Elimination of the A-square problem from cavity QED
aa r X i v : . [ qu a n t - ph ] O c t Elimination of the A-square problem from cavity QED
András Vukics, ∗ Tobias Grießer, and Peter Domokos Institute for Solid State Physics and Optics, Wigner Research Centre for Physics,Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary Institut für Theoretische Physik, Universität Innsbruck,Technikerstraße 25, 6020 Innsbruck, Austria
We generalize the Power–Zineau–Woolley transformation to obtain a canonical Hamiltonian of cavityquantum electrodynamics for arbitrary geometry of boundaries. This Hamiltonian is free from theA-square term and the instantaneous Coulomb interaction between distinct atoms. The single-modemodels of cavity QED (Dicke, Tavis–Cummings, Jaynes–Cummings) are justified by a term by termmapping to the proposed microscopic Hamiltonian. As one straightforward consequence, the basisof no-go argumentations concerning the Dicke phase transition with atoms in electromagnetic fieldsdissolves.
PACS numbers: 05.30.Rt,37.30.+i,42.50.Nn,42.50.Pq
The fundamental description of the interaction ofatomistic matter with the electromagnetic field in theCoulomb gauge is known to suffer from the presence ofan awkward term containing the square of the vector po-tential. In most of the practical cases, in the frameworkof a diluteness assumption for the atoms, this term canbe neglected and the observable effects are ultimately ac-counted for in terms of a simplified model, such as theJaynes–Cummings one, for example. In typical quantumoptical systems, such a phenomenological approach withproperly adjusted parameters usually gives a satisfactoryquantitative accuracy. However, there are situations inwhich even the qualitative behaviour of the system isquestionable because of the confusion around this term.A prominent example is the Dicke model, where the veryexistence of the predicted superradiant phase transitiondepends on the validity of the adopted effective model.[1–4] Further discrepancies due to the A-square term occurin relation with novel artificial systems in which the elec-tromagnetic field confined into a small volume is coupledto some kind of polarizable material in the so-called ul-trastrong coupling regime.[5, 6]In this Letter, we show that cavity quantum elec-trodynamics, i.e., when the field itself as well as thelight-matter interaction are significantly influenced bythe presence of boundaries, can be established at a fun-damental level on a Hamiltonian which eliminates theproblem of the A-square term. We present a canonicaltransformation which makes manifest that this term iscompensated by a dipole-dipole interaction term, and theremaining terms are of a simple linear form.[7] From ourapproach it follows, for example, that there is no prin-ciple that would prevent the superradiant phase tran-sition in the case of an ensemble of atomic dipoles ina cavity. The canonical transformation is analogous tothe Power–Zienau–Woolley (PZW) transformation in freespace, however, in our approach we allow for arbitrarygeometry, thereby treating general cavity QED system.All our vector fields are thus defined on a generic (pos- sibly even multiply connected) domain D in the three-dimensional real space bounded by (possibly several dis-junct) sufficiently smooth surfaces ∂ D , which consist of aperfect conductor. Overall, D is assumed to be bounded.Consider an arbitrary number of point charges cou-pled to the electromagnetic field confined into D . In theCoulomb (minimal-coupling) gauge, defined by ∇ · A = 0 , (1)the Hamiltonian of the system reads: H = X α [ p α − q α A ( r α )] m α + ε Z D d r ( ∇ U ) + H field , (2a)with U being the scalar potential, p α the canonical mo-mentum of particle α conjugate to its position r α , and H field = ε Z D d r "(cid:18) Π ε (cid:19) + c ( ∇ × A ) , (2b)with Π = ε ∂ t A being the momentum conjugate to A .An important observation is that, unlike in free space, the condition (1) does not fix the potentials completely.The remaining freedom of choosing the potentials withinthe Coulomb gauge amounts to a freedom in choosingdifferent constant values for U on each of the connectedcomponents of ∂ D , which will result in various configura-tions of condensator fields carried by U . Our choice herewill be to set U | ∂ D = 0 and A × n | ∂ D = 0 . (3)Together with Eq. (1), the latter condition makes up forthe vector potential satisfying both the electric and mag-netic boundary conditions.