Polarizability and the Optical Theorem for a Two-Level Atom with Radiative Broadening
PHYSICAL REVIEW A 74, 053816 ͑2006͒
Polarizability and the optical theorem for a two-level atom with radiative broadening
Paul R. Berman Michigan Center for Theoretical Physics, FOCUS Center, and Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
Robert W. Boyd The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
Peter W. Milonni 104 Sierra Vista Dr., Los Alamos, New Mexico 87544, USA ͑Received 3 August 2006; published 21 November 2006͒ The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these ͑frequency-dependent͒ imaginary parts must be included in both the resonant and antiresonant frequency denominators ͓P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 ͑2004͔͒; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schrödinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green’s-function approach.
DOI: 10.1103/PhysRevA.74.053816 PACS number͑s͒: 42.65.An, 32.10.Dk, 32.70.Jz, 32.80.Ϫt
I. INTRODUCTION ment is real. However, the situation is more complicated be- cause ␥ An atom in its ground state has a linear polarizability ͓1͔ j is, in general, frequency dependent, as has been emphasized recently in the case of radiative damping, where 1 1 1 ␥ ϰ3 ͓6͔. In that work the radiative damping was found not ␣͑͒ = ͚ ͉d ͉2ͩ + ͪ ͑1͒ j ប j 3 j j − j + to affect the antiresonant denominator. Of course, all these ␥ ͉ ͉ Ӷ correction terms are small if j / j ± 1; moreover, in if it is assumed that the field frequency is far removed this limit, the underlying atom-field interaction is Rayleigh from any absorption resonance. Here, dj and j are the elec- scattering. tric dipole matrix element and the angular transition fre- The question of the sign of the damping term in frequency quency, respectively, connecting the excited state j to the denominators arises also in the case of nonlinear susceptibili- ͑ ͓͒ ͔ ground state. This Kramers-Heisenberg 1 formula ignores ties. Long ͓7͔, for instance, discusses some historical aspects the effects of collisions, spontaneous emission, and other line of this question in the case of Raman scattering and advo- broadening phenomena that can be accounted for by adding cates the opposite-sign form. imaginary parts to the frequency denominators in Eq. ͑1͒,as The fact that antiresonant denominators are at issue in well as self-energy corrections that appear as additional real terms in both frequency denominators in Eq. ͑1͒. Surpris- these discussions means, of course, that derivations of the ingly enough, however, this familiar procedure is not applied polarizability, including relaxation processes, cannot be ͑ ͒ uniformly in the literature. Some authors ͓2͔ leave the based on the usual rotating-wave approximation RWA . The case of radiative damping is particularly complicated be- “antiresonant” denominator j + unaltered while replacing ␥ cause one must not only go beyond the RWA but must do so the “resonant” denominator j − by j − −i j, where ␥ ͑Ͼ ͒ within the Weisskopf-Wigner approximation or related ap- j 0 is the line width of the transition from the ground state to state j. Others advocate a similar change in the proximations that make the calculations tractable. On the antiresonant denominator, but there have been lively debates other hand, the case of radiative damping provides a simple as to whether the modified denominator should be of the test of the accuracy of the calculation of ␣͑͒, namely, that ␥ ͓ ͔ “same-sign” form j + −i j 3 or the “opposite-sign” form the polarizability satisfies the optical theorem. ␥ ͓ ͔ ␣͑͒ j + +i j 4,5 . The causality requirement that should Let us briefly recall the form and physical significance of be analytic in the upper half of the complex frequency the optical theorem in the context of an atom for which the plane would appear to rule out the same-sign form, and simi- only mechanism for energy loss in an isotropic environment larly, this form does not satisfy the “crossing relation,” is radiation. In this case, the optical theorem may be ex- ␣*͑͒=␣͑−͒, which guarantees that the induced dipole mo- pressed in the form
1050-2947/2006/74͑5͒/053816͑11͒ 053816-1 ©2006 The American Physical Society BERMAN, BOYD, AND MILONNI PHYSICAL REVIEW A 74, 053816 ͑2006͒
23 ␥͑͒ = ⌫ ͑͒ + ⌫ ͑͒, ͑9͒ ␣ ͑͒ ͉␣͉͑͒2 ͑ ͒ − + I = 3 , 2 3c P denotes principal part and ␣ ͑͒ ϱ where I is the imaginary part of the polarizability. This 2d2 ⌫ ͑͒ ͵ ⍀⍀3␦͑⍀ ͒͑͒ expression follows from the condition that the rate of change ± = d ± 10 3 ប c3 of the atomic ͑or field͒ energy is zero. Physically, it simply 0 states that the rate at which the atom’s energy increases due such that to absorption in the presence of a field of frequency must ⌫ ͑͒ ⌫ ͑͒ 23 ប 3 exactly balance the rate at which the atom loses energy by + =0; − =2d /3 c radiation, or, equivalently, the rate at which the field energy for Ͼ0. These results were obtained using the electric di- decreases due to absorption must equal the rate of increase of pole form of the atom interaction with the quantized field. field energy due to scattering. Thus, the ͑power͒ absorption ⌬ ͑͒ The divergent radiative level shifts ± must obviously be coefficient for the field in a dilute medium of N atoms per renormalized, but this was not of direct concern in MB ͓9͔ ͑͒ ͑ ͒ ␣ ͑͒ unit volume is a = 4 /c N I and, equating this to nor is it of concern here; the ͑mass͒ renormalization process ͑͒ N R , where in any event cannot be correctly carried out in the two-level 