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Comparison of Quantum and Classical Local-?eld E?ects on Two-Level Atoms in a Dielectric

Michael E. Crenshaw

AMSRD-AMR-WS-ST, US Army RDECOM, Aviation and Missile RDEC, Redstone Arsenal, AL 35898, USA (Dated: June 11, 2008) The macroscopic quantum theory of the electromagnetic ?eld in a dielectric medium interacting with embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-?eld condition. In order to retain the Lorentz condition, the macroscopic quantum electrodynamic theory must be supplanted by its more fundamental microscopic counterpart. We apply microscopic quantum electrodynamics and derive dielectric e?ects that are consistent with those obtained from the optical Bloch equations using the local-?eld condition.

PACS numbers: 42.50.Ct,42.50.Nn,03.50.De

arXiv:0805.2134v2 [quant-ph] 11 Jun 2008

I.

INTRODUCTION

The e?ect of a dielectric host medium on the spontaneous emission rate of a two-level atom remains an interesting and challenging problem in quantum optics with an importance that be?ts a rigorous test of our understanding of the interaction of light with matter. The essential characteristic of the problem is its multiscale nature with the atom being a creature of microscopic quantum electrodynamics and the dielectric manifesting in the realm of classical continuum electrodynamics. In practice, the spontaneous emission rate is calculated using a macroscopic quantum electrodynamic theory [1, 2, 3, 4, 5] in which the continuous dielectric medium is incorporated into a medium-assisted electromagnetic ?eld. Due to the renormalization of the ?eld operator and the density-of-states, the spontaneous emission rate of the embedded atom Γ = nΓ0 is enhanced by a factor of the refractive index n compared to the vacuum quantity Γ0 [4, 5, 6]. In principle, it is possible to represent both the atom and the dielectric microscopically in a quantum electrodynamic theory [7, 8, 9]. The microscopic Huttner–Barnett [9] model is a common starting point [10, 11, 12, 13], but the use of the continuum approximation in deriving the e?ective Hop?eld Hamiltonian [14] results in a macroscopic theory. Speci?cally, discrete Fourier transforms are evaluated with integrals in the transformation from coordinate space to reciprocal (wavenumber) space [15]. The experiments that have been performed, to date, do not provide an adequate basis for validating the macroscopic treatment of the spontaneous emission rate. The problem of a two-level atom embedded in a dielectric is distinct from a two-level atom in a spherical hole in the dielectric. A phenomenological factor is typically incorporated into the macroscopic result to account for local?eld e?ects and the local-?eld factor has a di?erent form if the atom is embedded in the dielectric (virtual cavity) or is placed in a real cavity. Although the virtual cavity and real cavity factors di?er in their dependence on the power of the refractive index, the experimental results have been inconclusive. Recently, Duan and Reid [16]

have re-analyzed much of the experimental record and found that the measurements of the index dependence of the spontaneous emission rate are typically being performed on real-cavity systems. The real-cavity result can be viewed as a consequence of selecting materials in which the separation of the atom from the host makes it possible, at the expense of introducing additional boundary conditions [17], to measure the spontaneous emission rate for di?erent dielectric hosts. Experiments of this type have been performed using ions placed in an organic ligand cage [18, 19] and in nanocrystals surrounded by a dielectric ?uid [23, 24, 25]. In dielectrics containing twolevel atoms as non-encapsulated impurities, the atom and host are closely associated making these materials poor candidates for measuring the dielectric dependence of the spontaneous emission rate by varying the properties of the host medium [19]. The purpose of this article is a theoretical exposition of the e?ects of a dielectric host on the electrodynamic properties of two-level atoms using the Lorentz local-?eld condition and the macroscopic and microscopic quantum electrodynamic theories. While the dielectric renormalization of the spontaneous emission rate of an embedded atom generates considerable interest, there are other consequences of embedding atoms in a host medium. We derive the e?ects of the dielectric on the dynamics of embedded atoms in a simple direct manner using i) the classical Lorentz local-?eld condition and ii) macroscopic quantum electrodynamics and we ?nd that these two well-established physical theories produce contradictory results and conclude that the macroscopic quantum theory violates the correspondence principle. Microscopic quantum electrodynamics is more fundamental than the macroscopic quantum theory. Applying the microscopic procedure that was described by Crenshaw and Bowden [20], we obtain results for the dielectric enhancement of the electromagnetic ?eld (Rabi frequency) and the enhancement of the near-dipole–dipole interaction between impurity atoms (Lorentz redshift) that are in full agreement with results that are based on the Lorentz local?eld condition [21, 22]. Speci?cally, the microscopic theory reproduces the local-?eld renormalization of the Rabi frequency and the Lorentz redshift by the same factor of

2 (n2 +2)/3, while the macroscopic theory respectively predicts factors of 1 and n. This result contraindicates the application of the macroscopic theory of quantum electrodynamics to two-level atoms embedded in linear media by speci?c examples in which the macroscopic theory fails to reproduce fundamental results. ?W i? =? ?t E? + 4π ? 4π ? P R21 ? E + P R21 3 3 (2.4b)

? γ (W ? Weq ).

