Recoil and Photon Momentum in a Dielectric P

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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 ' + ω = ---mv + ω , (6) 2 2 0 Garrison and Chiao [2] have recently discussed the implications of these forms for the photon momentum mv' = mv Ð k, (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 ω (up to terms of order 1/c) Doppler shift formula pc = n /c, (3) n()ω p v ()ω ω c2 ωω= 1 +,------v cosθ (8) A = g / , (4) 0 c and where θ is the angle between v and s and Ðk is the ()ω 2()ω ω 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 second and third are the Abraham and Minkowski forms, B × ()∇ × H + D × ()∇ + E Ð E()∇ ⋅ D respectively, that follow from the corresponding classi- ∂ (9) cal expressions when the field in the dielectric is quan- + -----()ρDB× = Ð[]EJB+ × tized (see, e.g., Eq. (42) below). Garrison and Chiao ∂t discuss the fact that the results of the JonesÐLeslie 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

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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 Ð 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 ω ω 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 Ð 1)ω/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 ++---ω σˆ z ---ω'∑σˆ zl (correct) Doppler shift formula (8) requires a reinter- 2 0 2 l pretation: k is not the momentum ( pM = pc) of the (12) emitted photon but rather the total momentum trans- ω † σ ⋅⋅ˆ ()ˆ σ ˆ ()ˆ ferred from the atom to the emitted photon and to the + ∑ kaˆ kλaˆ kλ Ð d ˆ x E R Ð ∑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 πω 1/2 of this paper is to explain this difference between p 2 k ikr⋅ † Ðik ⋅ r M Eˆ ()r = i∑ ------[]aˆ λe Ð 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 expectation 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 . We write σˆ x = σˆ + σˆ , † ()+ ()+ where σˆ and σˆ are, respectively, the lowering and Eˆ d ()r, ω = Eˆ s ()r, ω raising operators for the two-level source atom and ω i rrÐ/l c (19) ()+ ()Ð ()+ ()+ e Eˆ (R) = Eˆ (R) + Eˆ (R), where Eˆ (R) and + αω()∑ ∇× ∇ × Eˆ d ()r , ω ------l rrÐ ()Ð l Eˆ (R) are, respectively, the photon annihilation and l ω ˆ ()+ ˆ ()Ð for each frequency component of the field, where creation parts of the field. E (R) and E (R) are α(ω) is the polarizability at frequency ω. In the contin- given by uum approximation, we replace this equation by

