Light quantization for arbitrary scattering systems

Light quantization for arbitrary scattering systems

Salvatore Savasta, Omar Di Stefano, and Raffaello Girlanda

INFM and Dipartimento di Fisica della Materia e Tecnologie Fisiche Avanzate, Universita` di Messina, Salita Sperone 31, I-98166 Messina, Italy

We present a quantum theory of light scattering for the analysis of the quantum statistical and fluctuation properties of light scattered or emitted by micrometric and nanometric three-dimensional structures of arbitrary shape. We obtain general three-dimensional quantum-optical input-output relations providing the output photon operators in terms of the input photon operators and of the noise currents of the scattering system. These relations hold also for photon operators associated with evanescent fields, for anisotropic scattering systems and/or for media with a nonlocal susceptibility. We find that the commutation relations of the output photon operators, carrying all the information on the scattering and/or the emission process, result to be fixed by energy conservation and reciprocity. We prove that this quantization scheme is consistent with QED commu- tation rules by using a novel relationship between vacuum and thermal fluctuations. This theoretical framework has been applied to analyze the spectral density of light close to a point scatterer under different nonequilib- rium conditions.

DOI: 10.1103/PhysRevA.65.043801

I. INTRODUCTION

The study of the optical properties of systems with shapes and sizes varying on micrometric and nanometric scales is motivated by fundamental research and by possible applica- tions. In particular, the tailoring of electromagnetic modes allowed by microcavities and photonic bandgaps共PBGs兲 has given rise to a variety of striking phenomena observed in recent years关1,2兴 and it is expected to dramatically improve the performance of light-emitting devices. These develop- ments and recent continuous progress in scanning near-field optical microscopy have stimulated new theoretical ap- proaches for the analysis of a large class of problems dealing with three-dimensional 共3D兲 objects of arbitrary shape and dielectric functions 关3,4兴, and have renewed the interest in the classical theory of light scattering 关5,6兴.

In this paper we present a quantum generalization of the classical theory of light scattering based on Green’s dyadic technique. The theory presented here provides a general and unified basis for analyzing a large class of optical processes where quantum and/or thermal fluctuations play a role关7兴. It is expected to be adequate to analyze a wide range of optical phenomena and experiments such as precision measurements of Casimir forces关8兴, light emission from sources embedded in photonic systems关9兴, light fluctuations in finite inverted- population media with inclusion of spatial effects 关10兴, the spatial behavior of scattered and/or confined nonclassical light 关11,12兴, nanoscale radiative transfer 关13,14兴, etc.

Quantum electrodynamics in the presence of media started from the pioneering work by Agarwal 关15兴 who, ap- plying the fluctuation-dissipation theorem, developed the lin- ear response theory of spontaneous emission in presence of dielectrics and conductors. A more direct microscopic quan- tization procedure for light in a dispersive and absorptive homogeneous dielectric was first proposed by Huttner and Barnett 关16兴. Since this work, following the method of Langevin forces, light has been quantized in media of in- creasing generality 关17–23兴. Within this method quite gen-

eral results have been recently presented 关24,25兴. In particu- lar it has been proved that quantization of the Maxwell theory of the electromagnetic field in inhomogeneous three- dimensional, dispersive, and absorbing dielectric media of given causal permittivity is consistent with the fundamental equal-time commutation relations of QED 关25兴. A different general 3D quantization scheme that makes use of a set of auxiliary fields, followed by a canonical quantization proce- dure has been developed by Tip 关26兴. Recently the equiva- lence of the quantization schemes by Scheel et al. 关25兴 and by Tip关26兴 has been demonstrated 关27兴.

Here we generalize these results to media that can be anisotropic and/or with a nonlocal susceptibility. Further- more, we consider explicitly media that can have finite size.

This allows the analysis of quantized light scattering and allows us to derive general quantum-optical input-output re- lations relating the output photon operators to the input pho- ton operators and to the noise currents of the scattering sys- tem. These relations hold also for evanescent fields and thus allow us to define naturally output photon operators associ- ated with evanescent waves.

II. THE SCATTERING SYSTEM PROPERTIES Let us consider the most general nonmagnetic linear scat- tering system. It can be described by a causal and eventually nonlocal susceptibility tensor␹i, j(r,r⬘^,␻⁾ 关28兴. Thus we are considering a large class of material systems of arbitrary shape including anisotropic media and/or media driven by the electric field via a nonlocal susceptibility. Electronic states of semiconductors and semiconductor quantum struc- tures, and also all those systems with a spatially dispersive susceptibility are driven by the electric field via a nonlocal susceptibility关21兴.

In the following we will adopt the compact Dirac nota- tion. We introduce the operators L for⫺“⫻“⫻, e0for k²1, and the integral operator e_s, describing the effect of the scat- tering system. This operator applied to the electric field gives

具^r,i兩es兩E兲典^⫽k2

冕

^{d3r⬘␹i j共r,r⬘兲Ej共r⬘兲.}

Hence the wave equation relative to the material systems here considered, for the positive frequency components of the electric-field operator can be written in the compact Dirac notation as

共L⫹e0⫹es兲兩Eˆ^⫹典⫽i␻␮0兩jˆ典^, 共2.1兲 where the hat indicates quantum operators. The zero mean noise currents jˆ can be derived from the Heisenberg- Langevin equations for the material system 关29兴 and appear only if the susceptibility tensor is not real; they are a direct consequence of the fluctuation-dissipation theorem and obey the following commutation rules,

关 jˆi共r,␻兲, jˆj共r⬘^,␻兲兴⫽0, 共2.2兲 关 jˆi共r,␻兲, jˆj

†共r⬘^,␻兲兴⫽ ប

␲␮0

␻² c²兩␹i j

I共r,r⬘^,␻兲兩␦共␻⫺␻⬘兲, 共2.3兲

␹^I being the imaginary part of the susceptibility tensor.

