Elimination of the A-Square Problem from Cavity

Home , Electric dipole moment

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As concerning Hamiltonian. argumentations microscopic no-go of proposed the Jaynes–Cummi to Tavis–Cummings, mapping (Dicke, QED be cavity interaction of Coulomb models instantaneous the and term A-square 1 nttt o oi tt hsc n pis inrResear Wigner Optics, and Physics State Solid for Institute lmnto fteAsur rbe rmcvt QED cavity from problem A-square the of Elimination ugra cdm fSine,PO o 9 -55Budapest H-1525 49, Box P.O. Sciences, of Academy Hungarian nrsVukics, András 2 ntttfrTertsh hsk nvriä Innsbruck, Universität Physik, Theoretische für Institut ehiesrß 5 00Inbuk Austria Innsbruck, 6020 25, Technikerstraße 1, ∗ oisGrießer, Tobias with etmo particle of mentum il vnmlil once)domain connected) multiply even sibly hre om( eti ubro)sailyseparated, spatially point of) number individual (long-wavelength certain the (a 1 form that Step charges steps. assume two We in approximation): obtained be can mag- and conditions.[12] electric for boundary the up netic both makes satisfying condition potential latter vector the the (1), Eq. with Together ilb oset to be will with h aitna ftesse reads: system the of Hamiltonian the by defined gauge, (minimal-coupling) Coulomb h olm ag mut oafedmi choosing in freedom of components a for values to constant amounts different within gauge potentials completely. Coulomb the potentials the choosing the of fix freedom remaining not The does (1) condition the in fcnestrfilscridby carried fields condensator of tions ldt h lcrmgei edcnndinto confined field electromagnetic the to pled efc odco.Overall, conductor. perfect dis- surfaces several smooth (possibly sufficiently by junct) bounded space real dimensional H h lcrcdpl prxmto oti Hamiltonian this to approximation electric-dipole The nipratosraini that, is observation important An osdra rirr ubro on hre cou- charges point of number arbitrary an Consider 2 = H Π U n ee Domokos Peter and field X o ihaosi lcrmgei fields electromagnetic in atoms with ion banacnnclHmloino cavity of Hamiltonian canonical a obtain α en h clrpotential, scalar the being = we itntaos h single-mode The atoms. distinct tween e.Ti aitna sfe rmthe from free is Hamiltonian This ies. ε ihfradcneune h basis the consequence, aightforward = [ g)aejsie yatr yterm by term a by justified are ngs) p 0 ∂ α ε 2 t U 0 A − ∂ | D Z hCnr o Physics, for Centre ch ∂ D 2 q en h oetmcnuaeto conjugate momentum the being D m α , d A hc ilrsl nvrosconfigura- various in result will which 0 = α 3 α r ( r ojgt oisposition its to conjugate " · ∇ α and Hungary , )] ε Π 2 1 A 0 D + U A 0 = 2 sasmdt ebounded. be to assumed is ε 2 + nec fteconnected the of each on 0 × , D Z c n 2 ∂ p | d ∂ D ( α niei respace, free in unlike × ∇ D 3 U hc oss fa of consist which , r h aoia mo- canonical the 0 = u hiehere choice Our . ( ∇ D A U . ) ntethree- the in ) 2 2 # + D , r α H nthe In . and , field (2b) (2a) A (3) (1) , . 2 well-localized clusters, that is, atoms. Then, instead of which yields a displacement of the momenta

