Macroscopic quantum electrodynamics in nonlocal and nonreciprocal media

Macroscopic quantum electrodynamics in nonlocal and nonreciprocal media

Stefan Yoshi Buhmann and David T Butcher, Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London. Stefan Scheel, Imperial College London and Institut für Physik, Universität Rostock.

Received 6 July 2012; published 29 August 2012.

Abstract

We formulate macroscopic quantum electrodynamics in the most general linear, absorbing media. In particular, Onsager reciprocity is not assumed to hold. The field quantization is based on the source-quantity representation of the electromagnetic field in terms of the dyadic Green's tensor. For media with a nonlocal response, a description in terms of a complex conductivity tensor is employed. As an alternative description, we introduce the permittivity, permeability and magnetoelectric susceptibilities to obtain an explicitly duality-invariant scheme. We find that duality invariance only holds as a continuous symmetry when nonreciprocal responses are allowed for.

The displayed tensor equations of this paper are given below as named results in words, with the article's own equation numbers; the full algebra is at the source.

1. Introduction

The linear response of a macroscopic material to externally applied electromagnetic fields can go beyond the scope of simple descriptions via electric permittivities and magnetic permeabilities. In particular, cross-susceptibilities naturally arise in chiral metamaterials, topological insulators or moving media. In the latter case nonlocal responses arise with the additional complication that Onsager reciprocity fails to hold. Onsager reciprocity, the electrodynamic manifestation of time-reversal symmetry, would also be violated in Tellegen media, including the recently proposed perfect electromagnetic conductor that continuously interpolates between a perfect conductor and an infinitely permeable material.

Chiral metamaterials with cross-susceptibilities have been constructed based on nanoscale chiral objects, such as a helix. This leads to a discriminatory response of the medium to left- and right-circularly polarized light. This central feature of chiral media is important in biological systems due to the prevalence of left-handed objects in the processes crucial to life. Furthermore, chiral metamaterials have been discussed as candidates for repulsive Casimir forces. It should be noted that repulsive forces for magnetoelectric media were originally discussed for dielectric plates interacting with magnetic plates. To implement these effects with metamaterials, the anisotropic response of the medium needs to be taken into account.

Topological insulators are a novel class of materials which behave as insulators in their bulk phase but allow for conduction on the surface. Time reversal symmetry is an important feature which ensures an extremely high stability of the surface currents. The latter make topological insulators a promising candidate for quantum computing. It has recently been predicted that topological insulators, or materials with a Chern–Simons interaction, could be used to realize repulsive Casimir forces. A related phenomenon is the fractional quantum Hall effect where the Hall current takes fractional values due to electron–electron interactions. This medium can be nonlocal and in contrast to topological insulators it can violate time-reversal symmetry and hence Onsager reciprocity.

The impact of electric versus magnetic material properties can be studied in a systematic way by means of a duality transformation. It has recently been shown that macroscopic quantum electrodynamics in isotropic magnetoelectrics obeys a discrete duality symmetry. This has immediate consequences for dispersion forces in free space.

The successes in the realization of the above mentioned novel materials open the perspective on a range of new quantum phenomena related to photon-induced matter interactions, quantum dynamics and possibly irreversible quantum-light propagation. To make such studies possible, we will construct a quantum theory of the electromagnetic field in the most general linear absorbing media, including nonlocal, bianisotropic and Onsager reciprocity violating materials. A recent theory based on canonical quantization is a valuable step in this direction, which does not yet consider the most general nonlocal media, apart from the above mentioned moving media.

In addition, our theory shall answer the question under which circumstances duality can be realized as a continuous symmetry of the Maxwell equations in media; and it will shed light on the generalizations necessary to discuss moving media and quantum friction.

2. Field quantization in nonlocal media

We begin by recalling a quantization procedure of the electromagnetic field in the presence of an absorbing medium. In a linearly responding medium, the effect of an external electromagnetic field on the matter can be given by Ohm's law in its most general form, equation (1): the induced current density at a point and time is the integral over all delay times and over all other points of a conductivity tensor linking the two points, contracted with the electric field at the second point and the retarded time, plus a random noise current. Here the conductivity tensor is the two-point response function and the noise current is required to fulfil the fluctuation–dissipation theorem given below. Causality requires the conductivity to vanish whenever the delay time is shorter than the light travel time between the two points, and in particular for all negative delays.

In frequency space, Ohm's law takes the simpler form of equation (2), an integral over the second point only, with the frequency-domain conductivity defined by equation (3) as the one-sided Fourier transform of the time-domain conductivity. As a result of the causality requirement the conductivity obeys the Schwarz reflection principle, equation (4): its complex conjugate at one frequency equals its value at minus the complex conjugate frequency, for all pairs of points.

