Canonical quantum description of light propagation in dielectric media

E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 4

Canonical quantum description of light propagation in dielectric media* by

A. Luks and V Pefinova Laboratory of Quantum Optics and Research Center for Optics, Faculty of Natural Sciences, Palacky University, Tfida Svobody 26, 771 46 Olomouc, Czech Republic

This article is dedicated to Professor Jan Pei^ina on the occasion of his 65th birthday. 295

Contents

Page § 1. Introduction

297

§ 2.

Origin of the macroscopic approach

304

§ 3. Macroscopic theories and their appHcations

323

§ 4.

385

Microscopic theories

§ 5. Microscopic models as related to macroscopic concepts . . . .

414

§ 6.

Conclusions

424

Acknowledgments

425

References

426

296

§ 1. Introduction The importance of quantum optics cannot be denied at present. Part of the investigations in this field can be interpreted as a quantum theory of nonhnear optics. Nonhnear optical effects are proportional to the nonlinear optical susceptibility (Boyd [1999, 2000]). Theories which have been used to describe the interaction of a quantized electromagnetic field with a nonlinear dielectric medium are either phenomenological or derived by quantizing the macroscopic Maxwell equations. The microscopic approach has become established as an alternative. Justification of the Hamiltonians used in the quantum theory of nonlinear optics is an important part of placing the theory on a firmer basis. The phenomenological (effective) Hamiltonians which were previously studied in quantum optics and still have their importance can in fact be derived in a "pre-quantal" form using the Maxwell equations. The question arises whether the quantization can be performed on a more fundamental level, say, on that of the quantized Maxwell equations in the Heisenberg picture or on that of an appropriate description in the Schrodinger picture, so that one can equally well or perhaps better arrive at an effective Hamiltonian. From the historical viewpoint, the problem of quantizing the electromagnetic field in vacuo was solved long ago by Dirac [1927], and the quantization of a nonlinear theory is due to Born and Infeld [1934, 1935]. With respect to propagation in linear dielectric media it is appropriate to refer first to Jauch and Watson [1948]. The revived interest in this problem in the last twelve years can be traced to some dissatisfaction with the situation following the advent of the laser in 1958. The new optical effects are analyzed both by nonlinear optical methods belonging to classical physics and by quantum optics (Shen [1969]). The normal-mode-expansion approach used in quantum optics is well-suited to systems in optical cavities, such as an optical parametric oscillator, but it is not appropriate for open systems such as a parametric amplifier. In nonlinear optics (Bloembergen [1965], Shen [1984]), the Maxwell equations completed by the constitutive relations are solved utilizing the slowlyvarying-envelope approximation, and the resultant equations are sometimes simplified using parametric approximation. In one particular case, this led to a quantum-optical theory of counterpropagation without introducing an effective 297

298

Quantum description of light propagation in dielectric media

[4, § 1

Hamiltonian, which is usual in the case of copropagation (Milburn, Walls and Levenson [1984]). It has become standard practice in quantum optics to introduce phenomenological Hamiltonians without a quantitative connection to the classical equations describing nonlinear optical effects. This may be considered as unsatisfactory: while some processes occur in the classical regime, others in the quantum regime, the distinction is made only on the basis of the light intensity detected, and when the intensity changes continuously the boundary between the two domains cannot be determined. Philosophically, it would be desirable to replace any classical description with a quantum counterpart, but this goal has been considered either to be unreasonable or to have been achieved by the fathers of quantum theory in a form unsuitable for contemporary physicists. The quantization of the electromagnetic field in the presence of a dielectric is possible. This can be done in two ways: the macroscopic approach and the microscopic approach. In the macroscopic approach, the medium is completely described by its linear and nonlinear susceptibilities. No degrees of freedom of matter appear explicitly in this treatment. First a Lagrangian is sought which produces the macroscopic Maxwell equations for the field in a nonlinear medium; then the canonical momenta and the Hamiltonian are derived. Quantization is accomplished by imposing the standard equal-time commutation relations. In the microscopic approach, a model is constructed for the medium; degrees of freedom of both field and matter appear in the theory, and these are quantized. The result is a theory of mixed matter-field (polariton) modes, which are coupled by a nonlinear interaction. Hillery and Mlodinow [1984] used the electric displacement field as the canonical variable for nonlinear quantization, and they explored the macroscopic approach to the quantization of homogeneous nondispersive media. They pointed out that there is a difficulty in including the dispersion in the quantized macroscopic theory. The importance of a proper space-time description of squeezing has been recognized by Bialynicka-Birula and Bialynicki-Birula [1987]. The problem of a proper quantum-mechanical description of the operation of optical devices has been addressed by Knoll, Vogel and Welsch [1986, 1987]. In the past, many authors dealing with macroscopic quantum theories of light propagation wrote also on spatial displacements, shifts and translations of the electromagnetic field along with temporal displacements, shifts and translations, or simply the (time) evolution. Accordingly, they used the term "space evolution" in the former case. In the following, we will use the term space progression instead of space

4, § 1]

Introduction

299

evolution. Abram [1987] attempted to overcome the difficulties of conventional quantum optics by reformulating its assumptions. He based the formalism on the momentum operator for the radiation field and thus investigated not only the spatial progression of the electromagnetic wave, but also refraction and reflection. In addition to previous work devoted to the concept of quasinormal modes (Lang, Scully and Lamb Jr [1973], Barnett and Radmore [1988]), the modes of the universe have been used in the treatment of the spectrum of squeezing (Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a,b]). A quantum description of linear optical devices is of interest provided it includes refraction and reflection. Similarly, a quantum description of nonlinear devices should include effects such as solitons and their dynamics. The dispersion has been treated on the assumption of a narrow-band field (Drummond [1990]). Besides this, an attempt at formulating a quantum theory of the propagation of an optical wave in a lossless dispersive dielectric material has been made by Blow, Loudon, Phoenix and Sheperd [1990]. These applications made use of the fact that nonlinear quantum-optical processes are described quantum-optically in the parametric approximation with linear mathematical tools, so that quantization procedures and solutions of the dynamics need not face such immense difficulties as with a really nonlinear formalism (Huttner, Serulnik and Ben-Aryeh [1990]). An original approach to the description of a degenerate parametric amplifier (Deutsch and Garrison [1991a]) has been related to the theory of paraxial quantum propagation (Deutsch and Garrison [1991b]). A formalism for the macroscopic approach to quantization was developed by Abram and Cohen [1991]. The space-time displacement operators have been related to the elements of the energy-momentum tensor (Serulnik and Ben-Aryeh [1991]). The macroscopic quantization of the electromagnetic field has been applied to inhomogeneous media (Glauber and Lewenstein [1991]). Huttner and Barnett [1992a,b] presented a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on the Hopfield model of such a dielectric, where the matter is represented by a harmonic polarization field (Hopfield [1958]). A microscopic theory of an optical field in a lossy linear optical medium has been developed (Knoll and Leonhardt [1992]). S.-T. Ho and Kumar [1993] have contributed to the derivation of the macroscopic field operators. They discussed the questions of light propagation across a dielectric boundary and of squeezing in a linear dielectric medium. Abram and Cohen [1994] developed a traveling-wave formulation of the theory of quantum optics and applied it to quantum propagation of light in a Kerr medium. Theoretical methods for investigating propagation

300

Quantum description of light propagation in dielectric media

[4, § 1

in quantum optics in which the momentum operator is used along with the Hamiltonian were developed by Toren and Ben-Aryeh [1994]. Drummond has presented a review of his theory and its applications (Drummond [1994]). Jeffers and Barnett [1994] modeled the propagation of squeezed light through an absorbing dispersive dielectric medium. A quantum theory of a field in 1+1 dimensions coupled to localized oscillators was developed and an analytic solution to the Heisenberg equation was given by Boivin, Kartner and Haus [1994]. Multimode consideration of nonclassical effects is needed when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the propagation direction of the light beam (Kolobov [1999]). Matloob, Loudon, Barnett and Jeffers [1995] provided expressions for the electromagnetic field operators for three geometries: an infinite homogeneous dielectric, a semi-infinite dielectric and a dielectric slab. A microscopic derivation has shown that a canonical quantum theory of light at the dielectric-vacuum interface is possible (Barnett, Matloob and Loudon [1995]). Following Huttner and Barnett [1992a,b], Gruner and Welsch [1995] calculated the ground-state correlation of the quantum-mechanical intensity fluctuations. Dalton, Guerra and Knight [1996] dealt with the quantization of a field in dielectrics and applied it to the theory of atomic radiation in a one-dimensional Fabry-Perot resonator. Hradil [1996] considered "lossless" dispersive dielectrics, i.e., dielectrics with a thin absorption line. He formulated a canonical quantization of the electromagnetic field in a closed Fabry-Perot resonator with a dispersive slab. A simple quantum theory of the beamsplitter, which can be applied to a Fabry-Perot resonator, was introduced by Barnett, Gilson, Huttner and Imoto [1996] and developed by Barnett, Jeffers, Gatti and Loudon [1998]. Extensions of the previous work on the propagation in absorbing dielectrics took linear amplification into account (Jeffers, Barnett, Loudon, Matloob and Artoni [1996], Matloob, Loudon, Artoni, Barnett and Jeffers [1997], Artoni and Loudon [1998]). Artoni and Loudon [1997] applied the Huttner-Barnett scheme for quantization of the electromagnetic field in dispersive and absorbing dielectrics to calculations of the effects of perpendicular propagation in a dielectric slab and to the properties of the incident light pulse. Their approach has provided a deeper understanding of antibunching (Artoni and Loudon [1999]). Brun and Barnett [1998] considered an experimental set-up using a two-photon interferometer, where insertion of a dielectric into one or both arms of the interferometer is essential. Gruner and Welsch [1996a] performed an expansion of the field operators which is based on the Green fianction of the classical Maxwell equations and preserves the equal-time canonical commutation relation of the field. They found that the spatial progression

4, § 1]

Introduction

301

can be derived on the assumption of weak absorption. Dutra and Furuya [1997] considered a single-mode cavity filled with a medium consisting of twolevel atoms that are approximated by harmonic oscillators. They showed that macroscopic averaging of the dynamical variables can lead to a macroscopic description. Dutra and Furuya [1998a,b] observed that the (full) Huttner-Barnett model of a dielectric medium does not comprise all the dielectric permittivities of the medium expected from classical electrodynamics, although the field theory in linear dielectrics should have such a property. Schmidt, Jeffers, Barnett, Knoll and Welsch [1998] extended the microscopic approach to the quantum theory of light propagation to nonlinear media; a generalized nonlinear quantum Schrodinger equation well-known from the description of quantum solitons was derived for a dielectric with a Kerr nonlinearity Dung, Knoll and Welsch [1998] developed a quantization scheme for the electromagnetic field in a spatially varying three-dimensional linear dielectric which causes both dispersion and absorption. The well-known Green function was used for the case of a homogeneous dielectric, and it was shown that the indicated quantization scheme exactly preserves the fundamental equal-time commutation relations of quantum electrodynamics. The Green function has also been used in the more complicated case of two dielectric bodies with a common planar interface. The quantization of the full electromagnetic field in linear isotropic inhomogeneous Kramers-Kronig dielectrics based on the integral representation of the field with the Green tensor yields exactly the equal-time commutation relation of quantum electrodynamics (Scheel, Knoll and Welsch [1998]). Spontaneous decay of an excited atom in the presence of dispersing and absorbing bodies has been investigated using an extension of this formalism (Dung, Knoll and Welsch [2000]). Electromagnetic field quantization in an absorbing medium has been readdressed, and the Casimir effect both for two lossy dispersive dielectric slabs and between two conducting plates was analyzed by Matloob [1999a,b] and by Matloob, Keshavarz and Sedighi [1999]. A quantum scattering-theory approach to quantum-optical measurements has been expounded by Dalton, Barnett and Knight [1999a]. In addition to Lang, Scully and Lamb Jr [1973], and along with an independent work (K.C. Ho, Leung, Maassen van den Brink and Young [1998]) devoted to the concept of quasinormal modes, a quasimode theory of macroscopic canonical quantization was invented and applied by Dalton, Barnett and Knight [1999b,c,d]. A macroscopic canonical quantization of an electromagnetic field and a system of a radiating atoms, involving classical, linear optical devices, based on expanding the vector potential in terms of quasimode fimctions, was carried out by Dalton, Barnett and Knight

302

Quantum description of light propagation in dielectric media

[4, § 1

[1999b]. The relationship between the pure-mode and quasimode annihilation and creation operators was determined by Dalton, Barnett and Knight [1999c]. A quantum theory of the lossless beamsplitter is given in terms of the quasimode theory of macroscopic canonical quantization. The input and output operators that are related via the scattering operator are directly linked to multi-time quantum correlation functions (Dalton, Barnett and Knight [1999d]). Brown and Dalton [2001a] have generalized the quasimode theory of macroscopic quantization in quantum optics and cavity quantum electrodynamics developed by Dalton, Barnett and Knight [1999a,b]. This generalization admits the case where two or more quasipermittivities are introduced. The generalized form of quasimode theory has been applied to provide a fully quantum-theoretical derivation of the laws of reflection and refraction at a boundary (Brown and Dalton [2001b]). Using the microscopic approach, Hillery and Mlodinow [1997] devoted themselves to the standard optical interactions, and derived an effective Hamiltonian describing counterpropagating modes in a nonlinear medium. On considering multipolar coupled atoms interacting with an electromagnetic field, a quantum theory of dispersion has been obtained whose dispersion relations are equivalent to the standard Sellmeir equations for the description of a dispersive transparent medium (Drummond and Hillery [1999]). The Green-fijnction approach to the quantization of the phenomenological Maxwell theory was used by Gruner and Welsch [ 1996b] in the derivation of the quantum-optical input-output relations in the case of propagation in dispersive absorbing multilayer slabs. The behavior of short light pulses propagating in a dispersive absorbing linear dielectric, with a special attention to squeezed pulses, was studied by Schmidt, Knoll and Welsch [1996]. Scheel, Knoll, Welsch and Barnett [1999] found quantum local-field corrections appropriate for the spontaneous emission by an excited atom. Knoll, Scheel, Schmidt, Welsch and Chizhov [1999] investigated quantum-state transformation by dispersive and absorbing four-port devices. Under the usual assumptions on the dielectric permittivity, quantization of the Hamiltonian formalism of the electromagnetic field using a method close to the microscopic approach was performed by Tip [1998]. A proper definition of band gaps in the periodic case and a new continuity equation for energy flow was obtained, and an ^-matrix formalism for scattering from absorbing objects was worked out. In this way the generation of Cerenkov and transition radiation have been investigated. A path-integral formulation of quantum electrodynamics in a dispersive and absorbing dielectric medium has been presented by Bechler [1999], and has been applied on the microscopic level to the quantum theory of electromagnetic fields in dielectric

4, § 1]

Introduction

303

media. Results concerning quantum electrodynamics in dispersing and absorbing dielectric media have been reviewed by Knoll, Scheel and Welsch [2001]. Tip, Knoll, Scheel and Welsch [2001] have proven the equivalence of two methods for quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics: the Langevin noise current method and the auxiliary field method. In linear optical couplers both co- and counterpropagation have been considered, and an attempt at quantization was made by Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991]. A theory of the electromagnetic field with the time axis replaced by one of the spatial axes was outlined. This replacement corresponds to replacing the Hamiltonian by a momentum operator (BenAryeh, Luks and Pefinova [1992]). The flexibility of the classical canonical description and the obstacles to quantization in the case of counterpropagation in a nonlinear medium were analyzed by Luis and Pefina [1996]. Di Stefano, Savasta and Girlanda [1999] extended the field quantization to those material systems whose interaction with light is described, near a medium boundary, by a nonlocal susceptibility. Suggestive is a comparative study of fermion and boson beamsplitters (Loudon [1998]). Fermions can be studied in analogy with bosons (Cahill and Glauber [1999]). Independently, the theory of light propagation in a Bose-Einstein condensate and a zero-temperature noninteracting FermiDirac gas has been developed (Javanainen, Ruostekoski, Vestergaard and Francis [1999]). It is appropriate to mention here work concerning the photon wave function (Bialynicki-Birula [1996a,b, 1998], Inagaki [1998], Hawton [1999], Kobe [1999]), although it is relevant mainly to the electromagnetic field in vacuo. In this chapter we will review spatio-temporal descriptions of the electromagnetic field in linear and nonlinear dielectric media, applying macroscopic and microscopic theories. We will treat both macroscopic theories appropriately generalized from the free-space quantum electrodynamics and microscopic theories as related to the macroscopic descriptions. We will deal with the quasimode theory of macroscopic canonical quantization and the scattering operator theory in this chapter. We refer to an excellent review of linear and nonlinear couplers by Pefina Jr and Pefina [2000], where the restriction to mere spatial behavior of interesting optical fields has been accepted. We will use units following the original papers, and although the system of international (SI) units prevails, there are exceptions: some of the relations (2.14)-(2.47) and (3.192)-(3.216) are in Gaussian units, relations (3.217)-(3.273) are in rationalized cgs units, and relations (2.1)-(2.13), (3.13)-(3.93) and (3.274)(3.337) are in Heaviside-Lorentz units.