2 ͉n −1͉2 4 approximation, since the dipole sum rule does not hold in the ͑͒ = ͩ ͪ ͑3͒ restricted Hilbert space of the TLA. We can write R 3 N2 c ␦͑͒ ͓⌬ ͑͒ ⌬ ͔͑͒ ប ͑ ͒ =− E2 − E1 / , 11 is the cross section for Rayleigh scattering ͓8͔ and n−1 ⌬ ͑͒ ប⌬ ͑͒ ⌬ ͑͒ ប⌬ ͑͒ Х2N␣͑͒ is the refractive index, we obtain Eq. ͑2͒. where E2 =− − and E1 =− + are the Let us recall also the familiar example of a classical elec- level shifts of the upper ͉͑2͒͘ and lower ͉͑1͒͘ states, respec- ⌬ ͑ ͒ ⌬ ͑ ͒ tron oscillator with resonance frequency 0. Including the tively, of the TLA. E2 0 and E1 0 , when modified to radiation reaction force 2e2តx/3c3 in the equation for the elec- account for the fact that we have used the electric dipole tron displacement x in the case of a driving field of amplitude rather than minimal coupling form of the atom-field interac- E0 and frequency , we have tion, are the nonrelativistic, unrenormalized “Lamb shifts” of the TLA in the absence of an applied field ͓10͔. Note that 2e2 e 2 ត ͑ ͒ both the transition width and shift in ͑6͒ are evaluated at the mx¨ + x − x = E0 cos t, 4 0 3mc3 m applied field frequency rather than the atomic transition fre- quency. and, therefore, The purpose of MB ͓9͔ was to address the question of the e2/m damping term in the antiresonant denominator in the specific ␣͑͒ = , ͑5͒ 2 − 2 −2ie23/3mc3 case of radiative broadening. The main conclusions to be 0 drawn from the expression ͑6͒ are that ͑i͒ the radiative damp- and the optical theorem ͑2͒ is satisfied. Since a two-level ing rate appears in both the resonant and the antiresonant atom ͑TLA͒ that remains with high probability in its ground denominators, and ͑ii͒ this damping rate is consistent with state can for many purposes be approximated by such an the opposite-sign convention, albeit the rate is, in fact, fre- oscillator in which e2 /m is replaced by e2f /m, where f is the quency dependent. The expression ͑6͒ can be rewritten, as- ␥2͑͒Ӷ͉2 2͉ oscillator strength of the transition, it might be expected that suming 0 − and ignoring for the moment the the optical theorem for a TLA follows trivially from ͑5͒.We frequency shift ␦͑͒,as͓11͔ show that things are not so simple. 2 d2/ប 2 d2/ប In light of the recent controversy concerning the same- ␣͑͒ = 0 = 0 . 2 2 ␥͑͒ 2 2 24 ប 3 sign and opposite-sign forms, the polarizability for a two- 0 − −2i 0 − −2id /3 c ͓ ͔ level atom was recently revisited by Milonni and Boyd 9 , ͑12͒ hereafter referred to as MB, for the case of radiative damp- ing. It was found that The factor 4 in the damping term in the denominator con- 3 ͑ ͒ 2 2 trasts with the in the classical expression 5 . It follows d /ប d /ប ͑ ͒ ␣͑͒ = + , easily from Eq. 2 that a consequence of this difference is ␦͑͒ ␥͑͒ ␦͑͒ ␥͑͒ 0 − − − i 0 + + + i that ͑12͒ does not satisfy the optical theorem. ͑6͒ The major goal of this paper is to obtain a better approxi- mation than ͑6͒ for the TLA polarizability in the case of where d is the TLA transition dipole moment, which may be radiative broadening. The results we obtain satisfy the opti- taken to be real by an appropriate choice of the phases of the cal theorem within the order of validity of the approxima- lower- and upper-state wave functions, 0 is the TLA transi- tion. Although the TLA polarizability of interest here can tion frequency, and approximate that of a real atom in the case of a single ␦͑͒ = ⌬ ͑͒ − ⌬ ͑͒, ͑7͒ ground-to-excited-state transition with large oscillator − + strength, our primary interest here is in the calculation itself, ϱ which, as noted earlier, requires one to go beyond a 2d2 d⍀⍀3 ⌬ ͑͒ = P͵ , ͑8͒ Weisskopf-Wigner-type approximation, which in its standard ± ប 3 ⍀ 3 c 0 ± form involves the rotating-wave approximation as well as the
053816-2 POLARIZABILITY AND THE OPTICAL THEOREM FOR A… PHYSICAL REVIEW A 74, 053816 ͑2006͒
Markov approximation for which the line width and shift are d ͗˙ ͑t͒͘ =−i ͗͑t͒͘ − i E cos t͗ ͑t͒͘ frequency dependent. To gain additional insight into the dy- 0 ប 0 z namics responsible for the polarizability, we present calcula- t tions using both the Heisenberg and Schrödinger pictures. ͑ Ј ͒ 2͵ Ј͓͗ ͑ ͒͑ Ј͒͘ ͗ ͑ ͒†͑ Ј͔͒͘ i k t −t This is a rare case in quantum optics in which a Schrödinger- + Ck dt z t t + z t t e 0 picture calculation is no more difficult than a Heisenberg- t picture one ͓13͔; moreover, the Schrödinger-picture calcula- ͑ Ј ͒ 2͵ Ј͓͗†͑ Ј͒ ͑ ͒͘ ͗͑ Ј͒ ͑ ͔͒͘ −i k t −t tion is “exact” within certain limits, whereas the Heisenberg- − Ck dt t z t + t z t e . 0 picture approach is perturbative. It must be mentioned that a ͑ ͒ different approach based on time-dependent Green’s func- 14 ͓ ͔ tions has been taken by Loudon and Barnett 12 , who first Since the TLA is assumed to be initially in the lower state, obtained corrections to ͑6͒ that allow the optical theorem to ͗ ͑ ͒͘ we approximate z t in the second term on the right-hand be satisfied. Following the calculations of the following three side of ͑14͒ by −1; this is the familiar approximation in sections, we compare the assumptions and approximations which the atom responds to the field as a classical Lorentz made in these different approaches. oscillator but with the factor e2 /m replaced by e2f /m, where An important feature of the calculation is that it allows 2 2ប f =2m 0d /e is the oscillator strength. This approximation one to investigate the origin of the level shifts that enter. The assumes that the applied field frequency is far enough Lamb shifts of the levels of an isolated atom