II. CLASSICAL AND SEMICLASSICAL DIELECTRIC LOCAL-FIELD EFFECTS

In the view of Lorentz, classical continuum electrodynamics is better expressed in terms of an atomistic model of discrete particles embedded in the vacuum that interact with the microscopic electromagnetic ?eld at the point of the particle [17, 26]. The local ?eld that acts on that particle EL = E + 4π P 3 (2.1)

Here, ?elds are represented in the plane-wave limit by 1 envelope functions such that P = 2 (P e?iωt + c.c.), 1 1 ?iωt ?iωt + c.c.), and EL = 2 (EL e + c.c.). The E = 2 (E e macroscopic spatially averaged atomic variables in a rotating frame of reference are R21 = ρ21 eiωt sp , R12 = ρ12 e?iωt sp , and W = R22 ? R11 = ρ22 sp ? ρ11 sp , where · · · sp corresponds to a spatial average over a volume of the order of a resonance wavelength cubed and the ρij are the density matrix elements for a two-level system with a lower state |1 and an upper state |2 . Also, ? is the matrix element of the transition dipole moment, assumed real, γ⊥ is a phenomenological dipole dephasing rate, γ is a phenomenological population relaxation rate, and Weq is the population di?erence at equilibrium. Substituting the polarization envelope P= ε?1 ε+2 E+ 2N ?R21 4π 3 (2.5)

is comprised of the macroscopic Maxwell ?eld E and the reaction ?eld of all other particles, expressed in terms of the macroscopic polarization P. For a linearly polarizable material, the polarization P = pN EL is the product of the microscopic polarizability p, the number density N , and the local ?eld. Using the Lorentz local-?eld condition (2.1) to eliminate the microscopic local ?eld produces the Clausius–Mossotti–Lorentz–Lorenz relation 4π ε?1 pN = 3 ε+2 (2.2)

into the generalized Bloch equations (2.4) produces ?R21 4π = i ω ? ω0 ? N ?2 ?W ?t 3 R21 ? i? ?E W ? γ⊥ R21 2 (2.6a)

i ?W ? = ? [??E ? R21 ? ??E R21 ] ? γ (W ? Weq ). (2.6b) ?t For now, we assume that ? is real. Then, the local-?eld effect of the dielectric is simply an enhancement of the ?eld E , or Rabi frequency ?E / , and the inversion-dependent Lorentz redshift by ? [22]. The decay rates remain phenomenological.

III. MACROSCOPIC QUANTUM ELECTRODYNAMICS

between the polarizability and the macroscopic dielectric constant ε = 1 + 4πP/E . The local-?eld principle also applies to nonlinear media. Bloembergen [21] investigated nonlinear optics in the presence of a linear host medium and found that a local-?eld dielectric enhancement factor of ?= n2 + 2 3 (2.3)

accompanies each appearance of a macroscopic ?eld in the nonlinear susceptibility. In our notation, ε = n2 for a dielectric and ? refers to the speci?c quantity in the preceding equation and not to any other constant or variable representing a local-?eld factor. Bowden and co-workers [27, 28, 29] predicted intrinsic optical bistability in a dense collection of vacuum-embedded two-level atoms due to an inversion-dependent local-?eld shift of the resonance frequency. Later work [22] reported the e?ect of embedding the dense two-level systems in a host dielectric. The dynamics of the two-level systems are described by the generalized optical Bloch equations [30]: i? ?R21 = i(ω ? ω0 )R21 ? ?t 2 4π P W ? γ⊥ R21 3 (2.4a)

E+

Quantum electrodynamics can be viewed as the quantized version of Lorentzian electrodynamics in which discrete quantum particles interact with the vacuum ?eld modes. When applied to dielectrics, the practice has been to create a macroscopic version of quantum electrodynamics along the lines of a quantized version of continuum electrodynamics. The macroscopic theory can be derived either by quantizing the classical Maxwell ?elds or by applying a continuum approximation to the microscopic quantum electrodynamic Hamiltonian. Ginzburg [1] pioneered the procedure of canonical quantization of the ?eld in a dielectric and applied it to Cherenkov radiation. The macroscopic quantization procedure was limited to dielectrics with negligible dispersion and absorption. Jauch and Watson [2, 31] continued Ginzburg’s work and extended macroscopic quantization

3 to dispersive dielectrics [5, 32], while other researchers have treated absorption [33, 34] and nonlinear dielectrics [35, 36]. Knoester and Mukamel [7] and Huttner, Baumberg, and Barnett [37] start with the fundamental microscopic Hamiltonian and transform from coordinate space to wavenumber space in the continuum approximation to derive the macroscopic Hop?eld Hamiltonian [15]. Recent work [38, 39, 40, 41, 42] includes macroscopic quantization of ?elds in magnetodielectric media, including left-handed negative-index materials. The spontaneous emission rate of an atom in a dielectric is readily derived by the macroscopic quantum electrodynamic theory. In 1976, Nienhuis and Alkemade [6] used a macroscopic version of Fermi’s golden rule to derive the dielectrically enhanced spontaneous emission rate with the macroscopic Ginzburg ?elds. Huttner, Barnett, and Loudon [13], among others, have combined the macroscopic Fano–Hop?eld [14, 43] theory with Fermi’s golden rule to derive the dielectric renormalization of the spontaneous emission rate of an impurity atom, while other studies begin with macroscopic Green’s functions [44] or auxiliary ?elds [45]. For the most part, the collective attention is focused on the dielectric renormalization of the spontaneous emission rate. Knoester and Mukamel [7] obtained the dielectric e?ect on both the Lorentz redshift and the spontaneous decay rate by deriving operator equations of motion from the Hop?eld model. In this section, we derive equations of motion using Weisskopf–Wigner theory applied to the macroscopic Hamiltonian in terms of the Ginzburg ?eld operators [46]. A term-by-term comparison with the generalized Bloch equations that were derived in the preceding section under the Lorentz local-?eld condition exposes an extraordinary degree of disagreement between two known and accepted treatments of the e?ect of a dielectric host on the electrodynamics of two-level atoms. The principal product of the macroscopic quantization theory is the medium-assisted ?eld operator ? = i E n 2πωl ? kl λ , a ?l eikl ·r ? H.c. e V (3.1) ? renormalization of the electric ?eld operator (3.1) by 1/n, which is squared, and the D = n3 density-of-states factor. Local-?eld e?ects of a dielectric are suppressed in the macroscopic quantization procedure and such local-?eld e?ects must be introduced phenomenologically. The paradigm that emerged from the propagation studies of Hopf and Scully [47] and Bloembergen’s work [21] in nonlinear optics is that the e?ect of a dielectric host is to multiply each occurrence of the dipole moment of a twolevel atom by a local-?eld enhancement factor [3, 7, 48]. Using the Lorentz virtual cavity model of the local ?eld, the spontaneous emission rate Γ = n?2