304

Quantum description of light propagation in dielectric media

[4, § 2

§ 2. Origin of the macroscopic approach 2.1. Nondispersiue lossless homogeneous nonlinear dielectric The approach to a quantum theory of Hght propagation considered standard at least until the critique by Hillery and Mlodinow [1984], has been assessed by Drummond [1990, 1994] in a somewhat critical way. Concerning this approach, let us consider papers by Shen [1967, 1969]. The theory is restricted to steadystate propagation taking place in one dimension along the z-axis. On quantizing in a volume L^ and assuming that the field does not vary appreciably over a distance d large compared with the wavelength, and associating the discrete values of the wavevector k with d (instead of I), the localized annihilation and creation operators bk{z) and b\{z) have been proposed. An appropriate component of the vector potential operator has the expansion of the form

Mz,t) = cJ2J

2^^^13

{h(z)Qxp[-i{co^t - kz)] + H.c.Y

(2.1)

where z is the propagation distance, c is the speed of light, (O/, is the frequency, h is the Planck constant divided by 2jr, e^ = ^{(O/,) is the value of the dielectric function e at co^, and H.c. denotes the term Hermitian conjugate to the previous one. The annihilation and creation operators S^(z) and bl(z), respectively, satisfy the equal-space commutation relation

h(z)/bUz)

= 4A-'1.

(2.2)

The localized Hamiltonian and momentum operators H{z) and V(z) have been defined in terms of a Hamiltonian density H{z). It has not been specified whether these operators can always be written simply in terms of the localized operators bf,(z) and bjiz). Shen [1969] suggested that the momentum operator plays the role of the translation operator. The translation is interpreted as space "evolution" and appropriately described (cf eq. 2.29). We call this transformation a space progression. In analogy with the time-ordered product, the space-ordered product is introduced. An analogue of the Heisenberg and Schrodinger pictures is expounded. A localized density matrix (statistical operator) p(z) is defined. According to the pioneering paper of Hillery and Mlodinow [1984], the standard macroscopic quantum theory of electrodynamics in a nonlinear medium is due to Shen [1967] and has been elaborated upon by Tucker and Walls [1969]. Hillery and Mlodinow [1984] have pointed out some problems with the standard

4, § 2]

Origin of the macroscopic approach

305

theory, above all the fact that it is not consistent with the macroscopic Maxwell equations. One approach to the derivation of a macroscopic quantum theory would be to begin from a quantum-microscopic theory, as will be reviewed below. The other approach is to take the expression for the energy of the radiation in a nonlinear medium, which differs from the free-field Hamiltonian in part, and to keep interpreting the electric field (up to sign) as the variable canonically conjugated to the vector potential. (Note that this differs from Shen [1969].) In order to understand the problems of a previous theory, Hillery and Mlodinow examined its Hamiltonian formulation. This is the noncanonical Hamiltonian ^noncan(0 "^ ^ E M ( 0 + ^lnoncan(0?

(2-^)

where HEuit) is a free-field Hamiltonian and Anoncan(0 is an interaction Hamiltonian, ^EM(0 = \ j{E^+B^)d'x, ^lnoncan(0

=-

h p d ' x ,

(2.4) (2.5)

where E^ = E • E, with E = E{x, t) the magnitude of the electric field strength operator E{x, t), B^ = B B, with B = ^(jc, t) the magnitude of the magnetic induction field operator B{x,t), P = P{x,t) is the polarization operator of the medium, and we have used Heaviside-Lorentz units. The polarization of the medium is a fiinction of the electric field strength operator which may be written as a power series. It can be seen easily that, as an undesirable "quantum effect", we obtain an improper expression for the time derivative of the magnetic induction field operator B. Hillery and Mlodinow [1984] assumed that the medium is lossless, nondispersive and homogeneous. A Lagrangian is considered which gives proper equations of motion. The electric field E = E(x, t) and the magnetic induction field B = B{x, t) are expresssed in terms of the vector potential A ^ A{x, t) and the scalar potential AQ = Ao{x, t), dA E = ——-VAo, at

B^VxA.

(2.6)

The appropriate Lagrangian density depends on first partial derivatives of the four-vector A = A{xj) = {AQ,A). The momentum canonical to A is

Quantum description of light propagation in dielectric media

306

[4, §2

n = 77(jc, t) = (TTo, n), where TIQ = /7o(JC, t) = 0. The vanishing of i7o indicates that the system is constrained. It has been shown how to utiHze the quantization procedure developed by Dirac [1964] for constrained Hamiltonian systems. It can be derived that the canonical momentum is 77 = n(x,t) = -D, where D = D(x, t) is the electric displacement field. The canonical Hamiltonian has the form (2.7)

H(t) = HEM(t)^HM, where HEMit) = \

(2.8)

f(EE-^BB)d^x,

H,(t) - j E- \p-

j P{XE)(M

d^jc,

(2.9)

with P = P(jc, 0, P{E) = P[E{x, t),x, t] being the polarization of the medium. In order to simplify the quantization of the macroscopic Maxwell theory, the dual potential A = A(jc, t) has been introduced along with A = A(x, t) and Ao = Ao(jc, t)\ these we call the dual vector and scalar potentials, respectively. The relations in eq. (2.6) are replaced by Z) = V X A,

^^f.VAo.

(2.10)

It can be shown that the canonical momentum is 77^ = n^(x,t) = B. Finally, upon expressing the canonical Hamiltonian functional in terms of the electric displacement and magnetic induction fields the results are the same, H = H^(x, t) = H^. We can compare

AiixJXtlAx'A

=id^{x-x')\.

(2.11)

with A,(x,0,iT/(V,0

=id^{x-x')\.

(2.12)

where 6^{x) = d{x)6ij +

d" 1 dxidxj An\x\

(2.13)

are components of the transverse tensor-valued b function (Bjorken and Drell [1965]). Hillery and Mlodinow [1984] do not mention propagation, except in

4, § 2]

Origin of the macroscopic approach

307

a paragraph on interpretation problems, where they recommend to confine the medium to part of the quantization volume and to place the field source and the detector outside the medium, being aware that these require propagation to be taken into account. It is added that different diagonalizations indicated by the quadratic part of the total Hamiltonian generate different kinds of normal ordering. Doubt is expressed regarding the existence of a suitable ordering, and the microscopic approach is propounded. Dispersion is also considered a reason for contemplating a microscopic theory.

2.2. Nondispersive lossless linear dielectric 2.2.1. Momentum operator as translation operator In the late 1980s, the problem of propagation did not seem to be typical for quantum optics. Abram [1987] addressed the problem of light propagation through a linear nondispersive lossless medium. Although this model can be an appropriate limit of the Huttner-Bamett model, we expound the main ideas of (Abram [1987]). Abram criticized the modal Hamiltonian formalism, especially the inclusion of a linear polarization term in the Hamiltonian:

^(0-w-

8jr 7/ /(E'+H'-^4jTxE')dV,

(2.14)

where H^ = H - H, with H = H(x, t) the magnitude of the magnetic field strength operator H{x, t), x is the (linear) susceptibility of the material, and V is the quantization volume. This would lead to an incorrect result, mainly to a frequency change of modes which does not occur. Abram decided to extend the traditional theory of quantum optics in such a way that it could describe propagation phenomena without invoking the modal Hamiltonian; he observed that one of the propagation phenomena, refraction, suggests the momentum to be the appropriate concept for describing these phenomena. Quantum-mechanically, space and momentum are canonically conjugate variables. Huttner and Bamett [1992a,b] have demonstrated in a microscopic theory that a Hamiltonian including the light-matter interaction can be chosen. It is a good remedy against the idea that space and momentum are canonically conjugate variables like time and energy.

308

Quantum description of light propagation in dielectric media

[4, § 2

The propagation of the electromagnetic field is described by the Maxwell equations VxH

= - ^ , c at ^ ^ ^OB VxE =—^, c at V B =0,

(2.15) ,, ,,, (2.16) (2.17)

V-Z) = 0 ,

(2.18)

where the electric displacement field in the chosen system of units isD = E + 4jtP. It is assumed in the following that there are no free charges or currents and that we are dealing with nonmagnetic materials, so that B = H. For simplicity we consider only the case of plane waves propagating along the z-axis, with the electric field polarized along the x-axis, and the magnetic field along the 7-axis. This reduces the Maxwell equations to scalar differential equations, the directions of all vectors being implicit. Further it is assumed that light is propagating in a linear dielectric, where the induced polarization is at all times proportional to the incident electric field, P(z,t) = xE(z,t), where the susceptibility of the material is assumed for simplicity to be a scalar (neglecting its tensorial properties), independent of frequency (no dispersion). It is convenient to define also the dielectric function e of the material, e = 1 +4jrx, and the refi-active index, n = y^. The change in the total energy which is given by the integrated energy flux (the Poynting vector) over the surface of a body or volume is proposed in (Abram [1987]) as the proper quantum-mechanical Hamiltonian. The change in the total momentum is given as the integrated flux of the Maxwell stress tensor. The momentum is treated on the same footing as the Hamiltonian. However, the enigma of the Hamiltonian (2.14) is solved. It is possible to consider a square pulse which enters a dielectric. The total energy is conserved, but the energy density is increased by a factor of «, because the volume V reduces to F' = V/n. In volume V^ the wavelengths of the modes become A' = A/«, but the oscillator frequencies remain unchanged. It is interesting that in the absence of reflection, the electric and magnetic fields of the transmitted (T) waves in the dielectric are related to the corresponding incident (I) fields in free space by Ej(z, t) = -^Ei(z, t\

Hj(z, t) = V^mz,

t).

(2.19)

This change in the energy density implies a similar increase for the total momentum of the pulse, the components of which are always proportional to the

4, § 2]

Origin of the macroscopic approach

309

wavevectors of the excited modes. In propagation along the z-axis the Maxwell stress tensor is replaced by the energy density. When both directions of the propagation along the z-axis in free space are considered with the electric field polarized along the x-axis and the magnetic field along the>^-axis (x = 0,e= 1), the electromagnetic vector potential operator A = A(z, t) is usually written as i(z,0 = c Y , \ \ ^

(a/e'^'^'^-'^>" +a;e-*"'^^''-) ,

(2.20)

where Oj and hj are the creation and annihilation operators for a photon in the yth mode of wavevector kj (with k-j = -kj) and frequency (DJ = c\kj\ ftilfilling the boson commutation relations. In order to simplify notation, unit vectors are omitted. It is convenient to rearrange eq. (2.20) in a manner that is familiar to solid-state physicists, 2jr

i(z,0 = ^ E y ^

(a)e'^'^'^ + a,,e-'^'^'^) e-'^>-.

(2.21)

The electric and magnetic field operators may be obtained as

£(z, 0 = 4 | i ( . , 0 = E ^/ = - E V ^ ( ^ J - ^.')

(2.22)

H{z, t) = |i(z, 0 = E ^/ = -' E ^' V F ^*'^ ^ ^-'^'

^^-^^^

where Sj = sgny and bj = aj^'^'^'^-^'^K

(2.24)

When products of these operators are envisaged, it is supposed that they are symmetrized. The Hermiticity of the operators E = E{z, t) and H = H(z, t) can be verified using the relations e/ = e.j,

h] = Lj.

(2.25)

The energy density operator u = u(z, t) can be written as

'

'

= ^ E ^v {y]bi+^!,L,+i) = - ^ 0,, {b]bj+i i ) .

(2.26)

310

Quantum description of light propagation in dielectric media

[4, § 2

The energy densities u± = u±{z, t) due to the forward and backward waves alone can be expressed uniquely:

"^ = E f (b]k + iO '

«- = E f (b]k + ii) .

i(>0)

(2.27)

iX<0)

The total momentum operator G is then

J

Abram [1987] proposed the following equation of motion {h= 1): ^=-i[G,Ql

(2.29)

where Q is any operator. We would prefer to define the operator Q. Let us consider Q = Q(z, t) = Q[Eiz, t% Hiz, t)l

(2.30)

where |2[*, •] is a formal series in E and H. Since the differential operator ^ has the same formal algebraic properties as the superoperator -i[G, • ] , it suffices to verify the relation (2.29) for Q = E,H. This is true at least in the situations considered by Abram [1987]. Although the operators bj = bj(z,t) are studied using eq. (2.29), the Heisenberg equation of motion and the initial condition S,(0,0) = a„

(2.31)

as appropriate for any operator Q(z,t), we perceive that the operators do not obey our definition of the operator Q. We may calculate the Poynting vector operator as J

j

The Poynting vector operators due to the forward and backward waves alone can be expressed uniquely:

^ ^ = E f (^)^.-ii)' y(>0)

^ - - E f (^;^.-ii)-

(2.33)

i(
The total energy operator of the free field inside the volume of quantization is thus

n = u = ~(^s^-L^ = Y^ ^j {^h + f^)'

(2.34)

J Investigation of the case ^ ^ 0^ ^ ^ 1 does not lead to any new expansions of the field operators E and H. The individual components of the rearranged

4, § 2]

Origin of the macroscopic approach

311

electric and magnetic field operators according to eqs. (2.22) and (2.23) satisfy a modified operator algebra relative to that of the harmonic oscillator: [ej,ei] = [hjM

= 0,

[CjM = s^j ( ^ )

S.jjl

(2.35)

where djj is the Kronecker 6 function. The knowledge of these commutators and of the generalized total momentum operator G = Grefr obtained via appropriate modification of the relations (2.26) through (2.28), where, e.g., the relation (2.26) becomes

"=i(^^'^^^) = 8^E(-^-/-M-/),

(2.36)

enables one to derive the Maxwell equations via both the temporal derivatives and the spatial derivatives. The energy density operator (2.36) has been generalized to the quantization volume V that is entirely included in a homogeneous medium. In the expansion (2.26) we can set Uj = ii/refr, Uj refr

2V

[hytj + L/blj - 2JTX (b^ - b.^ (b\j - hi) ] .

(2.37)

The energy density operator iirefr may be diagonalized through a Bogoliubov transformation. To this end we introduce an anti-Hermitian operator R of the form

R = R{z, t) = Y, {bjb-j - b]P_^

(2.38)

J

and the operators Bj = Q-^%Q^^ = (cosh 7) bj - (sinh y) blj,

(2.39)

where y=\\ne={\nn.

(2.40)

Upon the substitution bj = Q^^BjQ-''^ = (cosh y)Bj + (sinh y)^!-,

(2.41)

312

Quantum description of light propagation in dielectric media

[4, § 2

the operator R takes the form R = Y,{BJB^J-BJB[,),

{IAD

j

and the energy density operator has the diagonal form «,refr = ^ ( 5 ; 4 + M - / ) -

(2-43)

The momentum operator is then given by G,efr = ^ (M+ - M-) = 5 1 ^/ (^/^/ + I ' ) '

^'^'^^^

J

with Kj = nkj, and the Hamikonian can be calculated as Wrefr = ^ 0 ; ; ( ^ / ^ , + i i ) .

(2.45)

By inserting eq. (2.41) into (2.22) and (2.23), respectively, we can obtain the electric and magnetic field operators inside the dielectric:

^(-'0 =-'E\/^(^/-^.')'

(2.46)

Hiz,t) = - i ^ . , Y ^ ( « t + ^ _ ^ ) .

(2.47)

Similarly as above, these relations can be interpreted as a result of the replacement bj \-^ Bj and a consequence of the quantized classical equations (2.19). For normal incidence on a sharp vacuum-dielectric interface, both reflection and diffraction occur. This more general case has also been treated by Abram [1987]. 2.2.2. Wave-functional description of Gaussian states Bialynicka-Birula and Bialynicki-Birula [1987] made the first attempt to define squeezing as a generalization of the standard definition for one mode of radiation. This definition can be reformulated referring to Bialynicki-Birula [2000]. The Riemann-Silberstein-Kramers complex vector has been introduced as

""•" = 7!

