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Long-lived exciton coherences have been recently observed in photosyntheticcomplexes via ultrafast spectroscopy, opening exciting possibilities for thestudy and design of coherent exciton transport. Yet, ambiguity in thespectroscopic signals has led to arguments for interpreting them in terms ofthe exciton dynamics, demanding more stringent tests. We propose a novelstrategy, Quantum Process Tomography (QPT) for ultrafast spectroscopy, toreconstruct the evolving quantum state of excitons in double-walledsupramolecular light-harvesting nanotubes at room temperature. The protocolcalls for eight transient grating experiments with varied pulse spectra. Ouranalysis reveals unidirectional energy transfer from the outer to the innerwall excitons, absence of nonsecular processes, and an unexpected coherencebetween those two states lasting about 150 femtoseconds, indicating weakelectronic coupling between the walls. Our work constitutes the firstexperimental QPT in a 'warm' and complex system, and provides an elegant schemeto maximize information from ultrafast spectroscopy experiments.

Coherent exciton dynamics in supramolecular light-harvesting nanotubes revealed by ultrafast quantum process tomography