3 4 ωb |?b |2 =n 3 c3

n2 + 2 3

2

Γ0

(3.3)

for atoms in a dielectric scales as n5 for large n. For an atom in a real cavity, the local-?eld enhancement factor is based on the Onsager model and the modi?ed spontaneous emission rate Γ=n 3 n2 2 n2 + 1

2

Γ0

(3.4)

scales as n. A study of local-?eld e?ects by de Vries and Lagendijk [49] found that the Lorentz virtual-cavity model is appropriate if the atom goes into a crystal substitutionally but that the Onsager real-cavity model should be used for interstitial impurities. The macroscopic quantum electrodynamic theory can be used to derive additional consequences of the dielectric host for two-level atoms. Taking the ?eld in a coherent state, the e?ective Hamiltonian is Hef f =

js

ωa j σ + 2 3

ωl a ?? ?l la

lλ

?

i n

j j ?ikl ·rj j j a ?? gl σ+ a ?l eikl ·rj ? gl l σ? e js lλ

?

lλ

where a ?? ?l are the macroscopic creation and destrucl and a tion operators for the ?eld modes and ωl is the frequency of the ?eld in the mode l. Also, V is the quantization ?kl λ is a unit vector in the direction of the povolume, e larization, and λ denotes the state of polarization. The spontaneous emission rate of an impurity atom in a dielectric can then be obtained by applying Fermi’s golden rule Γ= 2π | f |Hint |i |2 D (3.2)

i?a 2

j ? ?i(ωp t?kp ·rj ) ?? σ j ei(ωp t?kp ·rj ) . σ+ Ee ?E ? j j σ3 j σ±

?, to the e?ective interaction Hamiltonian Hint = ??a · E where i labels the initial state and f denotes all available ?nal states. As it is typically calculated, the dielectric renormalization of the vacuum spontaneous emission rate of an atom is found to be n [4, 5, 6] due to the dielectric

(3.5) For species a, is the inversion operator and are j the raising and lowering operators for the j th atom, gl = ?kl λ ) is the coupling between the (2πωl / V )1/2 ?a (? xj · e ? j is a unit atom at position rj and the radiation ?eld, x vector in the direction of the dipole moment at rj , ωa is the transition frequency, and ?a is the matrix element of the transition dipole moment. Except for coe?cients of 1/n, and macroscopic ?eldmode operators, the e?ective Hamiltonian is the same as the microscopic Hamiltonian for identical two-level atoms in the vacuum. Equations of motion can then be derived in the same manner, which is the primary reason for adopting the macroscopic formalism. The formal integral

4 of the Heisenberg equation of motion for the ?eld-mode operators is used to eliminate these operators from the remaining Heisenberg equations of motion. One obtains

j dσ? ?a j ? j = ?iωa σ? (t) + σ (t)E (t)e?i(ωp t?kp ·rj ) dt 2 3

IV.

MICROSCOPIC QUANTUM ELECTRODYNAMICS

+ 1 n2

1 n

j j gl σ3 (t)? al (0)e?i(ωl t?kl ·rj ) lλ t 0 t 0

+

j j gl σ3 (t) lλ

dt′e?iωl (t?t )

i=j,s

′

i i gl σ? (t′ )eikl ·(rj ?ri )

?

+

1 n2

j j gl σ3 (t) lλ

j j dt′ e?iωl (t?t ) gl σ? (t′ )

′

?

(3.6)

and a similar equation of motion for the inversion operator. The procedure to evaluate these terms for twolevel atoms in the vacuum is generally known and will be considered in detail in the next section. For now it is su?cient to note that the only di?erences from the vacuum case are the coe?cients of powers of n and the n3 renormalization of the density-of-states in a dielectric in which the sum over modes is evaluated as → n3

l

L 2πc

3 0

∞

2 dωl ωl

d?

(3.7)

in the mode continuum limit. Transforming to a rotating frame of reference and performing a local spatial average, one obtains the Bloch-like equations of motion 4π ?R21 = i ωp ? ωa ? N ?2 a nW ?t 3 ?n Γ0 R21 2 R21 ? i?a ? EW 2 (3.8a)

i ?W ? ?R21 ?? R21 ? ?a E ? nΓ0 (W + 1), (3.8b) = ? ?a E ?t where W = σ3 sp and R21 = ?iσ? sp . Further, local?eld factors of ? can be added phenomenologically in accordance with accepted practice [3, 7, 48]. In this case, Eqs. (3.8), become 4π ?R21 2 = i ωp ? ωa ? N ?2 a n? W ?t 3 ? n?2 Γ0 R21 2 R21 ? i?a ? ?E W 2 (3.9a)

In the preceding two sections, we derived generalized Bloch equations of motion for two-level atoms in a dielectric host by two well-known methods and obtained contradictory results. The most fundamental theoretical approach is to represent both the atoms and the dielectric microscopically [7, 8, 9, 10, 11, 12, 13, 35, 37, 50] and derive equations of motion from ?rst principles. We show that the results of microscopic quantum electrodynamics a?rm the Lorentz local-?eld theory with respect to the ?eld renormalization and the Lorentz redshift. The macroscopic quantum electrodynamic theory produces di?erent results for these e?ects and is therefore not valid. A dielectric host containing one or more two-level atoms is modeled quantum electromagnetically as a mixture of two species of atoms, a and b, embedded in the vacuum. To emphasize the symmetry of the local-?eld interaction, both species of atoms are initially treated as two-level systems. Species b is later taken in the harmonic oscillator limit that is associated with a relatively large detuning from resonance. Then, the total Hamiltonian is comprised of the Hamiltonians for the free atoms, the free-space quantized radiation ?eld, and the interaction of the two-level systems with the free-space quantized electromagnetic ?eld. The multipolar and minimalcoupling Hamiltonians are related by a canonical transformation and either can be used. However, due to the canonical transformation, the circumstances of the rotating-wave approximation (RWA) are di?erent for the two Hamiltonians [15]. In the typical derivation of the dielectric susceptibility from the minimal-coupling Hamiltonian, the RWA is invoked implicitly by replacing polariton eigenenergies with photon energies [7, 14, 43]. We take the direct route and use the multipolar Hamiltonian in the RWA. Using a plane-wave expansion of the electromagnetic ?eld E=i

