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PHYSICAL REVIEW A

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Reply to ‘‘Comment on ‘Phenomenological damping in optical response tensors’ ’’A. D. Buckingham and P. FischerDepartment of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom共Received 19 September 2000; published 16 March 2001兲We show that damping factors must not be incorporated in the perturbation of the ground state by a staticelectric field. If they are included, as in the theory of Stedman et al. 关preceding Comment, Phys. Rev. A 63,047801 共2001兲兴, then there would be an electric dipole in the y direction induced in a hydrogen atom in theMS⫽⫹12state by a static electric field in the x direction. Such a dipole is excluded by symmetry.DOI: 10.1103/PhysRevA.63.047802 PACS number共s兲: 42.50.Ct, 03.65.⫺w, 33.80.⫺b, 78.20.JqWe note that Stedman et al. 共SNAD兲关1兴 now agree thatoptical susceptibilities in the polarization formalism shouldhave the signs of damping factors as in Bloembergen 关2,3兴,Flytzanis 关4兴, Shen 关5兴, Butcher and Cotter 关6兴, and our paper关7兴. An important issue remaining, however, is the questionof whether damping is ‘‘inevitable from any perturbation,including a static field’’ 关1兴. Andrews et al. 关8兴 and SNADassert that it is inevitable. We shall address this importantissue by considering the response of a spherical atomto a static electric field and shall show that the assertion isfalse.Suppose that damping is necessary in the description ofthe first-order perturbed ground-state ket of an atom in astatic electric field F␤. Such a ket may then be written in theform兩gFdamp典⫽兩g典⫹兺j⫽g具j兩␮ˆ␤兩g典ប冉␻jg⫺i2⌫jg冊兩j典F␤, 共1兲where␮ˆis a dipole operator,␻jgis the transition angularfrequency between the upper level j and the ground state g,and ⌫jgis the associated damping factor. The summation isover all excited states j⫽g. The expectation value of the␣component of the atom’s dipole moment in the field wouldthen be␮␣⫽具gFdamp兩␮ˆ␣兩gFdamp典⫽兺j⫽g再具g兩␮ˆ␣兩j典具j兩␮ˆ␤兩g典ប冉␻jg⫺i2⌫jg冊⫹具g兩␮ˆ␤兩j典具j兩␮ˆ␣兩g典ប冉␻jg⫹i2⌫jg冊冎F␤⫽兺j⫽g2␻jgRe兵具g兩␮ˆ␣兩j典具j兩␮ˆ␤兩g典其⫺ ⌫jgIm兵具g兩␮ˆ␣兩j典具j兩␮ˆ␤兩g典其ប冉␻jg2⫹14⌫jg2冊F␤. 共2兲If兩g典is degenerate, as for a hydrogen or sodium atom in anMS⫽⫾12spin state, then兩g典is complex, and both the realand imaginary parts of具g兩␮ˆ␣兩j典具j兩␮ˆ␤兩g典are nonzero. Theimaginary part changes sign on interchanging␣and␤. Thiswould imply that a static field Fxinduces a dipole in the ydirection in a spherical atom in an MS⫽⫹12state. However,this is incompatible with quantum theory, for such a dipolemust be zero by symmetry 共it would be equal and oppositefor the MS⫽⫹12and ⫺12states and therefore change signunder time reversal兲.If兩g典is nondegenerate, as for a helium atom, then兩g典may, without loss of generality, be chosen to be real. Theinduced dipole would be␮x⫽兺j⫽g2␻jg円具g兩␮ˆx兩j典円2ប2冉␻jg2⫹14⌫jg2冊Fx⫽␣Fx, 共3兲which is an incorrect equation for the static polarizability␣.The correct equation for␣is obtained from Eq. 共3兲 by omit-ting all ⌫jg.We note that SNAD appear to have missed the notes inthe epilogue to Bloembergen’s book 关2兴; the section SNADrefer to has been replaced by the following statement: ‘‘Thecorrect limiting behavior for the case that either the electro-magnetic frequency or the material resonant frequency be-comes very small,␻→ 0or␻ng→ 0, respectively, requires amore careful treatment of the damping terms, as has beendiscussed in detail by Van Vleck and Weisskopf 关9兴.’’SNAD do not refute our argument that inclusion of damp-ing leads to a complex ket兩gFdamp典even for a nondegenerateground state, such that兩gFdamp典is linearly independent of兩gFdamp典*, which is incompatible with nondegeneracy.We conclude that damping factors must not be incorpo-rated in the perturbation of the ground state by a static field.PHYSICAL REVIEW A, VOLUME 63, 0478021050-2947/2001/63共4兲/047802共2兲/$20.00 ©2001 The American Physical Society63 047802-1关1兴 G. E. Stedman, S. Naguleswaran, D. L. Andrews, and L. C.Da´vila Romero, preceding Comment, Phys. Rev. A 63, 047801共2000兲.关2兴 N. Bloembergen, Nonlinear Optics, 4th ed. 共World Scientific,Singapore, 1996兲.关3兴 N. Bloembergen, H. Lotem, and R. T. Lynch, Indian J. PureAppl. Phys. 16, 151 共1978兲.关4兴 C. Flytzanis, in Quantum Electronics, edited by H. Rabin andC. L. Tang 共Academic, New York, 1975兲, Vol. 1, Pt. A.关5兴 Y. R. Shen, The Principles of Nonlinear Optics 共John Wiley &Sons, New York, 1984兲.关6兴 P. N. Butcher and D. Cotter, The Elements of Nonlinear Op-tics, Cambridge Studies in Modern Optics, reprint ed. 共Cam-bridge University Press, Cambridge, 1993兲.关7兴 A. D. Buckingham and P. Fischer, Phys. Rev. A 61, 035801共2000兲.关8兴 D. L. Andrews, S. Naguleswaran, and G. E. Stedman, Phys.Rev. A 57, 4925 共1998兲.关9兴 J. H. Van Vleck and V. F. Weisskopf, Rev. Mod. Phys. 17,227 共1945兲.COMMENTS PHYSICAL REVIEW A 63


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