#### Coherence maps reveal the pathways of energy flow in the LH2 complex

Coherence maps and instantaneous state-to-state rate (SSR) maps [*J. Phys. Chem*. B **126**, 9361-9375] offer an intuitive and very informative tool for investigating the quantum dynamics of energy transfer. We use coherence maps to observe the quantum superpositions that are created, sustained and damped by vibrational modes during the transfer of excitation energy from the B800 (outer) to the B850 (inner) ring of the light harvesting complex 2 (LH2) of purple bacteria with a variety of initial conditions. The reduced density matrix of the 24-pigment complex, where the ground and excited electronic states of each bacteriochlorophyll are explicitly coupled to 50 intramolecular vibrations at room temperature, is obtained from fully quantum mechanical small matrix path integral (SMatPI) calculations. The coherence maps show a very rapid localization within the outer ring, accompanied by the formation of inter-ring quantum superpositions that evolve to a partial quantum delocalization at equilibrium, and quantify in state-to-state detail the flow of energy within the complex.

The SSR map of LH2 consists of a 16×16 block that contains the instantaneous time derivatives of excited BChl populations within the inner (B850) ring, an 8×8 block that contains similar time derivatives within the outer (b800) ring, and two rectangular blocks that correspond to ring-to-ring transfer. Blue and white correspond to positive and negative rates, respectively.

The figure below shows the coherence and instantaneous SSR maps with the initial excitation placed on BChl 17. The left and middle columns show the real and imaginary parts of the RDM elements. Blue and green correspond to positive regions, while red and magenta correspond to negative regions. The right column shows the SSR map.