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pages, minor corrections"}],"earliest_date":"2008-11","refereed":true,"external_system_identifiers":[{"schema":"ADS","value":"2009PhRvL.102n0404B"},{"schema":"SPIRES","value":"SPIRES-8306168"}],"facet_author_name":["1946455_Stefan Yoshi Buhmann","2027630_Stefan Scheel"],"license":[{"license":"arXiv nonexclusive-distrib 1.0","material":"preprint","url":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/"}],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:809432","updated":"2026-02-06T10:23:41.559730"},"journal_title_variants":["Phys.Rev.Lett.","Phys. Rev. Lett."],"arxiv_eprints":[{"categories":["quant-ph"],"value":"0806.2211"}],"referenced_authors_bais":["C.Montonen.1","I.Klich.1","K.Joulain.2","O.Kenneth.1","C.Henkel.2","G.Y.Rainich.1","J.A.Wheeler.1","C.W.Misner.1","D.I.Olive.1"],"legacy_version":"20160328211312.0","inspire_categories":[{"term":"Other"},{"term":"Quantum Physics","source":"arxiv"}],"first_author":{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/041kmwe10"}],"full_name":"Buhmann, Stefan Yoshi","last_name":"Buhmann","first_name":"Stefan Yoshi","recid":1946455},"control_number":809432,"dois":[{"value":"10.1103/PhysRevLett.102.140404"},{"material":"publication","source":"arXiv","value":"10.1103/PhysRevLett.102.140404"}],"document_type":["article"],"texkeys":["Buhmann:2008ejy","Buhmann:2009zza"],"abstracts":[{"source":"arXiv","value":"We discuss under what conditions the duality between electric and magnetic fields is a valid symmetry of macroscopic quantum electrodynamics. It is shown that Maxwell's equations in the absence of free charges satisfy duality invariance on an operator level, whereas this is not true for Lorentz forces and atom--field couplings in general. We prove that derived quantities like Casimir forces, local-field corrected decay rates as well as van-der-Waals potentials are invariant with respect to a global exchange of electric and magnetic quantities. This exact symmetry can be used to deduce the physics of new configurations on the basis of already established ones.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"title":"Macroscopic quantum electrodynamics and duality"},{"title":"Macroscopic Quantum Electrodynamics and Duality"},{"source":"arXiv","title":"Macroscopic quantum electrodynamics and duality"}],"curated":true},"links":{"bibtex":"https://inspirehep.net/api/literature/809432?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/809432?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/809432?format=latex-us","json":"https://inspirehep.net/api/literature/809432?format=json","json-expanded":"https://inspirehep.net/api/literature/809432?format=json-expanded","cv":"https://inspirehep.net/api/literature/809432?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A809432"},"id":"809432","created":"2009-06-12T00:00:00+00:00","updated":"2026-02-06T10:23:41.559730+00:00"},{"metadata":{"publication_info":[{"journal_volume":"14","artid":"083034","year":2012,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1212465"},"journal_title":"New J.Phys."}],"documents":[{"filename":"njp12_8_083034.pdf","attachment":{"content":"T h e  o p e n – a c c e s s  j o u r n a l  f o r  p h y s i c s\n\nNew Journal of Physics\n\nMacroscopic quantum electrodynamics in nonlocal\nand nonreciprocal media\n\nStefan Yoshi Buhmann1,3, David T Butcher1 and Stefan Scheel1,2\n\n1 Quantum Optics and Laser Science, Blackett Laboratory, Imperial College\nLondon, Prince Consort Road, London SW7 2AZ, UK\n2 Institut für Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock,\nGermany\nE-mail: s.buhmann@imperial.ac.uk\n\nNew Journal of Physics 14 (2012) 083034 (12pp)\nReceived 6 July 2012\nPublished 29 August 2012\nOnline at http://www.njp.org/\ndoi:10.1088/1367-2630/14/8/083034\n\nAbstract. We formulate macroscopic quantum electrodynamics in the most\ngeneral linear, absorbing media. In particular, Onsager reciprocity is not assumed\nto hold. The field quantization is based on the source-quantity representation\nof the electromagnetic field in terms of the dyadic Green’s tensor. For media\nwith a nonlocal response, a description in terms of a complex conductivity\ntensor is employed. As an alternative description, we introduce the permittivity,\npermeability and magnetoelectric susceptibilities to obtain an explicitly duality-\ninvariant scheme. We find that duality invariance only holds as a continuous\nsymmetry when nonreciprocal responses are allowed for.\n\n3 Author to whom any correspondence should be addressed.\n\nContent from this work may be used under the terms of the Creative Commons Attribution-NonCommercial-\nShareAlike 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title\n\nof the work, journal citation and DOI.\n\nNew Journal of Physics 14 (2012) 083034\n1367-2630/12/083034+12$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft\n\nmailto:s.buhmann@imperial.ac.uk\nhttp://www.njp.org/\nhttp://creativecommons.org/licenses/by-nc-sa/3.0\nhttp://creativecommons.org/licenses/by-nc-sa/3.0\n\n\n2\n\nContents\n\n1. Introduction 2\n2. Field quantization in nonlocal media 3\n3. Field quantization in terms of electric and magnetic response functions 6\n4. Duality invariance 8\n5. Conclusion 10\nAcknowledgment 10\nAppendix. Integral relation for the Green tensor 10\nReferences 11\n\n1. Introduction\n\nThe linear response of a macroscopic material to externally applied electromagnetic fields\ncan go beyond the scope of simple descriptions via electric permittivities and magnetic\npermeabilities [1]. In particular, cross-susceptibilities naturally arise in chiral (meta-)\nmaterials [2], topological insulators [3] or moving media [4]. In the latter case nonlocal\nresponses arise [5] with the additional complication that Onsager reciprocity [6] fails to hold.\nOnsager reciprocity, the electrodynamic manifestation of time-reversal symmetry, would also\nbe violated in Tellegen media [7], including the recently proposed perfect electromagnetic\nconductor that continuously interpolates between a perfect conductor and an infinitely\npermeable material [8].\n\nChiral metamaterials with cross-susceptibilities have been constructed based on nanoscale\nchiral objects, such as a helix [9]. This leads to a discriminatory response of the medium to\nleft- and right-circularly polarized light. This central feature of chiral media is important in\nbiological systems due to the prevalence of left-handed objects in the processes crucial to\nlife [10]. Furthermore, chiral meta-materials have been discussed as candidates for repulsive\nCasimir forces [11]. It should be noted that repulsive forces for magnetoelectric media were\noriginally discussed for dielectric plates interacting with magnetic plates [12]. To implement\nthese effects with metamaterials, the anisotropic response of the medium needs to be taken into\naccount [13].\n\nTopological insulators are a novel class of materials which behave as insulators in their bulk\nphase but allow for conduction on the surface [14, 15]. Time reversal symmetry is an important\nfeature which ensures an extremely high stability of the surface currents. The latter make\ntopological insulators a promising candidate for quantum computing [16]. It has recently been\npredicted that topological insulators [3] (or materials with a Chern–Simons interaction [17])\ncould be used to realize repulsive Casimir forces. A related phenomenon is the fractional\nquantum hall effect where the Hall current takes fractional values due to electron–electron\ninteractions [18]. This medium can be nonlocal [19] and in contrast to topological insulators\nit can violate time-reversal symmetry [20] and hence Onsager reciprocity.