These equations show that a nonlocal susceptibility produces noise currents that are spatially correlated. These spontane- ous currents act as quantum Langevin forces. Their expecta- tion values determine the amounts of noise that are added to optical signals that propagate through the attenuating or am- plifying media. Moreover,具^jˆ†jˆ典 is the source term producing light emission. These noise currents are related to the Bosonic vector field describing the reservoir oscillators. The expectation values of noise currents depend on the specific state of the reservoir oscillators. We start by considering a system at a given temperature T. In this case the current’s correlation tensor is given by

具^jˆi

†共r,␻兲jˆj共r⬘^,␻兲典⫽ ប

␲␮0

␻² c²␹i j

I共r,r⬘^,␻兲N¯共␻^,T兲␦共␻⫺␻⬘兲, 共2.4兲 where N¯ (T) is the mode occupation described by Planck’s formula

N¯共␻^,T兲⫽ 1

exp共ប␻^/kBT兲⫺1. 共2.5兲 If we consider the medium composed of a collection of non- interacting two-level systems at thermal equilibrium, the ra- tio between the upper N_u- and the lower N_l-level occupa- tions associated with the dielectric response at frequency␻ is given by the Boltzmann distribution law

N_u

N_l⫽exp共⫺ប␻^kBT兲. 共2.6兲 From Eqs.共2.5兲 and 共2.6兲 we obtain

N¯共T兲⫽ N_u N_l⫺Nu

. 共2.7兲

Thermal equilibrium can be altered by, e.g., optical pumping.

In this case, Eq. 共2.7兲 can still be used considering T

⬅T(␻) as a frequency-dependent effective temperature.

Very high temperatures correspond to saturation of the tran- sition between the two levels. By taking a negative effective temperature, it is possible to describe also population inver- sion and hence amplifying media 关30兴. Equation 共2.4兲 can also be generalized to take into account a scattering system including media with different effective temperatures. We obtain

具^jˆi

†共r,␻兲jˆj共r⬘^,␻兲典⫽ ប

␲␮0

␻² c²

⫻

兺

m 共␹m

I 兲i j共r,r⬘^,␻兲N¯共Tm,␻兲

⫻␦共␻⫺␻⬘兲, 共2.8兲 where m labels the different media. We point out that the nonlocal susceptibility (␹m

I )_{i j}(r,r⬘^,␻) is different from zero only if r and r⬘ belong to the same medium m. The correla- tion 具^jˆi(r,␻^{) jˆ}j

†(r⬘^,␻⁾典 can be obtained from Eq. 共2.4兲 re- placing N¯ with N¯⫹1.

III. QUANTUM THEORY OF LIGHT SCATTERING In the absence of the scattering system, the electric-field operator can be derived following the well-known quantiza- tion schemes in vacuum. By using the angular spectrum of plane waves, the electric-field operator can be expanded in terms of photon operators as

Eˆ⁰共r,t兲⫽

冕

0

⬁

d␻^{e⫺i␻tEˆ0⫹}共r,␻兲⫹H.c.,

with

Eˆ⁰⫹共r,␻兲⫽i

冑

2^ប␧^␻0

兺

_␶,K ^{␾K␶共r,␻兲aˆK␶共␻兲, 共3.1兲}

where␶⫽⬎,⬍ indicates leftward and rightward propagating waves, and K⬅(K,␴) is a shortcut for the wave-vector pro- jection along the xy plane and the polarization direction␴^. aˆ_K^␶ are the photon operators obeying the usual Bosonic com- mutation rules,

关aˆK␶共␻兲,aˆ_K^␶⬘_⬘^†共␻⬘兲兴⫽␦␶,␶⬘␦K,K⬘␦共␻⫺␻⬘兲, 共3.2兲 关aˆK␶共␻兲,aˆ_K^␶⬘_⬘共␻⬘兲兴⫽0. 共3.3兲 The orthonormal set of vector fields is given by

␾K⬎/⬍共r,␻兲⫽␣Ke_K^⬎/⬍exp i共K•R⫾kzz兲, 共3.4兲

where r⫽(R,z), ek␶ is the polarization unit vector, k_z

⫽(␻^2/c2⫺K²)^1/2, and ␣K⫽(␻^/2␲^c2kzA)^1/2, A being the quantization surface. In compact notation, Eq. 共3.1兲 can be written as

兩Eˆ^0⫹典⫽i

冑

2^ប␧^␻0

兺

_␶,K ^{兩␾K␶典aˆK␶共␻兲. 共3.5兲}

We observe that the field operator in Eq.共3.1兲 verifies, in the same compact notation of Eq. 共2.1兲, the following wave equation:

共L⫹e0兲兩Eˆ^0⫹典⫽0. 共3.6兲 The Green operator associated with the complete system is defined by

共L⫹e0⫹es兲G⫽1. 共3.7兲

Adding Eq. 共3.6兲 to Eq. 共2.1兲 and using Eq. 共3.7兲 we obtain 兩Eˆ^⫹典⫽兩Eˆh⫹典⫹兩Eˆp⫹典^, 共3.8兲 with the particular solution Eˆ

p⫹given by

兩Eˆp⫹典^⫽i␻␮0G兩jˆ典^{, 共3.9兲} and the homogeneous solution

兩Eˆh⫹典⫽共1⫺Ges兲兩Eˆ^0⫹典^. 共3.10兲 In the r representation the obtained electric-field operator can be written as

Eˆ

hi⫹共r兲⫽Eˆi

0⫹共r兲⫺q²

冕

^{Gi j共r,r⬘兲␹jl共r⬘,r⬙兲Eˆl0⫹共r⬙兲dr⬘dr⬙,}

共3.11兲

Eˆ

pi⫹共r兲⫽i␻␮0

冕

^{Gi j共r,r⬘兲jˆj共r⬘兲dr⬘. 共3.12兲}

Introducing Eq.共3.1兲 into Eq. 共3.10兲, the homogeneous term can be expanded in terms of free-space photon operators as

兩Eˆ^⫹典^⫽i

冑

2^ប␧^␻0

兺

_␶,K ^{兩␺K␶典aˆK␶共␻兲, 共3.13兲}

where

兩␺K␶典⫽共1⫺Ges兲兩␾K␶典 共3.14兲 is the electric field arising from an input beam兩␾K␶典^scattered by the material system. Equation共3.8兲 gives the electric-field operator in terms of the input photon operators and the noise currents operators. Once the quantum states of the input light beams and of the scattering system are fixed, by using Eq.

