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Dyson Maps and Unitary Evolution for Maxwell Equations in Tensor Dielectric Media

A reformulation of Maxwell equations for an inhomogeneous, anisotropic, passive and non-dispersive medium results in a quantum-like Dirac equation that admits unitary time evolution. In contrast to other approaches, there is no a-priori introduction of the Riemann-Silberstein-Weber (RSW) vector but the Maxwell equations are considered in their standard fields, with given constitutive relations. From the electromagnetic conservation quantities a pseudo-Hermitian dynamics is found together with a Dyson map that recovers the full Hermicity of the dynamics in an extended Hilbert space that describes the physical notion of unitary evolution. As an example, a uniaxial tensor dielectric medium is considered, with the explicit Dyson map yielding an optimal representation in a set of generalized RSW vectors. In this newly discovered form, a Quantum Computing (QC) implementation for simulation of electromagnetic wave propagation in complex media can be made, with further extension into plasmas.


Introduction
The prospect that, for a range of problems, quantum computers could be be exponentially faster than conventional computers [1,2] has led to an enhanced interest in quantum computer sciences. For efficient use of quantum computers, it is necessary that the evolution equations for any physical system be expressed in terms of unitary operators [3]. There is no such requirement for classical computations. The tantalizing possibility of faster computations has been the motivation behind applying quantum information science to traditionally classical fields.
The propagation and scattering of electromagnetic waves in magneto-dielectric matter has been of considerable interest over many decades. The electromagnetic properties of a medium are included in Maxwell equations through constitutive relations that relate the electric displacement field and magnetic induction to the electric field and magnetic intensity, respectively. Most of these studies are classical -the de Broglie wavelengths being negligibly small compared to the wavelengths of the macroscopic fields. Thus, the implementation of Maxwell equations on quantum computers requires expressing a classical description in the language of quantum mechanics. The first step in this direction was taken by Laporte-Uhlenbeck [4] and Oppenheimer [5] casting Maxwell equations in vacuum into a form similar to the Dirac equation. More recent studies have been for a magneto-dielectric medium with scalar permittivity and permeability in the constitutive relations [6].
In this paper we formulate Maxwell equations for wave propagation in a dielectric medium such that they become amenable to quantum computations. The magneto-dielectric medium is assumed to be passive and non-dispersive for which both the permittivity and permeability can be a tensor. When the medium is spatially homogeneous, the Faraday-Ampere equations take on the form of a Dirac equation for spin 1 photons. The evolution operator is unitary and the state vector is a six-vector composed of the electric field and magnetic intensity. A unitarily similar representation is obtained for a state vector comprising of Riemann-Silberstein-Weber (RSW) vectors [7] -which represent the left and right hand polarizations of an electromagnetic field.
When the same prescription is extended to a spatially inhomogeneous medium, the evolution operator for the electromagnetic fields is not unitary anymore. We develop a pathway towards a unitary evolution equation through a two-step process. The first step is to identify the generator of dynamics -the Hamiltonian operatoras a pseudo-Hermitian operator. In a newly-defined weighted Hilbert space the Hamiltonian is Hermitian with respect to a weighed inner product structure. There has been a lot of interest in pseudo-Hermitian Hamiltonians in quantum mechanics, especially in the subset of P T -symmetric Hamiltonians [8][9][10]. The second step is to draw a connection between the physical Hilbert space and the initial Hilbert space that preserves the inner product structure of the two spaces. This is accomplished by constructing an appropriate isometric Dyson map. The end result is a fully unitary evolution equation with an explicit Hermitian Hamiltonian that could be implemented in a quantum computer. The fields evaluated from this evolution equation are directly related to the physical electromagnetic fields. This paper is organized as follows. In sections 2.1 and 2.2 we formulate the Faraday-Ampere equations in terms of a six-vector and in terms of RSW vectors, respectively, for a homogeneous medium. In section 2.3, we extend the description to allow for an inhomogeneous medium. From Poynting's theorem, as expected for a passive medium, we show that the total electromagnetic energy is conserved in a bounded medium subject to suitably chosen Dirichlet boundary conditions. In section 2.4, it is shown that the evolution generator of the previous section is not Hermitian due to spatial inhomogeneity. A physical Hilbert space is created in which the Hamiltonian is Hermitian. It is shown that, in this weighted Hilbert space, the norm of the state vector is the conserved energy that follows from Poynting's theorem. In section 2.5, we formulate three different forms of the Dyson map which lead to a Maxwell-Dirac equation with unitary evolution operator in the initial Hilbert space. In section 2.6, the entire formalism is applied to a uniaxial dielectric medium. There is a natural extension of the evolution equation to a set of spatially dependent RSW vectors which are a generalization of the RSW vectors in 2.2. The evolution equation can be implemented and tested on present-day classical computers using Qubit Lattice Algorithms (QLAs) [11][12][13][14][15]. This will help set the stage for implementing our unitary description on quantum computers.