D{r,t)

.B{r,t)

(2.48)

where the division by yJT^, yjjx^ is appropriate for SI units. It has been shown how the Green-function method can be used for solving linear equations for the

4, § 2]

Origin of the macroscopic approach

313

field operator F(r, t). This approach allows the medium under investigation to be inhomogeneous and time dependent. It has been suggested that the periodicity of the electric permittivity tensor e(r, /) or the magnetic permeability ^^(r, t) can be important for the generation of squeezed states. Only the dispersion of the medium has not been considered. It has been derived that photon-pair production is a necessary condition for squeezing. It is tempting to generalize the concept of a Gaussian state of the finitedimensional harmonic oscillator to the case of an infinite oscillator. BialynickaBirula and Bialynicki-Birula [1987] treat the time development of the Gaussian states in the free-field case. The Schrodinger picture is adopted and an analogue of the Schrodinger representation in quantum mechanics is introduced. Let us recall the quadrature representation in quantum optics. This representation is a wave functional W[A,t]. Let us observe that, contrary to the operator A{r, t), the argument A{r) of the wave functional does not depend on t, but the wave fianctional does depend on t. The Hamiltonian in this representation has the form ^ 7 1^ €o dA(ry ^io J Bialynicka-Birula and Bialynicki-Birula [1987] presented the wave functional of the vacuum state, i.e., the simplest Gaussian state of the electromagnetic field, as well as that of the "most general" Gaussian state. Thus, the exposition is confined to pure Gaussian states while it is possible to generalize it also to mixed Gaussian states of the electromagnetic field. The pure Gaussian state is determined by a complex matrix kernel, i.e., by two real matrix kernels. It is shown that the expectation values {B) = B and {D) = V (equivalently, {E) = £) evolve according to the free-field Maxwell equations; in addition, the equations for the matrix kernel W{r,r'\ t) can be found there. The entire electromagnetic field is treated as a huge infinite-dimensional harmonic oscillator. The wave function and the corresponding Wigner function then become fiinctionals of the field variables. Mrowczyhski and Miiller [1994] have considered only the scalar field. Bialynicki-Birula [2000] starts from the wave ftinctional (Misner, Thorne and Wheeler [1970]) <^oM = Cexp

^

^//fiW—L^.5(/)d^.dVl Ih J J \r-r'Y J

(2.50)

and from the wave functional (we change A -^ -D) ^ o [ - ^ ] = Cexp

1

[^

f

/•^(^)_L^.^(/)d3;.dV (2.51)

314

Quantum description of light propagation in dielectric media

[4, § 2

The normalization constant C is an issue not completely solved by BialynickiBirula [2000]. The analogy with the one-dimensional harmonic oscillator leads to other notions. The Wigner functional of the electromagnetic field in the ground state is Wo[A,-D] = Qxp{-2N[A,-D]},

(2.52)

where N[A,-D] = (2.53) d^rd^r Expression (2.53) also plays the role of a norm for the photon wave function (Bialynicki-Birula [1996a,b]). The Wigner functional for the thermal state of the electromagnetic field has been presented. This state is mixed and it even has infinitely many photons in the whole field. In each of the subsequent cases, the wave ftinctional and the Wigner fianctional have been introduced. The exception, the mixed state, has no wave functional. Let us remark that for (the statistical operator of) such a state a matrix element can be considered which is a functional of two arguments, A and A\ In particular, the Wigner functional for the coherent state of the electromagnetic field \A,-V) has been presented, where A(r) and V(r) are the vector potential and the electric displacement vector, respectively, which characterize the state. The exposition is related to the hot topic of the superposition of coherent states of the electromagnetic field. The exposition continues with the Wigner functional for the states of the electromagnetic field that describe a definite number of photons. An example of the functional for the one-photon state with the photon mode function/(r) is included. The norm (2.53) has not been related to any inner product of the photon wave functions, but these notions are connected. In contrast to (Bialynicka-Birula and Bialynicki-Birula [1987]), we introduce 1,[P] = jb(r,0)f(r)d'r,

(2.54)

X2[B] =

(2.55)

B(r,0)g(r)d'r,

- /*<•••'

(2.56) ^ W = JZJT I xr^T-—;^B(r')d^r\

(2.57)

4, § 2]

Origin of the macroscopic approach

315

The commutator of the X\ and X2 operators is [X, [VIMB]]

= ih jfir).

[V X gir)] d'r 1.

(2.58)

Let us note that the right-hand sides of eqs. (2.56) and (2.57) comprise the operator |V|"^ up to a certain factor (cf. Milbum, Walls and Levenson [1984]). Without resorting to this notation, we obtain that [X,[VIX2[B]] = ^7 f V(n)

AirOd'nl

(2.59)

We see easily that the usual commutator - ^ i l results from the field (T>,B) (or (A, -T>)) with the property

/•[-V(n)]'A(r0d'n=2k

(2.60)

We have not deepened the contrast by introducing the notation Xi [-V] andX2[^] on the left-hand sides of relations (2.54) and (2.55). Bialynicki-Birula [2000] presents the Wigner functional for the squeezed vacuum state, ^sq[^,-/>] = exp

—B • KsB ' B

^—DKODD'^B

KBD

' D' I dVd^f

Co

(2.61) where K^B, ^DD and A'BD are real matrix kernels. The kernel A'BD is not independent of A^BB and A^DD, but must obey a condition that is reminiscent of the Schrodinger-Robertson uncertainty relation (Bialynicki-Birula [1998]). The problem of time evolution is also discussed. Using a straightforward procedure, Mendonga, Martins and Guerreiro [2000] have quantized the linearized equations for an electromagnetic field in a plasma. They have determined an effective mass for the transverse photons. An extension of the quantization procedure leads to the definition of a photon charge operator. Zalesny [2001] has found that the influence of a medium on a photon can be described by some scalar and vector potentials. He extended the concept of the vector potential to relativistic velocities of the medium. He derived formulas for the photon mass in a resting and moving dielectric, and the velocity of the photon as a particle.

316

Quantum description of light propagation in dielectric media

[4, § 2

2.2.3. Source-field operator Knoll, Vogel and Welsch [1987] have compared the problem of quantummechanical treatment of action of optical devices with the input-output formalism (Collett and Gardiner [1984], Gardiner and CoUett [1985], Yamamoto and Imoto [1986], Nilsson, Yamamoto and Machida [1986], cf also Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a,b]), concluding that apart from the fact that only a very particular setup is considered in the input-output formalism, the theory does not take into account the full space-time structure of the field. Knoll, Vogel and Welsch [1987] have further elaborated an approach based on quantum field theory and applied to the problem of spectral filtering of light (Knoll, Vogel and Welsch [1986]). The only assumptions are that the interaction between sources and light is linear in the vector potential, that the optical system is lossless, and that the condition of sufficiently small dispersion is fiilfilled. First, the classical Maxwell equations with sources and optical devices are formulated and solved by the procedure of mode expansion, and a quantized version is derived. The classical Maxwell equations comprise the relative permittivity e{r) = rr{r), where n{r) is the space-dependent refractive index. The charge density and the current density are written in terms of point charges, Qa being the charge of the aih particle. Canonical structure is imposed on the field and the matter, starting from a Lagrangian including the masses of the particles, rua being the mass of the aih particle. The mode functions Ai{r) are introduced as the solutions of the following equation: V X (V X Ax{r)) - e{r)^Ax{r)

- 0,

(2.62)

with (ji)\ the separation constant for each A, from which the following gauge condition can be derived: V.(6(r)^,(r)) = a

(2.63)

It is assumed that these solutions are normalized and orthogonal as follows: / •

e{r)A,{r)-Ax'{r)dh

= d,A-

(2.64)

The vector potential can be decomposed in terms of these fianctions. The destruction and creation operators ai and a\ are defined in a standard way, with the properties [axraU = Su'l

[ax,ax'] = 6 = [alall

(2.65)

4, §2]

Origin of the macroscopic approach

317

Upon inserting the operators a^ and a\ into the decomposition of the vector potential, the operator of the vector potential A(r, t) is defined:

A{r,t) = Y,A,{r)\ax{t)

+ a\{t)

(2.66)

The source quantities, i.e. the position vectors r^ and the generalized momenta Pa, are related to point charges and taken into account as operators r^ and Pa obeying the standard commutation relations. [ha.Pk'a']

= ihdaa'^kk' 1,

[rka. ^kuA = 0 =

[pka.Pk'a'].

(2.67)

and the commutation relations [ha,

^A]

= 0 = [Vka, :^t 5 ! ],

[pka,

«A]

ti = 0 = [pka, Aa[]

(2.68)

The operator ^(r, t) can be used for the derivation of the electric field strength operator, which is associated with the radiation field by the relation

E{r,t) =

--A{rj\

(2.69)

and to the derivation of the magnetic induction field operator, B{r, 0 = V X A{r, t).

(2.70)

The mode fiinctions may approach the notion of photon waveftinctions if redefined so that they obey the normalization condition

/

t{r)Ax{r)Ai,(r)d'r 3„

-

le^ojx

5AA'.

(2.71)

The quantum-theoretical Hamiltonian is written as the sum of a field Hamiltonian, a source Hamiltonian and an interaction Hamiltonian. The latter is formulated in terms of appropriate operators, with the Hermitian operator

J{r, t) = T ^

Wr\ - ra)Pa + A.5(H - h) - QcAih) (5(H - h)

being the current density operator. The form of the normalization conditions (2.64) and (2.71) is tailored to realmode functions, and ways to modify some fundamental relations are commented

318

Quantum description of light propagation in dielectric media

[4, § 2

on by Knoll, Vogel and Welsch [1987]. All of these field operators may be written in the form F(r, t) = J2 [^AW ax(t) + Fl(r) al(t)

(2.73)

Dependent on the choice of the operator F(r,t), the fianctions Fx(r) can be derived fi-om the mode functions of the vector potential Ax(r). It is often convenient to decompose a given field operator F(r, t) into two parts by the relation F(r, 0 = F^^\r, t) + F^-\r, t\

{2.1 A)

where F^^\r, t) = ^F,(r)a,(t),

F^-\r, t) = F'^\r,t)

(2.75)

In the following, the vector components are introduced by mere labeling by an index k, and repeated indices indicate summation. Further, the Heisenberg equations of motion for the field operators are derived, so that the field operators can be expressed in terms of the free-field and source-field operators. It is typical of the approach of Knoll, Vogel and Welsch [1987] that any field operator F{^^ is decomposed into a free-field operator and a source-field operator as follows Fi%, t) = Fj;ljr,

t) + h s(/-, 0,

(2.76)

where ^IfreeC^O =Y.^kx{r)aur..{t\

(2.77)

A

Fks{r,t) = j j e{t-t')K,,>{r,t',r\t')Jk'{r'j')d'r'dt'.

(2.78)

The operator a A free (0 is defined relative to the condition ^Afree(OL = /o ^ ^A(^) for t = t{), and the dynamics for ^ ^ ^ can be found in (Knoll, Vogel and Welsch [1987]). In eq. (2.78), the kernel K^k' is defined by Kkk'{r,t'yj')

= - ^ ^ F a ( r ) ^ ^ ^ ( / ) exp[-ia;A(^-0].

(2.79)

4, § 2]

Origin of the macroscopic approach

319

Inserting eq. (2.78) into (2.76) yields the following representation of F^^ : ^ r V ^ 0 = J J e(t - t')K,k'{r. t; r\ t')Jk'{r\ t') &'r' dt' + F{;^,(r, 0(2.80) In particular, if F^^^ is identified with the vector potential Aj, , it holds that Fkx = Akx\ the kernel Kkw takes the form ^ , , . ( r , r ; / , O = -T^5]^^^W^^A(''0exp[-ia;A(/-O]-

(2-81)

A

Analogously, if one is interested in the electric field strength operator of the radiation E^j^\ the appropriate form of the kernel Kkk' is Kkk'{r.t-r\t')

-~\Y.

^xAkx{r)Al,,{r')

exp[-ia;A(^-^')].

(2.82)

A

Thus the symmetry relations ^;,,(r, n r\ t') = TK,'k(r\ t'; r, 0

(2.83)

are valid for Aj^ and E[^\ respectively. The information on the action of the optical instruments on the source field is contained in the space-time structure of the kernel K/^k', which may be regarded as the apparatus function, similar to what is used in classical optics. Further, the commutation relations for various combinations of field operators at different times are studied and relationships between field commutators and source-quantity commutators are derived. These commutation relations are used to express field correlation functions of free-field operators and source-field operators and to describe the effect of the optical system on the quantum properties of light fields. 2.2.4. Con tin uum frequency-space description Blow, Loudon, Phoenix and Sheperd [ 1990] have formulated a quantum theory of optical wave propagation without recourse to cavity quantization. This approach goes beyond the introduction of a box-related mode spacing and enables one to use a continuum frequency-space description. In two papers. Blow, Loudon, Phoenix and Sheperd [1990] and Blow, Loudon and Phoenix [1991] developed a continuous-mode quantum theory of the electromagnetic field. As usual in

320

Quantum description of light propagation in dielectric media

[4, § 2

quantum field theory, they considered box-related modes whose creation and destruction operators satisfy the usual independent boson commutation relations [ai,aj] = Sijl. Different modes of the cavity, labeled by z andy, have frequencies given by different integer multiples of the mode spacing A a;. The mode spectrum becomes continuous as Aoj —> 0, and in this limit the transformation to continuous-mode operators is convenient, a, -^ y/Acoa(a)). The authors considered a complete orthonormal set of functions which may describe states of finite energy. The set is numerable infinite, and a destruction operator is assigned to each function in it. Such operators have all properties usual for the operators of the monochromatic mode. Blow and colleagues treated additional specific states of the field, such as coherent states, number states, noise and squeezed states; with the use of noncontinuous operators, they proved a generalization of the single-mode normal-ordering theorem. They treated field quantization in a dielectric including the material dispersion, and the theory was applied to pulse propagation in an optical fiber. A comparison with results by Drummond [1990, 1994] would be in order. Let us consider the fields in a lossless dielectric material with real relative permittivity e((jo) and the refractive index n((ji)) related by e(co) = [n(a))]^. Let us recall the definition of the phase velocity, VFi(o) = ^ = - ^ , k

and that of the group velocity * VQ{W)

^^ dco

(2.84)

n{(jj) UQ{W),

1 ^ [^„(^)j c do)

(2.85)

The vector potential operator has been modified for the dispersive lossless medium, and compared with (Loudon [1963], Drummond [1990]). The positivefrequency part is

(2.86) where e(k. A) are the orthogonal polarization unit vectors and w=-^\kl Expression (2.86) can easily be converted to one-dimensional form.

(2.87)

4, § 2]

Origin of the macroscopic approach

321

It is shown how the formulation of the quantum field theory is modified for a one-dimensional optical system. The fields are defined in an infinite waveguide parallel to the z-axis, but of finite cross-sectional area A of rectangular form with sides parallel to the x- and j-axes. The x and y wavevector components are thus restricted to discrete values, and any three-dimensional integral over this spatial region is converted according to

On the assumption that modes with ky^ ^ 0 ox ky ^ 0 are vacuum modes, one can exploit a reduced Hilbert (namely Fock) space. The summation in eq. (2.88) can, therefore, be removed and, putting k- = k, the other conversions are ^(3)(^ _ k') ^ ^ S i k - k'),

a{k. A) - . ^a(k.

A).

(2.89)

The field operators are obtained in accordance with the relation E^^\z, t) = -l/"\z, t) B^-\z, t) = ^A'"\Z, t) at oz and with the expansion of the vector potential operator

(2.90)

^W(z,0= r / ^^°^^\ V e(k,X)a{k,X)tx^[-\{wt-kz)-\dk, J~oo V 4;reoccon(cL>M ^ f f ^ (2.91) which is taken to be oriented in the x-direction, A^^\z, t) = A (z, t) e^. Operators a(w) are introduced, whose normalization is fixed by requiring the normally ordered total energy density operator (/(z, t) to have a diagonal form: ^free(0 = ^ /

^(^, t)dz=

I hcoa\(o) a{co) dco.