()± ()± ()± ()+ ()+ Eˆ ()R = Eˆ 0 ()R + Eˆ d ()R , (15) Eˆ d ()r, ω = Eˆ s ()r, ω Σ ik rr ˆ 0 Ð ' (20) where E0 (R) is the solution of the homogeneous αω() 3 ∇∇× × ˆ ()+ (), ω e + N ∫d r' Ed r' ------, (source-free) wave equation for the field and Eˆ d (R) is rrÐ ' a the part of the field due to all the dipole sources. Writ- ω ing (14) in normal order for the field operators and tak- where k0 = /c. The integration is over the volume ing the expectation value in an initial state in which bounded by the surface Σ of the dielectric, and the sub- there are no photons in the field, we have script a on the integral sign indicates that a small sphere of radius a about r is excluded from the integration; () d † + a 0 after the integration. ----- 〈〉Pˆ = 2∇ 〈〉dσˆ ⋅ Eˆ d ()R . (16) dt Equation (20) is just a statement of the superposition Using the formal solution of the Heisenberg equa- principle: the total field at any point in the medium is the field from the source atom plus the field from the tion for aˆ kλ in Eq. (13) for the electric field operator, atoms making up the dielectric in which the source we obtain, of course, formally the same expression for atom is placed. It was derived by quantizing the field in the field as in classical electrodynamics for a collection vacuum; the field of each atom propagates with the vac- of dipoles [10]: uum speed of light c. This equation has exactly the same form as the classical integral equation that is the ˆ (), ˆ (), Ed r t = Es r t starting point in the proof of the EwaldÐOseen extinc- tion theorem given in Born and Wolf [11], except that, dσˆ ()t Ð ( rrÐ/c (17) + ∑∇ × ∇ × ------xl l , in the latter, the first term on the right-hand side is an rrÐ l externally applied field incident on the medium, l whereas in Eq. (20) it is the field from an atom inside or, in the rotating-wave approximation (RWA) [9] of the medium. If we assume a solution of Eq. (20) of the ˆ ()+ form replacing σˆ xl by σˆ l in Ed (r, t), ()+ ˆ iKr⋅ ()+ ()+ Eˆ d ()r, ω = Ae , (21) Eˆ d ()r, t ≅ Eˆ s ()r, t with K á Aˆ = 0, then, as shown in Born and Wolf, the dσˆ ()t Ð rrÐ/c (18) + ∑∇ × ∇ × ------l l , integral on the right-hand side of (20) is equal to rrÐ l π α ω 2 2 2 2 l [4 N ( )][K + 2k0 ]/[K Ð k0 ] plus a surface integral ˆ that must, for consistency, cancel the first term on the where Es (r, t) is the field from the source atom and the left-hand side. Therefore, second term on the right-hand side of (17) is the sum of 2 2 the fields of the dielectric atoms. ()+ 4πNαω()K + 2k0 ()+ Eˆ d ()r, ω = ------Eˆ d ()r, ω (22) The source atom is singled out in (17) and (18) 3 K 2 Ð k 2 because it is initially excited. The other atoms constitut- 0 ing the dielectric are initially unexcited, each of them or has a dipole moment induced by the field acting on it, ω2 and each frequency component of this dipole moment 2()ω 2()ω 2 2()ω K ==n k0 n ------, (23) is linearly proportional to the field at the same fre- c2 quency. In the RWA, at each field frequency σˆ l (t) is π α ω 2 ω 2 ω ω proportional to the photon annihilation part of the field where 4 N ( )/3 = [n ( ) Ð 1]/[n ( ) + 1], i.e., n( ) is at r . But because of normal ordering, only the source the refractive index. In other words, at each frequency l ω, the superposition of the fields from all the atoms of ˆ ()+ part of this field, Ed (rl, t), contributes to the expecta- the medium, each of which propagates at the velocity c,

LASER PHYSICS Vol. 15 No. 10 2005 RECOIL AND PHOTON MOMENTUM IN A DIELECTRIC 1435 results in the phase velocity c/n(ω) for the total field. that, with high probability, every dielectric atom This is all that will be required to infer that a photon in remains forever unexcited. Thus, the medium has a linear momentum of magnitude ω ω ú i ()+ n( ) /c. In particular, it is not necessary for this pur- σˆ ()R', t ≅ Ð i()σω' Ð iβ' ˆ ()R', t + ---dE⋅ ˆ s ()R', t . (27) pose to quantize the field in the dielectric rather than in vacuum. In the RWA, and with σˆú (t) ≅ Ðiω σˆ (t) and σúúˆ (t) ≅ Let us return now to the calculation of the recoil 0 −ω 2σ momentum of the source atom. The formal solution of 0 ˆ (t), we have aˆ λ t the Heisenberg equation of motion for k ( ) gives 2 1 ()+ d 3 ⋅ ˆ (), ≅ ()σˆ () ---dEs R' t -----k0 F R' t , (28) ˆ ()+ (), i 3 ω[]()⋅ 2 Ed R t = ------∫d k0 ddkÐ 0 k0/k0 4π2 where we now take R = 0 for the coordinate of the t (24) source atom and define ⋅ ⋅ ω () Ðik0 R Ðik0 Rl i k t' Ð t × dt' σˆ ()t' e + ∑σˆ ()t' e e , 2 ∫ l  1 ()dR⋅ F()R ------0 l = 1 Ð ------2 2 - k0 R d R which, in the dielectric continuum limit, is just another (29) 2  way of writing (20). The considerations leading from i 1 ()dR⋅ ik0 R + ------Ð13------Ð ------ e . (20) to (21) and (23) do not give us an explicit solution 2 2 3 3 2 2  ()+ k0 R k0 R d R for Eˆ d (R, t). But they determine how each frequency ()+ Then, component of Eˆ d (R, t) varies with R and, for the pur- 2 pose of calculating 〈〉dPˆ /dt as described below, allow d 3 σˆú ()R', t ≅ Ð i()σω' Ð iβ' ˆ ()R', t + i-----k F()σR' ˆ ()t (30) us to write (16) as 0