Quantum Representation of Maxwell Equations
The source-free Maxwell equations for a linear medium are, with the constitutive relations, where E is the electric field, B is the magnetic induction, D is the displacement field, H is the magnetic intensity, is the dielectric permittivity of the medium and µ is its magnetic permeability; and µ can be functions of space. In section 2.1, we express Maxwell equations in terms of a six-vector when and µ are independent of space and time. We show that the Faraday-Ampere equations (2) take on a form similar to the Dirac equation for spin 1 massless photon. In section 2.2, we rewrite the Faraday-Ampere system using the RSW vectors and draw similarities with the results in section 2.1.
In section 2.3, we assume that the medium is inhomogeneous in space, independent of time, and nondissipative -i.e., (r) and µ(r) are real functions. The Faraday-Ampere equations and the Poynting theorem are set up using the six-vector representation.

Six-vector formulation of Maxwell equations
The Faraday-Ampere equations (2) can be written in a compact form using a six-vector [16], where u = (E H) T is an ordered pair of three-vectors composed of the electromagnetic field, T indicates the transpose, I 3×3 is the 3 × 3 identity matrix, and 0 3×3 is the null matrix. The invertible, Hermitian matrix W operating on u yields the constitutive relations D(r, t) = E(r, t) and B(r, t) = µH(r, t) for a homogeneous medium. The Maxwell operator M is Hermitian in L 2 (R 3 , C) with the appropriate boundary conditions. We will discuss this further in sections 2.3 and 2.4. The generator of the evolution operator D in (4) is Hermitian since the Hermitian operators W −1 and M commute, where v = 1/ √ µ is the speed of light in the medium, U = W 1/2 u, the components of S = (S x , S y , S z ) are the spin 1 matrices, satisfying the commutator relation [S a , S b ] = i abc S c , p = −i∇ is equivalent to the quantum momentum operator for = 1, and the Pauli spin 1/2 matrices σ = (σ x , σ y , σ z ) are, Since the evolution operator D ρ is Hermitian, (6) is analogous to the Dirac equation for a spin 1 massless photon.

Riemann-Silberstein-Weber vectors and Maxwell equations
The RSW vectors F ± (r, t) are a re-expression of the electromagnetic fields in a form that is useful for a quantum-like formulation of Maxwell equations. They are defined as [7], For a homogeneous medium, the Faraday-Ampere equations take on the form [7,17], Equation (10) can be considered as a quantum representation of Maxwell equations with the RSW vectors as the photon wave function [7].
In the standard square-integrable Hilbert space H = L 2 (R 3 , C), the Hermitian Hamiltonian operator in (10), has eigenvalues E = ω, reflecting the monochromatic energy of a photon. This is analogous to the quantum definition of energy for = 1. The norm of the RSW vectors F ± is the electromagnetic energy of the macroscopic electromagnetic field, where † is the complex conjugate transpose of the vector. The evolution equation (10) is analogous to the Weyl equation for spin 1/2 massless particles, where ψ is the wave function composed of the two Weyl spinors. The analogy is not surprising since the two RSW vectors represent the two distinct polarizations of the electromagnetic field in a homogeneous, timeindependent medium. If we introduce a unitary transformation L : U → F where, then (6) takes the block diagonal form, which is exactly the form in (10). The six-vector form of the Faraday-Ampere equations is directly connected to the RSW vectors. In other words, the RSW transformation is a Weyl representation of the Dirac-type equation (6). Significantly, the representations (6) and (15) are equivalent due to the unitary nature of the transformation (14). In a homogeneous medium, the two field helicities are uncoupled as is the time evolution of the RSW vectors.