共3.8兲 in principle it is possible to compute the electric-field operator in the presence of the scattering system in all the space if the Green tensor G_{i j}(r,r⬘) is known. One efficient procedure to calculate Green tensors for complex 2D and 3D scattering objects is described in Ref.关4兴 and it is based on a

volume discretization procedure in close analogy with the discrete dipole approximation 关31,32兴. Recently the method of multipole expansion has been used to calculate 2D Green functions in photonic crystals 关33兴. A different scheme for the calculation of Green functions for photons propagating in complex dielectric structures based on an extension of the finite-difference time-domain method has been presented by Ward and Pendry关34兴.

The consistency of the quantization approach described in this section with the equal-time QED commutation relations is proved in Appendix A.

IV. QUANTUM-OPTICAL INPUT-OUTPUT RELATIONS Inside absorbing materials, owing to the presence of noise currents, it is not possible to define space-independent pho- ton operators as in free space 关18,29兴, however, we may at- tempt to find input and output photon operators outside the scattering system 关23兴. This would furnish useful input- output quantum-optical relations and it would imply that, just outside the scattering system, the light field, although carry- ing information on the scattering process, can be quantized as in free space. We proceed by bounding the scattering sys- tem with two planes at z⫽⫾L, thus separating space in three regions: the left region共I兲 (z⬍⫺L), the scattering region 共II兲 (⫺L⬍z⬍L), and the right region 共III兲 (z⬎L), as shown in Fig. 1. In the following we will show that it is possible to define space-independent photon operators outside the scat- tering region. We start from the Dyson equation

G⫽G⁰⫺G⁰e_sG, 共4.1兲 where G⁰ is the unperturbed free-space Green dyadic, obey- ing the following equation:

共L⫹e0兲G⁰⫽1. 共4.2兲

Using the angular plane-wave expansion, the free-space Green tensor can be written as

G_{i j}⁰共r,r⬘兲⫽⫺i␲^c2

␻

兺

_K ^{␾K,i⬎ 共r⬎兲␾⫺K, j⬍ 共r⬍兲⫹␦共z⫺z}_k^{2 ⬘兲zz}_z^2,

共4.3兲 FIG. 1. Scattering geometry and notation.

where r_⬎⫽r(r_⬍⫽r⬘^{) for z}⬎z⬘ ^{and r}⬎⫽r⬘^(r⬍⫽r) for z

⬍z⬘. Let us analyze the electric-field operator in the regions external to the scattering system共I and III兲. We start from the contribution共3.10兲 arising from the solution of the homoge- neous wave equation. We introduce the Dyson equation共4.1兲 into Eq.共3.10兲,

兩Eˆh⫹典^{⫽兩Eˆ0⫹}典^⫺G0共1⫺esG兲es兩Eˆ^0⫹典^{. 共4.4兲} Let us now consider the electric-field operator Eˆ

h

⫹(r) in re- gion III (z⬎L). In this case the free-space Green tensor in Eq.共4.4兲 appears always with z⬎z⬘ and thus can be written simply as

G_{i j}⁰共r,r⬘兲⫽⫺i␲^c2

␻

兺

_K ^{␾K,i⬎ 共r兲␾K⬍¯ , j共r⬘兲 共z⬎z⬘兲,}

共4.5兲 that in compact notation reads

G⁰⫽⫺i␲^c2

␻

兺

_K ^{兩␾K⬎典具␾K⬎兩, 共4.6兲}

where we have introduced the following definition:

具␾K⬎兩r典⬅具^r兩␾_⫺K^⬍ 典^. 共4.7兲 Introducing Eq.共4.6兲 into Eq. 共4.4兲 and using Eq. 共3.14兲 we obtain

兩Eˆh⫹典⫽兩Eˆ^0⫹典⫹i␲^c2

␻

兺

K 兩␾K⬎典具␺K⬎兩es兩Eˆ^0⫹典^. 共4.8兲 Following the same steps for Eˆ

p⫹ we obtain

兩Eˆp⫹典⫽i␻␮0

i␲^c2

␻

兺

_K ^{兩␾K⬎典具␺K⬎兩jˆ典. 共4.9兲}

By introducing Eq.共3.1兲 into Eq. 共4.8兲, the total electric-field operator in the region III can be written as

Eˆ⫹共r,␻兲⫽i

冑

2^ប␧^␻0

兺

_K ^{关␾K⬎共r,␻兲bˆK⬎共␻兲}

⫹␾K⬍共r,␻兲aˆK⬍共␻兲兴, 共4.10兲 with the space-independent output photon operators given by

bˆ_K^⬎⫽bˆK h⬎⫹bˆK

p⬎, 共4.11兲

with

bˆ_K^h⬎⫽aˆK⬎⫹␲^c2

␻

冑

²ប^␧␻⁰具␺K⬎兩es兩Eˆ^0⫹典 共4.12兲 and

bˆ_K^p⬎⫽i␲

␧0

冑

²ប^␧␻⁰具␺K⬎兩jˆ^⫹典^. 共4.13兲

Equation共4.11兲 with Eqs. 共4.12兲 and 共4.13兲 gives the output photon operator associated with a plane wave of given en- ergy and propagating along a fixed direction 共determined by K and␻⁾共see Fig. 1兲 in terms of the input photon operators and of the noise currents inside the material system. The integrals in Eqs.共4.12兲 and 共4.13兲 can be explicitly written in the r representation, the one in Eq. 共4.13兲 reads