2 and d d 3 2 2 2 ----- 〈〉Pˆ = Ð------ d k ωn()ω k []1 Ð ()dk⋅ /d k dt 2 ∫ 0 0 0 0 2 3 () 2π σ(), ≅ d k0 F R' σ() ˆ R' t ----------ω ω ˆ t , (31) t ' Ð 0 (25) × dt'[ 〈〉σˆ †()σt ˆ ()t' |ω ω | ∫ assuming that ' Ð 0 is large enough for this adiabatic 0 approximation and for β' to be neglected. In these Ðik ⋅ ()R' Ð R ω() approximations, (25) is replaced by + Nd∫ 3 R' 〈〉σˆ †()σt ˆ ()R', t' e 0 ]ei t' Ð t . 2 2 d d 3 ()dk⋅ ----- 〈〉Pˆ = Ð------ d k ωn()ω k 1 Ð ------0 - The Heisenberg equation of motion for σˆ (R', t) is dt π2 ∫ 0 0 2 2 2 d k0

i ()+ 2 ⋅ σú (), ω σ(), σ (), ⋅ ˆ (), Nd / 3 3 Ðik0 R ˆ R' t = Ði ' ˆ R' t Ð --- ˆ z R' t d E R' t . × () (32) 1 + ω------ω-k0 ∫d RF R e (26) ' Ð 0 t The second term on the right has contributions from the † iω()t' Ð t field of the dielectric at R' plus the fields of all the other × ∫dt' 〈〉σˆ ()σt ˆ ()t' e . atoms. The contribution from the atom at R', in the 0 Markovian or WeisskopfÐWigner approximation, is responsible for radiative damping of the dielectric atom The integral over time is evaluated in the standard Markovian or WeisskopfÐWigner approximation for at R'; i.e., it contributes Ðβ'(σˆ R', t), where β' = ω Ð1 2ω 3 π 3 t 0 : d ' /6 0 c . We ignore radiative shifts here, which for our purposes can be assumed to be included in the t ω() definitions of the transition frequencies. The effect of dt' 〈〉σˆ †()σt ˆ ()t' ei t' Ð t all the other atoms is complicated, and we approximate ∫ it by including only the effect of the source atom. For a 0 (33) dilute dielectric, the dielectric atoms make a relatively t i()ωω()t t ≅σ〈〉†()σ() Ð 0 ' Ð π ()δω() ω small contribution because they are unexcited; for this ˆ t ˆ t ∫dt'e p+ t Ð 0 , σ reason we also approximate ˆ z (R', t) by Ð1, assuming 0