Maxwell equations in an inhomogeneous, passive medium
The Faraday-Ampere equations (2) can be written as, where d(r, t) = (D, B) T is related to u(r, t) = (E, H) T by a linear constitutive operator L, The divergence equations (1) become, If at time t = 0, d 0 = d (r, 0) is such that, where u 0 = u (r, 0), then (16) ensures that ∇ · d (r, t) = 0 is for all times. We will assume that the medium is bounded by a perfect conductor so that, where n(r) is the outward pointing normal at the boundary, and u = (u 1 , u 2 ) T with u 1 and u 2 each being a three-vector. The boundary conditions are necessary for energy conservation and for ensuring that M remains Hermitian. The set of equations (16)- (20) are the complete mathematical description of electromagnetic waves in a Hilbert state space H = L 2 (Ω, R 6 ) u defined by the inner product [18], where u (r, t) and v (r, t) are two solutions within the bounded domain defined by Ω. A general form of the constitutive operator L has to satisfy five physical postulates [18]: determinism, linearity, causality, locality in space, and invariance under time translations. The form that is consistent with these postulates is [18], The first term on the right hand side in (22) corresponds to instantaneous optical response of the medium, and the second term with G as the susceptibility kernel is the dispersive response which includes memory effects.
For an anisotropic non-dispersive medium, (22) reduces to, where, In what follows, we will ignore dispersive effects. However, in general, a constitutive relation of the form (23) is an approximation to (22) which includes a non-local time-response function [19].
Since W is invertible [18], (16) takes on the form, where, In this representation, the Poynting theorem is [18], where S = E × H is the Poynting vector. Following [20], the electromagnetic energy density is, where W A = W − W † /2 is the anti-Hermitian part of W . For a passive medium [18,20], From (28) it follows that W must be Hermitian and semi-positive definite, The total integrated stored electromagnetic energy U Ω in a volume Ω is, Integrating (27) over Ω and making use of the divergence theorem, we obtain, where dA is an elemental area on the surface ∂Ω, and n is the outward pointing normal to ∂Ω. Since S = u 1 ×u 2 , it follows from (32) and the boundary condition (20) that U Ω (t) is constant in time, As expected for a passive medium, there is no net dissipation or generation of electromagnetic energy within Ω.

Pseudo-Hermitian Operators
In the Hilbert space H, where v = (v 1 , v 2 ) T , with v 1 , and v 2 each being three-vectors. In obtaining (34b) from (34a), we have made use of the vector identity ∇ · (a × b) = b · (∇ × a) − a · (∇ × b), and the divergence theorem. Since v * 1 ·( n × u 2 ) = −u 2 ·( n × v * 1 ), the surface integral in (34b) vanishes as a consequence of the boundary condition (20). It follows from (34) proving that M is Hermitian; i.e., M = M † . Even though W (r) and M are Hermitian, the operator D = W −1 M is not Hermitian. In contrast to a homogeneous medium, the commutator W −1 , M is non-zero for an inhomogeneous medium. Consequently, the operator on the right-hand side of the evolution equation (25) is non-unitary. In order to make Maxwell equations for an inhomogeneous medium suitable for quantum computing, we formulate a unitary representation that relies on D being a special kind of non-Hermitian operator -a pseudo-Hermitian operator.
A linear operator D in a Hilbert space H is pseudo-Hermitian if there exists an invertible Hermitian linear operator η in H with the property [21][22][23], For u| η|u > 0 for all nonzero states u, η is a positive definite metric operator, and we can define an inner product, with respect to a new weighted Hilbert space H η . From (26), Comparing with (36) we note that η = W . Furthermore, exploiting the Hermicity condition (37) of Maxwell operator M we obtain, v Du Thus, D † = D, i.e., D is Hermitian in the weighted Hilbert space H W . In H W , the inner product is as defined in (37) with η replaced by W , and the evolution equation (25) is unitary. Making use of (33), the square of the norm of u, is a constant independent of time. Consequently, the underlying conservation of the electromagnetic energy in the closed volume Ω is preserved in H W .

