Laser Physics, Vol. 15, No. 10, 2005, pp. 1432–1438. Original Text Copyright © 2005 by Astro, Ltd. Copyright © 2005 by MAIK “Nauka /Interperiodica” (Russia). Recoil and Photon Momentum in a Dielectric P. W. Milonni1, * and R. W. Boyd2 1 Theoretical Division (T-DOT), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2 Institute of Optics, University of Rochester, Rochester, New York 14627, USA *e-mail: [email protected] Received February 15, 2005 Abstract—We consider the recoil momentum of a spontaneously emitting atom in a dispersive dielectric, first from a microscopic approach, in which the dielectric is treated as a collection of atoms and the field is quantized in vacuum, and, then, from the macroscopic approach of quantizing the field in the dielectric. The atom’s recoil momentum differs from both the Abraham and Minkowski forms of the field momentum, and this difference is explained in terms of a dispersive contribution to the momentum acquired by the medium. 1. INTRODUCTION As discussed by Garrison and Chiao, the canonical photon momentum finds physical justification in treat- The problem of defining the electromagnetic ments of the Cerenkov and Doppler effects and in the momentum density has been considered by many phase-matching (momentum conservation) conditions authors [1]. Discussions typically center on compari- of nonlinear optics. Consider, for example, the Doppler sons of the Abraham and Minkowski forms, defined shift in the spontaneous radiation from an atom of mass respectively by ω m and transition frequency 0 with initial velocity v and × 2 final velocity v' after emission of a photon of frequency PA = EH/c (1) ω. Conservation of energy and linear momentum and require, in the nonrelativistic approximation [4], × 1 2 1 2 PM = DB. (2) ---mv ' + បω = ---mv + បω , (6) 2 2 0 Garrison and Chiao [2] have recently discussed the implications of these forms for the photon momentum mv' = mv – បk, (7) in a dispersive dielectric medium and, by quantizing the field in the dielectric, have identified three photon and, if we take k = [n(ω)ω/c]s, where s points in the momenta with magnitudes direction of the emitted photon, we obtain the correct ()ω បω (up to terms of order 1/c) Doppler shift formula pc = n /c, (3) n()ω p v ()ω បω c2 ωω= 1 +,------------v cosθ (8) A = g / , (4) 0 c and where θ is the angle between v and s and –បk is the ()ω 2()ω បω 2 recoil momentum of the atom. pM = v g n /c (5) It is useful for our purposes to summarize a few per- ω n ω v ω at frequency , where ( ) and g( ) are the refractive tinent remarks by Ginzburg [5]. From Maxwell’s equa- index and the group velocity, respectively. The first is tions for a nonmagnetic dielectric, one obtains the well- the canonical photon momentum, i.e., that associated known relation [5, 6] with the generator of space translations, while the sec- ond and third are the Abraham and Minkowski forms, B × ()∇ × H + D × ()∇ + E – E()∇ ⋅ D respectively, that follow from the corresponding classi- ∂ (9) cal expressions when the field in the dielectric is quan- + -----()ρDB× = –[]EJB+ × tized (see, e.g., Eq. (42) below). Garrison and Chiao ∂t discuss the fact that the results of the Jones–Leslie experiment [3], in which the radiation pressure on a in the usual notation. The right-hand side is the rate of mirror in a dielectric is measured, are consistent with change of the “mechanical momentum.” The last term the assignment of the canonical momentum to the pho- on the left is the time derivative of the Minkowski field tons. They find that the reported data deviate from the momentum density; however, it is the Abraham form Abraham and Minkowksi forms by 405 and 22 standard E × H/c2 that is generally regarded as the correct field deviations, respectively. momentum density [6]. For a nondispersive medium 1432 RECOIL AND PHOTON MOMENTUM IN A DIELECTRIC 1433 with dielectric constant ⑀, we can identify the Abraham tizing the macroscopic field in the dielectric. In Section 4 force density fA by writing we consider the momentum transferred to the bulk dielectric and discuss the relation of our approach to ∂ 1 ∂ A -----DB× = ---- -----EHf× + , (10) that of Loudon et al. Our conclusions are summarized ∂ 2 ∂ t c t in Section 5. i.e., fA = (n2 – 1)(∂/∂t)E × H/c2. fA can be interpreted as part of a force density [ρE + J × B + fA] acting on the 2. RECOIL OF AN EMITTER medium. Ginzburg remarks that “It is in reality impos- IN A DISPERSIVE DIELECTRIC sible to question this [Abraham] force notwithstanding We consider a model used recently in a