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

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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 interac- tions. 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 mul- tipolar 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 1050-2947/2003/68͑4͒/043811͑6͒/$20.0068 043811-1 ©2003 The American Physical Society JUZELIU¯ NAS, ROMERO, AND ANDREWS PHYSICAL REVIEW A 68, 043811 ͑2003͒ 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 cou- pling 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).
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