Eliminating Ground-State Dipole Moments in Quantum Optics Via Canonical Transformation

PHYSICAL REVIEW A 68, 043811 ͑2003͒

Eliminating ground-state dipole moments in quantum optics via canonical transformation

Gediminas Juzeliu¯nas,1 Luciana C. Da´vila Romero,2 and David L. Andrews2 1Institute of Theoretical Physics and Astronomy, Vilnius University, A. Gosˇtauto 12, 2600 Vilnius, Lithuania 2School of Chemical Sciences and Pharmacy, University of East Anglia, Norwich NR4 7TJ, United Kingdom ͑Received 29 April 2003; published 9 October 2003͒ By means of a canonical transformation it is shown how it is possible to recast the equations for molecular nonlinear optics to completely eliminate ground-state static dipole coupling terms. Such dipoles can certainly play a highly important role in nonlinear optical response—but equations derived by standard methods, in which these dipoles emerge only as special cases of transition moments, prove unnecessarily complex. It has been shown that the elimination of ground-state static dipoles in favor of dipole shifts results in a considerable simplification in form of the nonlinear optical susceptibilities. In a fully quantum theoretical treatment the validity of such a procedure has previously been verified using an expedient algorithm, whose defense was afforded only by a highly intricate proof. In this paper it is shown how a canonical transformation method entirely circumvents such an approach; it also affords insights into the formulation of quantum field interactions.

DOI: 10.1103/PhysRevA.68.043811 PACS number͑s͒: 42.65.An, 33.15.Kr, 42.50.Ct

I. INTRODUCTION mentation of a suitable canonical transformation on the multipolar form of the quantum optical Hamiltonian. Elucidating In recent years it has become increasingly evident that the quantum electrodynamical treatment in this way throws permanent ͑static͒ electric dipoles play a highly important light on a number of issues skirted over in the semiclassical role in the nonlinear optical response of molecular systems treatment, and it offers clear scope for extension to higher orders of multipole interaction. ͓1,2͔. Most molecules are intrinsically polar by nature, and Employment of the canonical transformation effects a calculation of their optical susceptibilities with regard only considerable simplification of the analysis of nonlinear opti- to transition dipole moments can produce results that are ͓ ͔ cal processes involving permanent dipole moments. For in- significantly in error 3,4 . In particular, many of the ‘‘push- stance, in the standard formulation, second-harmonic genera- pull’’ systems favored for their high degree of optical non- tion is represented by 3ϫ22ϭ12 different state sequences linearity are specifically those where permanent dipole ef- when a two-level molecular model is used. Each of these fects are the largest, through their designed juxtaposition of entails a product of three ‘‘transition’’ dipoles ͑one or more strongly electron-donating and electron-withdrawing func- of which may be permanent͒, divided by a product of two tional groups ͓5–7͔. energy factors. However, the new Hamiltonian will involve Whilst a number of groups have developed the theory to only three terms of simpler structure containing no explicit elicit permanent dipole contributions to nonlinear optical re- contributions due to the ground-state dipole moment, only sponse, the framework for most of this work has been semi- dipole shifts ͑i.e., differences between excited-state and classical ͓8,9͔. In such a context, where the molecular system ground-state moments͒—as we shall see in Sec. IV B. is treated with quantum-mechanical rigour but the radiation The paper is organized as follows. In the following sec- is treated as a classical oscillatory electric field, it has been tion the radiation matter Hamiltonian is introduced—taking demonstrated that a transformed ‘‘fluctuation dipole’’ Hamil- the leading electric dipole terms from the multipolar formu- tonian properly describes the optical interactions of the dy- lation of quantum electrodynamics ͑QED͒. In Sec. III A a namical system, and affords considerable calculational sim- canonical transformation is carried out to eliminate the cou- plification ͓4͔. However, the semiclassical treatment fails to pling between the quantized radiation field and permanent take into account the modifications associated with electro- dipoles of the molecules in ground electronic states. In Sec. magnetic field interactions—features that only emerge in a III B the Hamiltonian is reexpressed in a different represen- quantum electrodynamical treatment. Indeed, it has been re- tation, followed by an application to the study of nonlinear optical processes in Sec. IV. The example of second- marked that quantum electrodynamics affords the only com- ͑ ͒ pletely rigorous basis for descriptions of multipolar behavior harmonic generation SHG studied in detail in Sec. IV B ͓10,11͔. illustrates how the present formalism facilitates elimination When both the material and radiation parts of the system of the contributions due to the ground-state dipole in appro- are developed in fully quantized form, the transparency and priately time-ordered quantum channels for the molecule- correctness of deploying a transformed interaction Hamil- radiation processes. Extensions beyond the dipole approxi- tonian is potentially obscured ͓12͔. Both the conventional mation are considered in Sec. V, followed by concluding and transformed operators prove to lead to identical results remarks in Sec. VI. ͓ ͔ even at high orders of optical nonlinearity 13 , yet the II. THE MULTIPOLAR QED HAMILTONIAN equivalence of their predictions as a general principle has until now been established only through engagement in We begin with the Hamiltonian in multipolar form, for a proofs of considerable intricacy ͓14͔. It is our purpose in this system of molecules labeled X, interacting with a quantized paper to rectify this anomaly by demonstrating the imple- radiation field ͓15͔

