Macroscopic coherence as an emergent property in molecular nanotubes

Abstract

Nanotubular molecular self-aggregates are characterized by a high degree of symmetry and they are fundamental systems for light-harvesting and energy transport. While coherent effects are thought to be at the basis of their high efficiency, the relationship between structure, coherence and functionality is still an open problem. We analyse natural nanotubes present in Green Sulphur Bacteria. We show that they have the ability to support macroscopic coherent states, i.e. delocalized excitonic states coherently spread over many molecules, even at room temperature. Specifically, assuming a canonical thermal state we find, in natural structures, a large thermal coherence length, of the order of 1000 molecules. By comparing natural structures with other mathematical models, we show that this macroscopic coherence cannot be explained either by the magnitude of the nearest-neighbour coupling between the molecules, which would induce a thermal coherence length of the order of 10 molecules, nor by the presence of long-range interactions between the molecules. Indeed we prove that the existence of macroscopic coherent states is an emergent property of such structures due to the interplay between geometry and cooperativity (superradiance and super-transfer). In order to prove that, we give evidence that the lowest part of the spectrum of natural systems is determined by a cooperatively enhanced coupling (super-transfer) between the eigenstates of modular sub-units of the whole structure. Due to this enhanced coupling strength, the density of states is lowered close to the ground state, thus boosting the thermal coherence length. As a striking consequence of the lower density of states, an energy gap between the excitonic ground state and the first excited state emerges. Such energy gap increases with the length of the nanotube (instead of decreasing as one would expect), up to a critical system size which is close to the length of the natural complexes considered.

On the existence of superradiant excitonic states in microtubules

“…Our analysis shows that as the number of tubulin subunits considered grows, a superradiant state forms in the lowest exciton state of the system. This is exactly what happens in many photosynthetic antenna complexes, such as in green sulfur bacteria cylindrical antennas [19,20,37] and in self-assembled molecular nanotubes [38][39][40][41][42][43]. Superradiant states favor the absorption of photons by the microtubule.…”

“…Therefore, states with a large dipole strength will have a supertransfer coupling proportional to the dipole strength of the eigenstates. Note that the coupling between eigenstates with a small dipole strength can give rise to a subtransfer effect, which has been shown in [37]. In the lower panel of figure 5, the coupling between the lowest exciton states with a large dipole strength (blue circles) and between the most excited states with a very small dipole strength (green triangles) of two blocks of 13 spirals is compared with the average coupling between the molecules of each block (red squares).…”

“…The result in figure 5 clearly shows that the lowest exciton state of the whole structure mainly consists of a superposition of lowest exciton states of smaller blocks of spirals. This non-trivial result arises from the symmetry of the systems, see discussion in [37]. A very interesting consequence of this is that the total lowest exciton state emerges from coupling between the lowest exciton states of smaller blocks.…”

“…The above results show the relevance of native symmetry in excitonic energy transport through the microtubule. We would like to point out that several recent works have highlighted the role of symmetry in enhancing coherent [37] and transport properties of molecular networks [49]. Figure 6.…”

“…However, we note that robustness to disorder could also be connected to supertransfer, and not only to the long-range of the interaction. For further details see the discussion in [37] and in appendix C.…”