\n\nThe impact of electric versus magnetic material properties can be studied in a systematic\nway by means of a duality transformation [21]. It has recently been shown that macroscopic\nQED [22] in isotropic magnetoelectrics obeys a discrete duality symmetry [23]. This has\nimmediate consequences for dispersion forces in free space.\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n3\n\nThe successes in the realization of the above mentioned novel materials open the\nperspective on a range of new quantum phenomena related to photon-induced matter\ninteractions, quantum dynamics and (possibly irreversible) quantum-light propagation. To make\nsuch studies possible, we will construct a quantum theory of the electromagnetic field in the\nmost general linear absorbing media, including nonlocal, bianisotropic and Onsager reciprocity\nviolating materials. A recent theory based on canonical quantization is a valuable step in this\ndirection [24], which does not yet consider the most general nonlocal media (apart from the\nabove mentioned moving media).\n\nIn addition, our theory shall answer the question under which circumstances duality can be\nrealized as a continuous symmetry of the Maxwell equations in media; and it will shed light on\nthe generalizations necessary to discuss moving media and quantum friction.\n\n2. Field quantization in nonlocal media\n\nWe begin by recalling a quantization procedure of the electromagnetic field in the presence of\nan absorbing medium. For alternative methods for the quantization procedure, see [25–27] and\nreferences therein. In a linearly responding medium, the effect of an external electromagnetic\nfield on the matter can be given by Ohm’s law in its most general form\n\njin(r, t)=\n∫\n∞\n\n−∞\n\ndτ\n\n∫\nd3r ′Q(r, r′, τ ) ·E(r′, t − τ) + jN(r, t). (1)\n\nHere, Q(r, r′, τ ) is the conductivity tensor and jN(r, t) is the random noise current required\nto fulfil the fluctuation–dissipation theorem (17) as given below. Causality requires that\nQ(r, r′, τ )= 0 for cτ < |r− r′|, in particular for all τ < 0 [28]. In frequency space, Ohm’s law\ntakes the simpler form\n\njin(r, ω)=\n\n∫\nd3r ′Q(r, r′, ω) ·E(r′, ω) + jN(r, ω) (2)\n\nwith\n\nQ(r, r′, ω)=\n\n∫\n∞\n\n0\ndτ ei ωτ Q(r, r′, τ ). (3)\n\nAs a result of the causality requirement the conductivity obeys the Schwarz reflection principle,\n\nQ∗(r, r′, ω)= Q(r, r′,−ω∗) ∀ r, r′, ω. (4)\n\nQuantization is achieved by specifying the commutator\n\n[ĵN(r, ω), ĵ†\nN(r′, ω′)]=\n\nh̄ω\n\nπ\nRe[Q(r, r′, ω)]δ(ω−ω′), (5)\n\nwhere we have introduced generalized real and imaginary parts of a tensor field according to\n\nReT(r, r′)= 1\n2\n\n[\nT(r, r′) + T†(r′, r)\n\n]\n, (6)\n\nImT(r, r′)=\n1\n\n2i\n\n[\nT(r, r′)− T†(r′, r)\n\n]\n. (7)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n4\n\nThey reduce to ordinary real and imaginary parts for orthogonal tensor fields that are reciprocal,\ni.e. TT(r′, r)= T(r, r′). The other nontrivial current commutators follow alternatively via the\nrules [b̂, â]=−[â, b̂] or [â†, b̂†]=−[â, b̂]†;[\n\nĵ†\nN(r, ω), ĵN(r′, ω′)\n\n]\n=−\n\nh̄ω\n\nπ\nRe[Q∗(r, r′, ω)]δ(ω−ω′)\n\n=−\nh̄ω\n\nπ\nRe[QT(r′, r, ω)]δ(ω−ω′). (8)\n\nThe fact that the right-hand side of this expression is a Hermitian tensor field guarantees the\nconsistency of the commutation relations.