intermolecular coupling being mediated via the transverse ϭ ϩ ͑ ͒ϩ ͑ ͒ H Hrad ͚ Hmol X Hint , 1 radiation field. Specifically, each molecule is coupled to the X displacement field at the molecular site, as is evident in Eq. ͑6͒. where Hrad is the Hamiltonian for the free radiation field, each Hmol(X) is a Schro¨dinger operator for an isolated mol- III. TRANSFORMATION TO ANOTHER ecule, and Hint is a term representing a fully retarded coupling between the quantized radiation field and the molecular REPRESENTATION subsystem. A. Canonical transformation The first of these operators, H , is expressed as follows rad ␮(X) in terms of the transverse electric displacement field operator To eliminate the permanent ground-state dipole gg ,we dЌ(r) and magnetic-field operator b͑r͒: shall apply the following canonical transformation, which recasts operators but effects no change in any system observ- 1 Ϫ Ϫ ables: H ϭ ͵ ͕␧ 1͓dЌ͑r͔͒2ϩ␮ 1͓bЌ͑r͔͒2͖d3r ͑2͒ rad 2 0 0 uϭexp͑iS͒, SϭϪ͑1/ប͒ ͵ aЌ͑r͒ p ͑r͒d3r ͑8͒ with • g Ќ ͒ϭ␧ Ќ ͒ϩ Ќ ͒ ͑ ͒ d ͑r 0e ͑r p ͑r . 3 with In Eq. ͑3͒, eЌ(r) is the transverse part of the electric field, in ͑ ͒ϭ ␮͑X͒␦͑ Ϫ ͒ ͑ ͒ ͑ ͒ pg r ͚ gg r RX ; 9 the electric-dipole approximation, and p r is the polariza- X tion field of molecular origin, given by the latter signifies the polarization field produced by an as- p͑r͒ϭ͚ ␮͑X͒␦͑rϪR ͒, ͑4͒ sembly of ground-state molecules containing static dipoles, X ␮(X) X gg . The transformation does not alter the vector potential Ќ Ќ ͓ ͔ ␮ a (r), since the generator S commutes with it. Furthermore, p (r) being its transverse part 15 . Again, (X) is the di- ␮(X) since the dipole moment gg is a c-number characterized by pole operator of the molecule X positioned at RX , and the ͑ ͒ summations are taken over all the molecules of the system. a real value i.e., it is not an operator , the transformation Next, the Hamiltonian for molecule X is explicitly given by does not modify either the electron momenta pX,␣ or coordi- nates qX,␣ featured in the molecular Hamiltonian Hmol(X). 1 e2 1 The only canonical variable affected by the transforma- ͑ ͒ϭ 2 ϩ ͑ ͒ Hmol X ͚ pX,␣ ͚ , 5 tion is the electric displacement operator. From the commu- 2m ␣ 4␲␧ ␣ ␤ ͉q ␣Ϫq ␤͉ 0 , X, X, tator relation where p ␣ and q ␣ are, respectively, operators for the mo- Ќ Ќ Ќ X, X, ͓a ͑r͒,d ͑rЈ͔͒ϭϪiប␦ ͑rϪrЈ͒ ͑10͒ mentum and position of electron ␣. We note that in the stan- i j ij dard notation used here, care must be taken not to confuse we have ͑see the Appendix͒ ␣ the momentum of the electron , pX,␣ , with the polarization field p͑r͒ defined above. Finally the operator H , which ˜Ќ͑ ͒ϵ iS Ќ͑ ͒ ϪiSϭ Ќ͑ ͒Ϫ Ќ͑ ͒ ͑ ͒ int d r e d r e d r pg r , 11 describes the coupling between the molecular subsystem and the quantized radiation field, is expressible as i.e., the transformation effects a subtraction from the full displacement field dЌ(r) of the transverse part of the polar- Ϫ Ќ Ϫ Ќ ␮(X) ϭϪ͵ ␧ 1 ͑ ͒ ͑ ͒ 3 ϭϪ ␧ 1 ͑ ͒ ␮͑ ͒ ization field due to the ground-state dipoles, gg . Alterna- Hint 0 d r p r d r ͚ 0 d RX X . • X • tively, one can write ͑6͒ ˜Ќ ͒ϭ␧ Ќ ͒ϩ Ќ ͒ ͑ ͒ d ͑r 0e ͑r ˜p ͑r , 12 Although the field interaction is here cast in terms of the electric dipole approximation, our analysis can be extended where quite straightforwardly to incorporate higher multipole terms—this will be discussed in Sec. V. Lastly, the dipole ͒ϵ ͒Ϫ ͒ϭ ␮ ͒␦ Ϫ ͒ ͑ ͒ ˜p͑r p͑r pg͑r ͚ ˜ ͑X ͑r RX 13 operator ␮(X) can be cast in matrix form in terms of mo- X lecular dipole moments; is the polarization field excluding the ground-state dipoles, ␮͑ ͒ϭ ͉ ͑X͒͘␮͑X͒͗ ͑X͉͒ ͑ ͒ and X ͚ j jl l , 7 j,l ␮͑ ͒ϵ␮͑ ͒Ϫ␮͑X͒ϭ ͉ ͑X͒͘␮͑X͒͗ ͑X͉͒ ͑ ͒ (X) ˜ X X gg ͚ j ˜ ij l 14 where ͉j ͘ are the eigenvectors of the molecular Hamil- j,l tonian Hmol(X). In the multipolar QED formulation em- ␮(X)ϭ␮(X) ployed ͓15–17͔, the Hamiltonian of the system does not con- is the corresponding dipole operator, with ˜ ij ij Ϫ␮ (X)␦ tain any instantaneous dipole-dipole type interactions, all gg ij .