lλ

2πωl ? kl λ , al eikl ·rj ? H.c. e V

(4.1)

the multipolar RWA Hamiltonian is [20, 51] H=

js

ωa j σ + 2 3

ns

ωb n ? + 2 3

?

ω l a? l al

lλ

i ?W ? ?? R21 ? ?a ?E ?R21 ? n?2 Γ0 (W + 1). = ? ?a ?E ?t (3.9b) The equations of motion (3.8), or (3.9), that were derived using the macroscopic ?elds are inconsistent with the generalized Bloch equations (2.6) of the preceding section. Therefore, the application of quantized macroscopic ?elds to the electrodynamics of two-level atoms is contraindicated by Lorentz local-?eld theory.

?i

js lλ

j j ?ikl ·rj j j a? gl σ+ al eikl ·rj ? gl l σ? e

?i

ns lλ

? ? n ?ikl ·rn n ikl ·rn , (4.2) ? hn hn l al ?? e l ?+ al e

where a? l and al are the creation and destruction operators for the ?eld modes and ωl is the frequency of the ?eld

5

j in the mode l with kl = ωl /c. For species a, σ3 is the inj version operator and σ± are the raising and lowering opj ? kl λ ) erators for the j th atom, gl = (2πωl / V )1/2 ?a (? xj · e is the coupling between the atom at position rj and the ? j is a unit vector in the direction of radiation ?eld, x the dipole moment at rj , ωa is the transition frequency, n n and ?a is the dipole moment. For species b, ?3 , ?± , hn l , rn , ωb , and ?b , perform the same functions. Also, V is ?kl λ is the polarization vector, the quantization volume, e and λ denotes the state of polarization. The polarization indices on the variables have been suppressed for clarity. In the two-level approximation, the transitions s = ?m ∈ (?1, 0, +1) are treated separately and the operators need not carry a speci?c value for the magnetic sublevel [52]. Equations of motion for the material and ?eld-mode operators are developed in a straightforward manner from the Hamiltonian using the Heisenberg equation. We have t 0

dt′ e?i(ωl ?ωa )(t?t )

is

′

i i σ ?? (t′ )eikl ·(rj ?ri ) gl

?

+e?i(ωb ?ωa )t

lλ

j j gl σ3 (t)

t 0

dt′ e?i(ωl ?ωb )(t?t ) ×

′

? n ′ ikl ·(rj ?rn ) hn ?? (t )e l ? ns

(4.5)

in normal ordering with rji = rj ? ri . Likewise, one obtains

n d? ?? = dt n ikl ·rn al (0)e?i(ωl ?ωb )t hn l ?3 e lλ t 0

+ei(ωb ?ωa )t

lλ

n hn l ?3 (t)

dt′ e?i(ωl ?ωa )(t?t ) ×

′

dal = ?iωl al + dt

j ? j ?ikl ·rj gl σ? e js

+

ns

? n ?ikl ·rn hn l ?? e

(4.3a)

j dσ? j = ?iωa σ? + dt j j gl σ3 al eikl ·rj lλ

j j gl σ ?? (t′ )eikl ·(rn ?rj ) js t 0

?

(4.3b)

+

lλ

n hn l ?3 (t)

dt′ e?i(ωl ?ωb )(t?t ) ×

′

j dσ3 = ?2 dt

j j ?ikl ·rj j j a? gl σ+ al eikl ·rj + gl l σ? e lλ

?

(4.3c)

ms

? m ′ ikl ·(rn ?rm ) hm ?? (t )e l ?

(4.6)

n d?? n = ?iωb ?? + dt

n ikl ·rn hn l ?3 al e lλ

(4.3d)

n d?3 = ?2 dt

? ? n ?ikl ·rn n ikl ·rn . + hn hn l al ?? e l ?+ al e lλ

(4.3e) Bloch-like operator equations of motion are obtained by substituting the formal integral of the ?eld-mode operator equation (4.3a) al (t) = al (0)e?iωl t +

0 t

dt′ e?iωl (t?t ) × ?

′

? ?

j? j gl σ? (t′ )e?ikl ·rj js

+

ns

into the material operator equations of motion (4.3b)– (4.3e) [53]. We transform operator variables to di?erent j j iωa t = σ? e and rotating frames of reference in which σ ?? n n iωb t ? ?? = ?? e are slowly varying quantities. Performing the indicated substitution into Eq. (4.3b) produces

j dσ ?? = dt j j gl σ3 (t)eikl ·rj al (0)e?i(ωl ?ωa )t lλ

? n ′ ?ikl ·rn ? hn (4.4) l ?? (t )e

from Eq. (4.3d). Equations of motion for the inversion operators are obtained from Eqs. (4.3c) and (4.3e) in a similar fashion. The ?eld that drives an atom, Eq. (4.4), consists of the vacuum ?eld, the self-?eld, and the reaction ?eld and we can identify the terms on the right-hand side of Eqs. (4.5) and (4.6) with ?uctuations due to the vacuum ?eld, spontaneous decay from the self-?eld, near-dipole–dipole interactions between same-species atoms associated with the reaction ?eld, and near-dipole–dipole interactions of an atom with the atoms of the other species, also associated with the reaction ?eld. The usual procedure is to limit consideration to only the spontaneous decay rate of a single impurity atom by dropping the ?uctuations and the single-species interactions for both the bath and impurity atoms. These terms are retained here because they contain signi?cant information about local-?eld effects in dielectrics. For concreteness, we take species b to be the bath atoms and species a to be the two-level impurity atoms.

