Abstract
We employ another approach to quantize electromagnetic fields in the coordinate space, instead of the mode (or Fourier) space, such that local features of photons can be efficiently, physically, and more intuitively described. To do this, coordinate-ladder operators are defined from mode-ladder operators via the unitary transformation of systems involved in arbitrary inhomogeneous dielectric media. Then, one can expand electromagnetic field operators through the coordinate-ladder operators weighted by non-orthogonal and spatially-localized bases, which are propagators of initial quantum electromagnetic (complex-valued) field operators. Here, we call them QEM-CV-propagators. However, there are no general closed form solutions available for them. This inspires us to develop a quantum finite-difference time-domain (Q-FDTD) scheme to numerically time evolve QEM-CV-propagators. In order to check the validity of the proposed Q-FDTD scheme, we perform computer simulations to observe the Hong-Ou-Mandel effect resulting from the destructive interference of two photons in a 50/50 quantum beam splitter.
Highlights
The standard method to solve quantum Maxwell’s equations [1,2,3,4,5] is via canonical quantization where electromagnetic (EM) fields in the vacuum are quantized in the mode space [6], inspired by the motion of uncoupled harmonic oscillators and Hamiltonian mechanics, and many textbooks [4,7,8,9,10,11,12,13,14,15] explain the process in detail
The fundamental assumption in the above is that photon is treated as the smallest energy lump, which is carried in the form of EM fields, having a definite value of hω while being spatially indeterministic. These formulations correctly account for the anomalous observations including black-body radiation and photoelectric effects; it is possible, though, inefficient and, more importantly, physically less intuitive to characterize local features of photons observed in many quantum optics experiments
As an effective solution to such difficulty, for the first time, we recently proposed in Reference [36] the so-called numerical canonical quantization in which normal modes are numerically obtained by solving the Helmholtz wave equations for arbitrary inhomogeneous dielectric media through computational electromagnetic (CEM) tools such as finite-difference or -element methods
Summary
The standard method to solve quantum Maxwell’s equations [1,2,3,4,5] is via canonical quantization where electromagnetic (EM) fields in the vacuum are quantized in the mode (or Fourier) space [6], inspired by the motion of uncoupled harmonic oscillators and Hamiltonian mechanics, and many textbooks [4,7,8,9,10,11,12,13,14,15] explain the process in detail. The fundamental assumption in the above is that photon is treated as the smallest energy lump, which is carried in the form of EM fields, having a definite value of hω while being spatially indeterministic (for example, the expectation value of the energy density for a monochromatic photon in the vacuum is uniform over all space due to the characteristics of a plane wave) As a consequence, these formulations correctly account for the anomalous observations including black-body radiation and photoelectric effects; it is possible, though, inefficient and, more importantly, physically less intuitive to characterize local features of photons observed in many quantum optics experiments. The proposed Q-FDTD scheme would be a useful and accessible tool for theoretical/experimental scientists/laboratories in quantum optics
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