043811-2 ELIMINATING GROUND-STATE DIPOLE MOMENTS IN . . . PHYSICAL REVIEW A 68, 043811 ͑2003͒

B. Hamiltonian in the present representation molecular coupling is a direct consequence of eliminating Ќ ϭ˜Ќ ϩ Ќ ͑ ͒ ͑ ͒ the permanent ground-state dipoles from the ﬁeld-mediated Substituting d (r) d (r) pg (r) into Eqs. 2 and 6 ͑ ͒ molecule-molecule coupling in the reformulated multipolar the Hamiltonian of the system, Eq. 1 , can be reexpressed as QED representation.

ϭ ˜ ϩ ˜ ͑ ͒ϩ ˜ ͑ ͒ H Hrad ͚ Hmol X Hint , 15 IV. NONLINEAR OPTICAL PROCESSES X A. Introduction where With the Hamiltonian now reexpressed in a different representation, it is possible to apply it to the study of nonlinear ˜ ϭ Ϫ1 ͵ ͕␧Ϫ1͓ Ќ͑ ͔͒2ϩ␮Ϫ1͓ Ќ͑ ͔͒2͖ 3 ͑ ͒ Hrad 2 0 d r 0 b r d r 16 optical processes. The example of SHG to be studied in detail illustrates how this facilitates the elimination of ground- and state dipole terms in a rigorous QED treatment. This circumvents the highly intricate proof that previously afforded its ˜ ϭϪ͵ ␧Ϫ1 Ќ͑ ͒ ͑ ͒ 3 ϭϪ ␧Ϫ1˜Ќ͑ ͒ ␮͑X͒ only justiﬁcation in QED—and even then, only for two-level Hint 0 d r ˜p r d r ͚ 0 d RX ˜ ͓ ͔ • X • systems 13 . It is important to emphasize that the trans- ͑17͒ formed displacement ﬁeld operator and the radiation Hamil- tonian can be cast in the usual form ͓15͔: are the radiative and the interaction Hamiltonian in the ប ␧ 1/2 present representation. In this way, the radiation-molecule Ќ ck 0 ͑␭͒ ͑␭͒ ˜d ͑R͒ϭi͚ ͩ ͪ e ͑k͕͒a ͑k͒eik•R coupling is now represented in terms of the dipole moment k,␭ 2V operator ␮˜ (X)ϭ␮(X)Ϫ␮(X) which excludes a contribution as- gg ͑␭͒† Ϫik R sociated with the permanent ground-state dipole moment. Ϫa ͑k͒e • ͖ ͑20͒ The new molecular Hamiltonian can be written as and ˜ ͒ϭ ͒ϩ ͒ ͑ ͒ Hmol͑X Hmol͑X Vmol͑X , 18 ˜ ϭ ͑␭͒† ͒ ͑␭͒ ͒ϩ ប ͑ ͒ Hrad ͚ ͕a ͑k a ͑k 1/2͖ ck, 21 where the extra term reads k,␭

