Photosynthetic antenna complexes are responsible for absorbing energy from sunlight and transmitting it to remote locations where it can be stored. Recent experiments have found that this process involves long-lived quantum coherence between pigment molecules, called chromophores, which make up these complexes Expected to decay within 100 fs at room temperature, these coherences were instead found to persist for picosecond time scales, despite having no apparent isolation from the thermal environment of the cell. This paper derives a quantum master equation which describes the coherent evolution of a system in strong contact with a thermal environment. Conditions necessary for long coherence lifetimes are identi?ed, and the role of coherence in ef?cient energy transport is illuminated. Static spectra and exciton transfer rates for the PE545 complex of the cryptophyte algae CS24 are calculated and shown to have good agreement with experiment.

Fully analytic formulas, which do not involve any numerical integration, are derived for the discretized influence functionals of a very extensive assortment of spectral distributions. For Feynman integrals derived using the Trotter splitting and Strang splitting, we present general formulas for the discretized influence functionals in terms of proper integrals of the bath response function. When an analytic expression exists for the bath response function, these integrals can almost always be evaluated analytically. In cases where these proper integrals cannot be integrated analytically, numerically computing them is much faster and less error-prone than calculating the discretized influence functionals in the traditional way, which involves numerically calculating integrals whose bounds are both infinite. As an example, we present the analytic discretized influence functional for a bath response function of the form which is a natural form for many spectral distribution functions (including the very popular Lorentz-Drude/Debye function), and for other spectral distribution functions it is a form that is easily obtainable by a least-squares fit. Evaluating our analytic formulas for this example case is much faster and more rigorous than numerically calculating the discretized influence functional in the traditional way. In the appendix we provide analytic expressions for pj and Oj for a variety of spectral distribution forms, and as a second example we provide the analytic bath response function and analytic influence functionals for spectral distributions of the form The value of the analytic expression for this bath response function extends beyond its use for calculating Feynman integrals. We also provide open source MATLAB and Mathematica programs to make the results of this paper very easy to implement.