(2.92)

On noting that dco dk = — - - ,

a(k, A) = .yuGiaj)aico)

(2.93)

UG(CO)

and taking the polarization to be parallel to the x-axis, it follows from eq. (2.90) that the field operators are E^^\z,t) = i / W

^-—-a(co) expi -iw

n(co)z

do)

(2.94)

322

Quantum description of light propagation in dielectric media

[4, § 2

and 5'

*'<-•=•//lI^«»>-{-'"h=v^]}-

•-^'

Alternatively, the propagation constant can be expanded to the second order in frequency and a partial differential equation can be obtained (cf Drummond [1990]). Assuming a narrow bandwidth, the slowly varying field envelope can be represented by the operator a(z, t), which obeys the equation d k" d^ i - 5 ( z , 0 + y g^a(z, t) = 0,

(2.96)

where k^' is the second derivative with respect to the frequency of the propagation constant evaluated at the central frequency. The equation has been simplified by transforming the envelope into a frame moving with the group velocity, which is necessary for the envelope to be slowly varying. In classical nonlinear optics the stationary fields have envelopes too, but those are defined in a different way. The treatment of this problem in a noncontinuous basis proceeds from the replacement «(z,O = ^ 0 , ( z , O 9 ,

(2.97)

where 0j(z, t) form a complete orthonormal set of functions of z, and Cj are destruction operators obeying the usual commutation relations. The advantage of this treatment is that the functional dependence on z and t is contained in the c-number functions rather than the operators 5(z,/) as in the propagation equation (2.96) for instance. It is not emphasized by Blow, Loudon, Phoenix and Sheperd [1990] that the solution of eq. (2.96) preserves equal-space, not equaltime commutators. Similarly, the set of fijnctions 0/(z, t) enjoys orthonormality and completeness only as equal-space, not as equal-time properties. The propagation equation (2.96) now yields the following equations for the noncontinuous basis fiinctions: i - 0 , ( z , 0 + — ^ 0 ; ( z , t) = 0.

(2.98)

Finally, the process of photodetection in free space is considered and the results are applied to homodyne detection with both local oscillator and signal fields pulsed.

4, § 3]

Macroscopic theories and their applications

323

McDonald [2001] has considered a variation of the physical situation of "slow light" to show that the group velocity can be negative at central frequency. A Gaussian pulse can emerge from the far side of a slab earlier than it hits the near side and the pulse emission at the far side is accompanied by an antipulse emission, the antipulse propagating within the slab so as to annihilate the incident pulse at the near side.

§ 3. Macroscopic theories and their applications 3.1. Momentum-operator approach 3.1.1. Temporal modes and their application Huttner, Serulnik and Ben-Aryeh [1990] have developed a formalism that describes in a full quantum-mechanical way the propagation of light in a linear and in a nonlinear lossless dispersive medium. At first, they assume a similar situation as Abram [1987], i.e., they consider only the one-dimensional case restricting themselves even to fields propagating in the +z-direction only. They take for granted that quantum field theory has a generator for spatial progression, i.e., that relation (2.29) holds for any operator. They remark that the change in the quantization volume pointed out by Abram [1987] is not defined when the medium is dispersive, i.e., when the refractive index depends on the frequency, but they develop Abram's idea of the use of the energy flux not being dependent on the medium (cf. Caves and Crouch [1987]). In their opinion, the classical analysis of nonlinear optical processes shows that in order to obtain simple propagation equations it is usefial to introduce a photon-flux amplitude, i.e., a quantity whose square is proportional to the photon flux. At present we hesitate to accept the consequences of their approach (cf., however, Ben-Aryeh, Luks and Pefinova [1992]). Specifying the state at a given point (e.g., z = 0) and within a time period T cannot substitute for specifying the state at an initial time (t = 0) and within a quantization length L. Temporal modes of discrete frequencies a)„j, where a>,„ = 2mJt/T, cannot substitute for spatial modes. The equal-space commutation relations [a(z,Wi),a\z,ojj)]

= Syl

(3.1)

cannot substitute for the usual equal-time commutation relations. In comparison with (Abram [1987]), we see the following changes. In (Huttner, Serulnik and Ben-Aryeh [1990]), the MKSA (SI) system of units is

324

Quantum description of light propagation in dielectric media

[4, § 3

used. Instead of immediately reducing the Maxwell stress tensor to a single component, the tensor is first reduced to the Minkowski vector. The normal ordering is used where necessary. The notation ceases to express the dependence on both z and t, and states the dependence on z only. "The generalization" of the relation for the free-field momentum flux operator g(z, t) = c[b^-\z, t)B^^\z, t) + H.C.]

(3.2)

to the form g(z,t) = 0~\z,t)E^^\z,t) + Kc.]

(3.3)

is rather a modification, which remains correct in a dielectric medium. Its integration over T gives the momentum operator G{z)= r^

g{zj)dt

(3.4)

In the case of a linear dielectric medium, in contrast with (Abram [1987]), the electric field operator E{z, t) is dependent on the refractive index n(oJnj):

^

Y 2eQcTn{(On,) ^

^

while in (Abram [1987]) the operator is independent of the medium. The same paper does not present a pure Heisenberg picture, so that the equivalence of the two theories (for the refractive index n independent of o)) is not excluded. On substituting the relation (3.3) with appropriate b^~\z,t), D^^\z,t) into the relation (3.4), the momentum operator is obtained as Giin(z) = ^(hkn,) a\z, a),n) a(z, w„,),

(3.6)

m

where k^j = n((jo„j)(jOnj/c is the wavevector in the (linear) medium. The equalspace commutation relations are conserved. For such a medium, the equal-time commutation relations can be derived as A(z,t),-b(z\t)\ = ihd{z-z') i.

(3.7)

In an attempt at quantization in a nonlinear medium, Huttner, Serulnik and Ben-Aryeh [1990] concentrated on the propagation of light in a mukimode

4, § 3]

Macroscopic theories and their applications

325

degenerate parametric amplifier. The postulated relation (3.3) then leads to the nonlinear part of the momentum flux operator gnonlin(^, 0 = X^^^ U^"\z,

0 [E^'K^,

O ] ' + H.C. | ,

(3.8)

where £^^\z,t) = \£\Q-^i('¥-^P=) is the positive-frequency part of the pump field, with pump frequency Wp. From relations (3.4) and (3.8), the momentum operator is obtained as GnonvUz) = Y . ^ ^

[a\z,

COo + €,„) a\z,

COo - €,,) c'^-" + H.C.] ,

(3.9)

where e^ = o^m - COQ, a)o= ^-^, and X ( e „ . ) . ' - ^ , r ^ " - ^ - ' " '

(3.10)

is the coupling constant between different modes. It is assumed that the phase-matching condition at (JOQ, n((jOp) = n(a)o), is satisfied. It is found that the phase mismatch Akie^i) is proportional to e^,. As far as |AA:(e„;)| < A(€„,), the Bogoliubov transformation for squeezing emerges, and amplifying behavior can be recognized. When |AA:(6,„)| > A(e,„), the evolution is not essentially different from that in a linear medium: the squeezing effect is band-limited. For equality |AA:(e,„)| = ?i(e,„), amplification is present, but the increase is only linear, not exponential. For the nonlinear medium, the equal-time commutation relations are

i^"V,0,-I)^%',0] =

'~d{z-z')l

(3.11)

and relation (3.7) can be recovered only approximately. In relation to the experiment, a standard two-port homodyne detection scheme is assumed, where the light is mixed at a beamsplitter with a strong local oscillator e{z,t) of frequency COQ. For the correlation function gs{T) of the photocurrent difference and its Fourier transform y(r]) = j gs(r)c-'''dT, /

(3.12)

we refer to Huttner, Serulnik and Ben-Aryeh [1990]. It has been shown that the values of y(rj) can be minimized sufficiently uniformly by an adequate choice

326

Quantum description of light propagation in dielectric media

[4, § 3

of the local oscillator phase. The result is comparable with that of (Crouch [1988]), where the usual interpretation of homodyne detection in terms of the field quadratures is used. 3.1.2. Slowly-varying-amplitude momentum operator In spite of the above, there is a class of problems for which the modal approach is very convenient. This is cavity quantum electrodynamics. We mention its use in the development of the input-output formalism for nonlinear interactions in a cavity (Yurke [1984, 1985], Collett and Gardiner [1984], Gardiner and Collett [1985], Carmichael [1987]). The modal approach can describe many of the features of traveling-wave phenomena, but, in principle, it mixes effects related to spatial progression of the beam with the spectral manifestations of the nonlinearity. For example, for traveling-wave parametric generation (Tucker and Walls [1969]), wavevector mismatch appears as energy (frequency) nonconservation. Several authors have tried to reformulate the quantum-mechanical propagation in direct space. One technique (Drummond and Carter [1987]) involves partitioning the quantization box into finite cells. Another technique considers the spatial progression of the temporal Fourier components of the local electric field (Yurke, Grangier, Slusher and Potasek [1987], Caves and Crouch [1987]). The propagation of light in a magnetic (dielectric) medium is not usually considered in quantum optics. We proceed with the field inside an effective (linear or nonlinear) medium and the direct-space formulation of the theory of quantum optics as presented by Abram and Cohen [1991]. Their approach is an alternative to the conventional reciprocal-space approach to quantum optics; it relies on the electromagnetic momentum operator as well as on the Hamiltonian, and is restricted to a dispersionless lossless nonmagnetic dielectric medium. They derived an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. As an illustration, they applied the theory to the generation of squeezed light by parametric down-conversion of a short laser pulse. This approach does not use the conventional modal field description. The appeal of classical optical theory may be in the fact that it considers a material as a continuous dielectric characterized by a set of phenomenological constants. Classical nonlinear optics has given rise to the slowly-varyingamplitude approximation for the electromagnetic wave equation. An important simplification of quantum optics results when the microscopic description of the material is replaced by a macroscopic description, in terms of an effective linear or nonlinear polarization. In spite of the phenomenological treatment of the

4, § 3]

Macroscopic theories and their applications

327

medium, such an effective theory still permits a quantum-mechanical description of the field (Jauch and Watson [1948], Shen [1967], Glauber and Lewenstein [1989, 1991], Hillery and Mlodinow [1984], Drummond and Carter [1987]). In propagation problems, one examines the interactions undergone by a short pulse of light. Abram and Cohen [ 1991 ] use Heaviside-Lorentz units and take h = c = 1. They simplify the geometry for the electromagnetic field so that the electric field E is polarized along the jc-axis, the magnetic induction field B along the 7-axis, while propagation occurs along the z-axis. In this simple geometry, the Maxwell equations reduce to two scalar differential equations, d^^dB dz dt'

dB^dD dz dt '

where the electric displacement field D is defined by D = E +R

(3.14)

P is the polarization of the medium, which can be expressed as a converging power series in the electric field £", P = /^^E^x^^^E^ + • • -+/"^E" + . . . ,

(3.15)

where x^"^ is the «th-order susceptibility of the medium. The dispersion cannot be taken into account rigorously within a quantum-mechanical theory based on the effective (macroscopic) Hamiltonian formulation (Hillery and Mlodinow [1984]), but it can be introduced phenomenologically (Drummond and Carter [1987]). To impose canonical structure on the field, Abram and Cohen [1991] introduce the vector potential A and adopt the Coulomb gauge in which the scalar potential vanishes and^ is transverse. In the assumed geometry, the vector potential is polarized along the Jc-axis, A = Ae^, with Cv the unit vector in the +jc-direction, and is related to the electric field and the magnetic induction field by

The effective Lagrangian density has been chosen in the form (Hillery and Mlodinow [1984], Drummond and Carter [1987]) C = \{E^-B^) + {x^'^E^ + \x^^^E' + \x^'^E' + • • •,

(3.17)

which is known to provide the most general density dependent only on the electric field and having the gauge invariance.

328

Quantum description of light propagation in dielectric media

[4, § 3

Let us note that the theory with the effective Lagrangian density (3.17) is not renormaHzable (Power and Zienau [1959], Woolley [1971], Babiker and Loudon [1983], Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]). The canonically conjugated momentum of ^ with respect to the Lagrangian density (3.17) is the electric displacement n=--=-D. (3.18) oA The Lagrangian density is then transformed to some components of the energymomentum tensor of the electromagnetic field inside a nonlinear medium, O/^iy, namely, the energy density dA

e, =n—-c = \{B^ +A^) + ^/'^E'

(3.19) + l/'^E^ + ^^/'^E' + .. •,

(3.20)

and the momentum density dA er- = -n—=DB. (3.21) oz In setting up the Hamiltonian functional, the electric field E is to be expressed in terms of the electric displacement, which is the canonically conjugated momentum of A according to relation (3.21). It is assumed that E = P^^^D + P^^^D^^P^^^D^ + .. •,

(3.22)

where the coefficients 13^^^ may be expressed in terms of the susceptibilities x^"^ through definition (3.14) and relation (3.15) (Hillery and Mlodinow [1984]). The Hamiltonian functional is then written as

H= f Ond^r Jv

(3.23)

/'V. and the momentum has the form

G= I Ot,dV = I BDd^r,

(3.24)

where the integration is over the cavity obeying periodic boundary conditions, and the lower and upper limits extend to -oo and oo, respectively.

4, § 3]

Macroscopic theories and their applications

329

The field can now be quantized by replacing each field variable by the corresponding operator, and by replacing the Poisson bracket between the displacement D and the vector potential A by the (-i) multiple of the equal-time commutator [Z)(r, t\A{r\ t)] = id^ir - /) 1,

(3.25)

where the transverse delta fiinction d^Xr - r') reduces to the ordinary d ftmction, and where the three-dimensional position vector r can be replaced by the coordinate z. The vector potential A replaced by the operator A does not appear explicitly in the Hamiltonian (3.23) and momentum (3.24) operators, but rather in terms of its spatial derivative B. Taking the curl according to r' of both the sides of the canonical commutation relation (3.25) and using relation (3.16) in the simple geometry we obtain that [b{z,tXB{z',t)] = -id\z-z')

i,

(3.26)

where

b\z-z')=-b{z-z') oz

Chll)

is the derivative of the b fiinction. Ignoring divergencies, Abram and Cohen [1991] consider any product of noncommuting operators appearing in an expression to be fiilly symmetrized, i.e., to include all possible permutations of the individual field operators, such as, BD^ ^

.

(3.28)

3 In contrast, Abram and Cohen [1994] performed renormalization, i.e., normal ordering and elimination of divergencies. The description of propagative optical phenomena has been discussed within the framework of a direct-space formulation of quantum optics and the operatorial (or better commutator) equivalent of the Maxwell equations and the electromagnetic wave equation (Abram and Cohen [1991]). It is emphasized that in the Hamiltonian formulation of mechanics the time variable plays a prominent role.

330

Quantum description of light propagation in dielectric media

[4, § 3

The integrals in eqs. (3.23) and (3.24) and the equal-time commutator (3.25) correspond to the requirement that the field be specified over all space at one instant of time (e.g., at ^ = 0). Abram and Cohen [1991] use the Kubo [1962] notation for the commutator, or more exactly for a corresponding superoperator. The superoperator assigns operators to operators. Respecting this, the Heisenberg equation can be written as ^=i[H,Q]

= iH''Q,

(3.29)

where Q is any field operator and the superscript x denotes a superoperator, namely the commutator of the operator it indexes with another operator which follows. Equation (3.29) has the solution Q(t) = Qxp(itH'')Q(0) = e'^'^(0)e~'^^

(3.30)

The following Heisenberg-like equation involving the momentum can be considered: ^=-iG-Q.

(3.31)

This equation has the solution Q(z) = exp[-i(z-zo)G^]e(^o)-

(3.32)

Apart from the obvious similarity of eqs. (3.29) and (3.31), there is also a difference. The Hamiltonian of the electromagnetic field relates the desired spatial distribution of the field at an instant / + dMo its spatial distribution at t, but the momentum operator G relates the translated and non-translated fields only (at the same instant of time). In analogy with (3.13), two commutator equations can be derived: G''E = H''B,

G''B = H''b.

(3.33)

On the assumption that the medium is homogeneous, so that the Hamiltonian and momentum operators commute with each other, i.e. G^H = 0, relations (3.33) may be combined into the commutator equivalent of the electromagnetic wave equation: G''G''E = H''H''b.

(3.34)

Abram and Cohen [1991] illustrate the direct-space description of propagation (i.e., without passing over to a modal decomposition of a propagating pulse)

4, § 3]

Macroscopic theories and their applications

331

by examining the propagation of light (of a short light pulse) through a linear medium and through a vacuum-dielectric interface. For a linear medium, the commutator wave equation (3.34) reduces to 'H'')E = b,

CC'-eH

(3.35)

where 6 = I-^ X^^^ is the dielectric function of the medium. It is also convenient to define u = 1/ y/e, the velocity of an electromagnetic wave in a refractive medium. In the following exposition, the notation c will be used, because the convention c = 1 is not observed. The wave equation (3.35) enables one to rewrite Abram and Cohen's [1991] equations (4.2a) and (4.2b) in the form E(z, t) = cos\i{-\utG'')E(z, 0) - w sinh(-ii;^G^)^(z, 0),

(3.36)

B{z,t) = --sinh(-ii;^G^)£'(z,0) + cosh(-id;^G^)^(z,0).

(3.37)

V

Equations (3.36) and (3.37) indicate that the linear combination ^ ; ( z , 0 = ^(z, 0 + vB{z, t)

(3.38)

evolves in time as ^ ; ( z , 0 = exp(it;^G^) W^{z, 0) = ^ ; ( z - vt, 0).