V. NEAR-DIPOLE–DIPOLE INTERACTION

+

j σ3 (t) lλ

j gl ×

The near-dipole–dipole interaction is the basic mechanism of the action of the local ?eld. The Weisskopf– Wigner-based procedure to evaluate the dipole–dipole

6 interaction for a dense collection of identical two-level atoms was developed by Ben-Aryeh, Bowden and Englund [27], with corrections by Benedict, Malyshev, Trifonov, and Zaitsev [54], to investigate single-species intrinsic optical bistability. The results apply to both species of two-level atoms, individually, but we work with species b in order to maintain consistent notation when we take the harmonic oscillator limit of a two-level atom and derive the Lorentz local-?eld correction in a dielectric. We consider a dense collection of identical two-level atoms of species b in which the atoms are evenly distributed in the vacuum with a number density Nb . The same-species interaction I1 =

lλ n hn l ?3 (t) t 0

for ?m = 0 transitions, where βb =

3 2 ωb |?b |2 , 3 c3

(5.7)

R = kb rnm = ωb rnm /c, kb = |kb |, rnm = rn ? rm , and rnm = |rnm |. Performing the summation over the magnetic sublevels, the pairwise dipole–dipole interaction can be written as 3 dd n I1 = ?3 (t) βb 2 where Bnm = [(x ?m · x ?n ) ? (x ?m · n ?nm )(x ?n · n ? nm )] F1 (kb rnm )

m Bnm ? ?? (t ? rnm /c), m =n

(5.8)

dt′ e?i(ωl ?ωb )(t?t ) × + (x ?m · n ? nm )(x ?n · n ? nm )F2 (kb rnm ) (5.1) (5.9)

′

? m ′ ikl ·(rn ?rm ) hm ?? (t )e l ? ms

can be extracted from Eq. (4.6). The self-interaction of the nth atom with its own reaction ?eld is characterized by the term rm = rn in the n n interaction as a consequence of the relation ?3 (t)? ?? (t) = n (t) between Pauli spin operators for the same atom. ?? ?? Then

self I1 = lλ n hn l ?3 (t) t 0 ? n ′ dt′ e?i(ωl ?ωb )(t?t ) hn ?? (t ). l ?

′

and n ? nm = rnm /rnm is a unit vector in the direction of rnm = rn ? rm [27, 54]. Further, Bnm incorporates a view factor to account for the arrangement of the dipoles in the volume. The atoms are evenly distributed with a number density Nb . For the nth atom, the single-species dipole– dipole interaction is obtained in a summation over all other atoms of species b. In the region near rn , the interaction is evaluated by taking the location of dipoles as discrete, while the continuum approximation is applied elsewhere. Then 3 n dd (t) βb I1 = ?3 2 3 n + ?3 (t) βb Nb 2

m Bnm ? ?? (t ? rnm /c) m:rnm <δ

(5.2) Applying the typical Weisskopf–Wigner procedure [55, 56, 57] in the mode continuum limit, one obtains

self I1

2ω 3 |?b |2 n γb n =? b 3 ? ?? (t) = ? ? ? (t) 3 c 2 ?

(5.3)

B? ?? (t ? |r|/c)d3 r,

V ?Vδ

(5.10)

where γb =

3 4 ωb |?b |2 3 c3

(5.4)

is the spontaneous decay rate. For an atom of species b, initially in the excited state, γb is the spontaneous emission rate Γ0 into the vacuum. The pairwise interaction of atoms is carried in the remaining m = n part of the summation. In the Milonni– Knight [52] model of the interaction of two identical twolevel atoms, the strength of the interaction depends on the separation distance and the magnetic sublevel transition. Then [52], F1 (R) = e

iR

where δ is the radius of a small spherical volume Vδ , larger than a cubic wavelength, about the point rn . For cubic symmetry, the ?eld generated by the localized atoms rm = rn in the virtual cavity is zero at the center [17, 27, 54]. The atom n is located at the origin of a cylindrical volume of thickness L and radius R0 . The near-dipole– dipole interaction is obtained by evaluating the integral

dd I1 ≈

3 n βb Nb ?3 (t) 2 ρ r

2

2π

L/2

ρmax

dφ

0 ?L/2

dz

ρmin

ρdρ? ??

1? i 1 i ? + 3+ 2 R R R (5.5)

cos2 φ F1 +

ρ r

2

cos2 φF2 ,

(5.11)

for ?m = ±1 transitions and F2 (R) = eiR ? 2 2i ? 2 R3 R (5.6)

excluding a volume (4/3)πδ 3 about the origin from the range of integration, resulting in

dd I1 ≈ 3 ?4πi Nb ωb |?b |2 ikb δ n e ?3 (t) ? ?? (t ? |r|/c) 3 kb c3 sp .

