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We show that the Franck-Condon Factor (FCF) associated to a transitionbetween initial and final vibrational states in two different potential energysurfaces, having $N$ and $M$ vibrational quanta, respectively, is equivalent tocalculating the number of perfect matchings of a weighted graph with loops thathas $P = N+M$ vertices. This last quantity is the loop hafnian of the(symmetric) adjacency matrix of the graph which can be calculated in $O(P^32^{P/2})$ steps. In the limit of small numbers of vibrational quanta per normalmode our loop hafnian formula significantly improves the speed at which FCFscan be calculated. Our results more generally apply to the calculation of thematrix elements of a bosonic Gaussian unitary between two multimode Fock stateshaving $N$ and $M$ photons in total and provide a useful link between certaincalculations of quantum chemistry, quantum optics and graph theory.

Franck-Condon factors by counting perfect matchings of graphs with loops