can be calcu- removed from the absorption resonance that the atom re- lated in an unambiguous way; however, the level shifts that mains with high probability in the lower state. This approxi- appear in our calculation of the polarizability cannot be in- mation is made in MB ͓9͔, together with the “Markovian terpreted solely in terms of these Lamb shifts, since the shifts approximation” in the form associated with the excited state that we find depend on the ͗ ͑ ͒͑ ͒͘Х͗ ͑ ͒͑ ͒͘ ͗͑ ͒͘ ͑ ͒ dynamics of the atom-field interaction and are not the Lamb z t tЈ z tЈ tЈ =− tЈ 15 shifts of an isolated atom ͓14͔. This point is discussed in more detail at the end of Sec. IV. and likewise for the remaining three terms in the integrals on the right-hand side of ͑14͒. These approximations lead straightforwardly to the polarizability ͑6͒. ͑ ͒ II. HEISENBERG PICTURE I Here, we improve on the Markovian approximation 15 by using in ͑14͒ the formal solution of the Heisenberg equa- ͑ ͒ We use essentially the same Hamiltonian and notation as tion of motion for z t that follows from the Hamiltonian MB ͓9͔ for the interaction of a two-level atom with the field ͑13͒: in the electric dipole approximation, except that we treat the dE t applied field classically from the outset. For an applied field ͑t͒ = ͑tЈ͒ +2i 0 cos t͵ dtЉ͓͑tЉ͒ − †͑tЉ͔͒ z z ប E0 cos t polarized along the x direction, the Hamiltonian is tЈ then t ͵ Љ͕͓͑ Љ͒ †͑ Љ͔͒ ͑ Љ͒ +2Ck dt t − t ak t tЈ H = 1 ប + ប a†a − dE cos t͑ + †͒ 2 0 z k k k 0 †͑ Љ͓͒͑ Љ͒ †͑ Љ͔͖͒ ͑ ͒ − ak t t − t . 16 − i ប C ͑a + †a − a† − a††͒. ͑13͒ k k k k k Consider the first term in the first integral on the right- hand side of ͑14͒ when ͑16͒ and the equal-time operator † ͑ Ј͒͑ Ј͒ ͑ Ј͒ ak and ak are as usual the free-field photon annihilation and identity z t t =− t are used. We ignore terms of creation operators for mode k, where k for brevity denotes third- and higher-order in the atom field coupling Ck, which the wave vector k and the polarization index ͑=1,2͒ of a amounts to dropping the third term in ͑16͒ when ͑16͒ is used † ͑ ͒ free-space, plane-wave field mode, and , , and z are, in 14 . In this approximation, which we discuss in more respectively, the TLA lowering, raising, and population dif- detail later, ference operators. We use a summation convention in which t ͑ Ј ͒ repeated field indices are to be summed over on the right- 2͵ Ј͗ ͑ ͒͑ Ј͒͘ i k t −t Ck dt z t t e hand side of an equation unless they appear explicitly on the 0 ϵ͑ ͒ left-hand side. The coupling constant Ck d·ek t ϫ͑ ប ͒1/2 2 ͑ Ј ͒ 2 k / V , where ek is the polarization unit vector for Х ͵ Ј͗͑ Ј͒͘ i k t −t − Ck dt t e the mode ͑k,͒ and V is the quantization volume. Without 0 loss of generality for our purposes, we take Ck to be real and t t id ͑ Ј ͒ Љ Љ positive. As in MB ͓9͔, we write the Heisenberg equation of − E C2͵ dtЈei k t −t ͵ dtЉ͓e−i t + ei t ͔ ប 0 k motion for ͑t͒ and take expectation values over an initial 0 tЈ state in which the initial TLA state is the lower state and the ϫ͓͗͑tЉ͒͑tЈ͒͘ − ͗†͑tЉ͒͑tЈ͔͒͘. ͑17͒ initial field state is that in which all modes are unoccupied except for that corresponding to the external, classically de- To remain to second order in the atom-field coupling, we scribed field make the replacements
053816-3 BERMAN, BOYD, AND MILONNI PHYSICAL REVIEW A 74, 053816 ͑2006͒
−i ͑tЉ+tЈ͒ t ͗͑tЉ͒͑tЈ͒͘ → ͗͑0͒͑0͒͘e 0 =0, ͑ ͒͑ Ј ͒ 2͵ Ј i k± t −t ⌫ ͑͒ ⌬ ͑͒ ͑ ͒ Ck dt e = ± − i ± . 23 0 ͑ Љ Ј͒ ͗†͑tЉ͒͑tЈ͒͘ → ͗†͑0͒͑0͒͘ei 0 t −t =0. ͑18͒ ͑ ͓ ͔ ⌫ ͑͒ We depart here from the notation in MB 9 by using ± ␥ ͑͒ ␥͑͒ ⌫ ͑͒ ⌫ ͑͒ The first expression is identically zero, whereas the second is instead of ± and using to denote + + − ͓ ͑ ͔͒ ͑ ͒ zero under the assumption that the initial TLA state is the Eq. 9 . Then, 22 becomes lower state ͉1͑͑͘0͉͒1͘=0͒. Thus, t ͑ Ј ͒ 2͵ Ј͗ ͑ ͒†͑ Ј͒͘ i k t −t Ck dt z t t e t t 0 ͑ Ј ͒ ͑ Ј ͒ 2͵ Ј͗ ͑ ͒͑ Ј͒͘ i k t −t Х 2͵ Ј͗͑ Ј͒͘ i k t −t Ck dt z t t e − Ck dt t e , t −it 0 0 ͑ Ј ͒ dE0 e Х C2͵ dtЈ͗†͑tЈ͒͘ei k t −t + k ប ͑19͒ 0 + 0 ϫ͓⌫ ͑ ͒ ⌫ ͑͒ ⌬ ͑ ͒ ⌬ ͔͑͒ which is equivalent to the approximation ͑15͒ made in MB + 0 − − − i + 0 + i − ͓9͔ for the first term in the first integral on the right-hand side dE eit 0 ͓⌫ ͑ ͒ ⌫ ͑͒ ⌬ ͑ ͒ ⌬ ͔͑͒ of ͑14͒. − + 0 − + − i + 0 + i + . ប − Consider next the second term in the first integral on the 0 right-hand side of ͑14͒ when ͑16͒ and the identity ͑24͒ ͑ Ј͒†͑ Ј͒ †͑ Ј͒ z t t = t are used. Following the same approxi- We proceed in the same way to evaluate, approximately, ͑ ͒ mation leading to 17 , we make the replacement the second integral on the right-hand side of ͑14͒. Collecting t all the terms, we obtain the following approximate equation ͑ Ј ͒ 2͵ Ј͗ ͑ ͒†͑ Ј͒͘ i k t −t for ͗͑t͒͘: Ck dt z t t e 0 id ͗͑ ͒͘ ͗͑ ͒͘ ͓ −it it͔ t ˙ t =−i 0 t + E0 e + e ͑ Ј ͒ ប → 2͵ Ј͗†͑ Ј͒͘ i k t −t 2 ͚ Ck dt t e k 0 t ͑ Ј ͒ 2͵ Ј͓ ͗͑ Ј͒͘ ͗†͑ Ј͔͒͘ i k t −t t t + Ck dt − t + t e id ͑ Ј ͒ Љ Љ − E C2͵ dtЈei k t −t ͵ dtЉ͓e−i t + ei t ͔ 0 ប 0 k 0 tЈ t ͑ Ј ͒ 2͵ Ј͓ ͗†͑ Ј͒͘ ͗͑ Ј͔͒͘ −i k t −t ϫ͓͗͑ Љ͒†͑ Ј͒ †͑ Љ͒†͑ Ј͔͒͘ ͑ ͒ − Ck dt − t + t e t t − t t . 20 0 As in ͑18͒, we take d − ⌫ ͑͒ + i⌬ ͑͒ − i⌬ ͑ ͒ + E e−itͫ − − + 0 ប 0 ͑ Љ Ј͒ ͑ Љ Ј͒ + 0 ͗͑tЉ͒†͑tЈ͒͘Х͗͑0͒†͑0͒͘e−i 0 t −t = e−i 0 t −t , − ⌫ ͑͒ + i⌬ ͑ ͒ − i⌬ ͑͒ + + + 0 + ͬ ͑ Љ Ј͒ − ͗†͑tЉ͒†͑tЈ͒͘Х͗†͑0͒†͑0͒͘ei 0 t +t =0 ͑21͒ 0 d − ⌫ ͑͒ + i⌬ ͑ ͒ − i⌬ ͑͒ ͉ ͘ − E eitͫ − + 0 − for the initial TLA state 1 . Then, ប 0 + 0 t ͑ Ј ͒ − ⌫ ͑͒ + i⌬ ͑͒ − i⌬ ͑ ͒ C2͵ dtЈ͗ ͑t͒†͑tЈ͒͘ei k t −t + + + + 0 ͬ, ͑25͒ k z 0 − 0 t t t where we have used the fact that ⌫ ͑ ͒=0. 2 † i ͑tЈ−t͒ id 2 i ͑tЈ−t͒ + 0 Х C ͵ dtЈ͗ ͑tЈ͒͘e k − E C ͵ dtЈe k ͵ dtЉ −it it k ប 0 k The solution of ͑25͒ has the form ͗͑t͒͘=se +re , and 0 0 tЈ the induced dipole moment p=d͗+†͘=2d Re͓͑s ͑ ͒ −i tЉ i tЉ −i 0 tЉ−tЈ *͒ −i t͔ϵ ͓␣͑͒ −i t͔ ͑ ͒ ϫ͓e + e ͔e +r e Re E0e . Solving 25 for s and r,we t obtain 2 † i ͑tЈ−t͒ d = C ͵ dtЈ͗ ͑tЈ͒͘e k + E 2 k ប 0 2d 1 0 ␣͑͒Х 0 ប 2 2 ͓␥͑͒ ␦͔͑͒ t 0 − −2i − i 1 ͑ ͒͑ Ј ͒ ͑ ͒͑ Ј ͒ ϫ −i t 2͵ Ј͓ i k+ 0 t −t i k− t −t ͔ e Ck dt e − e ⌬ ͑͒ + i⌫ ͑͒ − ⌬ ͑ ͒ + ϫ ͫ − − + 0 0 0 1+2 + 0 t d 1 ͑ ͒͑ Ј ͒ ͑ ͒͑ Ј ͒ − E ei tC2͵ dtЈ͓ei k+ 0 t −t − ei k+ t −t ͔. ⌬ ͑ ͒ − ⌬ ͑͒ + i⌫ ͑͒ ប 0 k +2 + 0 + + ͬ ͑26͒ − 0 0 − 0 ͑22͒ for the TLA polarizability. This expression satisfies the cross- For the times tӷ1/ of interest, we use ing relation ␣*͑−͒=␣͑͒. If we retain only terms up to