(3.39)

Similarly, the linear combination W;{z,t) = E{zj)-vB{z,t)

(3.40)

evolves as W;{z, t) = Qxpi-iutG'') W-{z,0) = W;{z^vt,0).

(3.41)

To examine the problem of the interface, we could now consider two halfspaces such that the negative-z half-space is empty, while the positive-z halfspace consists of a transparent linear dielectric. We could then consider three waves: incident, W^{z, t), reflected, W~{z, t), and transmitted, W^^(^, t\ and the relations they obey. Abram and Cohen [1991] derive the commutator equivalent of the slowlyvarying-amplitude wave equation, on which the classical theory of nonlinear optics is based.

332

Quantum description of light propagation in dielectric media

[4, § 3

Not even classical optics provides a general solution to the problem of propagation of a short pulse in a nonlinear medium. In classical nonlinear optics, the assumption of a weak nonlinearity permits the slowly-varyingamplitude (SVA) approximation of the electromagnetic wave equation (Shen [1984]). Abram and Cohen [1991] examine a perturbative treatment of the time evolution of the field in a nonlinear medium that corresponds to propagation within the slowly-varying-amplitude approximation. For simplicity, a single nonlinear susceptibility x^"^ is considered. In the perturbative treatment of nonlinear propagation it is assumed that the optical nonlinearity of the medium is absent at ^ = -co, and is turned on adiabatically. In the absence of the nonlinearity, the electric strength and magnetic induction fields in the medium, EQ and ^o, as well as the displacement field Do, Do = CEQ, propagate under the Hamiltonian and momentum operators

Ho = \j

[BUIDI] 6

dV,

(3.42)

^3 = j Bobo dV,

(3.43)

respectively, of zeroth order in the nonlinear susceptibility ;^^^'^ Following standard perturbation theory (Itzykson and Zuber [1980]), the exact field operators in the nonlinear medium, D and B, can be derived from the zerothorder fields by the unitary transformation b{z,t)

= U-\t)bo(z,t)U(t\

(3.44)

B(z,t)

= U-\t)Bo(z,t)U(t).

(3.45)

Here tj(t) is the unitary operator which is the solution to the differential equation -U{t)

= -{Xh,(t)U{t),

(3.46)

with H\ the nonlinear interaction part of the Hamiltonian H,

A, = - ^

/yS'"'l);;+' dh =

^

/z""£'r' dV,

(3.47)

or f{dt) = expiitH,^)Hu

(3.48)

and obeys the initial condition U(t)\

= i. 1 / —>•

(3.49)

- o c

The Hamiltonian ^ i is of first order in the nonlinear susceptibility x^"'' which is expressed also by A, a dimensionless parameter introduced for the bookkeeping

4, § 3]

Macroscopic theories and their applications

333

of this and higher powers of x^'^^- The exact Hamihonian (3.23) can then be expressed perturbatively up to the first order in A as H = Ho + kHis^O(?i^X

(3.50)

where ^15 is the "diagonal part" ofH\,S with the Hnear Hamihonian HQ,

stands for stationary, which commutes

H,s=H\"^ = -^^js„,,Ah

(3.51)

with [f]

,

"^^ = 2""' E ^ / M O ,,^~"'Bl"'Er"", ^

(3.52)

{n - 2m)\{2m)\

where [JC] is the integer part of the number x. The neglect of O(A^) leads to decoupling of opposite-going fields. In the context of eqs. (3.51) and (3.52), a connection with the standard modal approach has been mentioned by Abram and Cohen [1991]. A method for deriving standard "effective Hamikonians" has been proposed by Sczaniecki [1983]. According to eq. (3.30), the time evolution of the displacement operator D can be written in the form D(z,0 - exp(iM^i^^)I)o(z,0 + AZ),(z,0,

(3.53)

where A = 1 and Dx{z,t)

i /

//i(r)dr

Do(z,0

(3.54)

is the first-order correction to the displacement field. The action of the superoperator on DQ in relation (3.53) can be compared with the mukiplication of the fast-varying ("carrier") wave by a slowly-varying-envelope fianction. On introducing the nonlinear polarization NL

-e^^^^Dl=x^'^El

(3.55)

it can be shown that the exact commutator wave equation (3.34) can be written up to order A^ as {G^G^-eH^H^)b, = b,

(3.56)

334

Quantum description of light propagation in dielectric media

[4, § 3

and that to order A^ leH^H^.bo = -eH,^H,^b, + G^ G^D, - G^ G^P^t-

(3.57)

The nonlinear polarization PNL consists of two parts, Pw and its complement, and Pw obeys the zeroth-order wave equation (G^G^-eH^H^)p^-b,

(3.58)

namely, Pw=P%^=/'%.

(3.59)

This partition again eliminates all terms that couple opposite-going waves in /*NL- Relying on the relation 6 = -eH^H^b, + G^G^b, -G^G^{P^L-PWI

(3.60)

we can derive the commutator equation leH^H^s^o = -G^G^Pw.

(3.61)

which has been compared to the classical slowly-varying-amplitude (SVA) wave equation, which is written as (Abram and Cohen [1991]) 2ik-E=-^P>r,

(3.62)

or, more often, in terms of the temporal Fourier components of E and Pw as ^E(w)=^P^(co),

(3.63)

where E is the envelope ftinction of the electric field. The connection between Piv and the modal approach has been shown by Abram and Cohen [1991]. The commutator equivalent of the slowly-varying-amplitude wave equation will be applied to the quantum-mechanical treatment of propagation in a nonlinear medium. Let us consider eq. (3.61) whose right-hand side, unlike the

4, § 3]

Macroscopic theories and their applications

335

left-hand side, does not contain DQ. This problem can be remedied by defining an effective "SVA" momentum operator obeying G^^^bo = {G^Pw

(3.64)

It follows that GsvA is the stationary part of the effective "interaction" momentum operator Gi = {jBopNLd'r,

(3.65)

namely, GsvA = G^svA = ^

IRn.x d^,

(3.66)

with [?]-!

Rn = 2 - ^ ' Y. ^

,

^-^ ntn e-'^^B'f^'Et'"^-\ {n - 2m - \)\{2m + \)\

(3.67)

Again, in the context of eqs. (3.66) and (3.67) for GSVA, the connection to the modal approach can be shown. With the definition (3.64), the commutator wave equation (3.61) can be written as (^S'VA^O'' + ^H^sH^ )bo = b.

(3.68)

In this form, the commutator SVA equation directly relates the slow component of the temporal evolution of a short pulse of the displacement field DQ to the long-scale modulation of its spatial progression. In order to clarify the role of eq. (3.68), the forward (+) and backward (-) polarization waves are defined in analogy with eqs. (3.38) and (3.40), V^=b±VeB,

(3.69)

which in the absence of the nonlinearity have the form Ko± = e r ± ,

(3.70)

where in accordance with perturbation theory the forward and backward electromagnetic waves are defined as W^=Eo±vh-

(3.71)

336

Quantum description of light propagation in dielectric media

[4, § 3

Relation (3.53) now becomes V^(z, t) ^ QxpiitH^s) ^^u(^^ 0 + VH^^ 0, V'(z, t) ^ exp(i/^,^) e^;(z, 0 + ^fC^, 0,

(3.72) (3.73)

where V^ are the first-order corrections to V given by equations analogous to (3.54). Equation (3.68) simplifies to VeH^sfV: =-G^^^w:,

(3.74)

VeH^sK =G^VAK,

(3.75)

for the forward-going and backward-going waves, respectively. These equations provide a simple rule for converting the temporal evolution of the modulation envelope to the spatial progression. That is, relations (3.72) and (3.73) can be written as F^(z, 0 = e expi-iutG^^^) W^(z - ut, 0) + F,^(z, /),

(3.76)

V-(z, t) = e cxpiiutG^y^) Wj(z + ut, 0) + V;{z, t).

(3.71)

In most practical situations, the first-order terms V(^ may be neglected and Wj^ can be introduced also on the left-hand sides of eqs. (3.76) and (3.11), using eq. (3.70). Nevertheless, V^ play an important role in that they incorporate the coupling to the wave going in the opposite direction and do give rise to the nonlinear reflection. As an illustration of the above quantum treatment, the traveling-wave generation of squeezed light by parametric down-conversion of a short pulse is examined. For the case of a classical pump, this problem was treated through a modal analysis by Tucker and Walls [1969]. More recently, Yurke, Grangier, Slusher and Potasek [1987] and Caves and Crouch [1987] treated this problem by using spatial differential equations for appropriately defined creation and annihilation operators. 3.1.3. Space-time displacement operators Serulnik and Ben-Aryeh [1991] have discussed the general problem of electromagnetic wave propagation through nonlinear nondispersive media. They have used a four-dimensional formalism of the field theory in order to develop an extension of the formalism introduced by Hillery and Mlodinow [1984]. The complications following from the common definitions for the vector and scalar

4, § 3]

Macroscopic theories and their applications

337

potentials are indicated. It is shown that the scalar potential can be neglected only by using alternative definitions. First, Serulnik and Ben-Aryeh [1991] show that the conventional approach that uses the standard potentials A and V is not appropriate for treating the general case of nonlinear polarization when V • P ^ 0, since in this case V does not vanish. As a solution to this problem they propose to use the vector potential ip, D = -Wxtp,

(3.78)

which fulfils the relation V • D = 0. This choice enables them to work in the new Coulomb gauge, where V • i/^ = 0, so that from the condition V • ^ = 0 it follows that the dual scalar potential $ obeys the equation V^§ = 0.

(3.79)

It is then consistent to assume ^ = 0 everywhere in a nonlinear medium, and the dual scalar potential need not be taken into account. Serulnik and Ben-Aryeh [1991] derive the Lagrangian and Hamikonian densities from the Maxwell equations by using nonconventional definitions for the scalar and vector potentials. The general form of the energy-momentum tensor is derived and explicit expression for its elements is given. The relation between this tensor and the space-time description of propagation is analyzed. Further the quantization is performed and the properties of space-time displacement operators are presented. Space-time is described by a Lie transform (Steinberg [1985]). Serulnik and Ben-Aryeh obtain the displacement operators from their energy-momentum tensor with an alternative definition for the vector potential. They are able to obtain explicit expressions for all the elements of the energy-momentum tensor and to discuss their physical meaning. In the following we will show that the actual relationship between the energymomentum tensor and the space-time description of propagation is different from that derived by Serulnik and Ben-Aryeh [1991]. Let us restrict ourselves to the usually treated one-dimensional case, where only the fields E\, D\, B2 and A2 are significant, and we use A2 = -^2 according to Drummond [1990, 1994]. The arguments of these fields are JC3 and ct. The corresponding quantum fields obey the commutation relations [A2(x3,cO,^2(y3,cO] =ihcd(x,-y,)l

(3.80)

[b,(x,,ct),B2(y3,ct)]

(3.81)

= -ihcd'ix, -y,) i.

Considering for this case the Hamikonian density

H=U-^b]+B\\,

(3.82)

Quantum description of light propagation in dielectric media

338

[4, §3

where the right-hand side is symmetrically ordered (cf. Abram and Cohen [1991]), we obtain the equations of motion in the Heisenberg picture:

'-•I"djci

dt dB2 dt

B

•I"

(3.83)

= CB2,

=

(3.84)

-C-

dX3

Relation (3.83) explains the role of the dual vector potential, and relation (3.84) is essentially the second of the Maxwell evolution equations. We could obtain the first of them as the equation of motion for the quantum field D\. It is a question whether the tensor element 1

(3.85)

c

is a correct quantum density for generation of the displacement as indicated by Serulnik and Ben-Aryeh [1991]. The presumable equations of the spatial progression are i

dA2 dx2

he

dB2 dx2

he

i

(3.86)

Al, • //<(£>,fi2)5 dx3 ^ •

«

.

/

(P,B2)s^^

_ dB2

(3.87)

Relation (3.86) expresses the role of the dual vector potential, and relation (3.87) is a mere tautology. The same is obtained for the quantum field D\. This failure of the application of the ordinary presentations of quantum field theory has been published by Ben-Aryeh and Serulnik [1991]. Considering, in contrast, the equal-space commutators [A2(x3,cO,I>i(x3,cO] = 6 ,

(3.88)

[A2(jC3,cO,^2(x3,cO] =ihed{et-et')\,

(3.89)

[B2{x3,et\B2{x3,et')\

(3.90)

= ihed\et - ct') 1,

[D,(x3,cO,^2(x3,cO] = 6 ,

(3.91)

[Z)i(x3,cO,Z>i(x3,cO] ='\heed'{et-et')\,

(3.92)

we obtain peculiar equations for the spatial progression: dA2 dx3

8X2

A2, f{b,B2hdt ^b,,j{b,B2)sdt

= -D^,

\dB2 ' c dt '

(3.93)

(3.94)

4, § 3]

Macroscopic theories and their applications

339

Relation (3.93) expresses the role of the dual vector potential, and relation (3.94) is essentially the second of the Maxwell evolution equations. We could obtain the first of them as the equation of the spatial progression for the quantum field B2. Since we often have to make a guess about previously unknown commutators, the above example is a warning against excessive trust in the spatial progression technique. For a medium with nonlinear polarization, the global nature of creation and annihilation operators is lost. By consistently following this idea, Serulnik and Ben-Aryeh [1991] have introduced the shift operators which, by their definition, are based on the energy-momentum tensor. They have followed in their treatment Peierls' solution of the problem of momentum conservation in matter (Peierls [1976, 1985]) by which the atoms or the bulk matter are considered to be at rest while the electromagnetic field is propagating. They show that it is always possible to relate the external field in front of the medium to that behind it by the use of the shift operators, that is by a Lie transformation. As we can see from relations (3.83), (3.84) and (3.93), (3.94), the transition from the so-called time-displacement operator to the displacement operator in the X3 direction must be accompanied by a change of integration variable from JC3 to ^ Leonhardt [2000] has determined an energy-momentum tensor of the electromagnetic fields in quantum dielectrics. The tensor is Abraham's [1909] plus the energy-momentum of the medium characterized by a dielectric pressure and enthalpy density. While the consistency of this picture with the theory of dielectrics has been demonstrated, a direct derivation from the first principles has been announced only.

3.1.4. Generator of spatial progression Theoretical methods for treating propagation in quantum optics have been developed in which the momentum operator is used in addition to the Hamiltonian. A successful quantum-mechanical analysis has been given for various physical systems which include amplification and coupling between electromagnetic modes by Toren and Ben-Aryeh [1994]. Distributed-feedback lasers have been described, but an overarching generalization of both successftil analyses has not been developed. The authors have paid attention to distributedfeedback lasers (Yariv and Yeh [1984], Yariv [1989]) in which contradirectional beams are amplified by an active medium and are coupled by a small periodic perturbation of a refractive index.

340

Quantum description of light propagation in dielectric media

[4, § 3

The energy and momentum properties of the electromagnetic field can be described, in four-dimensional form, by the energy-momentum tensor F^, where 7,A: = 0,1,2,3 (Roman [1969]), W g, gy g= JT/'^ =

Sx O^, 0,y O,.Sy Gy-, Oyy Oy-_

(3.95)

S- 0-v O-y 0--

The tensor element T^^ represents the energy density. The vector {gx,gy,gz) represents the density of the vectorial momentum (proportional to D x B). Let us take further, for example, the fourth row. The tensor element T^^ is the component of the Poynting vector representing the flux of energy in the z-direction. The vector (a-v, O-y, a--) refers to a flux of momentum in the propagation direction of z. In the conventional approach (Roman [1969]), the four-vector/?^^ is defined as /^=

f I f T^''(x,y,z,t)dxdydz.

(3.96)

The energy p^^ is used as the Hamiltonian for the description of time evolution; the momentum component/?^^ does not describe a real propagation equation (but cf eq.3.87). Ben-Aryeh and Serulnik [1991] have shown that for the description of the spatial progression, the four-vector p^^ can be used given as 3^ _ f f f T^k P = r\x,y,z,t)cdtdxdy.

(3.97)

We must assume that the commutators (3.88)-(3.92) have been simplified in analogy with the relations (3.80) and (3.81). The momentum component in the z-direction, p^^, can be used as the generator of the spatial progression, and the energy p^^ is expected to translate the field in time. Toren and Ben-Aryeh [1994] treat propagation problems by expanding the field operators in terms of mode operators associated with definite fi*equencies. Starting with (Caves and Crouch [1987]), the approach has been associated with the conservation of commutation relations for creation and annihilation operators, which are space dependent (cf. Huttner, Serulnik and Ben-Aryeh [1990]). Imoto [1989] has developed the basic equation of motion by using a modified procedure of canonical quantization in which time and space coordinates are interchanged in comparison with the conventional procedure.