\n\nCombining Ohm’s law with Maxwell’s equations [ i ωρ̂ in(r, ω)=∇ · ĵin(r, ω)]\n\n∇ · Ê(r, ω)=\nρ̂ in(r, ω)\n\nε0\n, ∇ × Ê(r, ω)− i ωB̂(r, ω)= 0, (9)\n\n∇ · B̂(r, ω)= 0, ∇ × B̂(r, ω) +\ni ω\n\nc2\nÊ(r, ω)= µ0ĵin(r, ω), (10)\n\none finds that the electric field obeys a generalized inhomogeneous Helmholtz equation of the\nform[\n∇ ×∇ ×−\n\nω2\n\nc2\n\n]\nÊ(r, ω)− i µ0ω\n\n∫\nd3r ′Q(r, r′, ω) · Ê(r′, ω)= i µ0ωĵN(r, ω). (11)\n\nWith the help of the Green function G(r, r′, ω) of the Helmholtz equation, defined by[\n∇ ×∇ ×−\n\nω2\n\nc2\n\n]\nG(r, r′, ω)− i µ0ω\n\n∫\nd3s Q(r, s, ω) ·G(s, r′, ω)= δ(r− r′), (12)\n\nwhere G(r, r′, ω)→ 0 for |r− r′| →∞, the formal solution to the integro-differential\nequation (11) reads\n\nÊ(r, ω)= i µ0ω[G(ω) ? ĵN(ω)](r). (13)\n\nHere, [G ? ĵN] is an abbreviation denoting the spatial convolution\n\n[G ? ĵN](r)≡\n∫\n\nd3r ′G(r, r′) · ĵN(r′).\n\nBy virtue of its definition (12), the Green tensor inherits the Schwarz reflection principle\nfrom the conductivity tensor (4),\n\nG∗(r, r′, ω)= G(r, r′,−ω∗) ∀ r, r′, ω. (14)\n\nHowever, as a major departure from previous treatments, we do not require the conductivity\nto obey reciprocity, i.e. the relation QT(r′, r, ω)= Q(r, r′, ω) does not necessarily hold. As a\nconsequence, the Green tensor will not obey the Onsager principle, i.e. the relation\n\nGT(r′, r, ω)= G(r, r′, ω) (15)\n\nwill not hold in general. Recall that the Onsager principle, applied to electromagnetic field\npropagation, states a reversibility of optical paths [6]. According to (13), the Green tensor\ngoverns the relation between a source current j at r′ along a direction e2 and the generated\nelectric field E at r along a direction e1. If (15) holds, then the Onsager principle states that the\nroles of source and field can be reversed. A source current j at r along a direction e1 would\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n5\n\nthen give rise to an electric field E at r′ along a direction e2. In a configuration involving\nnonreciprocal media, this is not necessarily the case. Despite the extension to nonreciprocal\nmedia, it is still possible to derive the useful integral relation (appendix)\n\nµ0ω[G(ω) ?ReQ(ω) ? G†(ω)](r, r′)= Im G(r, r′, ω). (16)\n\nIt generalizes the result from [29] to the case where Onsager reciprocity does not hold.\nThe theory thus far is analogous to classical electromagnetism in an absorbing medium\n\nunder the assumption of classical fluctuating current sources, ĵN 7→ jN. Their strengths are\ngoverned by the fluctuation–dissipation theorem in the classical (high-temperature) limit [30].\n\nIntroducing the ground state |{0}〉 of the medium-field system according to\nĵN(r, ω)|{0}〉 = 0, the currents satisfy the fluctuation–dissipation theorem as an immediate\nconsequence of (5),\n\n〈{1ĵN(r, ω),1ĵ†\nN(r′, ω′)}〉 =\n\nh̄\n\nπ\nIm[i ωQ(r, r′, ω)]δ(ω−ω′). (17)\n\nCombining (5) and (13), one finds that the fluctuations of the electric field are also consistent\nwith the fluctuation–dissipation theorem, as required:\n\n〈{1Ê(r, ω),1ʆ(r′, ω′)}〉 =\nh̄\n\nπ\nIm\n[\nµ0ω\n\n2G(r, r′, ω)\n]\nδ(ω−ω′). (18)\n\nIn order to verify the canonical equal-time commutation relations, we introduce the\nvector potential for the electromagnetic field in the Coulomb gauge, Â(r, ω)= Ê⊥(r, ω)/(i ω)\n\n(⊥: transverse part). Using (5) and (13), one finds[\nÊ(r, ω), †(r′, ω′)\n\n]\n=\n\nih̄µ0ω\n\nπ\nIm[G⊥(r, r′, ω)]δ(ω−ω′) (19)\n\nand hence [\nÊ(r), Â(r′)\n\n]\n=\n\nh̄µ0\n\n2π\n\n∫\n∞\n\n−∞\n\ndω ω\n[\nG⊥(r, r′, ω) + ⊥GT(r′, r, ω)\n\n]\n, (20)\n\nwhere the Schwarz reflection principle (14) has been used. Use has been made of the left- and\nright-sided transverse projections, which