053816-4 POLARIZABILITY AND THE OPTICAL THEOREM FOR A… PHYSICAL REVIEW A 74, 053816 ͑2006͒
fourth order in d ͑recall that all the widths and shifts are of The total ͑cycle-averaged͒ power radiated by the TLA is order d2͒, consistent with the approximations used in our ͑ ͒ 4 4d242͉͉2 derivation, then the optical theorem 2 is also satisfied to P = ͉␣͉2E2 = , ͑32͒ this order. We defer further discussion of this result to Sec. V. 3c3 0 3c3 In this section, we have shown how to generalize the re- sults of MB ͓9͔ to obtain an expression for the polarizability and the rate at which energy is lost from the external field is that is consistent with the optical theorem; however, it is calculated straightforwardly to be difficult to rigorously justify the factorization approxima- P =2ប 2 Im͓͔. ͑33͒ tions that were used in arriving at this result. We now give alternative derivations based a Schrödinger-picture approach The optical theorem is satisfied if these two expressions are and a Heisenberg picture, in which the approximations intro- equal duced are more transparent. Moreover, these alternative deri- 23d2 vations provide additional insight into the underlying physi- ͓͔ ͉͉2 ⌫ ͉͉͑͒2 ͑ ͒ Im = = − . 34 cal processes contributing to the polarizability. 3 ប c3
III. SCHRÖDINGER PICTURE In order to obtain , one is faced with the task of calcu- lating In many cases, it is easier, computationally, to evaluate ͑ ͒ expectation values of atomic Heisenberg operators, such as 21 = 2,0;1,0 + 2,k;1,k + 2,kkЈ;1,kkЈ + ¯ 35 the polarizability, using a Heisenberg- rather than + − Schrödinger-picture approach. In this case, however, since and extracting 21 and 21 from this quantity. To first order in we are working to first order in the external field amplitude, , it turns out that terms beyond the first two make contri- + − 2 ͑ ͒2 the Schrödinger approach is no more difficult than the butions to 21 and 21 of order Ck / 0 + k , that is, of order Heisenberg approach. Moreover, it is a simple matter to keep of the TLA “Lamb shift” divided by 0. Such terms, which track of the amplitudes that contribute and no factorization also determine the ͑non-RWA͒ excited-state population in the approximations are needed. Within certain approximations to absence of any applied fields, are systematically neglected in 2 ͑ ͒͑ ͒ be specified below, this approach is exact. this work, as are terms of order Ck / 0 + k + k : The state vector for the system can be written as C2 ͉͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ ͘ ͉ Ј͘ k Ӷ 1, ͑36a͒ = b1,0 1,0 + b2,0 2,0 + b1,k 1,k + b2,k 2,k + b1,kkЈ 1,kk ͑ ͒2 0 + k ͉ Ј͘ ͑ ͒ + b2,kkЈ 2,kk + ¯ . 27 C2 The field states, which are specified by the second label in k Ӷ 1. ͑36b͒ ͑ ͒͑ ͒ each ket, refer to vacuum field modes; as before, the exter- 0 + k + k nally applied field is treated classically. Thus, ͉1,0͘ and ͉2,0͘ As a consequence, the density matrix element ͑35͒ can be are, respectively, the states in which the TLA is in the lower approximated for our purposes as state and the upper state and the field state is the vacuum. ͉1,k͘ and ͉2,k͘ are states in which the TLA is in the lower Х ͑ ͒ 21 2,0;1,0 + 2,k;1,k. 37 state and the upper state, respectively, and there is a single photon in the field mode denoted by k, as in the preceding Instead of directly calculating density matrix elements, we section. Similarly, ͉1,kkЈ͘ and ͉2,kkЈ͘ are states in which the calculate probability amplitudes and form density matrix el- TLA is in the lower state and the upper state, respectively, ements from these amplitudes. This simplifies the calculation and there is a photon in mode k and a photon in mode kЈ. considerably and enables us to obtain an expression for ͑͒ The expectation value of the electric dipole moment is that is essentially exact to all orders in the vacuum coupling strength. An amplitude approach can be used in this problem −it it ͑ ͒ ͑ ͒ p = p+e + p−e = d 21 + 12 , 28 since the vacuum field states responsible for relaxation are * included explicitly in the state amplitudes. Such a procedure where 21= 12 is an off-diagonal density matrix element. Writing is practical in our case owing to the fact that the calculation is perturbative in the applied field—for a strong external field + −it − it ͑ ͒ 21 = 21e + 21e , 29 ͑Rabi frequency greater than decay rates͒, the number of we can express the complex polarizability as terms that enter would render the amplitude approach virtu- ally useless. p d2 ͑ ͒ ␣ = + = , ͑30͒ Owing to the definition 31 , we need only calculate 21 to ប E+ first order in . The equations for the probability amplitudes are easily obtained from the Hamiltonian ͑13͒ and the time- where dependent Schrödinger equation ͓15͔: ͓+ ͑− ͒*͔ lim 21 + 21 →0 ˙ ͑ it −it͒ ͑ ͒  = ͑31͒ b1,0 = i e + e b2,0 − Ckb2,k, 38a ប ˙ ͑ it −it͒ ͑ ͒ and the Rabi frequency is defined by =dE0 /2 . b2,0 = i e + e b1,0 − i 0b2,0 − Ckb1,k, 38b