具␺K⬎兩jˆ^⫹典⫽

冕

^␺K⬍¯共r兲•jˆ共r兲dr. 共4.14兲 Analogous results can be obtained for the electric-field op- erator in region I, that can be written as

Eˆ⫹共r,␻兲⫽i

冑

2^ប␧^␻0

兺

_K ^{关␾K⬎共r,␻兲aˆK⬎共␻兲}

⫹␾K⬍共r,␻兲bˆK⬍共␻兲兴, 共4.15兲

with bˆ_K^⬍(␻⁾⫽bˆK h⬍⫹bˆK

p⬍ given by

bˆ_K^h⬍⫽aˆK⬍⫹␲^c2

␻

冑

²ប^␧␻⁰具␺K⬍兩es兩Eˆ^0⫹典 ^共4.16兲 and

bˆ_K^p⬍⫽i␲

␧0

冑

²ប^␧␻⁰具␺K⬍兩jˆ典^{. 共4.17兲} Equations.共4.12兲 and 共4.16兲 can be further simplified evalu- ating the integrals. This can be done by using the Lippman- Schwinger equation

兩␺典⫽共1⫺G⁰e_s兲兩␺典^. 共4.18兲 By using the angular spectrum representation of the field 兩␺典, defined according to

␺共r兲⫽

兺

K ␺^K共z兲e^iK•R, 共4.19兲 we can project the Lippman-Schwinger equation as

兩␺^K典⫽兩␾^K典⫹i␲^c2

␻ ^兩␾K⬎(⬍),K典具␾K⬎(⬍)兩es兩␺典^, 共4.20兲 with␶⁽␶⬘) depending on which region共I or III兲 we are con- sidering. Introducing Eq. 共3.1兲 into Eq. 共4.12兲, using Eq.

共4.20兲, and observing that␾K,⬍,K␴(L)⫽eˆK,⬍␴exp关⫺ikzL兴, we ob- tain

具␺K⬎兩es兩Eˆ^0⫹典⫽具^Eˆ0⫹兩es兩␺_⫺K^⬍ 典

⫽

冑

^2ប⑀^␻0

兺

_K⬘ ^{␣K⬘2kz⬘A exp关⫺ikz⬘L兴}

⫻关兵␺K⬍,K¯ ,␴¯⬘共⫺L兲⫺␾_K^⬍,K¯⬘^,␴

¯⬘共⫺L兲其^eK^⬎⬘aˆ_K^⬎_⬘

⫹兵␺K⬍,K¯ ,␴¯⬘共L兲⫺␾_K^⬍,K¯⬘^,␴

¯⬘共L兲其^eK^⬍⬘aˆ_K^⬍_⬘兴.

共4.21兲 By using this equation, Eq.共4.12兲 can be written as

bˆ_K^h⬎⫽

兺

_K⬘ ^{关TKK⬘aˆK⬎}⬘⫹RK

K⬘aˆ_K^⬍_⬘兴, 共4.22兲

where

T_K^K⬘⫽e^⫺ikz⬘L

␣K⬘ ␺_K^⬍K¯¯⬘共⫺L兲•eK^⬎⬘ 共4.23兲 R_K^K⬘⫽e^⫺ikz⬘L

␣K⬘ 关␺_K¯⬍K¯⬘共L兲⫺␾⬍K_K¯¯⬘共L兲兴•eK^⬍⬘, 共4.24兲 where K¯⬅⫺K and K¯⫽(⫺K,␴). Analogously, we can ob- tain for the operators describing output in region I,

bˆ_K^h⬍⫽

兺

_K⬘ ^{关RKK⬘aˆK⬎}⬘⫹TK

K⬘aˆ_K^⬍_⬘兴 , 共4.25兲

where

RK

K⬘⫽e^⫺ikz⬘L

␣K⬘ 关␺_K¯⬎K¯⬘共⫺L兲⫺␾_K¯⬎K¯⬘共⫺L兲兴•eK^⬎⬘, 共4.26兲

TK

K⬘⫽e^⫺ikz⬘L

␣K⬘ ␺_K¯⬎K¯⬘共L兲•eK^⬍⬘. 共4.27兲 The obtained quantum-optical input-output relations relate the output operators bˆ_K⬅(bˆK⬎,bˆ_K^⬍) to the input photon op- erators aˆ_K⬅(aˆK⬎,aˆ_K^⬍) and to the noise currents jˆ(r) of the scattering system, according to

bˆ_K⫽

兺

_K⬘ ^SKK⬘aˆK⬘⫹FˆK, 共4.28兲 where S_K^K⬘is a 2⫻2 scattering matrix (S matrix兲,

S_K^K⬘⫽

冉

^{RTKKKK⬘⬘ RTKKKK⬘⬘}

冊

^{, 共4.29兲}

and Fˆ_K is a two-dimensional quantum noise vector,

Fˆ_K⫽i␲

␧0

冑

²ប^␧␻^0共具␺K⬎兩jˆ典^,具␺K⬍兩jˆ典兲.

If the quantum state of input radiation and of the material system is known, any output photon correlation can be di- rectly calculated by using these relations provided the clas- sical light modes ␺_K⬍¯ have been computed. Light modes for specific complex structures can be calculated using Eq.

共3.14兲 according to the scheme described in Ref. 关5兴. We

observe that, since we have not assumed any translation symmetry for our scattering system, in principle all the light modes ␺_K^␶_⬘arising from all possible input fields K⬘ ^{are ex-} pected to contribute to output waves propagating along a given direction determined by K and␻. Instead, due to reci- procity, Eqs.共4.12兲 and 共4.13兲 have a simpler structure show- ing that bˆ_K^⬎ depends on Eˆ^0⫹ and jˆ^⫹ only via the reciprocal mode ␺_K⬍¯ . The obtained input-output relations 共4.28兲 are based on the angular-spectrum representation. As it is well known, this representation describes explicitly also the eva- nescent waves 关6兴 that appears for K⬎k. These relations 共4.28兲 hold also for evanescent fields and define naturally output photon operators associated with evanescent waves.