4, § 3]

Macroscopic theories and their applications

341

Ben-Aryeh, Luks and Pefinova [1992] did the same by using a slightly different notation. Toren and Ben-Aryeh [1994] dissociate themselves from this approach, but they are not very explicit about whether the use of the integrals (3.97) is compatible with the canonical quantization in which the time coordinate plays the usual role. Linear amplification is treated by the use of momentum for space-dependent amplification. Traveling-wave attenuators and amplifiers can be treated as continuous limits of an array of beamsplitters (Jefifers, Imoto and Loudon [1993], Ban [1994]). According to Toren and Ben-Aryeh [1994], the propagating modes are coupled to a momentum reservoir. The Hamiltonian of this system is given by H = h (coa^a - ^

W/resi-/^)'

(3-98)

and the total momentum operator is G = h 0 a - Y, I^Ms'bjbj + Yji^i^^bj + >^;abj)

(3.99)

where the subscript res stands for the reservoir, K/ are appropriate coupUng constants, and a and bj represent (in the zeroth order) modes which are propagating in the positive direction of the z-axis. The equations of motion obtained from the momentum operator (3.99) are ^ = i[a, G] = i/5a + i 5 ^ Kjb], dz

(3.100)

db] ^ = -^\b], G] = -xK*a + ip,;J].

(3.101)

By using the spatial Wigner-Weisskopf approximation, the Heisenberg-Langevin equations can be obtained:

^ = [ i O S - 4 / 3 ) + i y ] « + Zt,

(3.102)

dz

where .«

y

_

vn

f

l'^(/^es)IV(fes)

= {2jr|/C(/^es)|V(/?res)}|/^^.^.,,,

.^

(3.103) (3.104)

342

Quantum description of light propagation in dielectric media

[4, § 3

with Vp. the principal value of the integral, p(ftes) the density function of the wave propagation constants ^yres in the reservoir, and

V = Y^iKjb]S^-\

(3.105)

The codirectional coupling is also analyzed. It is assumed that two modes are propagating in the same direction and they are coupled by a periodic change in the refractive index. For a classical description, we refer to Yariv and Yeh [1984] and Yariv [1989]. The Hamiltonian is given by Ho=H = ha) a\a\ +a\a2

(3.106)

where the classical relation a)\ = (O2 = (O has been used. The total momentum operator is G = h P\a\a\ + ^a\a2 + ka\a\ + k*a\a2

(3.107)

where ^\ and ft are components of the wavevectors of the two modes in the propagation direction of z and

;r = /rexp

/ im2jr \ —-z ,

^^ ,^^^ (3.108)

with K 3. coupling constant, m an integer, and A the "wavelength" of the spatial periodic change in the index of refraction (a perturbation in the dielectric constant). In this connection papers by Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] and Ben-Aryeh, Luks and Perinova [1992] are criticized by Toren and Ben-Aryeh [1994] for not taking account of the spatial dependence (3.108). The equations of motion obtained from the momentum operator (3.107) are ^ dz ^ dz

- k « i , 6 ] = iA«i+i^*«2, n = ^[a2,G] = i/cai+i/32a2. n

(3.109) (3.110)

We define slowly varying operators of the form ii(z) = al(z)e-^^^•^

A2(z) = a2(z)e-*^^\

(3.111)

4, § 3]

343

Macroscopic theories and their applications

Substituting the operators (3.111) into equations (3.109) and (3.110) we get d4i ~d7

(3.112) (3.113)

~d7 where

AI3 =

l3,-k-m^

(3.114)

is the mismatch. A "field" mismatch may be cancelled by a medium component. For the input-output relations we refer to Yariv and Yeh [1984], Yariv [1989] and Pefinova, Luks, Kfepelka, Sibiha and Bertolotti [1991]. Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] introduced m = 0 and 26 = -AI3 in an application to the codirectional coupler. In general, the solution to eqs. (3.112) and (3.113) coincides with the classical solution, Aj «-> AJ, J = 1,2, where A\ and A2 are the amplitudes of the waves propagating in the +z-direction. The counterdirectional coupling is analyzed in (Toren and Ben-Aryeh [1994]). The total Hamiltonian is given by ^ Pico \a\a\-a^a2

Hn=H

(3.115)

We consider the momentum operator in the form = h P\a\a\ - (hci2^2 + ^^1^2 "^ ^*^i^2 « 2 ^^

(3.116)

^T

It is reasonable that the Hamiltonian and the zeroth-order operator are related, respectively, to the flux of energy and that of the component of momentum in the z-direction. Compared to Toren and Ben-Aryeh [1994], we have interchanged the operators ^2 and al. Toren and Ben-Aryeh criticize our assumption [S2, ^2] - - 1 , which we obtained by this interchange (Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991]) fi*om the usual equal-space commutator [^2,^2] ^ ^- ^^ ^^ tempting to have the same alternation between the opposite-going modes as can be seen in comparison of eq. (3.303) with eq. (3.305) (cf Abram and Cohen

344

Quantum description of light propagation in dielectric media

[4,

[1994]). The equations of motion obtained from the operator (3.116) are given by dfli

i

"dF ~ li da2

i

"d7 ~ h

a\, G

= iP\a\ + iK*a2,

(3.117)

«2, G

= -ika\ + '\f^a2-

(3.118)

Using the slowly varying operators (3.111), in contrast to Toren and Ben-Aryeh [1994], we obtain that

"d7

^ i^*i2e-'^^^-".

- = -iKA,e^'''

(3.119)

with A^ defined by eq. (3.114). In the work of Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991], 26 = -AP still holds in an appHcation to the counterdirectional coupler. The solution to eqs. (3.119) coincides with the classical solution, Aj ^ AJ, j = 1,2, where A\ and A2 are the amplitudes of the waves propagating in the +z- and -z-directions. Yariv and Yeh [1984] obtained the solution to the corresponding classical equations for the boundary conditions A\(Z)\_^Q = A[(0), ^2(^)\- = i "= M{L)' First, however, one obtains the solution for the usual condition at z = 0. Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] obtained the output operators in terms of the input ones. While we simply determine the operators A\{L), ^2(0) from two equations for the operators ^i(O), ^2(0), A\{L), AiiL), we observe that in this procedure the equal-space commutator [Ai.Aj] - - 1 must depend on both z and Z in a complicated manner, and simplifies to [^2,^2] ^ 1 for z = 0,1. Since the commutators correspond to the Poisson brackets, much is illustrated by the appropriate classical theory (Luis and Pefina [1996]). One must be aware of the fact that in formulating the theory, Luis and Pefina [1996] avoided the above considerations on the z-coordinate and the generator of spatial progression, and they used in the main part of their paper the usual time dependence and the Hamiltonian fijnction. Although still obscure in the case of commutators, the situation is clear in the classical case, when the input-output transformation is characterized by the usual Poisson brackets and the solution for the usual boundary conditions at z = 0 requires the noncanonical transformation ai ^ cC{, with the complex amplitude ai. The richness of their theory is due to nonlinearities, whereas

4, § 3]

Macroscopic theories and their applications

345

it is shown that in the quantum case only a poor linear theory is possible. The difficulty lies in the formulation of an appropriate dynamical operator. Tarasov [2001] has defined a map of a dynamical nonlinear operator into a dynamical superoperator. He had in mind quantum dynamics of non-Hamiltonian and dissipative systems. A quantum-mechanical treatment of the distributed-feedback laser using the momentum operator in addition to the Hamiltonian has been developed by Toren and Ben-Aryeh [1994]. They start from the classical description based on two coupled equations M

= IvA,\yAi-iKA2&' -\i^J-.P^'^f' = — 1=

(3.120) ^=i/c*^,e-^/^'---i7^2,

where A \ and A2 are the amplitudes of the waves propagating in the +z- and -z-directions, respectively, K is the coupling constant, 7 is the amplification constant, and AjS is given by eq. (3.114), with /?i = ^, ft == -/S. The solution of the classical equations is well-known (Yariv and Yeh [1984], Yariv [1989]) and it shows that under special conditions the amplification becomes extremely large. The classical theory does not include the quantum noise which follows from the amplification process. Unfortunately, Toren and Ben-Aryeh [1994] did not develop an overarching generalization of the analysis of amplification and that of the counterdirectional coupling. To the best of our knowledge, such a quantum-mechanical theory is not in hand. The treatment of parametric down-conversion and parametric up-conversion by Dechoum, Marshall and Santos [2000] is interesting with its use of the Wigner representation of optical fields, but it starts just from the Maxwell equations for the field operators and the lossless neutral nonlinear dielectric medium. Using the common approximation of treating the laser pump as classical, they obtain classical equations of nonlinear optics. 3.2, Dispersive nonlinear dielectric 3.2.1. Lagrangian of narrow-band fields Drummond [1990] has presented a technique of canonical quantization in a general dispersive nonlinear dielectric medium. Contrary to Abram and Cohen [1991], Drummond creates an arbitrary number of slightly varying copies of the free electromagnetic field for the nonlinear dielectric medium; essentially, the

346

Quantum description of light propagation in dielectric media

[4, § 3

number required by the classical slowly-varying-amplitude approximation. But Abram and Cohen work with a single field. The paradox of both approaches being valid can be resolved only by a detailed microscopic theory. Drummond [1990] generalizes the treatment of a linear homogeneous dispersive medium (Schubert and Wilhelmi [1986]). Until 1990, the reference papers for the theory of inhomogeneous nondispersive linear dielectrics were Knoll, Vogel and Welsch [1987], Bialynicka-Birula and Bialynicki-Birula [1987] and Glauber and Lewenstein [1989]. Hillery and Mlodinow [1984] were attractive with their use of the idea due to Born and Infeld [1934] for the quantization of a homogeneous nonlinear nondispersive medium. Macroscopic quantization is a route to the simplest quantum theory compatible with known dielectric properties, unlike the microscopic derivation of the nonlinear quantum theory of electromagnetic propagation in a real dielectric. Drummond [1994] compares the quantum theory obtained via macroscopic quantization with the traditional quantum-field theory. He concedes that most model quantum field theories prove to be either tractable but unphysical, or physical but intractable. The tractable model quantum-field theory ceases to be unphysical when it is tested experimentally in quantum optics. An excellent example of this is provided by fiber-optical solitons whose quantization is given in detail by Drummond [1994]. In agreement with theoretical predictions (Carter, Drummond, Reid and Shelby [1987], Drummond and Carter [1987], Drummond, Carter and Shelby [1989], Shelby, Drummond and Carter [1990], Lai and Haus [1989], Haus and Lai [1990]), experiments (Rosenbluh and Shelby [1991]) led to evidence of quantum solitons. More recent experiments (Friberg, Machida and Yamamoto [1992]) demonstrate that solitons can be considered to be nonlinear bound states of a quantum field. In addition to the quadrature squeezing in Rosenbluh and Shelby [1991], quantum properties of soliton collisions were measured (Watanabe, Nakano, Honold and Yamamoto [1989], Haus, Watanabe and Yamamoto [1989]). Similar nonlinearities are encountered in photonic-bandgap theory (Yablonovitch and Gmitter [1987]), microcavity quantum electrodynamics (Hinds [1990]), pulsed squeezing (Slusher, Grangier, LaPorta, Yurke and Potasek [1987]) and quantum chaos (Toda, Adachi and Ikeda [1989]). It is advantageous to begin with the treatment of a classical dielectric by introducing the nonlinear response function in terms of the electric displacement field D. Contrary to the usual description (Bloembergen [1965]), which uses the dielectric permittivity tensors in a decomposition of this field, the inverse expansion is necessary here. For simplicity, the dielectric of interest is regarded as having uniform linear magnetic susceptibility. The charges are assumed

4, § 3]

Macroscopic theories and their applications

347

to occur only in the induced dipoles of polarization. The field equations are therefore V x : £ ( x .0 V x H{x

=

dB(x,t)

dD(x,t) dt ' = 0, =

,0 =

V D{x ,0

(3.121)

V B(x ,0 == 0, where D{x, t) = eoE(x, t) + P{x, t\ (3.122) B{x, t) = iiH{x, t). Here /»CX)

^"(^,0= /

X{x.T)'E{xJ-T)dT

Jo /•CXD

+ /

/»CXD

/

X^^\x,Tur2)

: E(x,t - T^)E(x,t - T2)dTidT2

Jo Jo /»(X)

/»(X)

/"CX)

+ / / / Jo Jo Jo

X^^^(x,ri,r2,r3):£'(Ac,/-ri)£'(A:,/-r2)£'(x,^-r3)dridr2dr3

+ •••, (3.123) where the tensor of rank 2 and, in general, the (n + l)th-rank susceptibility tensor read, respectively,

Xix,r)=^lx{x,(o)c"'"daj, X^"\x, Tu...,r„)=

QAJ-

• jt"\x,

w\...,

«")e-'<'"' ^' ^ - ^"'"^"' dw' • • • dw". (3.124)

348

Quantum description of light propagation in dielectric media

[4, § 3

After adding the vacuum electric displacement eoE(x,t) to both sides of eq. (3.123), we express the electric vector in the form

E(x,t)=

/ Jo

+ / / Jo Jo /»CXD

/»CXD

i(x,T)-D(x,t-T)dT

i^\x,ruT2)

: D{x,t - TOD(x,t - T2)dT, dT2

/>CXD

+

^^^\x,ruT2,T3y:D{x,t-Ti)D(x,t-T2)D(x,t-T3)dr^dT2dT3 Jo Jo Jo

+ •••, (3.125) where i(x,r)=^

JUx,CO)c-'^'^'do),

i^'\x, ri, . . . , T,)= (iy\j..p^\x,

co\...,

co'Oe-'^^'^'^' " -^^'^"^"Ma;^ • • -do;'',

(3.126) and the tensors in the right-hand sides of eqs. (3.126) are given by the recurrence relations e(x,w)'l(x,a))

= l,

e(x, 0)^ + 0)^) • l^^\x, a)\w^) + x^"\x, w*, w^) : Ux, co^) Ux, co^) = 0^^\ e(x, (0* + w^ + 0)^) • l^^\x, a)\ (XT, (X?) + 2x^'^\x,(o\o?

+ a?) :

l{x,o)^)l^'\x,a;^o?)

+ x^^\x, (D\ a;^ 0?): l{x, w^) l{x, a?) l{x, o?) = 0^^\

with e(x,(jo) = eol + Xix,(^) permittivity. In particular, l(x,co) = [e(x,co)r\

(3.127) the usual frequency-dependent tensor of

(3.128)

4, § 3]

Macroscopic theories and their applications

349

Here 1 and 0^"^ are the second-rank unit tensor and the {n + l)th-rank zero tensor, respectively. Introducing the Fourier transforms of the electric strength field and electric displacement field, respectively,

^(jc, CD) = I E{x, t)e'''' d^,

b{x, CO) = f D(x, t)e'''' d/,

(3.129)

and performing the Fourier transform of both sides of eq. (3.125), we obtain that E{x,

(D) = l(x,

-^

w) • ^(jc, (JO) ll^^\x,(o\w-w^):b(x,w^)b(x,co-(o^)dw^

+ / l^^\x,co\(o^,co-co^-w^)':b(x,(D^)b(x,co^)b{x,(jo-o)^-0)^)d(o^

&o?

+ • • •. (3.130) The inverse relation of b{x,w) involves the terms /(jc, co) = X^^\X,-(D\(JO), X^^\x, a)\(o-(o^) = X^^\x, -co; co\co-co^),.... A similar extension of notation is conceivable also in tensors l(x,co), l^^\x, co\co- co^),.... Let us note that Pefina's [1991] similar relation for P{x, co) comprises sums instead of the integrals. Relation (3.130) may be matched to relation (2.4) of Drummond [1990] on the condition that the integrals be replaced by the sums. Such a change does not only affect the meaning of the tensors /^"^ and l^"\ but also (and above all) the physical unit of their measurement. We will treat the time-averaged linear dispersive energy for a classical monochromatic field at nonzero frequency co. For a permittivity e(x, co), this can be written in terms of a complex amplitude £(x) (Bloembergen [1965], Landau and Lifshitz [1960], Bleany and Bleany [1985]),

{//) = J

U^x)

• ^[coe(x,co)]-£(x)^-^ ^ [coe(x, CO)]. £(x) + ^^ {B(x, t) • B(x, t)) ) d'x, (3.131)

where E(x, t) = 2 Re[f (jc)^'^'^']. It is important to distinguish the monochromatic case from the case of quasimonochromatic fields. In the more general case, the displacement D is expanded in terms of a series of complex (envelope) ftinctions, each of which

350

Quantum description of light propagation in dielectric media

[4, § 3

has a restricted bandwidth. The relevant non-zero central frequencies are then o;-^,...,a>^, thus yv

Z>(jc,0= 5^/)^(Jc,0,

(3.132)

v = -N

where D~^ = (Z)^)* and in the monochromatic case D\x,t)

= V\x)Q-'''^^.