053816-5 BERMAN, BOYD, AND MILONNI PHYSICAL REVIEW A 74, 053816 ͑2006͒
˙ ͑ it −it͒ 0=i − i͓⌬ + ⌬ ͑ ͔͒b+ − C b+ ͑42a͒ b1,k = Ckb2,0 − CkЈb2,kkЈ − i kb1,k + i e + e b2,k, + 0 2,0 k 1,k ͑38c͒ ͓⌬͑+͒ ⌬ ͑ ͔͒ − − ͑ ͒ 0=i − i + + 0 b2,0 − Ckb1,k 42b ˙ ͑ it −it͒ ͑ ͒ b2,k = i e + e b1,k − i 0 + k b2,k + Ckb1,0, ͑38d͒ + + ͓ ⌬ ͑ ͒ ⑀͔ + 0=Ckb2,0 − CkЈb2,kkЈ − i k − + + 0 − i b1,k ˙ ͑ ͒ ˜ ͑ ͒͑͒ b2,kkЈ =−i 0 + k + kЈ b2,kkЈ + CkЈb1,k + Ckb1,kЈ. + i b2,k no sum on k 42c ͑38e͒ We neglect ground-state amplitudes involving two or more − − ͓ ⌬ ͑ ͔͒ − 0=Ckb2,0 − CkЈb2,kkЈ − i k + + + 0 b1,k free-field photons and excited-state amplitudes involving ˜ ͑ ͒͑͒ three or more free-field photons, since they lead to correc- + i b2,k no sum on k 42d tions of order ͑36͒; the probability amplitudes we retain are the only ones that contribute to 21 to first order in the ex- ternal field. These probability amplitudes satisfy Eqs. ͑38͒, ͓ ⌬ ͑ ͔͒ + + 0=−i 0 − + k + kЈ + + 0 b + CkЈb which are to be solved to first order in and to all orders in 2,kkЈ 1,k the vacuum coupling C . + ͑ ͒͑͒ k + Ckb1,kЈ no sum 42e We are interested in steady-state solutions for the prob- ability amplitudes. From the structure of Eqs. ͑38͒,wede- ±it duce that b2,0, b1,k, and b2,kkЈ oscillate as e and depend ͓ ⌬ ͑ ͔͒ − − 0=−i 0 + + k + kЈ + + 0 b2,kkЈ + CkЈb1,k linearly on , while b1,0 and b2,k have no linear dependence on . To first order in , then, the solution of ͑38d͒ is − ͑ ͒ ͑ ͒ + Ckb1,kЈ no sum , 42f − iC b b Х k 1,0 , ͑39͒ 2,k 0 + k where and, substituting this back into ͑38a͒͑and neglecting the lead 2 ⌬ ⌬͑+͒ ͑ ͒ term, which is of order ͒, we obtain = 0 − ; = 0 + , 43
˙ Х ⌬ ͑ ͒ ͑ ͒ b1,0 i + 0 b1,0. 40 − iC In obtaining a solution to Eqs. ͑38͒ and ͑40͒ to first order ˜b Ϸ k . ͑44͒ ⌬ ͑ ͒ 2,k ±i t+i + 0 t + in , we find that b2,0, b1,k, and b2,kkЈ oscillate as e 0 k i⌬+͑0͒t while b1,0 and b2,k oscillate as e . Thus, we assume a ⑀ ͑ ͒ trial solution of the form The frequency k has been replaced by k −i in Eq. 42c , ⌬ ͑ ͒ where ⑀ is a positive quantity that tends toward zero, to en- i + 0 t ͑ ͒ b1,0 = e 41a + sure that a steady-state solution for b1,k exists. In terms of these parameters, it follows from Eqs. ͑29͒, ͑31͒, ͑37͒, and ⌬ ͑ ͒ b = ͑b+ e−i t + b− ei t͒ei + 0 t ͑41b͒ ͑ ͒ * 2,0 2,0 2,0 41 , and the fact that a,aЈ=b␣b␣Ј, that
⌬ ͑ ͒ ͑ + −i t − i t͒ i + 0 t ͑ ͒ b1,k = b e + b e e 41c 1,k 1,k b+ + ͑b− ͒* + ˜b ͑b− ͒* + ˜b* b+  = 2,0 2,0 2,k 1,k 2,k 1,k . ͑45͒ ⌬ ͑ ͒ − iCk ⌬ ͑ ͒ b = ˜b ei + 0 t = ei + 0 t ͑41d͒ 2,k 2,k ͓ ⌬ ͑ ͔͒ ͑ ͒ 0 + k + + 0 Equations 42 break up into two uncoupled sets of equa- + + ˜ + − − tions, one for b2,0, b1,k, b2,k, b2,kkЈ and the other for b2,0, b1,k, + − ⌬ ͑ ͒ b = ͑b e−i t + b ei t͒ei + 0 t. ͑41e͒ ˜ − 2,kkЈ 2,kkЈ 2,kkЈ b2,k, b2,kkЈ. However, it is sufficient to solve the first set only since Eqs. ͑42͒ imply that ͉ ͉ ͉ ͉ ͓ ⌬ ͑ ͔͒ Note that b1,0 =1 and b2,k =Ck / 0 + k + + 0 in the ͑ ͒ ͉ ͉ approximations implicit in 41 ; the fact that b2,k 0 results − ͑͒ + ͑ ͒ from fluctuations of the vacuum field that lead to a small ba,aЈ = ba,aЈ − steady-state value for this amplitude ͑but not to significant population or to measurable radiation in the absence of any applied field͒. ˜b ͑͒ = ˜b ͑− ͒͑46͒ When we substitute our trial solution into Eqs. ͑38͒, and 2,k 2,k keep terms only of first order in , we obtain the steady-state equations for any ͕a,aЈ͖. Therefore,
053816-6 POLARIZABILITY AND THE OPTICAL THEOREM FOR A… PHYSICAL REVIEW A 74, 053816 ͑2006͒
b+ ͑͒ + ͓b− ͔͑͒* + ˜b ͓͑͒b− ͔͑͒* + ˜b* ͑͒b+ ͑͒ ͑͒ = 2,0 2,0 2,k 1,k 2,k 1,k
b− ͑− ͒ + ͓b+ ͑− ͔͒* + ˜b ͑− ͓͒b+ ͑− ͔͒* + ˜b* ͑− ͒b− ͑− ͒ = 2,0 2,0 2,k 1,k 2,k 1,k = *͑− ͒,
i.e., the crossing relation is satisfied. 1 ˜␥ ͑ ͒ + ˜ b+ = ͫi − We now solve Eqs. 42 by eliminating b2,0, b2,k, and 2,0 ␥ ⌬ ⌬͑+͒ + ͑ ͒ + ˜ + i b2,kkЈ to obtain an integral equation for b1,k that is solved 2 + ˜ ͑ ͒ CkЈC b exactly using a Born series. We already have b2,k from 44 , k 1,kЈ ͬ ͑ ͒ + − i , 52 ͑ ͒ ͑ ͒͑ ⑀͒ and we can use 42e to eliminate b2,kkЈ 0 − + k + kЈ k − − i
+ + CkЈb1,k + Ckb1,kЈ where b+ = . ͑47͒ 2,kkЈ 0 − + k + kЈ ⌬͑+͒C2 i˜␥ = lim k = i⌫ ͑͒ + ⌬ ͑͒ − ⌬ ͑ ͒. ͑ ⑀͒͑ ͒ − − + 0 ͓In writing ͑44͒ and ͑47͒, and in what follows, we neglect ⑀→0 k − − i 0 + k ⌬ ͑ ͒ + 0 in all frequency denominators except those involving ͑53͒ ⌬ ͑ ͒ ͔ and k − , since it represents a correction of order 36 . Substituting this into ͑42c͒, and using ͑44͒, we obtain We have used
2 + + C b + CkЈCkb 1 1 + kЈ 1,k 1,kЈ lim = i␦͑x͒ + Pͩ ͪ, ͑54͒ 0=Ckb + i ⑀ 2,0 ⑀→0+ x − i x 0 − + k + kЈ ͓ ⌬ ͑ ͒ ⑀͔ + − i k − + + 0 − i b1,k where P again denotes the Cauchy principal part. Using the approximation C + k ͑no sum on k͒. ͑48͒ + 0 + k b1,kЈ + ͑ − + + ͒͑ − − i⑀͒ Since b1,k is sharply peaked at k = , we make the approxi- 0 k kЈ k mations + b1,kЈ 1 1 + + = ͫ − ͬ b b ⑀ 1,k Ϸ 1,k kЈ + 0 k − − i 0 − + k + kЈ ͑ ͒ 0 − + k + kЈ 0 + kЈ + b1,kЈ 1 1 Ϸ ͫ − ͬ, ͑55͒ ⑀ + + kЈ + 0 k − − i 0 + k b1,kЈ b1,kЈ Ϸ ͑49͒ ͑ ͒ ͑ ͒ 0 − + k + kЈ 0 + k we rewrite Eq. 52 as ͑ ͒ ␥ ␥ + with errors of order 36 . With this approximation, the term 1 ˜ ˜Ckb1,k 2 + ͫ ͬ ͑ ͒ + ͑ ͒ ⌬ ͑ ͒ b2,0 = i − ͑+͒ − i . 56 involving CkЈ in Eq. 48 is equal to i + 0 b1,k. This cancels ˜␥ + i⌬ ⌬ + ⌬ ͑ ͒ + k 0 the −i + 0 b1,k contribution from the second term on the right-hand side, giving When Eq. ͑56͒, in turn, is substituted back into Eq. ͑50͒ and ͑49͒ is used, we finally obtain the following ͑integral͒ equa- + + iC iCkЈb1,kЈ tion for b1,k: b+ = k ͫ− b+ − − ͬ. 1,k ⑀ 2,0 k − − i 0 + k 0 − + k + kЈ iC − i ˜␥ ͑ ͒ + k ͭͩ ͪ b = − + 50 1,k ⑀ ͑+͒ k − − i ˜␥ + i⌬ 0 + k ⌬ ͑˜␥ + i⌬͒ For future reference, we also note that + ˜␥ CkЈb ͩ ͪ 1,kЈ ͮ ͑ ͒ − i 1− . 57 1 1 1 1 1 ͑␥ ⌬͒ Ϸ ͫ − ͬ. ˜ + i 0 + kЈ ͑ ⑀͒ ͑ ͒ ⌬͑+͒ ⑀ k − − i 0 + k k − − i 0 + k This equation can be solved iteratively to evaluate the sums ͑51͒ over kЈ that occur in each iteration. In doing so, we can make When the solution ͑50͒ is substituted into ͑42a͒, we obtain the approximation
