Three-pulse photon echo of finite numbers of molecules: single-molecule traces

In conventional bulk nonlinear spectroscopy, the contribution from molecules with different environmental conditions sometimes conceals the properties of interest and prevents the assessment of the heterogeneity of complex systems. This is especially true when exploring mechanisms of coherence loss in multicomponent systems [Ishizaki and Fleming, J. Phys. Chem. B 2011, 115, 6227]. To avoid this drawback of ensemble measurements and evaluate single-molecule behavior, a quantum theory is proposed to study the three-pulse photon echo signal of a two-level system in a bath and reveal the fluctuations inherent to single molecules. The current method takes advantage of the coherent state representation to understand the photon echo experiment in a wave function formalism rather than the reduced density matrix. Information regarding the environmental degrees of freedom (DoF) is explicitly encoded in the initial state of the system plus bath. The thermal fluctuations of the initial states induce variation of the photon echo signal, which is clearly different from the ensemble average echo signal. We use our formalism to demonstrate the recovery of the conventional ensemble response signal from the single-molecule signal.