(5.12)

7 In the limit δ → 0, the near-dipole–dipole interaction

dd n I1 = ?iνb ?3 (t)? ?? n in the harmonic oscillator limit ?3 (t) → ?1, where the ?uctuating ?eld ? fb =

(5.13)

remains ?nite. Here 4π Nb |?b |2 νb = 3 (5.14)

(2πωl /V )1/2 (? xn · ek )eikl ·rn al (0)e?i(ωl ?ωb )t

lλ

is the strength of the near-dipole–dipole interaction and ? ?? = ? ?? (t ? |r|/c) sp represents a spatially averaged quantity. The details of this calculation can be found in the articles by Ben-Aryeh, Bowden, and Englund [27] and by Benedict, Malyshev, Trifonov and Zaitsev [54]. The atoms of species b can be treated as harmonic oscillators if all excitation frequencies are far from resonance with ωb . In this limit the atom essentially ren → ?1. Then mains in the ground state such that ?3 the near dipole–dipole interaction reduces to the Lorentz local-?eld correction, shifting the resonance frequency by 4πNb |?b |2 /(3 ). The microscopic result is in full agreement with the classical Lorentz local-?eld correction and has been experimentally validated [58, 59] by selective re?ection of Rb from a sapphire window. Finally, all of the results of this section can be applied to the other species of atom. Repeating for species a yields

self I1 =? 3 γa j 2 ωa |?a |2 j σ ?? (t) = ? σ ? (t) 3 c3 2 ?

is associated with the spontaneous decay rate by the Kramers–Kronig relations. The interspecies interaction in Eq. (6.1) describes how a speci?c host atom n interacts pairwise with each of the impurity atoms. The formal integral of the equation of motion of the host atoms, Eq. (6.1), is

n ? ?? (t) = ? t 0

dt′ e?iα(t?t )

′

?b

f? ?

0

t

dt′ e?iα(t?t )

′

hn l

lλ

t′ 0

dt′′ e?i(ωl ?ωa )(t ?t

′

′′

) is

i i σ ?? (t′′ )eikl ·rni , gl

?

(6.2) where α = ωb ? ωa ? νb ? iγb /2. Substituting Eq. (6.2) into Eq. (4.5), one obtains

j dσ ?? ?a j ? γa j j = σ3 f ? iνa σ3 σ ?? ? σ ? + I2 , dt 2 ?

(6.3)

(5.15) (5.16)

where I2 = ?

nslλ j j gl σ3 (t) t 0 ? dt′ e?i(ωl ?ωa )(t ?t) hn l ×

′

dd I1

≈

j ?iνa σ3 (t)

σ ?? (t ?|r|/c)

sp

=

j ?iνa σ3 (t)? σ? .

In addition,

3 4 ωa |?a |2 γa = 2βa = 3 c3

(5.17)

e

ikl ·rjn 0

t′

dt e

′′ ?iα(t′ ?t′′ ) l′ λ′

hn l′

t′′ 0

dt′′′ ×

and νa = are de?ned for later use.

VI. INTERSPECIES INTERACTION

4π Na |?a |2 3

(5.18)

e?i(ωl′ ?ωa )(t

′′

?t′′′ ) is′

i i gl ?? (t′′′ )eikl′ ·rni . ′ σ

?

(6.4)

The e?ect of the interspecies near-dipole–dipole interaction can also be evaluated microscopically using Weisskopf–Wigner theory. A single rotating frame of reference is used for both species of atoms by making the n n ?i(ωb ?ωa )t transformation ? ? ?? e . Applying the results ? = ? of the preceding section, we have

n d? ?? ?b ? γb n n = ?i(ωb ? ωa )? ?? ? fb + iνb ? ?? ? ? ? dt 2 ? t

?

lλ

hn l

dt′ e?i(ωl ?ωa )(t?t )

0 js

′

j j gl σ ?? (t′ )eikl ·(rn ?rj )

?

(6.1)

The term containing the ?uctuations has been dropped from consideration because the procedures presented here apply only to slowly varying quantities and because there will be no contribution from the random ?uctuations after averaging. The direct dipole–dipole interactions between impurity atoms was derived in Section V. Equation (6.4) contains two such dipole–dipole interactions between nonidentical atoms that are integrated over the di?erent subspaces corresponding to i) bath atoms and ii) twoj j level impurity atoms. Due to the relation σ3 (t)σ? (t) = j ?σ? (t) between Pauli spin operators, the term i = j is the special case that is associated with the renormalization of the spontaneous decay rate. This separates the self self ndd interaction I2 = I2 + I2 into I2 for the case i = j ndd and I2 for the summation over the rest of the impurity atoms. The two parts of the interaction will be considered separately.

8

A. Dielectric Mediated Dipole–Dipole Interaction

becomes

dd ′ I2 (t ) = ?

The nth atom of the dielectric interacts pairwise with every impurity atom. The term

dd A I2

4πi j χb νa σ3 σ ?? , 3

(6.11)

=

l′ λ′

hn l′

t′′ 0

dt e

′′′ ?i(ωl′ ?ωa )(t′′ ?t′′′ )

where χb is the linear susceptibility of species b. Adding the direct near-dipole–dipole interaction from Eq. (6.3), we obtain (6.5)

dd I2 ? iνa σ3 (t)? σ? (t) = ?i 1 +

i? i gl ?? (t′′′ )eikl′ ·(rn ?ri ) , ′ σ i=j,s

4π σ? (t) χb νa σ3 (t)? 3

extracted from Eq. (6.4), can be evaluated in the same manner as in Section V, except that the atoms are of di?erent species. The summation represents the e?ect of all the impurity atoms on a single atom of the host material. The sum over the i = j impurity atoms is performed i) in the near region by the discrete summation over the impurity atoms and ii) elsewhere by treating the impurity atoms in the continuum limit. For nonidentical atoms, the pairwise interaction goes as [60, 61]

dd I2 = A

= ?i

n2 + 2 νa σ3 (t)? σ? (t). 3

(6.12)

Comparison of Eq. (6.12) with the single species dipole– dipole interaction, Eq. (5.16), shows that the e?ect of the dielectric host is to enhance the interaction by a factor of ? = (n2 + 2)/3. Taking the local spatial average, W = σ3 sp and R21 = ?iσ? sp , one ?nds that the Lorentz red-shift n2 + 2 4 π N ?2 3 3 (6.13)

3 2

βa βb

i:rni <δ

i Bni σ ?? (t′′ ? rni /c)

+

3 2

βa βb N a

V ?Vδ

Bσ ?? (t′′ ? |r|/c)d3 r.