With the improvement in techniques based on measurement and control of evanescent waves, these relations should find application for the analysis of evanescent nonclassical fields, e.g., arising from the scattering of nonclassical input fields by nanometric objects.

V. COMMUTATION RELATIONS

The expansion in input and output photon operators per- formed above is consistent only if the output operators are true photon operators obeying Bosonic commutation rules.

Let us start looking at the commutator for the particular term of the rightward output operator. By using Eqs. 共2.3兲 and 共4.13兲, we obtain

关bˆK

p⬎共␻兲,bˆ_K^p⬎†_⬘ 共␻⬘兲兴⫽␲

␧0

A_K,K⬘共␻兲␦共␻⫺␻⬘兲, 共5.1兲 with

A_K,K⬘⫽

冕

^{dr关␺K¯⬍共r兲•J⬍¯K⬘*共r兲⫹JK¯⬍共r兲•␺K¯⬍⬘*}共r兲兴, 共5.2兲 where J(␻⁾⫽⫺i␻␧0␹␺⁽␻). We observe that A_K,K is the power loss of mode ␺_K^¯⬍(␻) due to the scattering system.

From the Maxwell equations, following the same steps as for the derivation of the Poynting theorem, we find that

A_K,K⬘⫹⌽K,K⬘⫽0, 共5.3兲 with

⌽K,K⬘⫽

冖

^{关␺K¯⬍共r兲⫻H⬍¯K⬘*共r兲⫹␺K¯⬍⬘*共r兲⫻HK⬍¯共r兲兴•nda,}

共5.4兲 where H⫽(1/i␻␮0)“⫻␺ is the corresponding magnetic field, the integration is over a surface bounding the scattering system and n is the unit vector normal to the surface.

⫺⌽K,K is the real power flowing into the scattering system.

Equation 共5.3兲 is a compact form for the Poynting theorem (K⫽K⬘) and for the Lorentz reciprocity theorem (K⫽K⬘^).

By manupulating the vector products, Eq. 共5.4兲 can be re- written as

⌽K,K⬘⫽ i

␻␮0

冕 冋

^{␺K¯⬍共r兲•⳵⳵n␺K¯⬍⬘*共r兲}

⫺␺_K¯^⬍⬘*_共r兲• ⳵

⳵ⁿ␺_K⬍¯共r兲

册

^{da. 共5.5兲}

This surface integral can be evaluated by choosing as bound- ing surface the two planes at z⫽⫾L and by using the angular-spectrum representation of the fields共4.19兲. The gra- dients in the direction normal to the surfaces can be easily evaluated by using Eq.共4.20兲. We obtain

⌽K,K⬘⫽⌽_K,K^out _⬘⫺⌽_K,Kⁱⁿ _⬘, 共5.6兲 with

⌽_K,K^out _⬘⫽␧0

␲

兺

_Q ^关RKQRKQ⬘*_⫹TK^QT_K

⬘

Q*_{兴, 共5.7兲}

⌽_K,Kⁱⁿ _⬘⫽␧0

␲ ␦^{K,K⬘. 共5.8兲}

Using Eq.共4.22兲, we also directly obtain

关bˆK

h⬎共␻兲,bˆK^h⬎†⬘ 共␻⬘兲兴⫽ ␲

␧0

⌽_K,K^out _⬘共␻兲␦共␻⫺␻⬘兲. 共5.9兲 Summing Eq. 共5.1兲 and Eq. 共5.9兲, the Boson commutation rules for the output operators are thus readily obtained. Re- sult of this is that the commutation relations for the output operators are determined by energy conservation (K⫽K⬘⁾ and by reciprocity (K⫽K⬘). In particular, reciprocity en- sures the independence of output operators with different wave vector or polarization (关bˆK⬎,bˆ_K^⬎†_⬘兴⫽0 for K⫽K⬘^{). It} would be violated if output operators with different wave vector or polarization are not independent as much as the nput operators are. So far we have discussed only the com- mutation relations for the output photon operators. The equal-time QED commutation relations between the funda- mental fields are shown in Appendix A.

VI. LIGHT EMISSION AND ELECTRIC-FIELD FLUCTUATIONS

In this section we analyze the fluctuation properties of the electromagnetic field in presence of absorbing and/or emit- ting media and present some examples of light propagation in nonequilibrium.

Let us start considering vacuum fluctuations in presence of a scattering system. As it is well known, vacuum fluctua- tions play a fundamental role in quantum-optical processes 关37兴. By using Eq. 共3.8兲 and the relation 共A4兲 we obtain

具^Eˆi共r1,␻兲Eˆj共r2,␻⬘兲典0,0⫽Si j

0共r1,r₂,␻兲␦共␻⫺␻⬘兲, 共6.1兲 with

S_{i j}⁰共r1,r₂,␻兲⫽⫺共ប␻^2/␧0␲^c2兲Gi j

I共r1,r₂,␻兲. 共6.2兲

In Eq. 共6.1兲, 具典a,b indicates the expectation value, where (a,b) labels respectively the state of input light and the state of the material system. In this case (0,0) indicates the vacuum state for both the Hilbert spaces. Equation 共6.2兲 agrees with results obtained by applying the fluctuation- dissipation theorem关15兴.

Let us now consider a scattering system with an effective uniform temperature T embedded in a vacuum at zero tem- perature. By using Eq.共3.8兲 we obtain

具^Eˆi⫺共r1,␻兲Eˆ⫹j 共r2,␻⬘兲典^{0,T⫽Wi j}共r1,r₂,␻兲␦共␻⫺␻⬘兲, 共6.3兲 with

W_{i j}共r1,r₂,␻兲⫽N共␻^,T兲共ប␻^2/␧0␲^c2兲A˜共r1,r₂,␻兲.