053816-7 BERMAN, BOYD, AND MILONNI PHYSICAL REVIEW A 74, 053816 ͑2006͒
C2 1 2 k ͩ ͪ b− = . ͑62͒ ⑀ 2,0 ⌬͑+͒ k − − i 0 + k C2 1 1 1 All other amplitudes contribute to  amounts that are smaller = k ͫ − ͬ ⌬͑+͒ ⑀ by factors of order ͑36͒. In this manner we obtain, to this k − − i 0 + k 0 + k order, the combination ͓˜b ͑b− ͒* +͑b− ͒*͔ needed in the 2 ␥ 2,k 1,k 2,0 Ck 1 1 i˜  Ϸ = , evaluation of as ⌬͑+͒ ⑀ ⌬͑+͒ k − − i 0 + k ͓˜b ͑b− ͒* + ͑b− ͒*͔Ϸ . ͑63͒ the omitted term being of order ͑36͒. In effect, we can re- 2,k 1,k 2,0 ⌬͑+͒ place /͑ + ͒ in ͑57͒ by /⌬͑+͒ since this leads only to 0 k ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ corrections of order ͑36͒ in calculating . Carrying out the Combining Eqs. 63 , 39 , 60 , 61 , and 45 , we arrive iteration using these approximations, we have at ⌬͑+͒ ϱ ͑͒ = ͑ ͒ i Ck ⌬⌬ + ␥ b+ =− ͑A + B͚͒ ͓i˜␥B͔n, ͑58͒ −2i 0˜ 1,k − − i⑀ k n=0 C2 * 2 * + ͫͩ k ͪ 0 ͬ ͑ ⑀͒͑ ͒ ⌬⌬͑+͒ ␥ where k − − i k + 0 +2i 0˜ 1 ⌬͑+͒ 2i ˜␥/⌬͑+͒ 1 i + = + 0 + A = ⌬͑+͒ ⌬⌬͑+͒ ␥ ⌬⌬͑+͒ ␥ ⌬͑+͒ ˜␥ + i⌬ −2i 0˜ −2i 0˜ 2 = 0 1 ˜␥ i⌬ ⌬⌬͑+͒ −2i ˜␥ B = ͩ1− ͪ = . 0 ⌬͑+͒ ͑␥ ⌬͒ ⌬͑+͒͑␥ ⌬͒ ˜ + i ˜ + i 2 = 0 . ͑64͒ ⌬⌬͑+͒ ͓⌬ ͑ ͒ ⌬ ͔͑͒ ⌫ ͑͒ Since +2 0 + 0 − − −2i 0 − This expression is exact, subject to the conditions ͑36͒, and it i˜␥⌬ ˜␥ ˜␥ ͉˜␥B͉ = ͯ ͯ Յ ͱͯ ͯ Յ ͱͯ ͯ Ͻ 1, is easily verified that it satisfies the optical theorem ͓Eq. ͑+͒ ⌬͑+͒ ⌬ ͑˜␥ + i⌬͒ 0 ͑34͔͒. The polarizability calculated in this Schrödinger- ͑59͒ picture approach is equivalent to that calculated in the Heisenberg picture when ͑36͒ holds and only contributions we can carry out the summation in Eq. ͑58͒ and obtain up to second order in d are retained. This can be seen by writing the factor in large brackets in ͑26͒ as 1+xХ1/͑1 ͑ ͒ − i Ck A + B − i Ck 2 0 −x͒, setting ⌫ ͑͒=0, and neglecting terms of order b+ = = , + 1,k ⑀ ␥ ⑀ ⌬⌬͑+͒ ␥ ͓⌬ ͑͒ ⌬ ͑ ͔͒ ⌬ ͑ ͒ k − − i 1−i˜B k − − i −2i 0˜ + − + 0 / , consistent with condition 36b . This Ӷ ͑ ͒ ͑60͒ would imply that the condition x 1, along with 36a ,is necessary for the validity of our factorization hypotheses. We ⌬ ⌬͑+͒ where the identity + =2 0 has been used. It then fol- now turn again to the Heisenberg picture, using an approxi- lows from Eqs. ͑52͒ and ͑60͒ that mation scheme that further supports this contention and helps to elucidate some aspects of the TLA polarizability. − ˜␥ + ͫ IV. HEISENBERG PICTURE II b = − i + ͑ ͒ 2,0 ˜␥ + i⌬ ⌬ + From the Hamiltonian ͑13͒ and the equal-time commuta- ␥ 2 ˜C 2 0 tion relations, we obtain + k ͬ ͑ ⑀͒͑ ͒ ⌬⌬͑+͒ ␥ k − − i k + 0 −2i 0˜ ͗͘ ͗͘ ͗ ͘ ͑͗ ͘ ͗ †͒͘ ˙ =−i 0 −2i cos t z + Ck ak z − zak , ␥ ␥2 − ˜ i˜ 2 0 ͑65͒ = ͫ− i + + ͬ ␥ ⌬ ⌬͑+͒ ⌬͑+͒ ⌬⌬͑+͒ ␥ ˜ + i −2i 0˜ ͗a˙ ͘ =−i ͗a ͘ + C ͑͗͘ + ͗†͒͘. ͑66͒ ⌬͑+͒ k k k k = . ͑61͒ ⌬⌬͑+͒ −2i ˜␥ Consistent with our previous approximations, we approxi- 0 ͗ ͘ ͑ ͒ mate z in the second term on the right-hand side of 65 This completes the calculation of the “+” terms. by −1, since we are neglecting any excited-state population Although Eq. ͑46͒ is exact, we cannot apply it directly to in the effect of the applied field on the atom. Thus, we work − − − with the approximate evolution equations obtain solutions for b2,0, b1,k, b2,kkЈ since we have assumed implicitly that Ͼ0 in obtaining the solution ͑61͒.Onthe ͗͘ ͗͘ ͑͗ ͘ ͗ ͘*͒ ˙ =−i 0 +2i cos t − Ck ak − ak other hand, it is easy to carry out an iterative solution to Eqs. ͑͗ ͘ ͗ † ͒͘ ͑ ͒ ͑42͒ to obtain, for instance, +2Ck ak 22 − ak 22 , 67a
053816-8 POLARIZABILITY AND THE OPTICAL THEOREM FOR A… PHYSICAL REVIEW A 74, 053816 ͑2006͒
͗a˙ ͘ =−i ͗a ͘ + C ͑͗͘ + ͗†͒͘, ͑67b͒ C* k k k k c =− k , ͑70b͒ k 0 + k ͗†͘ = ͗͘*, ͑67c͒ where the convergence parameter ⑀ has been reintroduced. ͗ †͘ ͗ ͘* ͑ ͒ The key point now is to assume trial solutions of the form ak = ak , 67d ͗͘ = se−i t + rei t ͑71a͒ where we have used z = 22− 11=2 22−1. It is now necessary to obtain equations of motion for ͗ ͘ ͗†͘ * −it * it ͑ ͒ ak 22 and its conjugate. This leads to an infinite set of = r e + s e 71b coupled equations; however, since we are working to first ͗ ͘ + −it − it ͑ ͒ order in , the equations can be truncated, just as in the ak = ak e + ak e 71c amplitude approach. In fact, we can be guided by the ampli- tude approach in determining which operators must be re- f = f+eit + f−e−it. ͑71d͒ ͗ ͘ * k k k tained. For example, ak 22 = 2k,2=b2kb2 and must be re- ͗ †͘ * 2 In terms of these variables, the polarizability ␣͑͒ and ͑͒, tained, whereas ak = 2,1k =b2b1k is of order and can be neglected. In this way, we obtain a closed set of operator introduced in Secs. II and III, respectively, are given by equations for the quantities 2d͑s + r*͒ ͑s + r*͒ ␣͑͒ = ; ͑͒ = . ͑72͒ ͗ ͑͒͘ fk = ak 22 68a E0
