Numerical quantization of interfering photons using liquid crystal retarder

Numerical quantization of interfering small photons using a liquid crystal (LC) retarder is investigated. A beam splitter can be considered a four-port device with the input-output relations in the Heisenberg picture using the creation and annihilation operator. We introduce the phase shift in one of the annihilation operators using the LC device, where, the amplitude of these operations is ultimately reduced. The alignment of an LC director is homogeneous and the director profile is calculated by solving the Euler-Lagrange equation. A total number of retardations can be calculated using the effective refractive index of the LC molecule integrating through cell thickness direction. As a result, a detail of the quantization operation is clarified.