α there appears A α∈A. We neglect all radiative δG2 ′ effects on the intra-atomic scale, that is, we set A(rα)= Π = = Π + RP, (5b) P P P δA A(rA), where rA is the position of that atom A which ∂G2 ′ ∂ 3 incorporates the charge α. Step 2: We assume that the pα = = pα + d r A · P. (5c) atoms have only electric dipole moment, that is, no net ∂rα ∂rα DZ charge and no further electric or magnetic moments. R Upon the first assumption, we split the Coulomb (elec- At this point, P is an arbitrary vector, and is part of trostatic) term into intra- and inter-atomic parts, and an orthogonal projector decomposition of the identity, take the intra-atomic part as identical to the one in free Q + R = idL2 . (6) space, under the assumption that the distance of atoms 0 from the boundary is much larger than the atomic radius. 2 2 R3 where L0 is the subspace of the Hilbert space L (D, ) The electric-dipole order of the Hamiltonian in Coulomb of square-integrable vector fields such that the elements gauge then reads: 2 of L0 satisfy the boundary condition that they are normal to the boundaries: 2 2 R3 2 R3 HED = HA − u pA · A(rA)+ v A (rA) L0(D, ) ≡ v ∈ L (D, ) v × n|∂D =0 , (7) A X which is of course nothing else than the boundary condi- dipole-self dipole-dipole + VCoulomb (rA)+ VCoulomb (rA−B ) + Hfield, tion on the electric field (and hence the vector potential) B X at a perfectly conducting surface. (4a) In order that the transformation (5) be canonical, R must be a projector onto the divergence-free subspace of 2 where u and v are constants composed of the mαs and L0: qαs. The single-atom Hamiltonian reads R 2 : L0 → ker(div0), (8) 2 pα qα qβ HA = + . (4b) because this ensures that A in Eq. (5a) can be treated 2mα 8πε |rα − rβ| α∈A 0 β∈A as unconstrained. Here, div0 (and curl0 below) are the X Xβ6=α divergence (and curl) operators over L2, with the do- 2 main restricted to L0. The notation ‘ker’ refers to the It is this Hamiltonian (4) that is usually taken as the kernel of the operator, that is, the set of such vectors as starting point of cavity QED. However, it is fraught with are mapped onto zero by the operator. Hence, both the the following problems: (i) the canonical momentum of Coulomb-gauge and the boundary conditions on A can the atoms does not equal their kinetical momentum; fur- be expressed by the single condition that RA = A. thermore, as we mentioned, (ii) the presence of the A- The crucial result for us to build upon here is the square term, which yields creation and annihilation of Helmholtz–Hodge decomposition of L2 [10, 11], which pairs of photons; and finally, (iii) there appears an instan- reads: taneous electrostatic interaction between remote atoms dipole-dipole ker(div) (VCoulomb ) and an interaction of a single dipole with dipole-self L2 , R3 H , its own induced surface charges (VCoulomb ). The former (D )= ran(grad0) ⊕ 2 ⊕ ran(curl) (9) is influenced, while the latter is created by the presence ker(curl ) z }| { of the boundaries [8]. 0 | {z } 2 R In free space, these weaknesses can be dissolved by where grad0 is the gradient operator over L (D, ) with performing the PZW transformation on the minimal cou- its domain restricted to such scalar fields v as van- pling Hamiltonian (2a) to the multipolar-coupling gauge ish on the boundaries: v|∂D = 0. The notation ‘ran’ (cf. Ref. [9] Chapter IV.C). Here, inspired by the free- refers to the range of the operator. In free space, (D = R3 space procedure, we elevate this transformation onto a ) ran(grad0) = ker(curl0) (longitudinal fields) and very general level, which allows for an arbitrary domain ran(curl)= ker(div) (transverse fields) holds, and the di- D and boundaries ∂D, i.e. for a general cavity QED sce- rect sum of the two makes up for the whole L2(R3, R3). nario. For general domains, however, the dimension of H2 is H The transformation that we adopt is canonical, defined non-zero. The elements of 2 are called cohomological by the Type-2 generating function fields, and, when the electric field is in question, also condensator fields. On the basis of Eq. (9), we can assert

3 ′ ′ that G2 ≡ d r A · Π + RP + rα · pα, (5a) 2 Z α L = ran(grad ) ⊕ ker(div0). (10) D X 0 0 3

From this equation, together with Eq. (8) it follows that particle α, eliminating problem (i) listed after Eq. (4). We in the decomposition of the identity in Eq. (6), the Q introduced the displacement field D ≡ ε0E + P, about projector must be defined as which, given that Π = ε0∂tA = −RE, it holds that Π′ = −RD = −D. The second equality holds because of Q : L2 → ran(grad ), (11) ′ 0 0 Eq. (14) and Gauss’s law. Hfield is formally equivalent to H , only with the transformed field momentum instead Q R field We recall that in free space [13] and [14] project of the Coulomb-gauge one. onto the longitudinal and transverse components of vec- We now move from the description of point charges to- tor fields, respectively. wards that of atoms in this picture. The polarization field The transformed Hamiltonian reads: is A PA, and since the atoms are spatially separated, 2 P ∂ 3 2 3 2 ′ 1 ′ 3 d r P = d r PA, (17) H = pα + d r A · P − qαA(rα) m ∂r A α 2 α  α  DZ DZ X DZ X  ε  + 0 d3r (∇U)2 therefore the first two terms of Hamiltonian (16) give 2 the internal energy of the atoms. In the electric-dipole Z D approximation of atoms 2 ε Π′ + RP + 0 d3r + c2 (∇× A)2 . (12) 2 " ε0 # < < DZ PA(r)= qαrα δ (r−rA) ≡ dA δ (r−rA), (18) α∈A ! So far, we have not specified P. Since according to Eq. (3) X dA being the electric dipole moment of atom A. The func- the scalar potential is an element of the domain of grad0, < Eq. (11) allows us to impose the condition on P that tion δ behaves as a delta function over a spatial scale that is larger than the size of the atoms, while on the