(6.6)

is consistent with the Lorentz local-?eld calculation, Eq. (2.6a), while the macroscopic quantum electrodynamic result, Eq. (3.9a) is not.

B. Dielectric-Enhanced Spontaneous Decay Rate

Then [27]

dd A I2

4πi Na ?? ?? (t′′ ? |r|/c) =? a ?b σ 3 =? 4πi Na ?? ?? . a ?b σ 3

sp

sp

(6.7)

The quantity σ ?? = σ ?? (t′′ ? |r|/c) time and the temporal integral

dd B ′ I2 (t ) t′

is slowly varying in

Most of the elements of the microscopic theory of the spontaneous decay rate of an atom in a dielectric are common to the treatment of the dipole-dipole interaction. In order to show this clearly, we consider an equivalent derivation of the dielectric mediated dipole–dipole interaction. Performing the temporal integrations ?rst, the interspecies interaction (6.4) can be written as I2 = ?

nslλ j j ? ikl ·(rj ?rn ) gl σ3 (t)πδ (ωl ? ωa )hn × l e

=

0

dt′′ e?iα(t ?t

′

′′

)

?4πi 3

Na ?? ?? (6.8) a ?b σ

can be performed in the adiabatic-following approximation. Repeatedly integrating Eq. (6.8) by parts [62], the series can be truncated at the ?rst term in the expansion if the time rate of change of σ ?? is much smaller than ασ ?? yielding

dd I2 (t′ ) = B

?i α

hn l′ πδ (ωl′ ? ωa )

is′ l′ λ′ is′

i i gl ?? (t′ )eikl′ ·(rn ?ri ) . ′ σ

?

?1 α

?4πi 3

dd I2

Na ?? ?? . a ?b σ

(6.9)

(6.14) Applying the Milonni-Knight interaction with a view factor results in I2 = ? ?i 9 j βa βb σ3 α 4 [(x ?j · n ?nj )(x ?n · n ?nj )]F2 (ka rnj )

n

integration is another The remaining part of the interspecies dipole–dipole interaction. In this case, the summation imparts the e?ect of all the atoms of the host dielectric, modi?ed by interspecies interaction with the impurity atoms, on the j th impurity atom. Combining terms,

dd ′ I2 (t ) =

+ [(x ?j · x ?n ) ? (x ?j · n ?nj )(x ?n · n ?nj )] F1 (ka rnj ) ×

?4πi 1 4πi j Nb ?? Na ?? ?? b ?a a ?b σ3 σ 3 α 3

(6.10)

[(x ?i · n ?ni )(x ?n · n ?ni )]F2 (ka rni )

i

9

i + [(x ?i · x ?n ) ? (x ?i · n ?ni )(x ?n · n ?ni )] F1 (ka rni ) σ? . (6.15)

Converting the sums, excluding i = j , to integrals and integrating over the subspace of two-level atoms and then over the subspace of oscillators is equivalent to the derivation of the dielectric mediated dipole–dipole interaction that was presented in the preceding subsection. The renormalization of the spontaneous decay rate of a dielectric-embedded two-level atom is derived from the interspecies interaction (6.4) in the same fashion by taking the target atom to be the same as the source atom. The summation over the two-level atoms is evaluated with the use of the delta-function δij , rather than the integration over the subspace of two-level atoms. Likewise, the summation over the magnetic sublevels invokes δss′ . Then,

self I2 =

as an addition to Eq. (6.3). Equation (7.2) contains the same type of interaction that was evaluated in the previous section. Performing the adiabatic-following approximation and the sum over polarizations, bath atoms, and magnetic sublevels in the mode continuum limit, we obtain I3 = n2 + 2 ?a j E σ3 . 3 2 (7.3)

The electromagnetic ?eld is enhanced by the same factor of ? as the reaction ?eld.

VIII.

OPTICAL BLOCH EQUATIONS FOR EMBEDDED ATOMS

?i 9 j βa βb σ? α 4

2 [(x ?j · n ?nj )(x ?n · n ?nj )]2 F2 (ka rnj ) n

The macroscopic optical Bloch equations can be derived from the quantum electrodynamic equations of motion in the limit of large numbers. Combining Eqs. (6.3), (6.11), and (7.2), one obtains

j dσ ?? ?a j j = ?i?νaσ3 σ ?? + ? E e?i(ωp ?ωa )t σ3 , dt 2

2 + [(x ?j · x ?n ) ? (x ?j · n ?nj )(x ?n · n ?nj )]2 F1 (ka rnj ) . (6.16)

The microscopic treatments of the spontaneous decay rate in a dielectric [20, 51] are missing elements of Eq. (6.16) and can neither a?rm nor contradict the macroscopic theory of quantum electrodynamics. Because the dielectric renormalization of the spontaneous decay rate does not have a classical local-?eld condition-based analogue, we do not consider it further.

VII. DIELECTRIC-ENHANCED FIELD

(8.1)

neglecting the Gaussian noise source with zero mean and absorption. The equation of motion for the inversion operator

j ?a j dσ3 j = 2 i?νa σ ?+ σ ?? ? σ ? ?E e?i(ωp ?ωa )t + H.c. dt 2 +

(8.2)

The dielectric has an e?ect on an applied electromagnetic ?eld that can also be evaluated microscopically. Taking the ?eld in a coherent state, the partial Hamiltonian is Hf = ? i 2

j ?i(ωp t?kp ·rj ) j i(ωp t?kp ·rj ) ?a σ+ e ? ?? a σ? e j

is derived in a similar manner. Optical Bloch equations of motion are obtained by transforming to a frame rotating at the frequency of the ?eld and taking a local-spatial average, as in Sec. III. We compare the optical Bloch equations ?R21 4π = i ωp ? ωa ? N ?2 a ?W ?t 3 R21 ? i?a ?E W 2 (8.3a) (8.3b)

?