(3.133)

The notation we use here differs slightly from that of Drummond [1990]. Again, the electric field vector can be expanded as N

E(x,t)=

^

E\x,t),

(3.134)

where E~^ = (E^y and in the monochromatic case E''(x,t) = £\x)Q-'"'^''.

(3.135)

In the case of quasimonochromatic fields, relations (3.133) and (3.135) should be replaced by 1

pC) +0

D\x, t)=:r-

b(x, 0)) e'"'" dco, *^"'"r'

E\x,

t)=—

(3.136) E(x, w) Q-''" d(o.

Bloembergen [1965] presented the relation (3.131) as being sufficiently accurate for this case. While relation (3.131) is exact for monochromatic fields, it must be modified for a quasimonochromatic field as follows:

mt')){t)=/ (i E ^"'(^'') • a^t'^'^^^' ^"^] • ^'(^' '^ (3.137) +

^^{B{x,t')B{x,t')){t)\d'x.

4, § 3]

Macroscopic theories and their applications

351

By modifying the summation, we obtain the energy integral in terms of the electric displacement fields

^"•(^'') • [^(^')-w''^i'{x,oj'')]-D\x,t) ^''

{H{t')){t) = IUY. •^ V \' = -N +

~{B(x,t')B(x,t'))(t)\A'x.

(3.138) To complete the description, we supplement relations (3.132) and (3.134) with the expansion of the magnetic induction field: N

B{x,t)= Y.

^"(^'')'

(3.139)

where B^ B\x,t)

= (By,

(3.140)

= — /

B(x,(D)Q-'"'d(D.

(3.141)

Next, ^(jc, a;) can be approximated near w = co^' by a quadratic Taylor polynomial, l(x, CO) ^ Ux) + wlUx) + {w'l[!(xl

(3.142)

so that ^d^(x,Oj)

^ix,co)-co ^y^

_ ^

^ ^

,,<://,

' ^ Ux)-\co^iy{x).

(3.143)

Using the notation D = (d/dt)D, we rewrite relation (3.138) in the form

(//(/'))(o=^E / v =

D\x,

t) • ly(x) • D'(x, t)

-N'

\b-\x,

t). ?:;(jc). b\x,

1 t) + ^,B\x, ^^

t) • B\x, t) d'x.

(3.144) Here we deviate slightly from Drummond [1990]. Drummond speaks of time averages, and he indicates time averaging on the left-hand side and partially on

352

Quantum description of light propagation in dielectric media

[4, § 3

the right-hand side in (3.138), but he does not remove the time dependence from the right-hand side. A canonical theory of linear dielectrics will be obtained using the causal local Lagrangian. Drummond [1990] considers a Lagrangian L[A~^,... ,A^], which is a functional of (components of) the dual vector potential. This is defined as A, for which Z)(jc, 0 = V X A(x, t),

Bix, t) = iiA{x, t).

(3.145)

We introduce also A{x,o))

= f A(x,t)e'''dt,

A"{x,t)

= ;^ / lit

1

(3.146)

MO +0

A{x,(D)e-""'do).

(3.147)

Each quasimonochromatic field obeys the Maxwell equations

V X E\x, t) = -B'(x, t), V X H\x, t) = b'(x, t), VD\x,t) = 0, V-B\x,t) = 0,

(3.148)

where E\x, t) = Ux) • D\x, t) + il',(x) • b'ix, t) - {l';(x) • b\x, 1

t), (3.149)

H\x,t)=-B\x,t). The components of the dual vector potential fulfil linear wave equations. On the basis of (3.144) we can infer a Hamiltonian fianction of the form W = Wo = ^ / ^

I [V X A-\x,

t)]. Ux) • [V X A\x, 0]

- i[V X A-\x,

+

t)] • ^(^(jc) • [V X A\x, t)]

(3.150)

IAA-\x,t)'A\x,t)[d^x.

In order to quantize the theory, a canonical Lagrangian must be found that corresponds to eq. (3.150) and generates the Maxwell equations (3.148) as Hamilton's equations. The linear Maxwell equations can be recast as a wave equation. Wave equations for wave functions are considered. It is next necessary

4, § 3]

Macroscopic theories and their applications

353

to derive a Lagrangian whose Lagrange's variational equations correspond to these wave equations and whose Hamiltonian corresponds to eq. (3.150). Since A^ can be specified to be transverse fields, the variations can also be restricted to be transverse. These restricted variations can be realized using transverse ftinctional derivatives (Power and Zienau [1959], Healey [1982]). Drummond [1994] derived the Lagrangian using the method of indeterminate coefficients in the form

N

-\

Y.

{^^'\x,t)A\xJ)-[V - i[V X

A-'(A:,

- i[V X A-\x,

X A-\xj)]

• Irix) • [V X A\x,t)]

0] • ?;(x). [V X A > , 0] 0] • ly{x)' [V X A\x, t)]} d^x. (3.151)

The canonical momenta are n(x,

t) = B\x,

0 - IV X i$;(jc) • D\x,

t) + i[[x)' D \x, t)

(3.152)

where for brevity we re-introduced the fields (3.145) again. Analogously, we can rewrite the Lagrangian of Drummond in the form

=\

C ^ 1 Y.{ -^'(^^ 0 • B\x, t) - D\x, - \D\x,

t) • l[ix). b\x,t)

t). Ux) • D(x, t)

t). l'^,{x). b{x,O} d^x. (3.153) The Legendre transformation, i.e., a substitution of JV' in the Hamiltonian (3.150) by an expression in W and A^\ was not performed by Drummond [1990]. The reason is that each JV' is to be found from (3.152) considered as a partial differential equation. Also, this theory simplifies a great deal if planewave one-dimensional propagation is considered. The local Lagrangian method is used as the foundation for a nonlinear canonical Lagrangian and Hamiltonian. The objective is the total Lagrangian and Hamiltonian of the form L = LQ- f U^(x,t)d^x,

- \b-\x,

n = HQ+ f U^(x,t)d^x,

(3.154)

354

Quantum description of light propagation in dielectric media

[4, § 3

where U^(x,t) is a nonlinear energy density, N

N

Vi =-N

N

V2= - / V V'3 = - A '

N

N

N

N

Vi = -N

V2 = -N

V3 = -N

V'4 =

-N

:Z)^'^(A:,0/>'-(A:,0^''(^,05-aV.,r,/'2W3+./'4 + • • ••

(3.155) In order to give an example, a one-dimensional case is treated and the nonlinear refractive index, being the lowest nonlinearity, is of most universal interest. For N = \, one has U''{x,t)=\l^'^\D\xJ)\\

(3.156)

Drummond [1990] has presented the quantization of the nonlinear medium using a treatment of modes defined relative to the new Lagrangian. The canonical momenta have the form (3.152) in the nonlinear case too. In the corresponding quantum theory, the field operators A^' and W are introduced, which obey transverse commutation relations of the form [Aj{xj\ni\x\t)] - ihd^{x-x')d,X

(3.157)

Since these operators are not Hermitian, it is also interesting to note that A-^ = (Ay,

ii'

= (my.

(3.158)

This entails that AJ and (fl-)^ commute. Then, a set of Fourier-transformed fields is defined and the annihilation operators a^ and b^ are introduced. The operators a^ correspond to the normal modes while b^ generate additional necessarily vacuum modes. This feature of the theory is due to the dependence of the Hamiltonian (3.150) on both the real and the imaginary parts of the components A^(x,t) (A^'(xJ)). 3.2.2. Propagation in one dimension and applications Drummond [1994] discusses in detail a simplified model of a one-dimensional dielectric, where l(x, (D) = l(co) 1, l^"\x, co\...,w") = l^"\co\..., co") 1^"\

4, § 3]

Macroscopic theories and their applications

355

with 1^"^ the (n + l)th-rank unit tensor for n odd and l^"\x,a)\...,co'') = 0^^^ for n even, « ^ 3. Instead of the time average of the energy (3.131), eq. (3.137), Drummond [1994] presents the total energy in the length L, I \\liH^{t)+ [ E(T)b(T)dT dx. Jo I J to

^(t)=

(3.159)

The Hillery-Mlodinow theory, which does not take account of dispersion (S.-T. Ho and Kumar [1993]), has the commutation relation of the electric field with the magnetic field modified from its free-field value. Drummond [1994] points out that this commutator problem is solved when one does take account of dispersion as an important physical property of a real dielectric. The ingenious analysis of the linear dielectric permittivity has not been generalized to the nonlinear dielectric permittivity. Traditionally, the description of the nonlinear medium assumes that the dispersion terms are negligible. Neglecting unphysical modes, the dual vector potential has the expansion

A\x, t) = y J

^5

ake''-\

where a^ and al, have the standard commutators [aA,5f|.,] = the solutions of a,, = J ^ ^ .

(3.160) 5A,A'1?

and Wi^ are

(3.161)

This enables one to write no = J2^^kalak

(3.162)

k

and (reintroducing D^) n = ^h(Dka\cik^k

[ U^(b^)dx.

(3.163)

J

When there is a nonlinear refractive index or a C^^^ term, the free particles interact via the Hamiltonian nonlinearity. It is this coupling that leads to soliton formation. It is also possible to involve other types of nonlinearity, such as Q^^ terms, that lead to second harmonic and parametric interactions.

356

Quantum description of light propagation in dielectric media

[4, § 3

With respect to practical applications, it is necessary to define photon-density and photon-flux amplitude fields. The photon-density amplitude field reminds us of the so-called detection operator (Mandel [1964], Mandel and Wolf [1995]). A polariton-density amplitude field is simply defined as

^(jc,o=4- y^e^^^"'^'^'"''^^/^-'

p-164)

where k^ = k{a)^) is the center wave number for the first envelope field. This field has an equal-time commutator of the form [W(xut), ^U^2,0] = 5(jci -JC2) i,

(3.165)

where S is an L-periodic version of the usual Dirac delta function

S(xi -X2) = jJ2^'^''^"'~"'\ ^

(3166)

Ak

where the range of Ak is equal to that of A: - A: ^ The total polariton number operator is « = /*.„,„*(.,-)d,.

(3.167,

The polariton-flux amplitude can also be approximately expressed as 0(x,O= A/|^e't<^'-^'^-^-^^'^'%,

(3.168)

where u is the central group velocity at the carrier frequency co^ Thisfluxhas an equal-time commutator of the form [0(xut), 0\x2,t)]

= u8{x, -X2) i.

(3.169)

Operationally, {0^(x,t)0(x,t)) is the photon-flux expectation value in units of photons/second. A common choice is to define the dimensionless field t/^(x, t) by the scaling ^(x,t)= W(xj)J—, (3.170) Vn where n is the photon number scale and to is a time scale, defined in such a way that {\l)\x,t)\l){xj)) is appropriate for the system. This scaling transformation

4, § 3]

Macroscopic theories and their applications

357

is accompanied by a change to a comoving coordinate frame. The first choice of an altered space variable gives

?.-V^,

T=-.

(3.171)

Here XQ is a spatial length scale introduced to scale the interaction times. An alternative moving frame transformation is

?--,

^. = ^ -

(3.172)

The quantization technique developed by Drummond [1990] was later applied to the case of a single-mode optical fiber (Drummond [1994]). On simplified assumptions, the nonlinear Hamiltonian is (cf. eq. 3.156)

n=

[hw(k)a\k)a(k)dk-i-\l^^^ hw(k)a\k)a(k)dk+\t^^^ / D\x)d^x. f D'

(3.173)

Here (o(k) are the angular frequencies of modes with wavevectors k describing the linear photon or polariton excitations in the fiber including dispersion. a(k) are corresponding annihilation operators defined so that, at equal times, [a(k'),a\k)] = d(k -k')l. In terms of modes of the wave guide and neglecting the modal dispersion, the electric displacement b{x) is expressed as

Dix) W = i iI /\/ /

fte(A:)fo(A;).^,_^__^,_ __^ ^,4^,

^'

' ' aik)uik,r)e"''dk

+ H.c.,

(3.174)

where

/ '

u{k,r)\'d'r=\.

(3.175)

Here u(k) is the group velocity and e{k) is the dielectric permittivity. The mode fianction u{k,r) is included here to show how the simplified one-dimensional quantum theory relates to the vector mode theory. When the interaction Hamiltonian describing the evolution of the polariton field W(x, t) in the slowly-

358

Quantum description of light propagation in dielectric media

[4, §3

varying-envelope and rotating-wave approximations is considered, the coupling constant Xe is introduced as

1V2 4e(A:')c

JHrtd'r.

(3.176)

After taking the free evolution into account, the following Heisenberg equation of motion for the field operator propagating in the +;c-direction can be found: d_ d_ 1'{x,t) = ^dx^ dt

2 dx^

+

\Xc'i'\x,t)W{x,t) V(x,t),

(3.177)

where u = v(k^) = dco/dk\^^j^,, CD" = 5^co/9A:^|^^^|. In a comoving reference frame, this reduces to the usual quantum-nonlinear Schrodinger equation:

— i:7-;^e^/(x„r)^,(x„0 ^ife,0, 2 ac2

(3.178)

where W\{xu,t) = ^(x^; + ut,t). In the case of anomalous dispersion which occurs at wavelength longer than 1.5 ^am, allowing solitons to form, the second derivative o)'' can be expressed as w'^ = h/m, where m is the effective mass of the particle. Similarly, the nonlinear term Xe describes an interaction potential

V(x„x'j = -X,S(x,-xly

(3.179)

This interaction potential is attractive when Xe is positive as it is in most Kerr media. It is known that this potential has bound states and is one of the simplest exactly soluble known quantum field theories (Ben-Aryeh [1999]). The repulsive and attractive cases were investigated by Yang [1967, 1968]. This theory is onedimensional and tractable and does not need renormalization, while two and three-dimensional versions do need renormalization. In calculations, it is preferable to substitute the flux amplitude operator

Wix,t)=^0(xj)

(3.180)

into equation (3.177). Drummond associates the idea of spatial progression with the flux amplitude operator. Upon modifying the time variable, he obtains an "unusual form" of the quantum-nonlinear Schrodinger equation, which he reduces to a more usual form again. Since the operators there have their standard meaning, they must have equal-time commutators. In contrast, the resulting

4, § 3]

Macroscopic theories and their applications

359

equation (Drummond [1994]) appears as the quantum-nonlinear Schrodinger equation with time and space interchanged. This means that the operators have equal-space commutators. The problem is whether these commutators are well defined. An important physical effect in propagation is the Raman scattering fi-om molecular excitations. For this reason, the nonlinear Schrodinger equation requires corrections due to refi-active-index fluctuations for pulses longer than about 1 ps, especially when high enough intensities are present, and fails for pulse durations much shorter than this. Korolkova, Loudon, Gardavsky, Hamilton and Leuchs [2001] have studied a quantum soliton in a Kerr medium. They have simplified, implicitly, the classical propagation equation for the slowly varying electric field envelope by introducing a new time measurement in dependence on a position. In changing to dimensionless variables they make the new time a "position" and the position a "time" variable and then get a classical nonlinear Schrodinger equation.

3.3. Modes of the universe and paraxial quantum propagation 3.3.1. Quasimode description of the spectrum of squeezing Toward the end of the 1980s it had become clear that the use of squeezed states (Walls [1983], Loudon and Knight [1987]) in interferometry can lead to enhanced signal-to-noise ratios. Milbum and Walls [1981] have shown that the cavity of a degenerate parametric oscillator admits only 50% squeezing (in the steady state). Yurke [1984] was the first to realize that this pessimistic conclusion does not hold, as the noise reduction in the transmitted field can be quite different from that in the intracavity field; the first step is to relate the field operators inside and outside the cavity. Whereas it was obvious at the time that the field operators inside the cavity remain the usual quantum-mechanical annihilation operators of one or a small number of harmonic oscillators, the connection of the field operators outside the cavity with the "Langevin-noise operators" was not established until the 1980s (Collett and Gardiner [1984], Gardiner and Collett [1985], Carmichael [1987]). These authors have clarified the relation between this subtle property of squeezed light and its generation and the concept of light propagation. Not only the interpretation, but also the derivation of the Langevin-like "noise" terms was presented by Lang and Scully [1973], after they had introduced and studied the "modes of the universe" (Lang, Scully and Lamb Jr [1973], Ujihara [1975, 1976, 1977]). Ley and Loudon [1987] studied the "mode-strength function", which exhibits peaks near the wavevectors

360

Quantum description of light propagation in dielectric media

[4, § 3

of the cavity quasimodes. Barnett and Radmore [1988] have defined a similar mode-strength function for the phenomenological coupling between the cavity quasimodes and the external environment and found that their function has peaks at the frequencies of the quasimodes. It is appropriate here to mention a book by Scully and Zubairy [1997], where the results of Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a] are expounded or formulated as exercises. The latter discuss the modes of the universe, which include the interior of the imperfect cavity of interest, and use them to define the intracavity quasimode, the incident external field mode and the output field mode. The mutual coupling of these modes emerges naturally in this formalism. Following Lang, Scully and Lamb Jr [1973], the one-sided empty cavity is described also by the relation t:J,(t) = r£W (t _ '^\ +lEl:\t),

(3.181)

where / is the cavity length, r is the real amplitude reflection coefficient and 1 is an appropriate transmission coefficient, 1 = \ / l - r ^ . Here £'•„ (0 is the positive-frequency part of the input field; it obeys the commutation relation El^\t),El;\s)\ =Kd(t-s)l,

(3.182)

where £|;>(0 = [El^\t)V and

with Q the quasimode frequency. For the full Fox-Li quasimode (Fox and Li [1961], Barnett and Radmore [1988]), a single-mode annihilation operator a(t) is defined: «(0 = y ^ ^ * : ^ ( O e ' " ' .

