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arXiv:2101.12184 · Physical Review A

Diagonalization of the Hamiltonian for finite-sized dispersive media: Canonical quantization with numerical mode decomposition

Dong-Yeop Na, Jie Zhu, Weng Cho Chew

Published Jun 10, 2021• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.103.063707

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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.
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.
In this work, we propose two approaches of investigating quantum entanglement. In the first approach, canonical quantization with numerical mode decomposition is applied to in-homogeneous dispersive media. Nonlocal dispersion cancellation effect of energy-time-entangled photon pair is demonstrated. In the second approach, we study the effect of scattering on spatial-entangled photon pair from spontaneous parametric down-conversion. Schmidt number is calculated numerically. Migration of entanglement between amplitude and phase is studied.
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.
In this work, we propose two approaches of numerical study of quantum entanglement. In the first approach, canonical quantization with numerical mode decomposition is applied to inhomogeneous dispersive media. The initial quantum state is assumed to be a cat state, which is an entangled state with coherent states. Results in2D2\mathrm{D}2Dmodeling will be presented. In the second approach, we study the effect of scattering on momentum-entangled photon pair from spontaneous parametric down-conversion. Scattered two-photon wave function is calculated using computational electromagnetics. Schmidt number is calculated numerically to investigate the enhancement of entanglement after scattering compared to the incident state.
Canonical quantization of the electromagnetic (EM) field is carried out for the situation where the total charge and current densities are the sum of contributions from neutral dielectric atoms whose effect is to be described purely classically in terms of spatially dependent electric permittivity and magnetic permeability functions, and neutral, stationary radiative atoms whose interaction with the EM field is to be treated quantum mechanically. The coefficients for the expansion of the vector potential in terms of mode functions determined from a generalized Helmholtz equation are chosen as independent generalized coordinates for the EM field. The spatially dependent electric permittivity and magnetic permeability appear in a generalized Helmholtz's equation and the former also occurs in the mode function orthogonality and normalization conditions. The quantum Hamiltonian is derived in a generalized multipolar form rather than the minimal coupling form obtained in other work. The radiative energy is the sum of quantum harmonic oscillator terms, one for each mode. The modes are independent in the present case of exact mode functions associated with the spatially dependent electric permittivity and magnetic permeability, there being no direct mode-mode coupling terms. In the electric dipole approximation the electric interaction energy contribution for each mode and radiative atom is proportional to the scalar product of the dipole operator with the mode function evaluated at the atom, times the annihilation operator, plus the Hermitian adjoint. This form has been widely used in studies of radiative processes for atomic systems in dielectric media, and it is justified here via the canonical quantization procedure. The results apply to the theoretical treatment of numerous quantum optical experiments involving such interactions in the presence of passive, lossless, dispersionless, linear classical optics devices such as resonator cavities, lenses, beam splitters, and so on. An illustrative application of the theory for atomic decay in a one-dimensional Fabry-Perot cavity is given. \textcopyright{} 1996 The American Physical Society.
A simple prescription for quantizing fields is developed and applied to the sytsem of interacting spins in an electromagnetic field. The classical equations of motion are written in the standard form $\ensuremath{-}i(\frac{d}{\mathrm{dt}})|V〉+\mathfrak{M}|V〉=0$, where the state vector ∣V〉|V〉∣V〉 has the state variables, such as the generalized coordinates and momenta, as its components, and the time development operator M\mathfrak{M}M contains derivative, integral, and matrix operators in general. By diagonalizing M\mathfrak{M}M and replacing the resulting classical variables with operators obeying boson commutation relations, the system is quantized, and the Hamiltonian and Heisenberg equations of motion are automatically cast into the harmonic-oscillator form. Thus the quantization is reduced to the simple problem of diagonalizing a matrix. This avoids the laborious process of finding a Lagrangian density (by trial and error in general), determining the canonical variables, forming the Hamiltonian density, and inventing transformations to creation and annihilation operators. Problems with zero-frequency modes, which can be annoying in the standard treatment, are automatically eliminated by the prescription. The bookkeeping of forward and backward waves, different branches of dispersion curves, etc., is simplified. In addition to affording a straightforward and simple method of quantizing fields, the method also gives the necessary and sufficient conditions for a Hamiltonian to be transformable into harmonic-oscillator form, gives a direct relation between the classical equations of motion and the Heisenberg quantum-mechanical equations, and gives a simple method of determining the constants of the motion.