1 where, in each expression, a sum is taken over radiation V ͑X͒ϭϪ ͚ mol modes characterized by wave vector k and unit polarization 2 XЈÞX vector e(␭)(k) ͑with ␭ denoting one of two polarization ͑ Ј͒ ͑ ͒ ͑ ͒ ͑ ͒ ␭ ␭ ␮ X ␮ X Ϫ3͑␮ X Rˆ ͒͑Rˆ ␮ X ͒ states͒; a( )†(k) and a( )(k) are the corresponding operators ϫ gg • gg gg • XXЈ XXЈ• gg 4␲␧ R3 for creation and annihilation of a photon, and V is an arbi- 0 XXЈ trarily large quantization volume. Note that the photon states ͑ Ј͒ ͑ ͒ ͑ Ј͒ ͑ ͒ ␮ X ␮˜ X Ϫ3͑␮ X Rˆ ͒͑Rˆ ␮˜ X ͒ are somewhat different from those emerging in the original Ϫ gg • gg • XXЈ XXЈ• Ќ ͚ 3 multipolar QED, because the canonical operator ˜d (R) dif- 4␲␧ R Ќ XЈÞX 0 XXЈ fers from the original displacement operator d (r) by the Ϫ Ќ ͑19͒ amount pg (r). For simplicity and clarity we dispense with refractive effects in the mode expansion of the displacement ϭ Ϫ Þ with RXXЈ RX RXЈ . The condition XЈ X ensures omis- field operator ͑20͒. Nonetheless, our method is amenable to sion of molecular self-interaction terms ͑otherwise known as the incorporation of such effects, the displacement field op- contact interaction terms͒. erator is then expanded in terms of photons fully dressed by The first term in Eq. ͑19͒ can be identified as the direct the molecular medium ͑i.e., polaritons͓͒18–21͔ rather than ͑Coulomb͒ interaction between the ground-state dipole of a the ‘‘bare’’ photons. particular molecule X and those of all other species XЈ, the From expression ͑20͒, the application of perturbation 1 factor of 2 preventing double counting when we sum over all theory for the study of optical processes is straightforward X centers—see Eqs. ͑15͒ and ͑18͒. Such a term leads to an and follows similar lines to those employed when the multi- overall shift in the molecular energy origin and hence does polar formalism is used ͓15͔. For a particular optical process, not contribute to linear or nonlinear optical processes. The a perturbative expansion of the transition amplitude SX , for second term in Eq. ͑19͒ can be identified as a contribution one center X, gives ͓22͔ due to Coulomb coupling between the transition dipoles of a ϱ p selected molecule X and the ground-state dipoles of sur- 1 rounding species XЈ. This term introduces an off-diagonal S ϰ͗ f͉ ͚ ͫ H˜ ͬ H˜ ͉i͘ , ͑22͒ X int int X pϭ0 ͑˜ Ϫ ˜ ͒ coupling in the molecular Hamiltonian, effecting a modifica- E0 H0 tion of the molecular energies in the second order of pertur- ˜ ˜ bation. Both types of molecule-molecule coupling are instan- where the unperturbed Hamiltonian H0 is given by Hrad ϩ͚ taneous, because they are associated with at least one XHmol(X) and the interaction Hamiltonian Hint is given permanent dipole. The emergence of the instantaneous inter- by Eq. ͑17͒. Here ͉i͘ and ͉f ͘ are the initial and final states of

043811-3 JUZELIU¯ NAS, ROMERO, AND ANDREWS PHYSICAL REVIEW A 68, 043811 ͑2003͒

FIG. 1. The three time-ordered diagrams for second-harmonic generation.

