two is plotted for the density matrix in (22) , with ρ = |0000011⟩⟨0000011|.

Image of FIG. 19.
FIG. 19.

A curve showing how the amplitude of the concurrence between sites 1 and 2 at 21 fs varies as a function of γ, with the density matrix from (22) , with ρ = |0000011⟩⟨0000011| and |α|2 = 0.5.

Tables

Generic image for table
Table I.

Subsystems and bipartite cuts relevant to the FMO system. One may take a subsystem reduced density matrix of any ⩽ 7 and consider all the bipartite cuts of each subsystem. This leads to a combinatoric explosion of different bipartite measures. Evidently it would be simpler to consider all cuts of the total system. We perform such convex roof calculations for the full seven chromophore system by restricting the convex roof to the single exciton manifold.

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/content/aip/journal/jcp/137/7/10.1063/1.4742333
2012-08-21
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Multipartite quantum entanglement evolution in photosynthetic complexes
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/7/10.1063/1.4742333
10.1063/1.4742333
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