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We study one- and two-photon scattering from a qubit embedded in aone-dimensional waveguide in the presence of modal dispersion. We use aresolvent based analysis and utilize techniques borrowed from the Lee modelstudies. Modal dispersion leads to atom-photon bound states which necessitatethe use of multichannel scattering theory. We present multichannel scatteringmatrix elements in terms of the solution of a Fredholm integral equation of thesecond kind. Through the use of the Lippmann-Schwinger equation, we derive aninfinite series of Feynman diagrams that represent the solution to the integralequation. We use the Feynman diagrams as vertex correction terms to come upwith closed form formulas that successfully predict the trapping rate of aphoton in the atom-photon bound state. We verify our formalism throughKrylov-subspace based numerical studies with pulsed excitations. Our resultsprovide the tools to calculate the complex correlations between scatteredphotons in a dispersive environment.

Effects of Modal Dispersion on Few Photon - Qubit Scattering in One-Dimensional Waveguides