LASER PHYSICS Vol. 15 No. 10 2005 1436 MILONNI, BOYD where we drop a Cauchy principal part term that gives consisting of N TLAs per unit volume, each having ω a “Lamb shift,” which we are ignoring, and p+(t) is the transition dipole moment d and transition frequency '. probability at time t that the source TLA is in the upper We have assumed that all the transition moments state. For our initially excited source atom, p+(t) = point in the same direction; whereas, to model an iso- exp(ÐA't), where A' is the spontaneous emission rate in tropic dielectric, we should allow the dielectric atoms the dielectric. Then, for t 1/A', to have transition moments pointing in all three direc- 2 2 tions. Doing this, and averaging over the relative orien- d 3 ()dk⋅ 〈〉Pˆ = Ð------ d kωn()ω k 1 Ð ------tations of the source- and dielectric-atom dipole π ∫ 2 2 2 A' d k moments, or, alternatively, carrying out the calculation 2 as above but allowing the dielectric atoms to make ⋅ ∆ ± × Nd / 3 3 ()Ðik R δω() ω m = 0, 1 transitions, for instance, we obtain [8] 1 + ω------ω-k0 ∫d RF R e Ð 0 ' Ð 0 3 3 2 7 (34) A[]1 + δnk d RF ()R = 1 + ---δn A, (39) d2ω 3 ()⋅ 2 0 ∫ 3 0 Ω []()ω dk0 = Ð------∫d k n 0 k0 1 Ð ------π 3 0 2 2 2 A' c d k0 which, as shown in [8], is the spontaneous emission rate 2 A of the source atom in the (dilute) dielectric; i.e., it is ik ⋅ R × Nd / 3 3 ()Ð 0 the small-δn approximation to nA[(n2 + 2)/3]2, the 1 +,ω------ω-k0 ∫d RF R e ' Ð 0 spontaneous emission rate, including the LorentzÐ Lorenz local field correction, in a dielectric. Thus, Ω []… where ∫d k is the integral over all solid angles 0 2ω 2 about k and now k = ω /c. Obviously the integral over 〈〉ˆ 2 2()ω 0 0 0 0 P = n 0 ------. (40) solid angles vanishes, as it must because the direction c2 of photon emission is equally likely in directions k0 and k The recoil momentum of the source atom has the Ð 0. However, it is straightforward to show, using the ω ω same approximations as above, that expected magnitude pc = n( 0) 0/c. Note that, although our derivation includes a local field correction d2ω 3 ()⋅ 2 at the source atom, this correction has no effect on the 〈〉ˆ 2 0 2()ω 2 2 Ω dk0 P = ------n 0 k0 ∫d k 1 Ð ------source atom’s recoil momentum. π 3 0 2 2 2 A' c d k0 (35) 2 ik ⋅ R 3. RECOIL CALCULATION BY FIELD × Nd / 3 3 ()Ð 0 1 +.ω------ω- k0 ∫d RF R e QUANTIZATION IN THE DIELECTRIC ' Ð 0 We now calculate the recoil momentum using the Now, single-atom Hamiltonian ()dk⋅ 2 π Ω 0 8 ˆ ˆ 2 1ω σ ∫d k 1 Ð ------= ------(36) H = P /2m + --- 0 ˆ z 0 2 2 3 2 d k0 (41) and ω † σ ⋅ ˆ ()ˆ + ∑ kaˆ kλaˆ kλ Ð d ˆ x E R 2 kλ ()dk⋅ Ðik ⋅ R dΩ 1 Ð ------0 - e 0 = 4πF ()R , (37) ∫ k0 2 2 i with the electric field quantized not in free space but in d k0 the dielectric [12]: where F (R) is the imaginary part of F(R). Therefore, i ω 1/2 ˆ (), () ikr⋅ 〈〉ˆ 2 E r t = i∑------ - aˆ kλ t ekλe + h.c., (42) P 2nng 0V kλ 2ω 3 2 4d 0 2 2 2 2πNd / 3 3 2 ω ω ω ω ω = ------n ()ω k 1 + ------k d RF ()R where k = n( ) /c, ng = n( ) + (dn/d ), and, of 3 0 0 ω ω 0 ∫ 3A'c ' Ð 0 course, the annihilation and creation operators now 2 (38) refer to the “dressed” photons of the dielectric. We A ω 3 3 2 = ----- n()ω ------0 []1 + δnk d RF ()R , assume that the field frequencies that will contribute to A' 0 c 0 ∫ recoil are far removed from any absorption resonances, so that the field is quantized in a lossless (but disper- where A is the spontaneous emission rate of the source sive) dielectric. Since the role of normal ordering and δ π 2 ω ω atom in free space and n = 2 Nd /[ ( ' Ð 0)] is the other aspects of the calculation here are straightforward ω refractive index at frequency 0 for a dilute dielectric and much the same as in the preceding section, we pro-

LASER PHYSICS Vol. 15 No. 10 2005 RECOIL AND PHOTON MOMENTUM IN A DIELECTRIC 1437 ceed immediately to the RWA expression for the rate of where χ(ω) is the susceptibility, which is taken to be ˆ 2 independent of r (no spatial dispersion). Assuming that change of the expectation value of P : ω j(r, t) is slowly varying compared with exp(Ði 0t), we ω have d 〈〉ˆ 2 2()⋅ 2 ----- P = 2 ∑------ - k dekλ ∞ dt 2nng 0V kλ (), ∆χ() ω P j r t = 0 d 0 t (43) ∫ ∞ × 〈〉σ†()σ() iω()t' Ð t Ð ∫dt' ˆ t ˆ t' e . ()ω ∆ Ði 0 + t (48) ∆χ ()…ω ˜ (), ∆ 0 + ' 0 + r e