i 2

n ?i(ωp t?kp ·rn ) n i(ωp t?kp ·rn ) , ?b ?+ e ? ?? a ?? e n

i ?W ? = ? [?a ?? E ? R21 ? ?a ?E R21 ], ?t

(7.1) where ωp is the nominal frequency of the ?eld and ?a = ?a E / and ?b = ?b E / are Rabi frequencies. The total Hamiltonian is now comprised of the Hamiltonians (7.1) and (4.2). Developing Heisenberg equations of motion and eliminating the ?eld-mode operators and the dielectric operators results in the appearance of I3 = ?a j σ (t)E (t) ? 2 3

j j gl σ3 (t) nslλ t′ 0 t 0

dt′ e?i(ωl ?ωp )(t?t ) ×

′

that were derived from ?rst principles, to the Lorentz local-?eld-based equations (2.6). Based on a favorable comparison of the local-?eld enhancement of the Lorentz redshift and the Rabi rate with the classically derived result, we can reasonably assert that the microscopic theory, unlike the macroscopic quantum electrodynamic theory, satis?es the correspondence principle. The microscopic theory allows us to consider the more general case of of a complex local-?eld enhancement factor. Separating the real and imaginary parts of ?, the optical Bloch equations can be written as ?R21 4π = i ωp ? ωa ? N ?2 a ?r W ?t 3 R21

? i(kl ?kp )·rjn hn l e

dt′′ eα(t ?t

′

′′

) ?a

2

E (t′′ )

(7.2)

10 ? i?a 4π ?E W ? N ?2 a ?i W R21 2 3 (8.4a) more general than the calculation of the spontaneous emission or decay rate. In Sec. III, we used the macroscopic theory to develop the optical Bloch equations for two-level atoms in a dielectric host medium. For comparative purposes, the optical Bloch equations were also derived using the classical Lorentz local-?eld condition 4π P 3

i ?W ? = ? [?a ?? E ? R21 ? ?a ?E R21 ] ? 4?i νa |R21 |2 (8.4b) ?t with ? = ?r + i?i . The microscopic theory justi?es the use of a complex refractive index in the classical Lorentz local-?eld condition. Then Eqs. (8.4), with phenomenological damping, can be derived by substituting the polarization (2.5) with complex n into the generalized Bloch equations (2.4). The imaginary part of the Lorentz redshift, derived in this manner, was found to be associated with an intrinsic cooperative decay for two-level atoms in an absorptive host [22]. This result is con?rmed by the microscopic theory. The optical Bloch equations (8.4) for dielectricembedded two-level atoms are derived from the microscopic description of quantum electrodynamics using vacuum-based ?elds that are known to satisfy the equaltime commutation relations. Because the ?eld-mode operators have been eliminated, the equal-time commutation relations cannot be discussed in the context of the optical Bloch equations (8.4) or Heisenberg equations (8.1) and (8.2). Instead, the optical Bloch equations, generalized for a dielectric host, must demonstrate conservation of probability. The total population is W 2 + 4|R21 |2 . Direct substitution from Eqs. (8.4) shows that the temporal derivative of this quantity is nil, as required, in the limit that absorption by the atoms and the host dielectric can be neglected.

IX. SUMMARY

EL = E +

in Sec. II. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. The di?erences in the Rabi frequencies can be reconciled with a phenomenological local-?eld factor applied in the macroscopic case, providing the virtual-cavity model is used. However, no such facile reconciliation occurs between the macroscopic and local-?eld predictions for the dielectric enhancement of the near-dipole–dipole interaction between impurity atoms. If we assume the validity of the Lorentz local?eld condition, then the macroscopic procedure is invalidated, in whole. Conversely, the validity of the macroscopic quantum electrodynamic theory would imply that the Lorentz local-?eld condition is incorrect. One deciding factor is that the Lorentz local-?eld correction has been validated experimentally [58, 59], while the experimental record for the macroscopic quantum theory has been inconclusive. The microscopic quantum electrodynamic theory is fundamental and we expect that optical Bloch equations derived therefrom would corroborate the local-?eld theory. Indeed, the microscopic Weisskopf–Wigner procedure that was described by Crenshaw and Bowden [20] was found to be consistent with the Lorentz local-?eld condition for the single-species local-?eld correction, for the Bloembergen enhancement of the electromagnetic ?elds, and for the Lorentz redshift of the impurity atoms. In conclusion, the interest in the dielectric renormalization of the spontaneous emission rate of an atom embedded in a dielectric material obscures the inconsistencies in the macroscopic theory of quantum electrodynamics. By considering all of the physical implications of the macroscopic theory of electromagnetic ?elds in dielectrics on the dynamics of atoms, we demonstrated that the correspondence principle is violated. We applied the more fundamental microscopic theory to the same problem and demonstrated complete agreement with classical theory for all e?ects that have classical local-?eld analogues in the optical Bloch equations. Therefore, the microscopic theory of quantum electrodynamics is favored over the macroscopic theory in describing the interaction of the electromagnetic ?eld with microscopic elements.

Microscopic quantum electrodynamics is generally regarded as the most fundamental theory of the interaction of ?elds with matter. There have been attempts to apply this cornerstone of electrodynamics to derive the e?ects of a dielectric on the electrodynamics of atoms from a microscopic model [7, 8, 9, 10, 11, 12, 63, 64]. In each case, an implicit continuum approximation was made in the transformation from coordinate space to reciprocal (wavenumber) space resulting in a macroscopic derivation of the dielectric e?ects. In the absence of a suitable microscopic quantum electrodynamic treatment for the spontaneous emission of an atom embedded in a host, the macroscopic theory has been used to predict a density-of-states-based enhancement by a factor of the refractive index n. The local-?eld e?ect has been applied phenomenologically using either a virtual-cavity or real-cavity model, without an experimental decision. The application of macroscopic quantum electrodynamic theory to dielectric-embedded two-level atoms is

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