Quantum control model for spatial propagation of electromagnetic fields in dielectrics

The design of many quantum optical devices is attributed to the determination of spatially distributed parameters (e.g., refraction index in photonic crystal), which calls for a theoretical description for the spatial propagation of quantum fields. This paper discusses the spatial evolution of quantized electromagnetic fields in waveguide with spatially variant dielectric parameters, which is in parallel with the time evolution in the standard input-output formalism [C. W. Gardiner and M. J. Collett, Phys.Rev.A 31, 3761 (1985)]. This formalism is non-trivial to quantum optics in nonlinear and dispersive medium, and provides a new perspective of modeling and analyzing the control of quantum fields by matter.