Going to the mode continuum limit and using dk = ∂ ω j Ði 0t (n /c)dω and (33), we have ≅ χE + i χ'------e , g 0 j 0 ∂t 2 d 2 n()ω ω χ χ χ ω ω ω ----- 〈〉Pˆ = n()ω A ------0 0 p ()t where and ' = d /d are evaluated at = 0. Then, dt 0 c + ∂ (44) E j ()ω ω 2 j------n 0 0 ÐA't ∂ = n()ω A ------e , xi 0 c (49) * * 1 ∂ 2 i ∂ j ∂ j ∂ j ∂ j = --- χ------E + --- χ' ------Ð ------2ω 3 π 3 0 ∂ 0 ∂ ∂ ∂ ∂ where again A = d 0 /3 0c is the free-space sponta- 2 xi 4 t xi t xi neous emission rate. Without any local field correction, ω upon cycle averaging. A' = n( 0)A, and (40) follows. If there is a local field correction such that A' = nA, it should be included In the case of a plane wave, iω t ik ⋅ r in the field, and, therefore, (44) should be multiplied by (), () Ð 0 0 ω E j r t = j0 t e e , (50) . In other words, the recoil momentum is n 0/c regardless of whether there is a local field correction, (49) becomes just as in the calculation in the preceding section. ∂ ∂ ∂* j 1 χ j0* j0 j------∂ - = --- 0 'k0i------∂ j0 + ------∂ j0 , (51) 4. MOMENTUM TRANSFER xi 4 t t TO THE DIELECTRIC which implies, from (45) and χ = n2 Ð 1,

Let us consider now the force on the dielectric 3 atoms. Since they are assumed to be unexcited, this dP 1 ∂ 2 ------= --- χ'k ----- ∑ V force must arise from the field of the source atom or, dt 4 0 0∂t j more generally, any applied field. The ith component of j = 1 (52) the force on the bulk dielectric is, in the continuum ω dnn 0 k0 ∂ 2 approximation and employing the summation conven- = n------E V. 0 dωc k ∂t tion for repeated indices, 0 d ∂E Therefore, the momentum density of the dielectric has i j 3 the magnitude Fi ==------∫P j------∂ d r, (45) dt xi 2ω n 0 dn 2 where i is the momentum imparted to the dielectric g = ------E . (53) 0c dω and Pj is the jth component of the macroscopic polarization in the dielectric. For simplicity we will formu- To phrase the discussion in terms of photons, we use late the discussion here classically to begin with. Eq. (42) to write, for a single plane-wave mode of the Since the field from the source atom is time-depen- field, dent, we write ω 2 0 † ∞ E = ------ aˆ aˆ (54) iω t iω t ∆ nng 0V (), (), Ð 0 Ð 0 ∆˜ (), ∆ Ði t E j r t = j r t e = e ∫ d j r e (46) after dropping the term associated with zero-point Ð∞ energy and momentum. Then, (53) implies that each and photon in the bulk dielectric imparts to the dielectric a ∞ momentum of magnitude i()ω ∆ t (), ∆χ() ω ∆ ˜ (), ∆ Ð 0 + n dn ω P j r t = 0 ∫ d 0 + r e , (47) ω p = ------ω------. (55) Ð∞ ng d c