are defined, respectively, as\n\n⊥T = δ⊥ ? T , T⊥ = T ? δ⊥. (21)\n\nClosing the integration contour in the upper half of the complex ω plane, where the Green’s\nfunction is analytic, and using the asymptote (ω2/c2)G(r, r′, ω)→−δ(r− r′) for |ω| →∞,\none finds the canonical commutation relation from free-space QED,[\n\nÊ(r), Â(r′)\n]\n=\n\nih̄\n\nε0\nδ⊥(r− r′), (22)\n\nas required. We now introduce the bosonic creation and annihilation operators of the matter-field\nsystem, f̂† and f̂ according to the prescription\n\nĵN(ω)=\n\n√\nh̄ω\n\nπ\nR(ω) ? f̂(ω), (23)\n\nwhere R is a square root of the positive definite tensor field Re[Q],\n\nR(ω) ? R†(ω)=Re[Q(ω)]. (24)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n6\n\nThis solution is only unique up to a unitary matrix which does not affect the physical results [29].\nTogether with equation (5), this ensures bosonic commutation relations,\n\n[f̂(r, ω), f̂†(r′, ω′)]= δ(r− r′)δ(ω−ω′). (25)\n\nThe Hamiltonian of the medium-field system is then\n\nĤF =\n\n∫\nd3r\n\n∫\n∞\n\n0\ndω h̄ω f̂†(r, ω) · f̂(r, ω). (26)\n\nIt leads to the free evolution of the dynamical variables as f̂(r, ω, t)= f̂(r, ω) e−i ωt ; hence\nMaxwell’s equations for the electromagnetic-field operators in the Heisenberg picture are valid\nby construction.\n\n3. Field quantization in terms of electric and magnetic response functions\n\nThe properties of media with spatially nonlocal or local responses can alternatively be described\nby their permittivity, permeability and magnetoelectric susceptibilities. To begin, it is convenient\nto cast the inhomogeneous Maxwell equations (10) into the forms (we drop the spatial and\nfrequency arguments from now on)\n\n∇ · D̂= 0, ∇ × Ĥ + i ωD̂= 0 (27)\n\nwith\n\nD̂= ε0Ê + P̂, Ĥ=\n1\n\nµ0\nB̂− M̂. (28)\n\nThe polarization and magnetization fields respond linearly to the electric and magnetic fields,\n\nP̂= ε0(ε− ξ ? µ−1 ? ζ − I) ? Ê + Z−1\n0 ξ ? µ−1 ? B̂ + P̂N, (29)\n\nM̂= Z−1\n0 µ−1 ? ζ ? Ê + µ−1\n\n0 (I−µ−1) ? B̂ + M̂N. (30)\n\nThe medium is characterized by its permittivity, ε(r, r′, ω), its permeability, µ(r, r′, ω)\n\nand its magnetoelectric susceptibilities, ξ(r, r′, ω) and ζ (r, r′, ω). P̂N(r, ω) and M̂N(r, ω)\n\ndenote the noise polarization and noise magnetization, respectively, and Z0 =\n√\n\nµ0/ε0 is\nthe vacuum impedance. In the case of nonlocal media the permittivity, permeability and\nmagnetoelectric susceptibilities are functions of two independent spatial variables, whereas in\na locally responding media they read ε(r, r′, ω)= ε(r, ω)δ(r− r′), µ(r, r′, ω)= µ(r, ω)δ(r−\nr′), ξ(r, r′, ω)= ξ(r, ω)δ(r− r′) and ζ (r, r′, ω)= ζ (r, ω)δ(r− r′). By combining (28)–(30),\nthe constitutive relations can be given in the more familiar form (see [31] for the nonconducting\ncase)\n\nD̂= ε0ε ? Ê + c−1ξ ? Ĥ + P̂N + c−1ξ ? M̂N, (31)\n\nB̂= c−1ζ ? Ê + µ0µ ? Ĥ + µ0µ ? M̂N, (32)\n\nwhere the notational distinction between locally and nonlocally responding media as given\nabove applies.\n\nIn order to distinguish reciprocal magnetoelectric susceptibilities from nonreciprocal ones,\nas previously discussed in the nondispersive case [32], one commonly writes ξ = χT\n\n− iκT and\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n7\n\nζ = χ + iκ . The chirality tensor κ = (ζ − ξT\n)/(2i) represents the reciprocal magnetoelectric\n\nresponse; whereas the nonreciprocal magnetoelectric tensor χ = (ζ + ξT\n)/2 vanishes for a\n\nreciprocal medium.