Quantization is achieved by specifying the commutator, equation (5): the commutator of the noise current operator with its Hermitian conjugate at a second point and frequency equals the reduced Planck constant times the frequency over pi, times the generalized real part of the conductivity tensor, times a delta function in frequency. Here we have introduced generalized real and imaginary parts of a tensor field according to equations (6) and (7): the generalized real part is one half of the sum of the tensor and the Hermitian conjugate of the tensor with its two arguments exchanged, and the generalized imaginary part is that same construction with a difference and a factor of one over twice the imaginary unit.

They reduce to ordinary real and imaginary parts for orthogonal tensor fields that are reciprocal, that is, when the transposed tensor with arguments exchanged equals the original tensor. The other nontrivial current commutators follow from the ordinary commutator rules, equation (8), and the fact that the right-hand side of this expression is a Hermitian tensor field guarantees the consistency of the commutation relations.

Combining Ohm's law with Maxwell's equations, equations (9) and (10), one finds that the electric field obeys a generalized inhomogeneous Helmholtz equation, equation (11): the double curl minus the squared frequency over the squared speed of light, acting on the field, minus the frequency-weighted convolution of the conductivity with the field, equals the frequency-weighted noise current. With the help of the Green function of the Helmholtz equation, defined by equation (12) with the boundary condition that the Green tensor vanishes at infinite separation, the formal solution to the integro-differential equation reads equation (13): the electric field is the permeability of free space times the imaginary unit times the frequency, times the spatial convolution of the Green tensor with the noise current.

By virtue of its definition, the Green tensor inherits the Schwarz reflection principle from the conductivity tensor, equation (14). However, as a major departure from previous treatments, we do not require the conductivity to obey reciprocity. As a consequence, the Green tensor will not obey the Onsager principle, equation (15), in general. Recall that the Onsager principle, applied to electromagnetic field propagation, states a reversibility of optical paths. The Green tensor governs the relation between a source current at one point along one direction and the generated electric field at another point along a second direction. If the Onsager principle holds, then the roles of source and field can be reversed: a source current at the second point along the second direction would give rise to an electric field at the first point along the first direction. In a configuration involving nonreciprocal media, this is not necessarily the case.

Despite the extension to nonreciprocal media, it is still possible to derive the useful integral relation, equation (16): the permeability of free space times the frequency times the Green tensor convolved with the generalized real part of the conductivity and then with the Hermitian conjugate Green tensor equals the generalized imaginary part of the Green tensor. It generalizes the known result to the case where Onsager reciprocity does not hold.

The theory thus far is analogous to classical electromagnetism in an absorbing medium under the assumption of classical fluctuating current sources. Their strengths are governed by the fluctuation–dissipation theorem in the classical, high-temperature limit. Introducing the ground state of the medium-field system as the state annihilated by the noise current operator, the currents satisfy the fluctuation–dissipation theorem as an immediate consequence of the commutator, equation (17): the symmetrised correlation of the noise-current fluctuations equals the reduced Planck constant over pi times the generalized imaginary part of the frequency-weighted conductivity, times a delta function in frequency. Combining the commutator with the formal solution, one finds that the fluctuations of the electric field are also consistent with the fluctuation–dissipation theorem, as required, equation (18), with the Green tensor now carrying the frequency-squared weight.

In order to verify the canonical equal-time commutation relations, we introduce the vector potential for the electromagnetic field in the Coulomb gauge, as the transverse part of the electric field divided by the imaginary unit times the frequency. One finds equation (19), the commutator of the electric field with the conjugate vector potential at a second point and frequency, expressed through the transverse projection of the generalized imaginary part of the Green tensor, and hence equation (20), the equal-time commutator written as a frequency integral over the transverse projections of the Green tensor and of its argument-exchanged transpose, where the Schwarz reflection principle has been used. Use has been made of the left- and right-sided transverse projections defined in equation (21).

Closing the integration contour in the upper half of the complex frequency plane, where the Green function is analytic, and using the high-frequency asymptote in which the frequency-weighted Green tensor tends to minus a delta function, one finds the canonical commutation relation from free-space quantum electrodynamics, equation (22): the commutator of the electric field and the vector potential is the imaginary unit times the reduced Planck constant over the permittivity of free space, times the transverse delta function — as required.

We now introduce the bosonic creation and annihilation operators of the matter-field system according to the prescription of equation (23), in which the noise current is the square root of the reduced Planck constant times the frequency over pi, times the convolution of a tensor field with the annihilation operator, where that tensor field is a square root of the positive definite generalized real part of the conductivity, equation (24). This solution is only unique up to a unitary matrix which does not affect the physical results. Together with the current commutator, this ensures bosonic commutation relations, equation (25). The Hamiltonian of the medium-field system is then equation (26), the integral over all space and all positive frequencies of the reduced Planck constant times the frequency times the number density of these bosonic excitations. It leads to the free evolution of the dynamical variables as a simple harmonic phase factor; hence Maxwell's equations for the electromagnetic-field operators in the Heisenberg picture are valid by construction.