† Using the trial solution in the original equations, one finds y = ͗a a ͑͘68b͒ kkЈ kЈ k steady-state equations ͗ ͑͒͘ ⌬ ͓ + ͑ −͒*͔ ͓ + ͑ −͒*͔ zkkЈ = akЈak 68c 0=−i s + i − Ckak − Ck ak − Ckfk − Ck fk ͑73a͒ ͗ ͑͒͘ ck = ak 68d ⌬͑+͒ * ͓ + ͑ −͒*͔ ͓ + ͑ −͒*͔ 0=i r − i + Ckak − Ck ak + Ckfk − Ck fk x = ͗a† a a ͑͘68e͒ kkЈkЉ kЉ kЈ k 22 ͑73b͒ of the form 0=−i͑ − − i⑀͒a+ + C ͓s + r*͔͑73c͒ df k k k k =−i kfk + CkЈykЈk + CkЈzkkЈ + CkЈykkЈ dt ͑ ͒͑ −͒* ͓ *͔͑͒ 0=i k + ak + Ck s + r 73d ͗†͑͒͘ −2i ck cos t + Ck 69a ͑ ⑀͒ + * ͑ ͒ 0=−i k − − i fk − i ck + Ckr 73e dykkЈ =−i͑ + − ͒y + C x − C x* 0 k kЈ kkЈ kЉ kkЉkЈ kЉ kЈkЉk ͑ ͒͑ −͒* * ͑ ͒ dt 0=i k + fk + i ck + Cks 73f ͗ † ͑͒͘ + CkЈfk + Ck a 69b kЈ iC c =− k . ͑73g͒ k dz 0 + k kkЈ ͑ ͒ ͗ ͘ ͗ ͘ =−i 0 + k + kЈ zkkЈ − Ck akЈ + CkЈ ak − CkЉxkkЈkЉ dt The solution of these equations is now straightforward, with ͑69c͒ − iC ͑s + r*͒ a+ = k ͑74a͒ k − − i⑀ C k c =−i k ; ͑69d͒ k 0 + k iC ͑s + r*͒ ͑a−͒* = k ͑74b͒ k + dx k kkЈkЉ ͑ ͒ =−i k + kЈ − kЉ xkkЈkЉ − CkЉzkkЈ + CkЈykkЉ dt and ͑ ͒ ͓ + ͑ −͒*͔ ͑͒ ␥ ͑͒ * ⌬ ͑͒ ͑ ͒ − CkykЈkЉ, 69e − i Ckfk − Ck fk = g − i − r − + s, 75 where It can be shown that the contributions from the xkkЈkЉ, zkkЈ, and ykkЉ terms are down by order of radiative shifts or widths ␥ ͑͒ ⌬ ͑ ͒ ⌬ ͑ ͒ − ⌬ ͑͒ g͑͒ = i − − + 0 − + 0 + ͑76͒ divided by 0, and therefore can be neglected, consistent ⌬͑+͒ ⌬͑+͒ ⌬ with condition ͑36a͒. Thus, the relevant equations are df and k =−i͑ − i⑀͒f −2ic cos t + C ͗†͑͘70a͒ k k k k ␥ ͑͒ ⌫ ͑͒ ⌬ ͑͒ dt − = − − i − .
053816-9 BERMAN, BOYD, AND MILONNI PHYSICAL REVIEW A 74, 053816 ͑2006͒
When these results are substituted into Eqs. ͑73a͒ and ͑73b͒, 0=i⌬͑+͒r* − i + g͑͒͑s + r*͒ one finds ͓ ͑͒ ␥ ͑͒ * ⌬ ͑͒ ͔ ͑ ͒ −2i g − i − r − + s , 78 0=−i⌬s + i − g͑͒͑s + r*͒ ͓ ͑͒ ␥ ͑͒ * ⌬ ͑͒ ͔ ͑ ͒ +2i g − i − r − + s , 77 which can be solved to give
i⌫ ͑͒ ⌬ ͑ ͒ − ⌬ ͑͒ ⌬ ͑ ͒ − ⌬ ͑͒ 2 ͩ1+2 − −2 + 0 − −2 + 0 + ͪ 0 ⌬͑+͒ ⌬͑+͒ ⌬  = . ͑79͒ ⌬⌬͑+͒ ͓ ⌫ ͑͒ ⌬ ͑͒ ⌬ ͔͑͒ −2 i − + − − +
Note that the last term in the numerator does not diverge as nant and antiresonant components of the driving field, such ⌬ goes to zero owing to the definition ͑8͒. Equation ͑79͒ as that implied in an expression given by Sakurai ͓2͔͑ne- coincides exactly with the polarizability ͑26͒ obtained in Sec. glecting level shifts͒, II, since ⌫ ͑͒=0 for Ͼ0. Moreover, since all the fractions + 1 1 in the numerator are assumed to be small, we can move them  = + ͑81͒ 1 ⌬ ⌫ ͑͒ ⌬͑+͒ to the denominator. In this way, using Eq. ͑43͒, one obtains − i − leads to a result, 2  0 = ͑+͒  2 ⌬⌬ +2 ͓− i⌫ ͑͒ − ⌬ ͑͒ − ⌬ ͑͒ +2⌬ ͑ ͔͒ Im 1 2 0 0 − − + + 0 Ϸ ͩ ͪ , ⌫ ͉͑͒ ͉2 ⌬͑+͒ ͑80͒ − 1 which is consistent with the optical theorem ͓Im͓͔ in agreement with the result of the amplitude approach ͑64͒ 2 =⌫ ͉͉͑͒2͔ only near resonance, when 2 /⌬͑+͒ = 0 Ϸ1. if one sets ͓⌬ ͑͒−⌬ ͑ ͔͒/⌬Ϸ0, consistent with condi- − 0 0+ + + 0 ͓ ͑ ͔͒ ͑ ͒ tions ͑36b͒. In contrast, our expression Eq. 80 neglecting level shifts Thus, it appears that the validity conditions for the 2 /⌬͑+͒ Heisenberg approach are that ͑36a͒ holds and that the ratios  = 0 ⌬ ͑ ⌬ ͒⌫ ͑͒ in the numerator of Eq. ͑79͒ are much less than unity. On the − i 2 0/ + − other hand, the Schrödinger approach is valid provided both is consistent with the optical theorem for any atom-field de- conditions ͑36a͒ and ͑36b͒ are satisfied. It can be shown that tuning. Clearly, if one uses Eq. ͑33͕͒P=2ប2 Im͓͔͖, for ͑ ͒ Eq. 80 agrees exactly with a perturbative solution of Eqs. the energy absorbed from the field, the use of Eq. ͑81͒ leads ͑ ͒ 38 carried out to second order in the vacuum coupling. to considerable errors for large detunings. For example, if There are two features of Eq. ͑80͒ that merit some discus- ⌬ ⌬͑+͒ / 0 =1/2, / 0 =3/2, then sion. First, we see that the excited state Lamb shift, ប⌬ ͑ ͒ ⌫ ͑͒ − − 0 , that would enter the calculation is replaced by a  Ϸ − ⌬ ͑͒ Im 1 2 “Lamb shift” evaluated at rather than 0. In fact, − ⌬ must be viewed as a level shift resulting from the combined dynamics of the applied and vacuum field. This result im- while plies that Lamb shifts associated with excited states must be ⌫ ͑͒ 2 2 9 ⌫ ͑͒  Ϸ − ͩ 0 ͪ − calculated within the context of a given problem. For ex- Im ͑ ͒ = . ⌬2 ⌬ + 4 ⌬2 ample, in spontaneous emission from an atom prepared in ⌬ ͑ ͒ the excited state, − 0 is the relevant shift parameter asso- ciated with the excited state ͓16͔; however, in the case we V. CONCLUDING REMARKS consider here of an atom driven by an external field the rel- ⌬ ͑͒ evant parameter is − . The frequency denominator in Eq. The expression ͑26͓͒or ͑80͔͒ for the polarizability coin- ͑ ͒ ͓⌬ ͑͒ ⌬ ͑ ͔͒ ͓⌬ ͑ ͒ 80 contains the differences − − + 0 and − − cides, with one small difference, with that derived by Lou- ⌬ ͑ ͔͒ ͓ ͔ − + 0 , which can be interpreted as the differences be- don and Barnett 12 . The difference is that the factor in ͑ ͒ 2 ͑ ͒2 tween the excited state level shifts associated with the reso- large brackets in 26 is missing a term −2Ck / k + 0 that nant and antiresonant components of the driving field, minus appears in the Loudon-Barnett expression. In our approach, the ground state Lamb shift. The ground state Lamb shift is we have ignored terms of this order by consistently invoking the only shift that is independent of the atom-field dynamics. ͑36a͓͒17͔. The second point to note is that the use of an “intuitive”- As noted in the Introduction, a TLA that remains with type expression for the polarizability resulting from the reso- high probability in its ground state is often approximated by
053816-10 POLARIZABILITY AND THE OPTICAL THEOREM FOR A… PHYSICAL REVIEW A 74, 053816 ͑2006͒