共6.4兲 where A˜ (r₁,r₂,␻) is defined in Appendix A. Equation共6.4兲 is very similar to the expression used in Ref.关6兴 to calculate the cross-spectral density tensor of the near field thermally emitted into free space by an opaque planar source. Using Eq. 共A4兲, Eq. 共6.4兲 can be written in the form

W共r,r⬘^,␻兲⫽N¯共␻^,T兲

冋

^{S0共r,r⬘␻兲⫺2ប␧␻0˜␳共r,r⬘,␻兲}

册

^.

共6.5兲 This equation establishes a general relationship between the spatial variations of the second-order coherence tensors for vacuum fluctuations and spontaneous light emission. We ob- serve that, while vacuum fluctuations originate from both the scattering system and the input light modes, light emission in a zero-temperature free space comes only from the scattering system. This explains why the spatial variation of the tensor describing light emission can be obtained by subtracting from the contribution due to the vacuum fluctuation the con- tribution originating from the input light modes ^{˜ (r,r}␳ ⬘^,␻⁾ and eventually reflected by the thermal source.

The noise properties of the electromagnetic field are manifested by electric-field fluctuation spectrum in the ab- sence of any input signal. Let us consider a material system at a given uniform temperature. The electric-field correlation spectrum is defined by

具^Eˆ共r,␻兲Eˆ共r⬘^,␻⬘兲典^0,T⫽具^Eˆh共r,␻兲Eˆh共r⬘^,␻⬘兲典⁰

⫹具^Eˆp共r,␻兲Eˆp共r⬘^,␻⬘兲典T

⫽S共r,r⬘^,␻兲␦共␻⫺␻⬘兲. 共6.6兲 By inserting the expression for the electric-field operator de- rived in Sec. III, we obtain

具^Eˆh共r,␻兲Eˆh共r⬘^,␻⬘兲典0⫽具^Eˆh⫹共r,␻兲Eˆ⫺h共r⬘^,␻⬘兲典0

⫽ប␻

2⑀0^˜␳共r,r⬘^,␻兲␦共␻⫺␻⬘兲, 共6.7兲

具^Eˆp共r,␻兲Eˆp共r⬘^,␻⬘兲典T⫽2N共␻^,T兲⫹1 N共␻^,T兲

⫻W共r,r⬘^,␻兲␦共␻⫺␻⬘兲, 共6.8兲 hence Eq. 共6.6兲 can be written as

S共r,r⬘^,␻兲⫽S⁰共r,r⬘^,␻兲⫹2W共r,r⬘^,␻兲. 共6.9兲 We observe that both S⁰ and W for a specific system can be directly calculated once the Green tensor has been derived.

The power spectrum S(r,␻) of the electric-field fluctuations at position r is obtained by taking the trace of Eq. 共6.6兲.

These power spectra are usually obtained using the fluctuation-dissipation theorem 关18,19兴. Fluctuation- dissipation theorems have also been derived for amplifying media关38,39兴. However, this approach cannot be used when the whole system is not in thermal equilibrium as in the present example. In this case we are considering an attenu- ating or amplifying medium at a given effective temperature embedded in free space at zero temperature. In Fig. 2, we display the electric-field fluctuations S(r,␻) as a function of the radial distance for a pointlike scattering object embedded in free space. In Appendix B, the Green tensor for this el- ementary scattering system is derived. Figure 2 shows S(r,␻⁾ 共normalized with respect to the free-space value兲 for T⫽0 and for an effective temperature such that N(␻^,T)

⫽3. We have considered a point scatterer of radius a

⫽15 nm with complex permittivity ␧⫽6⫹0.8i. The wave- length of the radiation is 600 nm.

Let us now analyze another nonequilibrium physical situ- ation. We consider a scattering system with an effective uni- form temperature T₁ with mode occupation N₁ embedded in a thermal free space that is the cavity of a black body at temperature T₂ and mode occupation N₂ with walls very far from the scattering system. In this case the electric-field cross-spectral density tensor W⬘^,

具^Eˆi⫺共r1,␻兲Eˆ⫹j 共r2,␻⬘兲典^T₂^,T₁^⫽Wi j⬘共r1,r₂,␻兲␦共␻⫺␻⬘兲, 共6.10兲 is given by the following expression:

W_{i j}⬘共r1,r₂,␻兲⫽N1S_{i j}⁰共r1,r₂,␻兲

⫹ប␻

2␧0␳i j共r1,r₂,␻兲关N2⫺N1兴. 共6.11兲 We observe that the spatial variations of this correlation function change continuously as a function of T₁and T₂. At equilibrium (T₁⫽T2) the spatial behavior of W⬘ ^coincides with that of G^I and hence with that of the tensor S describing vacuum fluctuations. As expected G^I describes the electromagnetic-field fluctuations at equilibrium 关15兴. The light intensity as a function of frequency is proportional to I⬘^(r,␻⁾⫽Tr W⬘^(r,r,␻). We obtain

I⬘共r,␻兲⫽N1Tr S⁰共r,r,␻兲⫹ប␻

2␧0␳共r,␻兲关N2⫺N1兴, 共6.12兲 where ␳^(r,␻⁾⫽Tr^˜(r,r,␳ ␻); as it can be inferred from Eq.