3. Field quantization in terms of electric and magnetic response functions

The properties of media with spatially nonlocal or local responses can alternatively be described by their permittivity, permeability and magnetoelectric susceptibilities. To begin, it is convenient to cast the inhomogeneous Maxwell equations into the forms of equation (27), with the displacement and magnetic field strength defined in the usual way from the polarization and magnetization, equation (28).

The polarization and magnetization fields respond linearly to the electric and magnetic fields, equations (29) and (30). The medium is characterized by its permittivity, its permeability and its two magnetoelectric susceptibilities, each a function of two independent spatial variables in the nonlocal case and proportional to a delta function of the separation in the local case. The noise polarization and noise magnetization appear as the added stochastic terms, and the vacuum impedance is the square root of the ratio of the permeability to the permittivity of free space. By combining these relations, the constitutive relations can be given in the more familiar form of equations (31) and (32), in which the displacement field is built from the permittivity convolved with the electric field plus the first magnetoelectric susceptibility convolved with the magnetic field strength, plus the noise terms, and the magnetic flux density from the second magnetoelectric susceptibility convolved with the electric field plus the permeability convolved with the magnetic field strength, plus the noise magnetization.

In order to distinguish reciprocal magnetoelectric susceptibilities from nonreciprocal ones, one commonly writes the two susceptibilities as a symmetric part plus or minus an imaginary antisymmetric part. The chirality tensor, one half of the difference of the second susceptibility and the transpose of the first divided by the imaginary unit, represents the reciprocal magnetoelectric response; whereas the nonreciprocal magnetoelectric tensor, one half of their sum, vanishes for a reciprocal medium.

By combining Maxwell's equations with the constitutive relations, we note that the respective Green tensor is the solution to the defining equation with the conductivity replaced by the expression in equation (33), built from curls of the inverse permeability and of the magnetoelectric susceptibilities together with the permittivity term, while the noise current becomes equation (34), minus the imaginary unit times the frequency times the noise polarization plus the curl of the noise magnetization. The Green tensor for the electric field then solves equation (35), the generalized Helmholtz equation written directly in terms of permittivity, permeability and the magnetoelectric susceptibilities.

The commutation relations for the noise polarization and noise magnetization can be deduced by substituting the real parts of these expressions into the fundamental current commutator, giving equations (36) to (39): the polarization–polarization commutator is set by the generalized imaginary part of the permittivity corrected by the magnetoelectric term; the two cross commutators, polarization with magnetization and magnetization with polarization, are set by combinations of the susceptibilities with the inverse permeability; and the magnetization–magnetization commutator is set by the generalized imaginary part of the inverse permeability. We now introduce bosonic creation and annihilation operators with the usual commutation relations, equation (40), labelled by an electric and a magnetic index, according to equation (41), in which the pair of noise fields is a six-by-six matrix root, equation (42), acting on the pair of bosonic operators. The Hamiltonian of the body–field system is again quadratic and diagonal in the bosonic variables, equation (43).

Note that the constitutive relations imply a separation of the internal current density into electric and magnetic parts. This separation and the resulting explicit field quantization is not unique in spatially dispersive, that is nonlocal, media, as the magnetization field can be absorbed into the transverse part of the polarization field. While a local magnetoelectric medium can always be described in terms of a nonlocal conductivity without reference to magnetic properties, the equivalent description in terms of a local permittivity, permeability and cross-susceptibility is much more accessible. These parameters are often known experimentally and they allow for a classification of electromagnetic responses.

4. Duality invariance

An electromagnetic system separated into distinct electric and magnetic causes and effects can be subject to a duality transformation operation, that is, a global exchange of the electric and magnetic properties. A system invariant under such an operation is said to possess duality invariance as a symmetry. This symmetry can be exploited in order to simplify the computation of dispersion forces involving, say, magnetizable media, from known dispersion forces between polarizable media.

By introducing dual-pair notation, pairing the electric field with the impedance-weighted magnetic field strength and the impedance-weighted displacement field with the magnetic flux density, we may write the Maxwell equations in the compact forms of equations (44) and (45), and the constitutive relations in the condensed form of equation (46), with the auxiliary matrix defined in equation (47).

Maxwell's equations are invariant under duality transformations, equation (48), in which the dual pair is rotated by an angle through an ordinary two-by-two rotation matrix, because that matrix is symplectic. From the constitutive relations we find the transformed medium response functions, equation (49): the four response functions — permittivity, the two magnetoelectric susceptibilities and permeability — rotate as a set under the tensor product of the rotation matrix with itself, equation (50). The noise polarization and magnetization transform as in equation (51), through the auxiliary matrix and its inverse, equation (52).