˜ The optical process can be represented by three time-ordered the radiation matter system, E0 being the corresponding en- ˜ diagrams ͓12,13͔͑see also Fig. 1͒. Each of these time- ergy of the initial state. This energy E0 is modified by the static dipoles of the surrounding medium via the coupling ordered diagrams, in turn, contributes various terms depend- ͑ ͒ 1 ing on the number of molecular states involved in the optical term Vmol(X) featured in Eq. 18 . For dilute gases such a modification does not play an important role. In Eq. ͑22͒ the process. In the conventional formulation, the number of terms can be fairly large. For example, if the molecules are leading term for an m-photon process is, as expected, p represented by a two-level model, the transition matrix will ϭm Ϫ1, as in the untransformed representation. 1 possess 3ϫ22ϭ12 contributions—each a product of three Before continuing with an example application, we make ‘‘transition’’ dipoles ͑one or more of which may be perma- one observation on utilizing the dilute gas approximation. nent͒, divided by a product of two energy factors. However, The mode expansion of the displacement field operator ͑20͒ the new Hamiltonian involves only three simpler terms con- is essentially based on such an approximation, since local taining only contributions directly associated with dipole field effects associated with transition dipoles of the sur- shifts. In the more general case, from Eqs. ͑22͒ and ͑23͒, and rounding medium are not included. Therefore, although mo- lifting the two-level approximation, the transition amplitude lecular energy shifts due to the static dipoles of the surround- for one center reads in the present representation, ing species feature in the molecular energies within our formalism, it is inappropriate to engage in detailed consider- ϳ͗͑ Ϫ ͒͑ ␭͒ ͑ Ј ␭͒ ͉ ation of the molecular energy shifts due to the surrounding SX n 2 k, ,1 k , ;X,i static dipoles without recasting the displacement field opera- 1 1 ͓ ͔ ϫ ˜ ˜ ˜ ͉ ͑ ␭͒ tor in terms of polaritons 18–21 . The considerable increase ͭ Hint Hint Hintͮ n k, ;X,i͘, ˜ Ϫ ˜ ͒ ˜ Ϫ ˜ ͒ in complexity that results invites detailed consideration in a ͑E0 H0 ͑E0 H0 future piece of work. ͑24͒

˜ ϭ˜ ϩ ប␻ B. Second-harmonic generation where E0 Ei n is the energy of the initial state of the radiation matter system. Second-harmonic generation serves to illustrate the use of ͑ ͒ ͑ ͒ ͑ ͒ When Eqs. 16 and 20 are used the transition matrix new Hamiltonian H, Eq. 15 . This well-known optical pro- ͑24͒ can be expressed as cess can be described as fundamentally involving the annihilation of two photons of a certain frequency ␻ and the បc 3/2 creation of one photon of double the frequency, 2␻. The S ϭϪiͩ ͪ ͑k2kЈ͒1/2͕n͑nϪ1͖͒1/2eie e ␤ , ͑25͒ X 2␧ V j k ijk molecular centers are initially in their ground state: ͉i(X)͘ 0 ϵ͉g(X)͘. Since SHG is an elastic process, the final molecular ␤ where ijk is the hyperpolarisability tensor given by states are identical to the initial ones. The initial state of the ͉ ␭ ͘ ir rs is ir rs is radiation field, n(k, ) , contains n photon in a particular ␮˜ ␮˜ ␮˜ ␮˜ ␮˜ ␮˜ electromagnetic mode (k,␭), while the final radiative state is ␤ ϭ͚ ͫ i j k ϩ j i k ijk r,s ˜ ϩ ប␻͒ ˜ ϩប␻͒ ˜ Ϫប␻͒ ˜ ϩប␻͒ ͉nϪ2(k,␭); 1(kЈ,␭)͘, with ͉kЈ͉ϭ2͉k͉. In summary, the ini- ͑Eir 2 ͑Eis ͑Eir ͑Eis tial and final states of the system for SHG are ␮ir␮rs␮is ˜ j ˜ k ˜ i ͉i ͘ϭ͉n͑k,␭͒;X,i͘, ϩ ͬ, ͑26͒ X ˜ Ϫប␻͒ ˜ Ϫ ប␻͒ ͑Eir ͑Eis 2

͉ ϭ͉ Ϫ ͒ ␭͒ ␭͒ ͑ ͒ ␻ϭ ˜ ϵ˜ Ϫ˜ f X͘ ͑n 2 ͑k, ;l͑kЈ, ;X,i͘. 23 where ck and Eir Ei Er are the molecular transition energies. In passing we note that in any application the ␤ ϵ ␤ ϩ␤ index-symmetrized form, i( jk) 1/2( ijk ikj), would 1Note that the tilde designation here has the speciﬁc connotation necessarily be invoked because of the corresponding symme- of a dipole-induced shift, not to be confused with its use in other try in the radiation tensor—with which it is eventually con- work by the authors to signify the inclusion of damping. tracted to give a result for the signal. The result given by Eq.