[12]The electric-dipole approximation to this Hamiltoniancan be obtained in two steps. Step 1 (long-wavelengthapproximation): We assume that the individual pointcharges form (a certain number of) spatially separated,well-localized clusters, that is, atoms. Then, instead of P α there appears P A P α ∈ A . We neglect all radiativeeffects on the intra-atomic scale, that is, we set A ( r α ) = A ( r A ) , where r A is the position of that atom A whichincorporates the charge α . Step 2: We assume that theatoms have only electric dipole moment, that is, no netcharge and no further electric or magnetic moments.Upon the first assumption, we split the Coulomb (elec-trostatic) term into intra- and inter-atomic parts, andtake the intra-atomic part as identical to the one in freespace, under the assumption that the distance of atomsfrom the boundary is much larger than the atomic radius.The electric-dipole order of the Hamiltonian in Coulombgauge then reads: H ED = X A (cid:20) H A − u p A · A ( r A ) + v A ( r A )+ V dipole-selfCoulomb ( r A ) + X B V dipole-dipoleCoulomb ( r A − B ) (cid:21) + H field , (4a)where u and v are constants composed of the m α s and q α s. The single-atom Hamiltonian reads H A = X α ∈ A (cid:18) p α m α + q α πε X β ∈ A β = α q β | r α − r β | (cid:19) . (4b)It is this Hamiltonian (4) that is usually taken as thestarting point of cavity QED. However, it is fraught withthe following problems: (i) the canonical momentum ofthe atoms does not equal their kinetical momentum; fur-thermore, as we mentioned, (ii) the presence of the A-square term, which yields creation and annihilation ofpairs of photons; and finally, (iii) there appears an instan-taneous electrostatic interaction between remote atoms( V dipole-dipoleCoulomb ) and an interaction of a single dipole withits own induced surface charges ( V dipole-selfCoulomb ). The formeris influenced, while the latter is created by the presenceof the boundaries [8].In free space, these weaknesses can be dissolved byperforming the PZW transformation on the minimal cou-pling Hamiltonian (2a) to the multipolar-coupling gauge(cf. Ref. [9] Chapter IV.C). Here, inspired by the free-space procedure, we elevate this transformation onto avery general level, which allows for an arbitrary domain D and boundaries ∂ D , i.e. for a general cavity QED sce-nario.The transformation that we adopt is canonical, definedby the Type-2 generating function G ≡ Z D d r A · (cid:0) Π ′ + R P (cid:1) + X α r α · p ′ α , (5a) which yields a displacement of the momenta Π = δG δ A = Π ′ + R P , (5b) p α = ∂G ∂ r α = p ′ α + ∂∂ r α Z D d r A · P . (5c)At this point, P is an arbitrary vector, and R is part ofan orthogonal projector decomposition of the identity, Q + R = id L . (6)where L is the subspace of the Hilbert space L ( D , R ) of square-integrable vector fields such that the elementsof L satisfy the boundary condition that they are normalto the boundaries: L ( D , R ) ≡ (cid:8) v ∈ L ( D , R ) (cid:12)(cid:12) v × n | ∂ D = 0 (cid:9) , (7)which is of course nothing else than the boundary condi-tion on the electric field (and hence the vector potential)at a perfectly conducting surface.In order that the transformation (5) be canonical, R must be a projector onto the divergence-free subspace of L : R : L → ker ( div ) , (8)because this ensures that A in Eq. (5a) can be treatedas unconstrained. Here, div (and curl below) are thedivergence (and curl) operators over L , with the do-main restricted to L . The notation ‘ker’ refers to thekernel of the operator, that is, the set of such vectors asare mapped onto zero by the operator. Hence, both theCoulomb-gauge and the boundary conditions on A canbe expressed by the single condition that R A = A .The crucial result for us to build upon here is theHelmholtz–Hodge decomposition of L [10, 11], whichreads: L ( D , R ) = | {z } ker ( curl ) ran ( grad ) ⊕ ker ( div ) z }| { H ⊕ ran ( curl ) , (9)where grad is the gradient operator over L ( D , R ) withits domain restricted to such scalar fields v as van-ish on the boundaries: v | ∂ D = 0 . The notation ‘ran’refers to the range of the operator. In free space, ( D = R ) ran ( grad ) = ker ( curl ) (longitudinal fields) andran ( curl ) = ker ( div ) (transverse fields) holds, and the di-rect sum of the two makes up for the whole L ( R , R ) .For general domains, however, the dimension of H isnon-zero. The elements of H are called cohomologicalfields, and, when the electric field is in question, also condensator fields . On the basis of Eq. (9), we can assertthat L = ran ( grad ) ⊕ ker ( div ) . (10)From this equation, together with Eq. (8) it follows thatin the decomposition of the identity in Eq. (6), the Q projector must be defined as Q : L → ran ( grad ) , (11)We recall that in free space Q [13] and R [14] projectonto the longitudinal and transverse components of vec-tor fields, respectively.The transformed Hamiltonian reads: H ′ = X α m α p ′ α + ∂∂ r α Z D d r A · P − q α A ( r α ) + ε Z D d r ( ∇ U ) + ε Z D d r "(cid:18) Π ′ + R P ε (cid:19) + c ( ∇ × A ) . (12)So far, we have not specified P . Since according to Eq. (3)the scalar potential is an element of the domain of grad ,Eq. (11) allows us to impose the condition on P that ε ∇ U = Q P . (13)Hence, on account of Eq. (6) the electrostatic term in thesecond line of Eq. (12) and the term containing P in thethird line combine to give ε R D d r P .Condition (13) is equivalent to [15] ∇ · P = − ρ, (14)which motivates us to identify the vector field P , so far in-troduced on purely mathematical grounds, with the phys-ical notion of the polarization density.Besides the condition (13), the following condition onthe other orthogonal component of P , ∂∂ r α Z D d r A · R P = q α A ( r α ) , (15)would make the first term of H ′ simplify. However, itis not known whether the conditions (13) and (15) canbe simultaneously met in general. Nevertheless, we showthat in the special case of the electric-dipole approxima-tion to be performed in the next step, both conditionscan be satisfied.At this point, we summarize that under the condi-tion (15), the Hamiltonian would have the form H ′ = X α p ′ α m α + 12 ε Z D d r P − ε Z D d r D · P + H ′ field , (16)where the kinetic term manifests the coincidence of thecanonical momentum p ′ α with the kinetic momentum of particle α , eliminating problem (i) listed after Eq. (4). Weintroduced the displacement field D ≡ ε E + P , aboutwhich, given that Π = ε ∂ t A = − R E , it holds that Π ′ = − R D = − D . The second equality holds because ofEq. (14) and Gauss’s law. H ′ field is formally equivalent to H field , only with the transformed field momentum insteadof the Coulomb-gauge one.We now move from the description of point charges to-wards that of atoms in this picture. The polarization fieldis P A P A , and since the atoms are spatially separated, Z D d r P = X A Z D d r P A , (17)therefore the first two terms of Hamiltonian (16) givethe internal energy of the atoms. In the electric-dipoleapproximation of atoms P A ( r ) = X α ∈ A q α r α ! δ < ( r − r A ) ≡ d A δ < ( r − r A ) , (18) d A being the electric dipole moment of atom A . The func-tion δ < behaves as a delta function over a spatial scalethat is larger than the size of the atoms, while on theintra-atomic scale it is defined such that condition (14)be satisfied (clearly, for a nonzero dipole moment, thecharges cannot be at exactly the same position). Withthis definition, condition (15) is met under our assump-tion that A ( r α ) = A ( r A ) .With the two conditions being satisfied, we can pro-ceed from Hamiltonian (16) to obtain the electric-dipoleHamiltonian in this picture: H ′ ED = X A (cid:18) H ′ A − d A · D ( r A ) ε (cid:19) + H ′ field , (19a)where the single-atom Hamiltonian has the form: H ′ A = X α ∈ A p ′ α m α + Z supp ( P A ) d r P A . (19b)In the second term, the domain of the integration canbe restricted to the support of P A , so that unless theatom is very close to any of the boundary surfaces, thesingle-atom Hamiltonian is not at all affected by the pres-ence of the boundaries. The intra-atomic Coulomb term(equivalent to the second term of the Hamiltonian (4b))can be recovered from this same term, whereupon theremainder gives what is usually termed the dipole self-energy in this picture. This, however, does not concernus here because our agenda is to define the atomic lev-els in this picture simply on the basis of the full single-atom Hamiltonian (19b). For all practical purposes, thedescription of atoms is restricted to a few selected dis-crete energy levels, which can be taken phenomenologi-cally from spectroscopic data. We note that the “atom”is not a gauge-invariant concept. The phenomenologicalreplacement of the atom with a simple level structure(two-level, lambda, etc.) can be safely performed in thegauge of the new Hamiltonian (19), because it is freefrom the problems listed above. Here, (i) the canonicalmomentum coincides with the kinetic one, (ii) the awk-ward A-square term has disappeared, as have (iii) thetwo Coulomb terms, describing atom-atom and atom-boundary interaction. In H ′ ED , the boundary enters onlyvia the displacement field D , hence the atoms interactonly via the retarded radiation field.For quantizing the theory, we introduce the transversemodes as solutions to the constraint vectorial Helmholtzequation [16]: ∇×∇× ϕ λ = ω λ c ϕ λ , with ∇· ϕ λ = 0 and ϕ λ × n | ∂ D = 0 . (20)The vector potential A can be expanded in terms of thesemodes: A = 1 ε X λ (cid:16) ϕ λ a λ + ϕ ∗ λ a † λ (cid:17) , (21a)where a λ is the annihilation operator of the correspond-ing mode, and this expansion was left invariant with re-spect to the Coulomb