It is worth discussing a few special cases of bianisotropic media, their characteristic features and behaviour under duality transformations.

• Local media. Every response function is proportional to a delta function of the separation, so convolution operators reduce to ordinary matrix products; compatible with all other special cases below.
• Isotropic media. Permittivity and permeability are scalars times the identity and the magnetoelectric susceptibilities vanish: Onsager reciprocity holds; the noise polarization and the conjugate noise magnetization commute; generalized real and imaginary parts reduce to ordinary ones; discrete duality symmetry.
• Bi-isotropic media. All four response functions are scalars times the identity: generalized real and imaginary parts reduce to ordinary ones; continuous duality symmetry.
• Anisotropic media. The magnetoelectric susceptibilities vanish: the noise polarization and the conjugate noise magnetization commute; discrete duality symmetry.
• Reciprocal media. The permittivity and permeability are symmetric and the two magnetoelectric susceptibilities are minus each other's transpose: the Onsager relation holds; generalized real and imaginary parts reduce to ordinary ones; discrete duality symmetry.

Here, discrete duality symmetry means that the rotation angle is restricted to whole multiples of a quarter turn. Note that duality is only realized as a continuous symmetry when Onsager violation is allowed for. Notably, a reduction in reciprocity symmetry leads to an enhancement of duality symmetry.

In order to derive transformation laws for the Green tensor, we combine the Maxwell equations, the Helmholtz equation and the constitutive relations to write equation (53), which expresses the dual field pair through a matrix of Green tensors acting on the dual noise pair, with the auxiliary matrices given in equations (54) and (55). The four blocks of that matrix are the electric–electric Green tensor, the electric–magnetic and magnetic–electric tensors formed by acting with a curl on one argument, and the magnetic–magnetic tensor formed by acting with a curl on both. The transformed Green tensors follow by applying duality transformations on both sides of this equation, equations (56) and (57). In the special case of both points being in free space, the auxiliary matrices are the identity, so that the Green tensor simply conjugates with the rotation, equation (58), and the Green tensors then transform like the medium response functions.

5. Conclusion

Based on the general Ohm's law, we have quantized the electromagnetic field in the presence of nonlocal, nonreciprocal media which satisfies, first, the canonical commutation relations from free-space quantum electrodynamics; second, the linear fluctuation–dissipation theorem; and third, the macroscopic Maxwell equations. A key feature of the scheme is the symmetrization of tensor fields via generalized real and imaginary parts, which is necessary whenever Onsager reciprocity does not hold. Their presence in the fluctuation–dissipation theorem is the key to avoiding restrictions on the allowed medium response.

For nonlocal and local bianisotropic media we have shown that quantization can alternatively be performed by the introduction of permittivity, permeability and magnetoelectric susceptibilities. When the latter do not vanish, the noise polarization and magnetization do not commute. We have explicitly determined the behaviour of the fields and response functions under duality transformations. The full continuous transformation group applies for bianisotropic and bi-isotropic media, but reduces to a discrete symmetry for isotropic, anisotropic and reciprocal media.

The scheme lays the foundation for exact studies of quantum phenomena such as dispersion forces, Förster energy transfer or environment-assisted molecular transition rates in the presence of motion or novel media with chiral or nonreciprocal properties. Moving media are a prime example for the occurrence of nonreciprocal material properties which have to be thoroughly accounted for in order to understand, for example, quantum friction. Charge-parity violation in atoms or molecules is manifest in their nonreciprocal cross-polarizability. The Curie principle, stating that certain interactions between two partners — atoms, molecules, bodies and so on — require them to possess similar properties, then allows for a detection of that violation via atom–surface interactions provided the surface exhibits a corresponding nonreciprocity.

Acknowledgment

This work was supported by the UK Engineering and Physical Sciences Research Council.

Appendix. Integral relation for the Green tensor

To derive the integral relation, equation (16), for the Green tensor, we write the Helmholtz equation in operator form as equation (A.1), the Helmholtz operator acting on the Green operator giving the identity, where the position matrix elements of the Green operator are the Green tensor and those of the Helmholtz operator are the double curl minus the frequency-squared term, minus the frequency-weighted conductivity. The Green operator is the right-inverse and, within any group of invertible operators, also the left-inverse of the Helmholtz operator, equation (A.2). From this relation and its Hermitian conjugate we find equation (A.3): the Green operator acting on the difference between the Helmholtz operator and its Hermitian conjugate, acting in turn on the conjugate Green operator, equals the difference between the conjugate Green operator and the Green operator. In coordinate space this relation reads equation (A.4), a double spatial integral of the Green tensor with the generalized real part of the conductivity and the conjugate Green tensor, weighted by the permeability of free space and the frequency, equalling the generalized imaginary part of the Green tensor — which in convolution notation takes the form of equation (16).

(The reference list of the published article is omitted here; the complete text is at the source.)