043811-4 ELIMINATING GROUND-STATE DIPOLE MOMENTS IN . . . PHYSICAL REVIEW A 68, 043811 ͑2003͒

͑26͒ is correct for molecular cases far from resonance. If nent ground-state dipole moment. An additional instanta- near-resonant terms are considered then phenomenological neous intermolecular interaction appears in the transformed damping factors are introduced. By adopting the convention Hamiltonian, representing changes in the molecular eigen- of a constant sign for the damping ͓23͔, as recently con- states and corresponding energies due to a surrounding polar firmed ͓24͔, the result coincides exactly with earlier work, medium. The emergence of instantaneous intermolecular and now without any need to assume that the linewidth is coupling is a direct consequence of eliminating, in the field- small compared to the harmonic frequency. mediated molecule-molecule coupling, the permanent ground-state dipoles in favor of dipole shifts. V. EXTENSION BEYOND THE DIPOLE APPROXIMATION The present canonical transformation method concisely circumvents a highly intricate proof that previously afforded The above analysis can be extended beyond the dipole the only rigorous justification. Moreover, the quantum elec- approximation. For this purpose one needs to add nondipole Ќ trodynamical treatment elucidates a number of issues skirted contributions to the polarization field p (r) featured in the over in the semiclassical treatment. The QED method is di- ͑ ͒ displacement field given by Eq. 3 . Furthermore, one needs rectly amenable to the inclusion of higher rank multipole to include the nondipole terms in the multipolar Hamiltonian moments; furthermore it extends previous work in permitting ͓ ͔ 15 . Subsequently, one can transform such a full multipolar application to molecules with an arbitrary number of mo- Hamiltonian via the canonical transformation of the form of lecular states. This allows the proper representation of non- ͑ ͒ Eq. 8 that excludes the polarization field not only due to resonant optical processes, and is fully consistent with the ␮(X) static dipoles gg but also the corresponding higher-order constant sign convention for phenomenological damping. electric and magnetic multipoles. This will lead to the Hamil- The method leads directly to susceptibility results cast in the tonian of the same form as Eq. ͑15͒, in which the interaction simplest possible form, as has been illustrated with the ex- ˜ Hamiltonian Hint now accommodates the full multipolar example of second-harmonic generation. pansion of the radiation matter coupling ͑as presented explicitly in Ref. ͓15͔͒, subject to replacement not only of the ACKNOWLEDGMENTS dipole operator ␮(X) by ␮˜ (X), but also with corresponding We are grateful to Guy Woolley and Mohamed Babiker modifications to the higher-order moments. Furthermore, for helpful comments. G.J. acknowledges the Royal Society there will be an additional contribution in the operator for funding his visit to the United Kingdom. D.L.A. and V (X) due to the coupling between the multipoles of a mol L.C.D.R. acknowledge financial support from the Engineer- specific molecule X and the static multipoles of the remain- ing and Physical Sciences Research Council. ing species. In this way electrostatic interactions due to the permanent multipoles of the ground state are included in the ˜ Æ iS ÀiS Hamiltonian for the electrostatic intermolecular coupling. As APPENDIX: CALCULATION OF d„r… e d„r…e such, the method extends the applicability of the calcula- Consider the auxiliary function tional algorithm ͓13͔, previously directed only at electric dipole interactions, offering further scope for simplifying the f͑r͒ϭei␣SdЌ͑r͒eϪi␣S, ͑A1͒ formulation of theory for optical processes involving perma- Ќ nent multipoles. This will be the subject of a future study. where f␣(r)ϵ˜d (r) for ␣ϭ1. Differentiating Eq. ͑A1͒ with respect to the parameter ␣, one has VI. CONCLUDING REMARKS Ј͑ ͒ϭ i␣S͓ Ќ͑ ͔͒ Ϫi␣SϭϪ Ќ͑ ͒ ͑ ͒ f␣ r ie S,d r e pg r , A2 A canonical transformation has been introduced to com- ͓ Ќ ͔ pletely eliminate ground-state dipole coupling terms in the Here the use has been made of the relationship S,d (r) ϭϪ Ќ ͑ ͒ multipolar formulation of quantum electrodynamics. The ipg (r), which follows from Eq. 8 combined with the Ќ ͑ ͒ commutation relationship given in Eq. 10 . Since f0(r) transformation does not alter the vector potential a (r), yet Ќ it effects a subtraction from the full displacement field dЌ(r) ϵd (r), the solution to Eq. ͑2͒ reads of the transverse part of the polarization field due to the ͑ ͒ϭ Ќ͑ ͒Ϫ␣ Ќ͑ ͒ ͑ ͒ ␮(X) f␣ r d r pg r , A3 ground-state dipoles, gg . The radiation-molecule coupling is then represented in terms of the dipole moment operator which, with ␣ϭ1, leads to the required result given in ␮(X)ϭ␮(X)Ϫ␮(X) ͑ ͒ ˜ gg excluding a contribution due to the perma- Eq. 11 .