gauge. D is simply the canonicalconjugate: D = − Π ′ = iε X λ (cid:16) ϕ λ a λ − ϕ ∗ λ a † λ (cid:17) . (21b)We are now ready to systematically introduce thesingle-mode approximation, which is fundamental tothe standard models of cavity QED (Dicke, Tavis–Cummings, Jaynes–Cummings). Our analysis has shownthat even in the case of boundaries, when the possibil-ity of a single-mode approximation arises at all, we stillneed the full mode expansion (20) for the cancellationof the A-square and the dipole-dipole interaction terms.Once this is done, in the new picture we can safely pickout one of the modes ϕ λ . This is at variance with theapproaches of Refs. [2, 7]. For example, when the atomscan be treated as two-level systems, we obtain the Dickemodel: H Dicke = X A (cid:16) ω A σ ( A ) z + g A (cid:0) a + a † (cid:1) σ ( A ) x (cid:17) + ω a † a, (22)where the three terms correspond one by one to the termsof the exact microscopic Hamiltonian (19) in the sameorder. We can thus conclude that these simplified modelsare better than generally expected.This work was supported by the EU FP7 (ITN,CCQED-264666), the Hungarian National Office for Re-search and Technology under the contract ERC_HU_09OPTOMECH, and the Hungarian Academy of Sciences(Lendület Program, LP2011-016). A. V. acknowledges support from the János Bolyai Research Scholarship ofthe Hungarian Academy of Sciences. ∗ Electronic address: [email protected][1] K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phys.Rev. Lett. , 432 (1975).[2] J. M. Knight, Y. Aharonov, and G. T. C. Hsieh, Phys.Rev. A , 1454 (1978).[3] K. Rza¸zewski and K. Wódkiewicz, Phys. Rev. A , 593(1991).[4] P. Nataf and C. Ciuti, Nature Comm. (2010).[5] C. Ciuti, G. Bastard, and I. Carusotto, Phys. Rev. B ,115303 (2005).[6] Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liber-ato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Phys.Rev. Lett. (2010).[7] J. Keeling, J. Phys.: Cond. Mat. , 295213 (2007).[8] A. Vukics and P. Domokos, Physical Review A - Atomic,Molecular, and Optical Physics (2012).[9] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms (Wiley-Interscience, 1997).[10] R. Dautray and J.-L. Lions,
Mathematical Analysis andNumerical Methods for Science and Technology , vol. 3(Springer, 1990).[11] E. Binz and R. Alfred, Journal of Physics: ConferenceSeries , 012006 (2010).[12] The freedom of choosing the potentials within theCoulomb gauge is equivalent also to a freedom of fixinghow the inclusion of the cohomological fields introducedlater in Eq. (9), is shared between the scalar or the vec-tor potential. With our fixing of the potentials within theCoulomb gauge, what we attain is that U ∈ dom ( grad ) and A ∈ ker ( div ) , that is, the electrostatic and radiative parts of the dy-namics take place in the two distinct orthogonal sub-spaces listed later in Eq. (10), the cohomological com-ponents of E (condensator fields) being attributed solelyto A . Note that the form of the Hamiltonian (2a) de-pends on this decomposition result, since this ensuresthat there are separate electrostatic and radiative termsin the Hamiltonian, with no overlap between the two.[13] Since the explicit form of Q is not needed for our deriva-tion, we merely note that on the subspace of those v ∈ L whose divergence exists, it can be written as ( Q v ) ( r ) ≡ −∇ Z D d r ′ (cid:0) ∇ ′ · v (cid:0) r ′ (cid:1)(cid:1) G (cid:0) r , r ′ (cid:1) , where G is the Dirichlet Green’s function of the problem: ∆ G ( r , r ′ ) ≡ δ ( r − r ′ ) within D , and G| ∂ D = 0 . [14] The explicit form of R will not be used, so we merely notethat it can be expressed with the full set of transversemodes (20) as R = X λ ϕ λ ⊗ ϕ λ . [15] To prove the equivalence, we first prove (13) = ⇒ (14): − ρ = ε ∆ U = ∇ · Q P = ∇ · ( Q + R ) P = ∇ · P , where the first equality is the Poisson equation, the sec-ond is obtained by applying the ∇ operator on both sidesof Eq. (13), the third is on account of ∇ · R P = 0 , whilethe fourth reflects Eq. (6). To prove (14) = ⇒ (13) weproceed as ∇ · ( ε ∇ U − P ) = ∇ · ( ε ∇ U − Q P ) , where the first equality follows from Eq. (14) and thePoisson equation, while in the second we applied again ∇ · R P = 0 . It follows that the vector in parenthesis onthe right-hand side is both in ran ( grad ) = ran ( Q ) , andker ( div ) , which, on account of Eq. (10), cannot be truebut for the zero vector, so that ε ∇ U = Q P must hold.[16] It can be proven that the set of the transverse modes,that is, the eigenvectors corresponding to non-negativeeigenvalues span ker ( div ) , and that the subspace of zero-frequency modes coincides with H , that is, ω λ = 0 if andonly if ϕ λ ∈ H . Hence, on this degenerate finite dimen-sional subspace H2