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Equivalence of Interaction Hamiltonians in the Electric Dipole Approximation*

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Equivalence of Interaction Hamiltonians in the Electric Dipole Approximation* journal of modern optics, 20 may 2004 vol. 51, no. 8, 1137–1147 Equivalence of interaction Hamiltonians in the electric dipole approximation* K. RZAZZEWSKI_ y and R. W. BOYD**z yDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA and Center for Theoretical Physics, PAN, Al. Lotnikow 32/46 02-668 Warsaw, Poland zInstitute of Optics, University of Rochester, Rochester, New York 14627, USA (Received 6 August 2003; revision received 30 September 2003) Abstract. The interaction of an atomic system with an externally applied electromagnetic field can be treated in the electric dipole approximation by means of either the minimal coupling ðp Á AÞ or direct coupling ðd Á EÞ Hamiltonian. It is shown that both methods lead to identical and unambiguous predictions for observable quantities as long as the atomic wavefunctions are transformed when used in the minimal-coupling formulation. The physical meaning of kinetic momentum is used to show that the atomic states must be described by wavefunctions calculated in the absence of an electromagnetic field when using the d Á E (but not the p Á A) form of the interaction Hamiltonian. When, however, observables are calculated using the common approximations of resonance atomic physics – the two-level approximation and the rotating- wave approximation – the two formulations can lead to measurably different results. This point is illustrated by calculating the induced polarization (and hence the refractive index) of an atomic system for the two exactly soluble cases of the harmonic oscillator and the hydrogen atom. 1. Introduction There are two competing ways of describing the interaction of a quantum mechanical system with an electromagnetic radiation field, the characteristic wavelengths of which are assumed much greater than the dimensions of the system. One method involves replacing p with p ðe=cÞA in the field-free Hamiltonian, and is referred to as the ‘minimal coupling’ procedure or as the p Á A form of the interaction. The other method, which is conceptually somewhat simpler, involves introducing an interaction Hamiltonian of the form d Á E, and is referred to as the ‘direct coupling’ of atomic dipole transition moment d to the applied electric field strength E. Many authors refer to these formulations as the velocity gauge and the length gauge, respectively. The question of whether the two descriptions always lead to the same predictions for observable quantities has been addressed by many authors [1–10], but there has been considerable disagreement regarding the answer to this question. In particular, references [2, 3, 6, 7] state *Reviewing of this paper was handled by a Member of the Editorial Board. **Author for correspondence; e-mail: [email protected] Journal of Modern Optics ISSN 0950–0340 print/ISSN 1362–3044 online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09500340310001631635 1138 K. Rzazzewski_ and R. W. Boyd that there are no measurable differences between the predictions of the two descriptions, whereas references [1, 5, 8, 10, 11] state that such differences can exist. We note that Aharanov and Au [11] have presented an eloquent argument that an additional phase factor has to be added to the atomic wavefunction when used to calculate the interaction in the p Á A but not the d Á E form of the interaction Hamiltonian. This line of reasoning suggests that the d Á E form is the simpler to use in that correction factors need not be included when using this form. We also note that conclusions similar to those reached in the present paper have been presented by Leubner and Zoller [12] for the specific case of multiphoton ionization probabilities. It has also been pointed out [13] that direct numerical integration of the Schro¨dinger equation for strong-field ionization if performed with the p Á A form yields consistent results only if the distinction between canonical and kinetic momentum is properly treated. In treating large-scale atomic structure calculations, Deb et al. [14] point out that results obtained using the two forms of the interaction Hamiltonian are numerically very similar, which is taken to be a self-consistency check of their results. Cormier and Lampropoulis [15] state that the p Á A form is more convenient for calculations in strong-field ionization. However, Xie et al. [16] claim that the d Á E form gives results that more closely agree with experiment than does the p Á A form. Also, Kuan et al. [17] suggest that different forms of the interaction Hamiltonian lead to different predictions. In this paper, we restrict our attention to the case in which the electromagnetic field is treated