LASER PHYSICS Vol. 15 No. 10 2005 1438 MILONNI, BOYD

Now, consider the conservation of momentum when the Note added in proof: Campbell et al. [Phys. Rev. source atom emits a photon and recoils. If we take the Lett. 94, 170403 (2005)] have recently reported the direction of the emitted photon to be positive, the results of an experiment conﬁrming that an atom in a source atom’s recoil momentum is Ðhnω /c, as shown dispersive medium acquires a recoil momentum of 0 ω ω earlier. The momentum of the emitted ﬁeld is given by magnitude ( 0/c)n( 0) when it absorbs a photon of ω ω the Abraham momentum 0/ngc, while the medium frequency 0, in agreement with the conclusions of this 2 ω picks up a momentum (n Ð 1) 0/ngc corresponding to paper. the Abraham force (n2 Ð 1)(∂/∂t)(E × B)/c2 [5]. Because of dispersion, the medium also picks up the momentum ACKNOWLEDGMENTS p. Thus, the conservation of linear momentum can be expressed in the form We thank Professor Eberly for innumerable discussions about topics relating to this work, which we 2 ω 1 n Ð 1 n ω dn 0 present here partly in the hope of stimulating more dis- Ðn ++------+------ω------= 0, (56) ng ng ng d c cussions in the future. We also thank J.C. Garrison and R.Y. Chiao for correspondence concerning the Abra- which can be written equivalently as ham and Minkowski momenta and P.R. Berman for insights into radiative processes in dielectrics. One of Ð p ++p p = 0. (57) c M us (RWB) gratefully acknowledges support by the Note that the momentum densities gM or g can be neg- National Science Foundation and by the DARPA/DSO ative but that their sum is always positive for n > 0. slow-light program.

5. DISCUSSION REFERENCES Equation (56) explains why neither the Abraham 1. I. Brevik, Phys. Rep. 52, 133 (1979); V. L. Ginzburg, nor the Minkowski momenta for a photon give the Theoretical Physics and Astrophysics (Pergamon, momentum imparted to an object by a ﬁeld in a disper- Oxford, 1979), Chap. XII. sive dielectric medium [2]. Both of these momenta are 2. J. C. Garrison and R. Y. Chiao, Phys. Rev. A 70, 053 826 part of a momentum conservation condition that must (2004). also include the dispersive component p of the 3. R. V. Jones and B. Leslie, Proc. R. Soc. London A 360, momentum picked up by the medium; the latter, plus 347 (1978). the Minkowski momentum, is equal in magnitude to the 4. E. Fermi, Rev. Mod. Phys. 4, 87 (1932). canonical momentum p . The sum p + p (=p ) may be c M c 5. V. L. Ginzburg, Theoretical Physics and Astrophysics regarded in our example as the momentum imparted to (Pergamon, Oxford, 1979), pp. 281Ð288. the ﬁeld and the bulk dielectric. 6. J. D. Jackson, Classical Electrodynamics (Wiley, N. Y., In their treatment of the propagation characteristics 1975), p. 240. of energy and momentum in dispersive (and absorbing) 7. R. Loudon, J. Mod. Opt. 49, 821 (2002); R. Loudon, dielectrics, Loudon, Allen, and Nelson have also L. Allen, and D. F. Nelson, Phys. Rev. E 55, 1071 obtained, in a rather different way, a momentum of (1997); D. F. Nelson, Phys. Rev. A 44, 3985 (1991). form (53), as did Nelson earlier [7]. These treatments, 8. P. R. Berman and P. W. Milonni, Phys. Rev. Lett. 92, based on a Lagrangian formulation, arrive at such a 053601 (2004). form as a dispersive contribution to the pseudomomentum, a quantity that is conserved (in the absence of dis- 9. L. Allen and J. H. Eberly, Optical Resonance and Two- sipation) when the dielectric is homogeneous. The Level Atoms (Wiley, New York, 1975). pseudomomentum combines with the momentum to 10. M. Born and E. Wolf, Principles of Optics (Pergamon, give the “wave momentum,” which, in the terminology Oxford, 1969; Nauka, Moscow, 1973), p. 104. of this paper, is the canonical momentum of Garrison 11. M. Born and E. Wolf, Principles of Optics (Pergamon, and Chiao [2]. In essence this dispersive component of Oxford, 1969; Nauka, Moscow, 1973), p. 105, Eq. (4). the pseudomomentum is a dispersive contribution to 12. K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shep- the Minkowksi momentum. erd, Phys. Rev. A 42, 4102 (1990).

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