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Quantum Reports

Quantum Electromagnetic Finite-Difference Time-Domain Solver

Dong-Yeop Na, Weng Cho Chew

Published Apr 10, 2020• 13 citations
Publisher

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.

DOI
10.3390/quantum2020016

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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.
We investigate quantum information preserving computational electromagnetics. To do so, we propose two different methods which are (1) numerical canonical quantization and (2) quantum finite-difference time-domain scheme. The proposed algorithms find numerical solutions of quantum Maxwell's equations involved in arbitrary inhomogeneous dielectric media based on the macroscopic theory on quantum electrodynamics; hence, they are useful to study the quantum nature of electromagnetic fields in arbitrary passive and lossless quantum optical instruments. We provide a numerical example associated with two photon interference occurring in a quantum beam splitter, which is the underlying principle of recent quantum technologies.
We present a computational framework for canonical quantization in arbitrary inhomogeneous dielectric media by incorporating quantum electromagnetic effects into complex solutions of quantum Maxwell's equations. To do so, the proposed algorithm integrates and performs (1) numerical computation of normal modes and (2) evaluation of arbitrary products of ladder operators acting on multimode Fock states. The former is associated with Hermitian-Helmholtz linear systems using finite-element or finite-difference methods; consequently, the complete set of numerical normal modes diagonalizes the Hamiltonian operators up to floating-point precision. Its Hermiticity is retained, allowing its quantization. Then, we perform quantum numerical simulations of two-photon interference occurring in a 50:50 beam splitter to observe the Hong-Ou-Mandel effect. Our prototype model is useful for numerical analyses on various narrow-band quantum-optical multiphoton systems such as quantum metasurfaces, quantum-optical filters, and quantum electrodynamics in open optical cavities.
Abstract Physics systems are becoming increasingly complex and require more and more computing time. Quantum computing, which has shown its efficiency on some problems, such as the factorisation of a number with Shor's algorithm, may be the solution to reduce these computation times. Here, the authors propose two quantum numerical schemes for the simulation of physics phenomena, based on the finite difference method. The aim is to see if quantum versions of standard numerical schemes offer an advantage over their classical counterparts, either in accuracy, stability or computation time. First, the authors will present the different phenomena studied as well as the classical solution methods chosen. The authors will then describe the implementation of the quantum numerical schemes and present some results obtained on the different physics phenomena beforehand and then compare both approaches, classical and quantum.
We present a math-physics modeling approach called canonical quantization with numerical mode decomposition for capturing the physics of how incoming photons interact with finite-sized dispersive media, which is not describable by the previous Fano-diagonalization methods. The main procedure is to (1) study a system where electromagnetic fields are coupled to nonuniformly distributed Lorentz oscillators in Hamiltonian mechanics, (2) derive a generalized Hermitian eigenvalue problem for conjugate pairs in coordinate space, (3) apply computational electromagnetics methods to find a countably finite set of time-harmonic eigenmodes that diagonalizes the Hamiltonian, and (4) perform the subsequent canonical quantization with mode decomposition. Moreover, we provide several numerical simulations that capture the physics of full quantum effects, impossible by classical Maxwell's equations, such as nonlocal dispersion cancellation of an entangled photon pair and the Hong-Ou-Mandel effect in a dispersive beam splitter.
Two-photon interference effects arise because photons are indistinguishable particles. In the wellknown Hong-Ou-Mandel (HOM) effect, the transmission of two photons at a beam splitter interferes destructively with the reflection of both photons, requiring both photons to "bunch up" by leaving the beam splitter on the same side. Here, we show that the interference between locally propagating photons and photons exchanged by a mode swap can be implemented by post-selecting spatially separated photon outputs of a four-path interferometer. Even though the photons detected at spatially separated locations must have travelled along paths that never met up at the same beam splitter, the Hong-Ou-Mandel effect can be observed in correlations between the output ports that originate from the association of detection events with non-local output modes defined by the two single photon inputs. Local phase shifts can be used to map out non-classical correlations between the photons detected at different output locations, clarifying the role of linear optics in generating entanglement between spatially separated photons. Our work thus establishes a fundamental relation between multiphoton interference and entanglement, opening the door to new possibilities in optical quantum technologies.