\nBy combining Maxwell’s equations with the constitutive relations (31) and (32), we note\n\nthat the respective Green tensor is the solution to equation (12) with\n\nQ= (i µ0ω)−1\n∇ × (µ−1\n\n− I)×\n←−\n∇ + Z−1\n\n0 ∇ ×µ−1 ? ζ + Z−1\n0 ξ ? µ−1\n\n×\n←−\n∇\n\n−iε0ω(ε− ξ ? µ−1 ? ζ − I) (33)\n\nand\n\nĵN =−i ωP̂N +∇ × M̂N, (34)\n\nwhere [T×\n←−\n∇ ]i j(r, r′)= ε jkl∂\n\n′\n\nl Tik(r, r′) denotes a derivative acting on the second argument of\na tensor function. The Green tensor for the electric field (13) solves[\n∇ ×µ−1 ?∇ ×−\n\ni ω\n\nc\n∇ ×µ−1 ? ζ +\n\ni ω\n\nc\nξ ? µ−1 ?∇ −\n\nω2\n\nc2\n(ε− ξ ? µ−1 ? ζ )\n\n]\n? G= δ. (35)\n\nThe commutation relations for P̂N and M̂N can be deduced by substituting the real parts of (33)\nand (34) into (5),[\nP̂N(r, ω), P̂†\n\nN(r′, ω′)\n]\n=\n\nε0h̄\n\nπ\nIm\n\n{\nε(ω)− [ξ(ω) ?µ−1(ω) ? ζ (ω)]\n\n}\n(r, r′)δ(ω−ω′), (36)\n\n[\nP̂N(r, ω), M̂†\n\nN(r′, ω′)\n]\n=\n\nh̄\n\n2π iZ0\n\n{\n[ξ(ω) ?µ−1(ω)]− [ζ †(ω) ?µ−1†(ω)]\n\n}\n(r, r′)δ(ω−ω′), (37)\n\n[\nM̂N(r, ω), P̂†\n\nN(r′, ω′)\n]\n=\n\nh̄\n\n2π iZ0\n\n{\n[µ−1(ω) ? ζ (ω)]− [µ−1†(ω) ? ξ †\n\n(ω)]\n}\n(r, r′)δ(ω−ω′), (38)[\n\nM̂N(r, ω), M̂†\nN(r′, ω′)\n\n]\n=−\n\nh̄\n\nπµ0\nIm[µ−1(r, r′, ω)]δ(ω−ω′). (39)\n\nWe now introduce the bosonic creation and annihilation operators with commutation relations[\nf̂λ(r, ω), f̂†\n\nλ′(r\n′, ω′)\n\n]\n= δλλ′δ(r− r′)δ(ω−ω′), (λ, λ′ = e, m) (40)\n\naccording to (\nP̂N\n\nM̂N\n\n)\n=\n\n√\nh̄\n\nπ\nR ?\n\n(\nf̂e\n\nf̂m\n\n)\n, (41)\n\nwhere the (6× 6)-matrix R is a root of\n\nR ?R†\n=\n\nε0Im[ε− ξ ? µ−1 ? ζ ]\nξ ? µ−1\n\n− ζ † ? µ−1†\n\n2iZ0\nµ−1 ? ζ −µ−1† ? ξ †\n\n2iZ0\n−\nIm[µ−1]\n\nµ0\n\n. (42)\n\nThe Hamiltonian of the body–field system is again quadratic and diagonal in the bosonic\nvariables,\n\nĤF =\n\n∑\nλ=e,m\n\n∫\nd3r\n\n∫\n∞\n\n0\ndω h̄ω f̂†\n\nλ(r, ω) · f̂λ(r, ω). (43)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n8\n\nNote that (28)–(30) imply a separation of the internal current density into electric and magnetic\nparts, ĵin =−i ωP̂ +∇ × M̂. This separation and the resulting explicit field quantization is not\nunique in spatially dispersive (i.e. nonlocal) media, as the magnetization field can be absorbed\ninto the transverse part of the polarization field [28]. While a local magnetoelectric medium\ncan always be described in terms of a (nonlocal) conductivity without reference to magnetic\nproperties, the equivalent description in terms of a local permittivity, permeability and cross-\nsusceptibility is much more accessible. These parameters are often known experimentally and\nthey allow for a classification of electromagnetic responses.\n\n4. Duality invariance\n\nAn electromagnetic system separated into distinct electric and magnetic causes and effects can\nbe subject to a duality transformation operation, that is, a global exchange of the electric and\nmagnetic properties. A system invariant under such an operation is said to possess duality\ninvariance as a symmetry [21]. This symmetry can be exploited in order to simplify the\ncomputation of dispersion forces involving, say magnetizable media, from known dispersion\nforces between polariable media [23].\n\nBy introducing dual-pair notation (ÊT, Z0ĤT)T, (Z0D̂T, B̂T)T, we may write the Maxwell\nequations (9) and (27) in the compact form\n\n∇ ·\n\n(\nZ0D̂\n\nB̂\n\n)\n=\n\n(\n0\n\n0\n\n)\n, (44)\n\n∇ ×\n\n(\nÊ\n\nZ0Ĥ\n\n)\n− i ω\n\n(\n0 1\n\n−1 0\n\n)(\nZ0D̂\n\nB̂\n\n)\n=\n\n(\n0\n\n0\n\n)\n. (45)\n\nThe constitutive relations (31) and (32) in condensed form read(\nZ0D̂\n\nB̂\n\n)\n=\n\n1\n\nc\n\n(\nε ξ\n\nζ µ\n\n)\n?\n\n(\nÊ\n\nZ0Ĥ\n\n)\n+A ?\n\n(\nZ0P̂N\n\nµ0M̂N\n\n)\n(46)\n\nwith\n\nA=\n\n(\nI ξ\n\n0 µ\n\n)\n. (47)\n\nMaxwell’s equations are invariant under duality transformations(\nx\ny\n\n)~\n= D(θ)\n\n(\nx\ny\n\n)\n, D(θ)=\n\n(\ncos θ sin θ\n\n− sin θ cos θ\n\n)\n, (48)\n\nbecause D(θ) is a symplectic matrix. From the constitutive relations, as shown in (46), we find\nthe transformed medium response functions\n\nε\n\nζ\n\nξ\n\nµ\n\n\n~\n\n=D(θ)\n\n\nε\n\nζ\n\nξ\n\nµ\n\n (49)\n\nwith\n\nD(θ)= D(θ)⊗ D(θ) (50)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n9\n\nas well as (\nZ0P̂N\n\nµ0M̂N\n\n)~\n=A\n\n~−1 ? D(θ)A ?