# Quantum Electromagnetic Finite-Difference Time-Domain Solver

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Canonical Quantization

## 3. Quantization of Electromagnetic Fields in the Coordinate Space

#### 3.1. Relation between Mode- and Coordinate-Ladder Operators

#### 3.2. Hamiltonian Operator in the Coordinate Space

#### 3.3. Electric Field Operator in the Coordinate Space

## 4. Quantum Finite-Difference Time-Domain Scheme

## 5. Initial Quantum States for Few Photons

## 6. Initial Conditions of Quantum Finite-Difference Time-Domain Scheme

## 7. Numerical Simulations of Quantum Beam Splitter

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example 2-D problem geometry. In a periodic vacuum box, there is an arbitrarily-shaped inhomogeneous dielectric object. To extract traveling-wave normal modes of the system, we use Bloch-periodic boundary condition instead of periodic boundary condition under which only standing-wave normal modes can exist [36].

**Figure 2.**Example of the temporal behaviors of quantum electromagnetic (complex-valued) (QEM-CV)-propagator and computational electromagnetic (complex-valued) (CEM-CV)-propagator when ${x}_{n}=0$.

**Figure 3.**Schematic of computer simulations for a quantum beam splitter to observe Hong-Ou- Mandel effect.

**Figure 4.**The Hong-Ou-Mandel effects are numerically evaluated by using (1) numerical canonical quantization (solid line), (2) Quantum finite-difference time-domain (Q-FDTD) with exact initialization (round markers), and (3) Q-FDTD with approximate initialization (asterisk markers).

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**MDPI and ACS Style**

Na, D.-Y.; Chew, W.C. Quantum Electromagnetic Finite-Difference Time-Domain Solver. *Quantum Rep.* **2020**, *2*, 253-265.
https://doi.org/10.3390/quantum2020016

**AMA Style**

Na D-Y, Chew WC. Quantum Electromagnetic Finite-Difference Time-Domain Solver. *Quantum Reports*. 2020; 2(2):253-265.
https://doi.org/10.3390/quantum2020016

**Chicago/Turabian Style**

Na, Dong-Yeop, and Weng Cho Chew. 2020. "Quantum Electromagnetic Finite-Difference Time-Domain Solver" *Quantum Reports* 2, no. 2: 253-265.
https://doi.org/10.3390/quantum2020016