共A3兲, ␳^(r,␻) describes the local optical density of states 共DOS兲. It gives the intensity of light at r due to incoherent illumination, i.e., with input light modes arriving from all the spatial directions and it is currently used to characterize the optical properties of PBG structures关4兴 and more generally of dielectric systems 关35,36兴. Before presenting some nu- merical results, we observe that when N₁ equals N₂, a situ- ation of thermal equilibrium is recovered and the spatial variation of light intensity is the same of vacuum fluctuations and is determined by the trace of the imaginary part of the Green tensor as prescribed by the fluctuation-dissipation theorem. Out of equilibrium the fluctuation-dissipation theo- rem does not hold. If N₁⫽0, which means that the medium is in its ground state and does not emit light, the spatial variation of I⬘^(r,␻) is determined by the local optical den- sity of states␳^(r,␻). In the opposite limit N₂⫽0, there is no input light and the spatial variation of I⬘^(r,␻) describes the emission pattern of the medium that is given by the trace of Eq. 共6.4兲.

Figure 3 displays I⬘^(r,␻^)/关(N1⫹N2)Tr G₀^I兴 for the same pointlike scattering object of Fig. 2. We consider different ratios N₂/N₁. Figure 3共a兲 obtained with N2/N₁⫽0 displays the emission pattern of the pointlike scatterer. Figure 3共c兲 calculated at equilibrium (N₁⫽N2) displays Tr G^I/Tr G₀^I , Fig. 3共e兲 displays the normalized local DOS. The other two panels describe intermediate situations. We point out that the oscillations observed in Figs. 3共b兲–3共e兲 originate from the interference between the input and the reflected light fields.

These oscillations are absent in Fig. 3共a兲 because in this case there is only emission from the scattering object. Figure 3 shows that oscillations increase when increasing the ratio N₂/N₁. This is the consequence of the definite phase relation between the input and the scattered lights共the input and the scattered lights are proportional to N₂), on the contrary the emitted light⬀N1does not interfere with input light. We also FIG. 2. Normalized 共with respect to free space兲 electric-field

fluctuations for a resonant-point scatterer at zero temperature共con- tinuous line兲 and at a given effective temperature 共dotted line兲 as a function of distance from the scatterer. Parameters are given in the text.

observe that Figs. 3共c兲 and 3共e兲 display a different spatial behavior showing that in the presence of absorption the well- known relationship between the Green tensor and the local DOS,

␳共r,␻兲⫽⫺ 2␻

␲^{c2Tr G}

I共r,r,␻兲 共6.13兲

is not correct.

VII. CONCLUSIONS

In conclusion, we have presented a general quantum theory of light scattering for 3D systems of arbitrary geom- etry, providing a unified basis for analyzing a large class of optical processes where quantum and/or thermal fluctuations play a role. We have derived general 3D quantum-optical input-output relations providing the output photon operators in terms of the input photon operators and of the noise cur- rents of the scattering system. These relations hold also for photon operators associated with evanescent fields and thus can be applied to the analysis of evanescent nonclassical fields, e.g., arising from the scattering of nonclassical input light by nanometric objects. The theory puts forward the con- nection between general theorems of classical electrodynam- ics and commutation relations for the output photon opera- tors carrying all the information on the scattering and/or emission process. We have shown that this theory satisfies QED commutation rules by using a novel relationship be- tween vacuum and thermal fluctuations. Applications involv- ing scattering from complex nanometric scattering objects

are under current development.

ACKNOWLEDGMENT

We acknowledge the support of INFM through the Project NanoSNOM.

APPENDIX A: QED COMMUTATION RELATIONS In order to ensure overall consistency of this treatment, we show how fundamental equal-time commutation relations of QED are preserved. This consistency check has been proved for a quite general three-dimensional dielectric with a local and scalar permittivity, not including the homogeneous solution of the electric-field operator关25兴, i.e., by expressing the electric-field operator via the Green tensor as a function of only the noise currents. Here we generalize this result showing that equal-time commutation relations of QED are preserved also for more general systems that can be aniso- tropic or even driven by a nonlocal susceptibility. Moreover, we also include the homogeneous solution of Eq. 共2.1兲, thus considering explicitly the scattered fields from bounded scat- tering systems. Let us consider the共equal-time兲 commutation relations between the fundamental fields Eˆ (r,t) and Bˆ(r,t).

Using the expression for the homogeneous electric-field op- erator and that i␻^Bˆ⫹(␻⁾⫽“⫻Eˆ^⫹(␻), we obtain

关Eˆi共r兲,Bˆl共r⬘兲兴⫽ iប 2␧0⑀lm j⳵m

r⬘

冕

0

⬁

d␻

冋

^{␳˜i, j共r,r⬘兲}

⫹2␻

␲^{c2A˜i j共r,r}⬘兲

册

^{⫺c.c., 共A1兲}

where

A˜⫽Ges

IG* _共A2兲

and

␳

˜_{i, j}共r,r⬘兲⫽

兺

_␶,K ^{␺␶K,i共r兲␺K, j␶*共r⬘兲. 共A3兲}

Equation共A1兲 can be simplified using the following relation:

␲^c2

2␻ ␳^{˜i, j共r,r}⬘兲⫽⫺Gi j

I共r,r⬘兲⫺A˜i j共r,r⬘兲. 共A4兲 This equation can be proved using the mode expansion共4.3兲 of G⁰ and applying the Dyson equation. Equation共A4兲 has been demonstrated for particular cases 关21,23兴. Its general derivation will be presented elsewhere. By using Eq. 共A4兲, Eq. 共A1兲 reduces to

关Eˆi共r兲,Bˆl共r⬘兲兴⫽⫺ iប

␲␧0⑀lm j⳵m r⬘

冕

0

⬁

d␻␻

c²G_{i j}^I共r,r⬘^,␻兲⫺c.c.

共A5兲 FIG. 3. Normalized light intensity I⬘^(r,␻)/关(N1⫹N2)Tr G₀^I兴 as

a function of distance from a resonant-point scatterer under differ- ent mode occupations N1and N2. Parameters are given in the text.