͓1͔ D. L. Andrews and W. J. Meath, J. Phys. B 26, 4633 ͑1993͒. ͓5͔ C. Castiglioni, M. Del Zoppo, and G. Zerbi, Phys. Rev. B 53, ͓2͔ G. B. Hadjichristov, M. D. Stamova, and P. P. Kircheva, J. 13 319 ͑1996͒. Phys. B 28, 3441 ͑1995͒. ͓6͔ M. Barzoukas, C. Runser, A. Fort, and M. Blanchard Desce, ͓3͔ J. P. Lavoine, C. Hoerner, and A. A. Villaeys, Phys. Rev. A 44, Chem. Phys. Lett. 257, 531 ͑1996͒. 5947 ͑1991͒. ͓7͔ K. Clays and B. J. Coe, Chem. Mater. 15, 642 ͑2003͒. ͓4͔ D. M. Bishop, J. Chem. Phys. 100, 6535 ͑1994͒. ͓8͔ W. J. Meath and E. A. Power, Mol. Phys. 51, 585 ͑1984͒.

043811-5 JUZELIU¯ NAS, ROMERO, AND ANDREWS PHYSICAL REVIEW A 68, 043811 ͑2003͒

͓9͔ P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics A 251, 427 ͑1959͒. ͑Cambridge University Press, Cambridge, 1990͒. ͓17͔ R. G. Woolley, Proc. R. Soc. London, Ser. A 321, 557 ͑1971͒. ͓10͔ R. G. Woolley, Int. J. Quantum Chem. 74, 531 ͑1999͒. ͓18͔ G. Juzeliu¯nas, Chem. Phys. 198, 145 ͑1995͒. ͓11͔ R. G. Woolley, Proc. R. Soc. London, Ser. A 456, 1803 ͑2000͒. ͓19͔ G. Juzeliu¯nas, Phys. Rev. A 53, 3543 ͑1996͒. ͓12͔ D. L. Andrews and P. Allcock, Optical Harmonics in Molecu- ͓20͔ G. Juzeliu¯nas, Phys. Rev. A 55, 929 ͑1997͒. lar Systems ͑Wiley-VCH, Weinheim, 2002͒. ͓21͔ G. Juzeliu¯nas and D. L. Andrews, Adv. Chem. Phys. 112, 357 ͓13͔ D. L. Andrews, L. C. Da´vila Romero, and W. J. Meath, J. ͑2000͒. Phys. B 32,1͑1999͒. ͓22͔ C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Pho- ͓14͔ L. C. Da´vila Romero and D. L. Andrews, J. Phys. B 32, 2277 tons and Atoms: Introduction to Quantum Electrodynamics ͑1999͒. ͑Wiley, New York, 1989͒. ͓15͔ D. P. Craig and T. Thirunamachandran, Molecular Quantum ͓23͔ D. L. Andrews, S. Naguleswaran, and G. E. Stedman, Phys. Electrodynamics: An Introduction to Radiation-Molecule In- Rev. A 57, 4925 ͑1998͒. teractions ͑Academic Press, London, 1984͒. ͓24͔ D. L. Andrews, L. C. Da´vila Romero, and G. E. Stedman, ͓16͔ E. A. Power and S. Zienau, Philos. Trans. R. Soc. London, Ser. Phys. Rev. A 67, 055801 ͑2003͒.

043811-6