͗ ͑ ͒͘ a harmonic, Lorentzian oscillator obtained by replacing z proximate 22 t by 0 in the interaction of the atom with ͑ ͒ everywhere by −1. The polarizability for this model is then the applied field, but we cannot replace the operator 22 t ͑ ͒ obtained using Eq. 14 , replacing z by −1, and following by 0 when it appears in operator-product expectation values, ͗ ͑ ͒ ͑ ͒͘ exactly the same approach as in Sec. II except that no “fac- such as ak t 22 t . To first order in the external field am- torization” approximations, such as ͑15͒ or ͑18͒, are needed plitude, corrections to ͗ ͑t͒͘=0 are of order ͑36a͒͑the ex- ͗͑ ͒͘ 22 because the model is now linear in t . One easily obtains cited state population in the absence of any applied fields, ͒ 2d2 1 resulting solely from the vacuum field , whereas contribu- ␣͑͒ = 0 , ͑82͒ tions arising from f =͗a ͑t͒ ͑t͒͘ ͓Eq. ͑75͔͒ are of order ប 2 2 ͓␥͑−͒͑͒ ␦͑+͔͒͑͒ k k z 0 − −2i 0 − i ␥͑͒/⌬͑+͒͑͒ and must be included. As noted previously, ␥͑−͒͑͒ ⌫ ͑͒ ⌫ ͑͒ ␦͑+͒͑͒ ⌬ ͑͒ ⌬ ͑͒ ͗ ͘ * with = − − + and = − + + . Al- ak 22 = 2k,2=b2kb2; this expectation value arises from in- though this expression satisfies the optical theorem, it has the terference between processes associated with the excitation unphysical feature that the radiative frequency shift ␦͑+͒͑͒ of the atom by the external field and the antiresonant com- involves the sum of the radiative level shifts. The radiative ponent of the vacuum field. frequency shift ␦͑͒ in the polarizability ͑26͒, on the other hand, involves the expected difference in the two radiative ACKNOWLEDGMENTS level shifts. This point explains one of the reasons for includ- ing the radiative shifts in our calculations, even though, as We thank S. M. Barnett and R. Loudon for sharing their noted earlier, these shifts cannot be properly renormalized in insights on this problem and for providing us with a preprint a two-level model to obtain physically meaningful Lamb of their paper. We also thank G. W. Ford and J. E. Sipe for shifts. brief but helpful remarks relating to this work. This research The familiar Lorentzian oscillator approximation to a is supported in part by the National Science Foundation TLA therefore fails to provide a physically satisfactory ex- through Grants No. PHY0244841 and No. ECS-0355206, pression for the polarizability, even when the TLA has neg- and the FOCUS Grant, and by the US Army Research Office ligible excitation probability. The reason for this is clear through a MURI grant, and by the Michigan Center for The- from Eq. ͑14͒: for a TLA near its ground state, we can ap- oretical Physics.
͓1͔ H. A. Kramers and W. Heisenberg, Zeits. f. Physik. 31, 681 ͓9͔ P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 ͑1925͒. An English translation of this paper, and a discussion ͑2004͒. There is a typographical error in Eq. ͑25͒: V in that of its importance in the development of quantum theory, may equation should be replaced by −V. be found in B. L. van der Waerden, Sources of Quantum Me- ͓10͔ See, for instance, P. W. Milonni, The Quantum Vacuum: An ͑ ͒ chanics Dover, New York, 1968 . Introduction to Quantum Electrodynamics ͑Academic, San Di- ͓2͔ See, for instance, J. J. Sakurai, Advanced Quantum Mechanics ego, 1994͒. ͑Addison-Wesley, Reading, MA, 1976͒, Eq. ͑2.188͒. Sakurai ͓11͔ Equation ͑12͒ follows directly from Eqs. ͑25͒ and ͑26͒ of MB notes that the optical theorem is exact, but that his expression ͑ ͒ for the scattering amplitude is perturbative and satisfies the when the approximation noted after Eq. 26 is not made. ͓ ͔ ͑ ͒ optical theorem only in the vicinity of resonance. 12 R. Loudon and S. M. Barnett, J. Phys. B 39, S555 2006 . ͓3͔ See, for instance, D. L. Andrews, S. Naguleswaran, and G. E. ͓13͔ See, for example, P. R. Berman and B. Dubetsky, Phys. Rev. A Stedman, Phys. Rev. A 57, 4925 ͑1998͒; G. E. Stedman, S. 55, 4060 ͑1997͒, and references therein. Naguleswaran, D. L. Andrews, and L. C. Dávila Romero, ibid. ͓14͔ The level shifts do not include light shifts, proportional to the 63, 047801 ͑2001͒; D. L. Andrews, L. C. Dávila Romero, and intensity of the external field, which are neglected in this work. G. E. Stedman, ibid. 67, 055801 ͑2003͒. ͓15͔ The upper- and lower-state energy levels are now taken to be ͓ ͔ ប 1 ប 4 See for instance, A. D. Buckingham and P. Fischer, Phys. Rev. 0 and 0, respectively, rather than ± 2 0 as implied by the A 61, 035801 ͑2000͒; Phys. Rev. A 63, 047802 ͑ c4 c4a c4b Hamiltonian ͑13͒. c4cR. W. Boyd, Nonlinear Optics, 2nd ed. ͑Academic Press, ͓16͔ G. S. Agarwal, Phys. Rev. A 7, 1195 ͑1973͒. New York, 158. ͓17͔ It might be worth noting, however, that we can obtain a result ͓5͔ The relevance of the damping terms to the question of the in exact agreement with Loudon and Barnett Ref. ͓12͔͓an 2 ͑ ͒2 existence of a linear electro-optic effect in isotropic chiral me- extra term equal to −2Ck / k + 0 in the bracketed factor of dia is discussed by R. W. Boyd, J. E. Sipe, and P. W. Milonni, ͑26͔͒, if we turn on the vacuum field adiabatically and keep J. Opt. A, Pure Appl. Opt. 6, S14 ͑2004͒. terms of this order. In effect, this extra term would correspond ͓6͔ G. S. Agarwal and R. W. Boyd, Phys. Rev. A 67, 043821 to the fact that there is a population difference between the ͑2003͒. “bare” ground and excited states when these states are dressed ͓7͔ D. A. Long, The Raman Effect ͑Wiley, New York, 2001͒. by the vacuum field. By neglecting terms of this order, we ͓8͔ See, for instance, J. D. Jackson, Classical Electrodynamics avoid corrections related to nonvanishing excited state popula- ͑Wiley, New York, 1975͒, p. 423. tions in the absence of any external field.
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