Furthermore, from Eq. 共3.7兲 and the relation ␹*⁽␻⁾

⫽␹⁽⫺␻^{) it follows that G}i j*(r,r⬘^,␻⁾⫽Gi j(r,r⬘^,⫺␻^), Eq. 共A5兲 becomes

关Eˆi共r兲,Bˆl共r⬘兲兴⫽⫺ ប

␲␧0⑀lm j⳵m r⬘

冕

⫺⬁

⬁

d␻␻

c²G_{i j}共r,r⬘^,␻兲.

共A6兲 Although, owing to generalizations, the starting point 共A1兲 was quite different from Eq. 共26兲 of Ref. 关24兴, after using Eq. 共A4兲 and some manipulation we arrived at Eq.

共A6兲 that coincides with the corresponding findings in Refs.

关11,24兴. From now the canonical commutation relations can be demonstrated using a machinery analogous to that of Ref.

关11兴. In particular equal-time commutation relations of QED are preserved if the following relation holds:

⑀lm j⳵m r⬘

冕

⫺⬁

⬁

d␻␻

c²G_{i j}共r,r⬘^,␻兲⫽⫺i␲⑀lm j⳵m

r⬘␦i j␦共r⫺r⬘兲.

共A7兲 For the sake of completeness and also because we adopt a medium susceptibility with a more complex structure, in the following we provide a concise demonstration of Eq. 共A7兲.

First we observe that the Kramers-Kronig relations imply that the causal-complex-valued susceptibility tensor

␹i j(r,r⬘^,␻) is a holomorfic function of␻ in the upper com- plex plane. Moreover, Kramers-Kronig relations imply that for兩␻兩→⬁, ␹i j(r,r⬘^,␻⁾→0 at least as␻^⫺1. Also the Green tensor for causality requirements is a holomorfic function of

␻ in the upper complex plane. This can also be derived ex- plicitly from applying iteratively the Dyson equation and ob- serving that both ␹i j and G⁰ are holomorfic in the upper complex plane. The analytical properties of G⁰ can be di- rectly inspected, with its analytical expression being known.

The free-space Green tensor can be expanded in plane waves as

G_{i j}⁰共r,r⬘^,␻兲⫽ 1

共2␲兲³

冕

^{Gi j0共p,␻兲eip•(r⫺r⬘)dp, 共A8兲}

with

G_{i j}⁰共p,␻兲⫽ 1

k²⫺p²

冉

^{␦i j⫺ppip2j}

冊

^{⫹k12 ppip2j. 共A9兲}

From these expansion it follows that for 兩␻兩→⬁, G_{i j}⁰(r,r⬘^,␻⁾ approaches zero as ␻^⫺2. We note that G_{i j}⁰(r,r⬘^,␻⁾ 关see Eq. 共A9兲兴 is singular at␻⫽0.

By introducing the Dyson equation into the integral on the right-hand side共rhs兲 of Eq. 共A11兲, the expression inside the integral becomes

␻ c²G⫽␻

c²G⁰⫹␻

c² _n

兺

_⫽1^{⬁ 关G0es兴nG0, 共A10兲}

where we have developed by iteration the Dyson equation.

Owing to the last term in Eq.共A9兲, the two terms on the rhs

of Eq. 共A10兲 are singular at␻⫽0. Nevertheless, these sin- gularities, that have to be treated as principal values, does not contribute to the integral being odd functions (⬀␻^{⫺(2n⫹1)) of}␻. Now we observe that the summation on the left-hand side of Eq. 共A10兲 for 兩␻兩→⬁ approaches zero at least as ␻^⫺2, thus this summation does not contribute to the integral as can be evaluated by performing the integration on the upper complex plane. As a consequence, we obtain

冕

⫺⬁

⬁

d␻␻

c²G_{i j}共r,r⬘^,␻兲⫽

冕

⫺⬁

⬁

d␻␻

c²G_{i j}⁰共r,r⬘^,␻兲.

共A11兲 The expression for the G⁰ in the real space is

G_{i j}⁰共r,r⬘^,␻兲⫽⫺

冉

^{␦i j⫺k12⳵ir⳵jr⬘}

冊

^{g0共r⫺r⬘,␻兲, 共A12兲}

where

g⁰共r⫺r⬘^,␻兲⫽ e^{ik兩r⫺r⬘兩} 4␲兩r⫺r⬘兩.

Inserting Eq 共A12兲 in Eq. 共A6兲 and recalling that

⑀lm j⳵m r⬘⳵j

r⬘兵•••其⫽0 we obtain 关Eˆi共r兲,Bˆl共r⬘兲兴⫽ ប

␲␧0⑀lm j⳵m r⬘␦i j

冕

⫺⬁

⬁

d␻␻

c²g⁰共r⫺r⬘^,␻兲.

共A13兲 Using the known relation 关40兴

冕

⫺⬁

⬁

d␻␻

c²g⁰共r⫺r⬘^,␻兲⫽i␲␦共r⫺r⬘兲, 共A14兲 we finally obtain

关Eˆi共r兲,Bˆl共r⬘兲兴⫽⫺iប

␧0⑀lm j⳵m

r␦i j␦共r⫺r⬘兲. 共A15兲

Similarly, it can be shown that 关Eˆi共r兲,Eˆl共r⬘兲兴⫽0, 关Bˆi共r兲,Bˆl共r⬘兲兴⫽0.

We also point out that Eq.共A7兲 proved here is also the con- dition for obtaining the correct commutation relations for the potentials and canonically conjugated momenta 关11,24兴.

APPENDIX B: THE GREEN TENSOR FOR A POINTLIKE SCATTERING OBJECT

Let us consider an absorbing pointlike scattering object. It can be regarded as the building block of much more compli- cated scattering objects. It has been shown how to calculate the Green tensor of complex nanopatterned scattering objects by discretizing them in terms of these building blocks 关4兴.

Following the approach by de Vries et al.关35兴, it is possible to obtain an analytical expression for the Green tensor of this very simple 3D system.

As it is well known, the Green tensor at r⫽0 has a sin- gular behavior. Performing a regularization procedure 关35兴