as an external, prescribed, nondynamic, c-number function. We believe that in this simple case there is no ambiguity, and that either description, if used in accordance with the principle of gauge invariance, leads to the same results. In section 2 of this paper, we demonstrate the general equivalence of these two descriptions, and point out the necessity of transforming the initial states of the system when describing the dynamics through the use of a gauge in which the vector potential A is nonzero. In section 3, we illustrate these general results by applying them to a calculation of the refractive index of a collection of atoms. We show that while the exact results of either description lead to the same prediction for the refractive index, the commonly used approximations of resonance atomic physics, i.e., the two-level approximation and the rotating-wave approximation (RWA), lead to measurably large discrepancies between the predictions of the two descriptions. We suspect that some of the past confusion regarding the possible observable consequences of the choice of the interaction Hamiltonian has resulted from a failure to make a distinction between exact and approximate results. Since it is often necessary to make the two-level and/or rotating-wave approximations in practical problems in quantum optics, section 3 also presents a numerical compar- ison of the predictions based on these approximations with the exact results for two cases – the harmonic oscillator and the hydrogen atom – where exact results can be obtained. 2. General equivalence of the two formulations Before turning to our main problem, let us consider the following simple example which illustrates the influence of the choice of gauge for a Equivalence of interaction Hamiltonians 1139 problem in classical Hamiltonian mechanics. The Hamiltonian describing the dynamics of a particle of charge e ¼jej in a constant, uniform magnetic field is given by hi 1 e 2 H ¼ p À AðrÞ ; ð2:1Þ 2m c where, however, there is no unique choice for the vector potential AðrÞ. Two (out of infinitely many) of the most commonly used forms are given by 1 A ðrÞ¼ ðBy; Bx; 0Þ; ð2:2Þ 1 2 and A2ðrÞ¼ðÀBy; 0; 0Þ; ð2:3Þ for a magnetic field given by B ¼ð0; 0; BÞ. One can easily verify that Lz ¼ ðÞr  p z commutes with the Hamiltonian (2.1) if (2.2) is substituted as the vector potential, but that it does not if (2.3) is used. The question then arises: is the z-component of the angular momentum a conserved quantity for a charged particle moving in a constant z-directed magnetic field? The paradox is easily resolved by deriving a formula for the kinetic momentum p from (2.1): hi @H e p mr_ ¼ m ¼ p À AðrÞ : ð2:4Þ @p c Since velocity has a well-defined operational meaning and the vector potential can be chosen in different ways, we conclude that the canonical momentum p is not a unique dynamical variable, and in fact depends upon the choice of gauge. Therefore, together with the two different vector potentials A1ðrÞ and A2ðrÞ we also have two different canonical momenta p1 and p2 and consequently two different canonical angular momentum L1z and L2z. Incidentally, neither of these coincides with the kinetic angular momentum r  p. The distinction between different canonical momenta is well understood in classical mechanics, but is often forgotten in quantum mechanics because we are used to representing canonical variables in the coordinate-space Schro¨dinger representation, where canonical momentum is always represented by differentiation: @ p^ !i~ : ð2:5Þ @r The primary intent of our paper is to consider the interaction of an atomic system with an external electromagnetic field. In the electric-dipole approximation and using the minimal-coupling formulation of the interaction, the evolution of the atomic system is governed by the Schro¨dinger equation hi @ðr; tÞ 1 e 2 i~ ¼ p À AðtÞ þVðrÞ ðr; tÞ; ð2:6Þ @t 2m c where VðrÞ is the scalar potential and where the vector potential is restricted only by the requirement 1 @A EðtÞ¼À ; ð2:7Þ c @t i.e., that its time derivative must reproduce the applied electric field. 1140 K. Rzazzewski_ and R. W. Boyd Now, making the substitution ðr; tÞ¼eiðe=~cÞAðtÞr ðr; tÞ; ð2:8Þ which was known already to Pauli [18] in the early 1940s, we obtain @ ðr; tÞ 1 i~ ¼ p2 þ VðrÞer Á EðtÞ ðr; tÞ; ð2:9Þ @t 2m which constitutes another equally legitimate formulation of the same dynamical problem, and is governed by the Hamiltonian: p2 H ¼ À er Á EðtÞþVðrÞ: ð2:10Þ 2m This formulation is conceptually simpler than that of equation (2.6), because by (2.4) its canonical momentum coincides with the kinetic momentum of the charged particle. For this reason we chose p to denote the standard differential operator appearing in (2.9) and (2.10). Note, however, that going beyond the dipole approximation is simpler in the p Á A formulation.
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