\n\n(\nZ0P̂N\n\nµ0M̂N\n\n)\n, (51)\n\nwhere\n\nA\n−1\n=\n\n(\nI −ξ ·µ−1\n\n0 µ−1\n\n)\n. (52)\n\nIt is worth discussing a few special cases of bianisotropic media, their characteristic features\nand behaviour under duality transformations:\n\nLocal media [ε(r, r′, ω)= ε(r, ω)δ(r− r′), similarly for the other response functions]:\nConvolution operators reduce to ordinary matrix products, compatible with all other special\ncases below.\nIsotropic media (ε = εI, µ= µI, ξ = ζ = 0): Onsager reciprocity (15) holds; P̂N and M̂†\n\nN\ncommute; generalized real and imaginary parts reduce to ordinary ones; discrete duality\nsymmetry.\nBi-isotropic media (ε = εI, µ= µI, ξ = ξ I, ζ = ζ I): generalized real and imaginary parts\nin (36) and (39) reduce to ordinary ones; continuous duality symmetry.\n\nAnisotropic media (ξ = ζ = 0): P̂N and M̂†\nN commute; discrete duality symmetry.\n\nReciprocal media (εT\n= ε, ξ T\n\n=−ζ , µT\n= µ): (15) holds; generalized real and imaginary\n\nparts reduce to ordinary ones; discrete duality symmetry.\n\nHere, discrete duality symmetry means that the rotation angle is restricted to values θ =\n\nnπ/2 with n ∈ Z. Note that duality is only realized as a continuous symmetry when Onsager-\nviolation is allowed for. Notably, a reduction in reciprocity symmetry leads to an enhancement\nof duality symmetry.\n\nIn order to derive transformation laws for the Green tensor, we combine (9), (11), (28), (30)\nand (34) to write(\n\nÊ\nZ0Ĥ\n\n)\n=−cB ?G ?\n\n(\nZ0P̂N\n\nµ0M̂N\n\n)\n, (53)\n\nB=\n\n(\nI 0\n\n−µ−1 ? ζ µ−1\n\n)\n, (54)\n\nG=\n\n(\nGee Gem\n\nGme Gmm + µ\n\n)\n, (55)\n\nwhere we have introduced the shorthand notations Gee = (i ω/c)G(i ω/c), Gem = (i ω/c)G×\n←−\n∇\n′, Gme =∇ ×G(i ω/c) and Gmm =∇ ×G×\n\n←−\n∇\n′. The transformed Green’s tensors follow by\n\napplying duality transformations on both sides of this equation,\n\nG\n~\n= B\n\n~−1 ? D(θ)B ?G ?A−1 ? D−1(θ)A~, (56)\n\nwhere\n\nB\n−1\n=\n\n(\nI 0\n\n−µ−1 ? ζ µ−1\n\n)\n. (57)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n10\n\nIn the special case of r and r′ being in free space, we have A,B= I, so that G~ = D(θ)G ?\n\nD−1(θ) and hence\nGee\n\nGem\n\nGem\n\nGmm + δ\n\n\n~\n\n=D(θ)\n\n\nGee\n\nGem\n\nGem\n\nGmm + δ\n\n. (58)\n\nThe Green tensors then transform like the medium response functions with (50).\n\n5. Conclusion\n\nBased on the general Ohm’s law, we have quantized the electromagnetic field in the presence of\nnonlocal, nonreciprocal media which satisfies (i) the canonical commutation relations from free-\nspace QED; (ii) the linear fluctuation–dissipation theorem; and (iii) the macroscopic Maxwell\nequations. A key feature of the scheme is the symmetrization of tensor fields via generalized\nreal and imaginary parts, which is necessary whenever Onsager reciprocity does not hold. Their\npresence in the fluctuation–dissipation theorem is the key to avoiding restrictions on the allowed\nmedium response.\n\nFor nonlocal and local bianisotropic media we have shown that quantization can\nalternatively be performed by the introduction of permittivity, permeability and magnetoelectric\nsusceptibilities. When the latter do not vanish, the noise polarization and magnetization\ndo not commute. We have explicitly determined the behaviour of the fields and response\nfunctions under duality transformations. The full continuous transformation group applies\nfor bianisotropic and bi-isotropic media, but reduces to a discrete symmetry for isotropic,\nanisotropic and/or reciprocal media.