microscopic that it has as yet not been reliably measured directly. In theory of spontaneous emission in a dispersive dielec- that way the problem [of choosing between the tric [8]. Here, however, the source atom is taken to be a Minkowski and Abraham forms] would be solved “in two-level atom (TLA) [9] with mass m, transition fre- favor” of the Abraham tensor.” For a nondispersive quency ω , and transition dipole moment d, and the medium, Abraham and Minkowski photon momenta (4) 0 បω បω dielectric consists of N identical TLAs per unit volume, and (5) become pA = /nc and pM = n /c, while the each having a transition frequency ω' and, for simplic- momentum pA associated with the Abraham force is ity here, the same transition dipole moment d as the (n2 – 1)បω/nc [5]. Thus, source atom. The Hamiltonian in the dipole approxima- A tion is pM ==pA + p pc. (11) From this point of view, the simple derivation of the ˆ 2 1 1 Hˆ = P /2m ++---បω σˆ z ---បω'∑σˆ zl (correct) Doppler shift formula (8) requires a reinter- 2 0 2 ប l pretation: k is not the momentum ( pM = pc) of the (12) emitted photon but rather the total momentum trans- បω † σ ⋅⋅ˆ ()ˆ σ ˆ ()ˆ ferred from the atom to the emitted photon and to the + ∑ kaˆ kλaˆ kλ – d ˆ x E R – ∑d ˆ xl E Rl , medium. In the case of a dispersive medium, however, kλ l p ≠ p , as noted by Garrison and Chiao. One purpose M c πបω 1/2 of this paper is to explain this difference between p 2 k ikr⋅ † –ik ⋅ r M Eˆ ()r = i∑ ---------------- []aˆ λe – aˆ λe e λ, (13) and the momentum p delivered by a photon to an V k k k c λ object in the medium, or, in the example we consider, k the momentum associated with recoil in spontaneous where operators are indicated by carets (∧). We are emission. In particular, we attempt to explain why the using the standard notation involving the Pauli two- Minkowski momentum only gives the correct recoil of state operators (σˆ x , σˆ y , σˆ z for the source atom and an atom—or, more generally, the force on an object in σˆ , σˆ , σˆ for the lth dielectric atom) and the photon the dielectric—in the case in which dispersion is xl yl zl † ignored. annihilation (aˆ kλ ) and creation (aˆ kλ ) operators. V is the λ The large literature concerned with the definition of quantization volume, and ekλ ( = 1, 2) is a polarization electromagnetic field momentum has dealt mainly with unit vector for the mode with wave vector k and polar- λ | | classical fields and nondispersive media. The more ization . ekλ and d are taken to be real, and k = k = recent literature includes detailed classical and quan- ω k/c; i.e., the field is quantized in free space. Effects of tized-field analyses for dispersive and absorbing dielec- the dielectric on the rate of emission and the recoil of trics by Loudon et al. [7], which have clarified consid- the source atom are due explicitly to the atoms consti- erably the difference between field momentum and the tuting the dielectric, not to “dressed” photons defined momentum actually transferred from an incoming field to an object in the medium. In particular, Loudon advo- by quantizing the field in the dielectric. Pˆ is the linear cates a direct calculation of the Lorentz force to deter- momentum operator for the center-of-mass motion of mine the forces in and on dielectrics. the source atom. We ignore, to begin with, any recoil of Given the fundamental importance of the subject, it the dielectric atoms, which in effect are taken to be infi- is an interesting academic exercise to obtain the recoil nitely massive. momentum of a spontaneously emitting atom in a From the Hamiltonian (12), we obtain the Heisen- dielectric without explicit quantization of the electro- berg equation of motion for the linear momentum Pˆ of magnetic field in the dielectric, i.e., by quantizing the the source atom: field in free space and deducing the recoil momentum as a consequence of the fact that, with each atom of the dPˆ ------- = ∇σ[]ˆ dE⋅ ˆ ()R . (14) dielectric, a field propagating at the vacuum speed of dt x light c is associated. Such an exercise is carried out in the following section, and, in Section 3, we outline the In the dipole approximation, the spatial variations of straightforward calculation of the same result by quan- the field are small over the dimensions of the atom, so LASER PHYSICS Vol. 15 No. 10 2005 1434 MILONNI, BOYD that R in Eq. (14) is regarded in effect as the expecta- tion value in (16), so that, in effect, we can ignore the ˆ † source-free (vacuum) field and write the equation tion value of the operator R .