The Dyson map for Maxwell equations
Even though D is Hermitian in the new Hilbert space H W , in the Maxwell-Dirac equation (25) there is no change except that u ∈ H W . We need to connect the original, physical, Hilbert space H, in which D is not Hermitian, to H W by an isometric transformation that preserves the inner product structure between the two Hilbert spaces. Such an invertible transformation ρ(r) : H W → H between equivalent descriptions of a physical system is referred to as a Dyson map [21][22][23].
Our derivation of the Dyson map ρ(r) is based on the factorization of the metric operator η = W , which preserves the inner product structure since, where v, u ∈ H W and φ, ψ ∈ H. For Maxwell equations, there can be three different factorization forms of W in (24) [24].
For the spectral decomposition (43), ∆(r) = λ i (r)δ ij (there is no implied summation over repeated indices) with λ(r) > 0 and ∆(r) = λ i (r)δ ij . For the Cholesky decomposition (47), the T matrix is an upper triangular matrix with positive diagonal elements. The particular choice of a Dyson map is based on the decomposition scheme which leads to a sparse ρ. The Dyson map leads to a Hermitian form for Maxwell equations in H. Multiplying (25) by ρ gives, where ψ = ρu ∈ H, and D ρ = ρ(r) D ρ −1 (r) is Hermitian in H The unitary evolution of ψ (r, t) is, where ψ 0 (r) is the initial condition at time t = 0. In general, any operator A η : H η → H η is related to its counterpart A in H through a similarity transformation, The Dyson map ρ connecting H to H η can be schematically represented in Figure 1. This diagram illustrates Thus, all other dynamics preserving transformations are equivalent once the Dyson map ρ is established. Indeed, this holds for the formulation in terms of RSW vectors F = L W 1/2 u.

Application to a uniaxial dielectric medium
As an illustration of the formalism developed in section 2.5, we consider a non-magnetic, uniaxial dielectric medium, A useful choice for a sparse Dyson map is, where, Then, D ρ = ρ D ρ −1 is, where, in terms of the refractive index n i (r) = i (r)/ 0 , the components of Z = (Z x , Z y , Z z ) are, with n x = x / 0 and n z = z / 0 being the indices of refraction in the x and z directions, respectively, and, as before, p = −i∇.
Applying the unitary operator L in (14) to (49) gives, Upon defining, the unitary evolution equation (58) takes on the form, where the superscripts H and A represent the Hermitian and the anti-Hermitian parts of the operator, respectively. The definition in (59) for an inhomogeneous medium is a generalization of the RSW vectors (9) for a homogeneous medium. For a non-dissipative medium, the anti-Hermitian part of Z in (60) is zero, and the time evolution of the two RSW vectors F ± decouples. It is straightforward to show that for a homogeneous non-dissipative medium, (60) reduces to (15).

Conclusions
The propagation and scattering of electromagnetic waves in a dielectric medium is governed by classical Maxwell equations. The enticing possibility of an exponential reduction in computational time on quantum computers has led to recasting some topics in classical physics into the framework of quantum information science. We have expressed the Faraday-Ampere equations for a passive, non-dispersive dielectric medium in a form that is similar to the Dirac equation for a massless spin 1 particle. For a medium homogeneous in space, the permittivity and permeability are scalars. In the Dirac-type evolution equation, the electromagnetic fields are expressed either as a six-vector or as Riemann-Silberstein-Weber vectors. When the permittivity and permeability of the medium are functions of space then, in contrast to a homogeneous medium, the Maxwell operator M and the operator for the constitutive relations W do not commute. Even though both operators are Hermitian, their product is not. A remedy is to construct a weighted Hilbert space H W in which the generator of dynamics is pseudo-Hermitian. The Hermitian, positive definite metric operator that defines the inner product within H W is W . The norm of a state vector in H W is the electromagnetic energy which, from Poynting's theorem, is conserved. The connection between the original Hilbert space and H W is established through a Dyson map. Significantly, the Dyson map is a fundamental way to construct a unitary evolution of Maxwell equations for wave propagation in a complex medium. Any other representation, preserving the dynamics, is generated through a unitary transformation of the Dyson map. There are three different Dyson maps that are suitable for connecting the two Hilbert spaces. The preferred Dyson map could be guided by the sparseness of the associated matrix operators. Regardless of the choice, the final form of the Faraday-Ampere equations comprises unitary evolution operator. The formal development of Maxwell equations into a unitary evolution equation is applied to a uniaxial, inhomogeneous, dielectric medium. The resulting equation is suitable for implementing in a quantum lattice algorithm for numerical simulations of electromagnetic wave propagation.