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Abstract We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.
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We consider the propagation of light in a random medium of two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a two-photon state and show that at long times and large distances, the corresponding average probability densities can be determined from the solutions to a system of kinetic equations.more » « less
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We consider the quantum electrodynamics of single photons in arrays of one-way waveguides, each containing many atoms. We investigate both chiral and antichiral arrays, in which the group velocities of the waveguides are the same or alternate in sign, respectively. We find that in the continuum limit, the one-photon amplitude obeys a Dirac equation. In the chiral case, the Dirac equation is hyperbolic, while in the antichiral case it is elliptic. This distinction has implications for the nature of photon transport in waveguide arrays. Our results are illustrated by numerical simulations.more » « less
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We investigate the inverse scattering problem for scalar waves. We report conditions under which the terms in the inverse Born series cancel in pairs, leaving only one term at each order. We refer to the resulting expansion as the reduced inverse Born series. The reduced series can also be derived from a nonperturbative inversion formula. Our results are illustrated by numerical simulations that compare the performance of the reduced series to the full inverse Born series and the Newton–Kantorovich method.more » « less
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Abstract We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach spaces. An application to the inverse scattering problem with diffuse waves is described.more » « less
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We consider the theory of spontaneous emission for a random medium of stationary two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a one-photon state of the system. At long times and large distances, we show that the corresponding average probability densities can be determined from the solutions to a pair of kinetic equations.more » « less