\n\nThe scheme lays the foundation for exact studies of quantum phenomena such as dispersion\nforces, Förster energy transfer or environment-assisted molecular transition rates in the presence\nof motion or novel media with chiral or nonreciprocal properties. Moving media are a\nprime example for the occurrence of nonreciprocal material properties [28] which have to be\nthoroughly accounted for in order to understand, e.g. quantum friction. CP violation in atoms or\nmolecules is manifest in their nonreciprocal cross-polarizability. The Curie principle, stating\nthat certain interactions between two partners (atoms, molecules, bodies, etc) require them\nto possess similar properties, then allows for a detection of CP violation via atom–surface\ninteractions provided the surface exhibits a corresponding nonreciprocity.\n\nAcknowledgment\n\nThis work was supported by the UK Engineering and Physical Sciences Research Council\n(EPSRC).\n\nAppendix. Integral relation for the Green tensor\n\nTo derive the integral relation (16) for the Green tensor, we write the Helmholtz equation (12)\nas\n\nĤ · Ĝ= Î, (A.1)\n\nNew Journal of Physics 14 (2012) 083034 (http://www.njp.org/)\n\nhttp://www.njp.org/\n\n\n11\n\nwhere 〈r|Ĝ|r′〉 = G(r, r′, ω) and 〈r|Ĥ|r′〉 = [∇ ×∇ ×−ω2/c2]δ(r− r′)− i µ0ωQ(r, r′, ω). The\nGreen operator is the right-inverse and, within any group of invertible operators, also the left-\ninverse of the Helmholtz operator,\n\nĜ · Ĥ= Î. (A.2)\n\nFrom this relation and its Hermitian conjugate we find that\n\nĜ · (Ĥ− Ĥ†) · Ĝ†\n= Ĝ†\n\n− Ĝ. (A.3)\n\nIn coordinate space this relation reads\n\nµ0ω\n\n∫\nd3s\n\n∫\nd3s ′G(r, s, ω) · ReQ(s, s′, ω) · G†(r′, s′, ω)= Im G(r, r′, ω) (A.4)\n\nwhich in convolution notation takes the form of equation (16).\n\nReferences\n\n[1] Lindell L V, Sihvola A H, Tretyakov S A and Viitanen A J 1994 Electromagnetic Waves in Chiral and\nBi-Isotropic Media (Norwood, MA: Artech House)\n\nHehl F H and Obukhov Y N 2003 Foundations of Classical Electrodynamics (Boston, MA: Birkhäuser)\nHehl F H and Obukhov Y N 2005 Phys. Lett. A 334 249\n\n[2] Ishimaru A, Lee S W, Kuga Y and Jandhyala V 2003 IEEE Trans. Antennas Propag. 51 2550\nYannopapas V 2006 J. Phys.: Condens. Matter 18 6883\nThiel M, Rill M S, von Freymann G and Wegener M 2009 Adv. Mater. 21 4680\n\n[3] Grushin A G and Cortijo A 2011 Phys. Rev. Lett. 106 020403\n[4] McKenzie J F 1967 Proc. Phys. Soc. 91 532\n[5] Horsley S A R 2012 Phys. Rev. A 86 023830\n[6] Onsager L 1931 Phys. Rev. 37 405\n[7] Tellegen B D H 1948 Philips Res. 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Field quantization in terms of electric and magnetic response functions\n\t4. Duality invariance\n\t5. Conclusion\n\tAcknowledgment\n\tAppendix. Integral relation for the Green tensor\n\tReferences"},"fulltext":true,"key":"84faa482be8466d0a1bb71845f2b52c7","url":"https://inspirehep.net/files/84faa482be8466d0a1bb71845f2b52c7"}],"authors":[{"raw_affiliations":[{"value":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom"}],"full_name_unicode_normalized":"buhmann, stefan yoshi","full_name":"Buhmann, Stefan Yoshi","record":{"$ref":"https://inspirehep.net/api/authors/1946455"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/902868"},"value":"Imperial Coll., London"}],"last_name":"Buhmann","ids":[{"schema":"INSPIRE BAI","value":"S.Y.Buhmann.3"}],"signature_block":"BANANs","first_name":"Stefan Yoshi","uuid":"455006d4-7ae7-4534-a854-0bb4243433eb","recid":1946455},{"raw_affiliations":[{"value":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United 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