ε0∇U = QP. (13) intra-atomic scale it is defined such that condition (14) be satisfied (clearly, for a nonzero dipole moment, the Hence, on account of Eq. (6) the electrostatic term in the charges cannot be at exactly the same position). With second line of Eq. (12) and the term containing P2 in the this definition, condition (15) is met under our assump- 1 d3r P2 third line combine to give 2ε0 D . tion that A(rα)= A(rA). Condition (13) is equivalent to [15] With the two conditions being satisfied, we can pro- R ceed from Hamiltonian (16) to obtain the electric-dipole ∇ · P = −ρ, (14) Hamiltonian in this picture: which motivates us to identify the vector field P, so far in- D(r ) H′ = H′ − d · A + H′ , (19a) troduced on purely mathematical grounds, with the phys- ED A A ε field A 0 ical notion of the polarization density. X Besides the condition (13), the following condition on where the single-atom Hamiltonian has the form: the other orthogonal component of P, ′ 2 ′ pα 3 2 ∂ 3 HA = + d r PA. (19b) d r A · RP = qαA(rα), (15) 2mα ∂r α∈A ZP α X supp( A) DZ In the second term, the domain of the integration can would make the first term of H′ simplify. However, it be restricted to the support of P , so that unless the is not known whether the conditions (13) and (15) can A atom is very close to any of the boundary surfaces, the be simultaneously met in general. Nevertheless, we show single-atom Hamiltonian is not at all affected by the pres- that in the special case of the electric-dipole approxima- ence of the boundaries. The intra-atomic Coulomb term tion to be performed in the next step, both conditions (equivalent to the second term of the Hamiltonian (4b)) can be satisfied. can be recovered from this same term, whereupon the At this point, we summarize that under the condi- remainder gives what is usually termed the dipole self- tion (15), the Hamiltonian would have the form energy in this picture. This, however, does not concern p′ 2 1 1 us here because our agenda is to define the atomic lev- H′ = α + d3r P2 − d3r D · P + H′ , els in this picture simply on the basis of the full single- 2m 2ε ε field α α 0 0 atom Hamiltonian (19b). For all practical purposes, the X DZ DZ (16) description of atoms is restricted to a few selected dis- where the kinetic term manifests the coincidence of the crete energy levels, which can be taken phenomenologi- ′ canonical momentum pα with the kinetic momentum of cally from spectroscopic data. We note that the “atom” 4 is not a gauge-invariant concept. The phenomenological support from the János Bolyai Research Scholarship of replacement of the atom with a simple level structure the Hungarian Academy of Sciences. (two-level, lambda, etc.) can be safely performed in the gauge of the new Hamiltonian (19), because it is free from the problems listed above. Here, (i) the canonical momentum coincides with the kinetic one, (ii) the awk- ∗ ward A-square term has disappeared, as have (iii) the Electronic address: [email protected] [1] K. Rzażewski, K. Wódkiewicz, and W. Żakowicz, Phys. two Coulomb terms, describing atom-atom and atom- 35 ′ Rev. Lett. , 432 (1975). boundary interaction. In HED, the boundary enters only [2] J. M. Knight, Y. Aharonov, and G. T. C. Hsieh, Phys. via the displacement field D, hence the atoms interact Rev. A 17, 1454 (1978). only via the retarded radiation field. [3] K. Rza¸zewski and K. Wódkiewicz, Phys. Rev. A 43, 593 For quantizing the theory, we introduce the transverse (1991). modes as solutions to the constraint vectorial Helmholtz [4] P. Nataf and C. Ciuti, Nature Comm. 1 (2010). 72 equation [16]: [5] C. Ciuti, G. Bastard, and I. Carusotto, Phys. Rev. B , 115303 (2005). ω2 [6] Y. Todorov, A. M. Andrews, R. Colombelli, S. De Liber- ϕ λ ϕ , ϕ ϕ n . ato, C. Ciuti, P. Klang, G. Strasser, and C. Sirtori, Phys. ∇×∇× λ = 2 λ with ∇· λ =0 and λ × |∂D =0 c Rev. Lett. 105 (2010). (20) [7] J. Keeling, J. Phys.: Cond. Mat. 19, 295213 (2007). The vector potential A can be expanded in terms of these [8] A. Vukics and P. Domokos, Physical Review A - Atomic, modes: Molecular, and Optical Physics 86 (2012). [9] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, 1 A = ϕ a + ϕ∗ a† , (21a) Photons and Atoms (Wiley-Interscience, 1997). ε λ λ λ λ 0 λ [10] R. Dautray and J.-L. Lions, Mathematical Analysis and X Numerical Methods for Science and Technology, vol. 3 where aλ is the annihilation operator of the correspond- (Springer, 1990). ing mode, and this expansion was left invariant with re- [11] E. Binz and R. Alfred, Journal of Physics: Conference 237 spect to the Coulomb gauge. D is simply the canonical Series , 012006 (2010). [12] The freedom of choosing the potentials within the conjugate: Coulomb gauge is equivalent also to a freedom of fixing how the inclusion of the cohomological fields introduced ′ ∗ † D = −Π = iε0 ϕλaλ − ϕλaλ . (21b) later in Eq. (9), is shared between the scalar or the vec- λ tor potential. With our fixing of the potentials within the X Coulomb gauge, what we attain is that We are now ready to systematically introduce the single-mode approximation, which is fundamental to U ∈ dom(grad0) and A ∈ ker(div0), the standard models of cavity QED (Dicke, Tavis– Cummings, Jaynes–Cummings). Our analysis has shown that is, the electrostatic and radiative parts of the dy- that even in the case of boundaries, when the possibil- namics take place in the two distinct orthogonal sub- spaces listed later in Eq. (10), the cohomological com- ity of a single-mode approximation arises at all, we still ponents of E (condensator fields) being attributed solely need the full mode expansion (20) for the cancellation to A. Note that the form of the Hamiltonian (2a) de- of the A-square and the dipole-dipole interaction terms. pends on this decomposition result, since this ensures Once this is done, in the new picture we can safely pick that there are separate electrostatic and radiative terms out one of the modes ϕλ. This is at variance with the in the Hamiltonian, with no overlap between the two. approaches of Refs. [2, 7]. For example, when the atoms [13] Since the explicit form of Q is not needed for our deriva- v 2 can be treated as two-level systems, we obtain the Dicke tion, we merely note that on the subspace of those ∈ L whose divergence exists, it can be written as model:

3 ′ ′ ′ ′ (A) † (A) † (Qv)(r) ≡ −∇ d r ∇ · v r G r, r , HDicke = ωA σz + gA a + a σx +ωa a, (22) Z A D X where the three terms correspond one by one to the terms where G is the Dirichlet Green’s function of the problem: of the exact microscopic Hamiltonian (19) in the same r r′ r r′ order. We can thus conclude that these simpliﬁed models ∆G( , ) ≡ δ( − ) within D, and G|∂D = 0. are better than generally expected. [14] The explicit form of R will not be used, so we merely note This work was supported by the EU FP7 (ITN, that it can be expressed with the full set of transverse CCQED-264666), the Hungarian National Oﬃce for Re- modes (20) as search and Technology under the contract ERC_HU_09

OPTOMECH, and the Hungarian Academy of Sciences R = ϕλ ⊗ ϕλ. (Lendület Program, LP2011-016). A. V. acknowledges Xλ 5

[15] To prove the equivalence, we first prove (13) =⇒ (14): ∇ · RP = 0. It follows that the vector in parenthesis on the right-hand side is both in ran(grad0) = ran(Q), and P P P −ρ = ε0∆U = ∇ · Q = ∇ · (Q + R) = ∇ · , ker(div0), which, on account of Eq. (10), cannot be true but for the zero vector, so that ε0∇U = QP must hold. where the first equality is the Poisson equation, the sec- [16] It can be proven that the set of the transverse modes, ond is obtained by applying the ∇ operator on both sides that is, the eigenvectors corresponding to non-negative of Eq. (13), the third is on account of ∇ · RP = 0, while eigenvalues span ker(div0), and that the subspace of zero- the fourth reflects Eq. (6). To prove (14) =⇒ (13) we frequency modes coincides with H2, that is, ωλ = 0 if and proceed as only if ϕλ ∈ H2. Hence, on this degenerate finite dimensional subspace H2, an arbitrary basis can be chosen. 0= ∇ · (ε0∇U − P)= ∇ · (ε0∇U − QP) ,

where the ﬁrst equality follows from Eq. (14) and the Poisson equation, while in the second we applied again