(3.184)

It is convenient to use the slowly varying amplitudes £l^\t), fout(0 ^^^ fcav(0 for the input, output and cavity fields related to the cavity frequency Q, respectively: ^''\t)

= £J^\t)Q''^^\

7 = in, out, cav.

(3.185)

4, § 3]

Macroscopic theories and their applications

361

In the situation where it holds that

(,_|)»(,_|;.),(„_|1„„

(3.186)

in the short cavity round-trip time limit, we get a quantum Langevin equation

^a(0 = - r a ( 0 + y ^ C ( 0 ,

(3.187)

where

^ = 'i ^

(3.188)

Further, Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a] define, for arbitrary measurement times, the spectrum of squeezing of the output field via the quadrature variances. They present a microscopic effective Hamiltonian model of balanced homodyne detection. They refer to the fundamental papers of Collett and Gardiner [1984], Gardiner and Collett [1985], Caves and Schumaker [1985] and Yurke [1985], where this concept of spectral squeezing was originally treated. As an approximation, the following operator is introduced: ^out(6a;) =

1 /KT

I ^0

^-'out (Oe'^^^'^^d/,

(3.189)

where bo) is the (level) separation between two nondegenerate ground states of the atom, and T is the measurement time or the inverse of the detection bandwidth. As shown also by Yurke [1985] and Carmichael [1987], with a balanced homodyne detector one measures the combinations with appropriate phase shift 6: ^OUtfAO

2

e'^^C(0 + e-'^^C(0

(3.190)

where £oJ(t) = [oouliOY^ ^^^ fr^i^i this the natural generalization of the singlemode quadrature concept is ^out,(8w) •

e'AU^aj)^Q'''^Aa-^co)

(3.191)

One might wonder why a non-Hermitian operator is taken for such a generalization of the Hermitian operator. Finally, the connection between single quasimode

362

Quantum description of light propagation in dielectric media

[4, § 3

squeezing and spectral squeezing is explored and the difference in the noise reduction inside and outside the cavity is clarified in a way that lends itself to simple visualization. Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990b] have first analyzed measurements of small phase or frequency changes for an ordinary laser, and calculated the extracavity phase noise for a phase-locked laser. These analyses are based on the mean values and normally ordered variances of quantum operators for which classical Langevin equations may be written down. The classical Langevin formalism is fiarther replaced by the alternative Fokker-Planck formalism for the calculation of the spectrum of squeezing. This general Fokker-Planck formalism has been applied to the twophoton correlated-spontaneous-emission laser. It was seen that without onephoton resonance and initial atomic coherences involving the middle level, the maximum squeezing of the intracavity mode is 50%, while the detected field can be almost perfectly squeezed. Almost the exact reverse holds, however, if one-photon resonance and initial atomic coherences involving the middle level are present. In particular, the intracavity field may be perfectly squeezed while the outside field shows almost no squeezing in the same quadrature, but still has, in fact, increased noise in the conjugate quadrature. Finally, the effect of finite measurement time on the quadrature variances has been briefly analyzed. Dutra and Nienhuis [2000] have unified the concept of normal modes used in quantum optics and that of Fox-Li modes from semiclassical laser physics. Their one-dimensional theory solves the problem of how to describe the quantized radiation field in a leaky cavity using Fox-Li modes. In this theory, unlike conventional models, system and reservoir operators no longer commute with each other, as a consequence of the use of natural cavity modes. Aiello [2000] has derived simple relations for an electromagnetic field inside and outside an optical cavity, limiting himself to one- and two-photon states of the field. He has expressed input-output relations using a nonunitary transformation between intracavity and output operators. 3.3,2. Steady-state propagation Deutsch and Garrison [1991a]) assumed that in the case of an ampHfier, one is usually interested in the spatial dependence of temporally steady-state fields. They do not attempt to reformulate one-dimensional propagation, cf Abram and Cohen [1991], where the temporal evolution by the Hamiltonian is supplemented by the spatial progression with the momentum operator. The

4, § 3]

Macroscopic theories and their applications

363

alternative proposal is made that the quantum-mechanical equivalent of the classical steady-state condition is the description of the system by a stationary state of a suitable Hamiltonian. There is a formal resemblance to nonrelativistic many-body theory for a complex scalar field (Deutsch and Garrison [1991b]), which helps determine the Hamiltonian. In this theory a non-Hermitian envelopefield operator W{z, t) with the property W{z,t\ W\z',t)\ = 6{z-z') i

(3.192)

is introduced. In the application to the optical field, the vector potential operator (or electric-field-strength operator in the lowest order) corresponding to a carrier plane wave of a given polarization unit vector e is expressed as

t:\z,t) = eJj^^^W(z,t)

exp[i(^z-a;0],

(3.193)

where A is the beam area and n(a)) is the dispersive index of refi-action. In contrast to Deutsch and Garrison [1991a], we make a simplification, i.e., we will not consider a carrier-wave Hamiltonian. For a single wave interacting nonlinearly with matter, the total Hamiltonian can be written as

^ ( 0 =4nv(0 + ^mt(0,

(3.194)

where Hem(t) is the Hamiltonian governing the free progression of the envelope and Hmi(t) is a general interaction Hamiltonian. In fact, the generality will not be exercised and we will treat only the vacuum input and the case of a degenerate parametric amplifier. In the standard Heisenberg picture, the equation of motion for the envelopefield operator reads dW{z,t) _ i 'nz,t),H(t) dt h

(3.196)

or for a linear medium dW(z,t) _ c• d4^(z,t) dt n(w) dz '

(3 197) y' J

364

Quantum description of light propagation in dielectric media

[4, §3

The solution is ct

(3.198)

•^(^'^^=^f^-;K^'«

In the standard Schrodinger picture, the state |
dt

(3.199)

H,n,(t) + Hm(t) | 0 ) .

Introducing the carrier-wave Hamiltonian has invoked the consideration of the Schrodinger picture (Deutsch and Garrison [1991b]), along with the envelope picture which we have confined ourselves to. Relation (3.193) is the positivefrequency component of the electric field in the envelope picture similarly as relation (4.5b) of Caves and Schumaker [1985] is this component in the interaction picture. The envelope picture is essentially the modulation picture of Caves and Schumaker [1985]. For the application under consideration, the steady state (ss) solutions are identified with the stationary solutions to eq. (3.199),

J^env ' ^ i n

(3.200)

l^)ss = ^l<^)s

For the stationary solutions, the label (ss) will be omitted. In the case of the degenerate parametric amplifier, the interaction Hamiltonian can be written as (Hillery and Mlodinow [1984])

//int(0

- ^ / / / ( ^

/'\z)£;(z)cxp[-i(k,z

a)^t)][El:\z,t)f

+ }lx.}

dxdydz,

(3.201) where a^p is the pump frequency, (J0p = 2co, X^^\^) is the second-order susceptibility coupling the pump to the degenerate signal and idler fields, £p(z) is the pump •(+)/ M+) 7(+)/ amplitude, and [Elo\z,t)Y = £';,7(z,0 • E\o (z,t\ with E\o\z,t) given by the 7( + )/ formula E^^\z,t) = e- E\,\z, t). Substituting for E'^^z.0 from relation (3.193) gives the interaction Hamiltonian in the envelope picture: ih c H\r^x — 2 n{w)

I

K''(Z)W\Z)-K(Z)W^-(Z)

dz.

(3.202)

4, §3]

Macroscopic theories and their applications

365

with (3.203)

J^(^) = ^go(^)exp[i0(z)],

(3.204)

n((x))c (l>(z)

Jt

(3.205)

+ Mz^l3(zy

Here go(z) is the standard power gain coupHng constant (Yariv [1985]), Ak = 2k- kp is the phase mismatch at the degenerate frequency, and I3(z) is the remaining phase originating from the product /^^^(z)f*(z). To solve the time-independent Schrodinger equation, Deutsch and Garrison [1991a] assume that the eigenstate is a squeezed vacuum state corresponding to a two-photon wave function. They define a functional squeezing operator S[^] = exp

0/

^iz)W^\z)-r{z)W\z)

dz

(3.206)

with z-dependent squeezing parameter ^(z) = -r{z) exp[i6{z)]. The squeezed vacuum is defined as |0){s}=5[|]|0).

(3.207)

For the squeezed vacuum to be a stationary solution, ^(z) and A should obey ^^[?](4nv -f^int) m

|0) = A |0).

(3.208)

Applying the operator W(z) to both sides of eq. (3.208), and taking into account that A«l^(z)|0)=0 (3.209) =

S^[^](^H,,,+H,r,,)s[^W(z)\0),

we rewrite the eigenvalue problem in the A-independent form [S^[^]{H,,,+H,n,^

S[^l W(z)\ |0)^.j = 0 = ^ |0),

(3.210)

366

Quantum description of light propagation in dielectric media

[4, § 3

where the commutator C is n(co) { - exp[i0(z)] ( exp[-i0(z)] cosh[r(z)] ^ (exp[i0(z)] sinh[r(z)]) d - -— (cosh[r(z)]) sinh[r(z)] dz + /f*(z)exp[i0(z)]sinh^[r(z)] - K(z) exp[-i0(z)] cosh^[r(z)]) W\z) (3.211) ( d + cosh[r(z)]-—(cosh[r(z)]) Mz V dz - exp[i0(z)] sinh[r(z)]-^ (exp[-i0(z)] sinh[r(z)]) dz + /c*(z) exp[i0(z)] sinh[r(z)] cosh[r(z)] - K(z) exp[-i0(z)] sinh[r(z)] cosh[r(z)]^ W{z)

+ dz^^(^)}. The eigenvalue condition requires that the real and imaginary parts of the coefficient of ^ ^ vanish, yielding the desired propagation equations £

= igocos(0-0),

%- =-goCOth(2r)sin(0-0). dz

(3.212) (3.213)

Upon introducing the complex amplitude C(z) = - exp[ie(z)] tanh[r(z)],

(3.214)

we can write a propagation equation for it in the compact form ^±&-

= K{z)-K*{z)l;\z),

(3.215)

which may be useful for guessing the boundary condition r(z)Uo = 0,

0(z)Uo = 0(O).

(3.216)

When /3(z) = 0 and the phase difference Q(z) - 0(z) is small, the squeezing parameter r(z) is a weighted integral of the experimental parameter ^go(^0? ^' ^ [^,2:];

4, § 3]

Macroscopic theories and their applications

367

when moreover go{oo) > \Ak\, the squeezing parameter 6{z) converges to a function of the experimental parameters {0(z)-arcsin[A^go(oo)]}. Direct solution of eq. (3.208) requires that the real and imaginary parts of the coefficient of W\z) vanish, yielding the propagation equations (3.212) and (3.213) again. The presence of the singular operator W{z)^f\z) indicates that A generally has no finite value. 3.3.3. Slowly-uarying-enuelope approximation A macroscopic approach to the quantum propagation aims at a quantum version of the slowly-varying-envelope approximation. Such an envelope implies that the wave is paraxial and monochromatic. The problem of quantum propagation of paraxial fields was considered first by Graham and Haken [1968]. Revived interest is indicated by Kennedy and Wright [1988]. Deutsch and Garrison [1991b] begin with generalizing the results of Lax, Louisell and McKnight [1974], which develop the classical theory of a strictly monochromatic wave in an inhomogeneous nonlinear (perhaps amplifying) medium. The generalization is made only to a quasimonochromatic wave, and the quantum theory is presented in the simplest system of codirectional propagation considering only the freefield dynamics. In the Coulomb gauge, the positive-frequency component of the vector potential satisfies the free-field wave equation W'A^-\xJ)-\^A^^\x,t) = 0.

(3.217)

The approximation of the slowly varying envelope is introduced by expressing A^^\x,t) as an envelope modulating a carrier plane wave propagating in the z-direction with wave number ko and frequency COQ ^ ck^, A^'-\x, t) = Ao W(x, t) exp[i(^oz - oj^t)].

(3.218)

Here, ^{x, t) is a vector-valued function, henceforth referred to as an envelope field, and ^o is a normalization constant, which we will specify above relation (3.226). The initial positive-frequency component can be expressed as "'*'*^^'' = ^ ^ = r 2 i ^ / r a

5]^A(*)^AWe'*-M^ft.

(3.219)

Here ex{k) are the orthogonal polarization unit vectors, and the reduced Planck constant is introduced in view of the possible later quantization. Tx{k) are thus momentum-space wave functions.

368

Quantum description of light propagation in dielectric media

[4, § 3

The intuitive notion of a paraxial field is that it is composed of rays making small angles with the main propagation axis. In other words, a paraxial wave function {Tx(k)} is concentrated in a small neighborhood of the wavevector ko = koei, of the carrier wave. We define/A(^) by the relation fx{q) = ^x{q^h\

(3.220)

where q is the relative wavevector. Let us observe that q = {qj,qz), where qj is the transverse part of ^, ^ = ^j + ^r^3. In contrast to Deutsch and Garrison [1991b], we stress that we express the concentration in a small neighborhood of qo = 0, by letting the wave fiinction {fx(q)} depend on a small positive parameter 0\f}{q) ^fx(q, 0). Let us assume that /.(,„,..)= V / V 7 , ( ^ , ^ , e ) ,

(3.221)

where

and we have introduced the notational convention that an overbar indicates a dimensionless fianction of the scaled variables (and perhaps 6). This relates to defining a dimensionless "momentum" vector i; == (i^, r/-), where

The fimctions of interest are those that have a convergent power-series expansion in 6, oc

7,(i/,0) = ^ 0 " / r ( i j ) .

(3.224)

n=0

In contrast to Deutsch and Garrison [1991b], we note that

7,(t;,0) =7f(i;).

(3.225)

Relations (3.223) and (3.225) lead to the wave ftinction being ^-dependent, different from Deutsch and Garrison [1991b].

4, § 3]

Macroscopic theories and their applications

369

Substituting the integral (3.219) for the envelope field defined by eq. (3.218) at ^ = 0 by the momentum-space wave fianction given by eq. (3.220), and choosing Ao = \/hc/2ko, we find that

•^(-.^ - 0,0) = ^

/ 7 ^ »

E ..(. +W.(.,a)e-'d3,. A — 1.2

(3.226) Here, the parameter 6 has been introduced, which is not present in the integral (3.219), where Tx(k) = ^A(^, ^), and A^^\x, t) = ^^^^(jc, t, 6). Deutsch and Garrison [1991b] investigated the integro-differential form of the wave equation for A^^\x, t, 6): i-^'^»(jc, t, 6) = c(-V-)''- A^*\x, t, 0),

(3.227)

where (-V^)''^^ is an integral operator defined by

{-VY'F(x) = ^^_

J \k\ F(k) e'*- d^*,

(3.228)

with F(k) the Fourier transform. Let us note that V = (Vj, §:). Substituting from eq. (3.218) into (3.227) gives i— W(x, t, d) = {cQ - (i)Q)W{x, t, e\ at where

(3.229)

with the transverse Laplacian

The scaled configuration-space variables | = (|T, 0 are §T = Okoxj,

t = d^h)Z,

(3.232)

and the dimensionless time variable is T = e^oMit.

(3.233)

370

Quantum description of light propagation in dielectric media

[4, § 3

After expressing the envelope field in the form W(x, t, 6) = W(xj,z, t, 6) = -^WiOkoXj, vV

e^koz, d^coot, 0),

(3.234)

we can rewrite relation (3.229) as

i—W(iT,e) = n(e)V(iT,e),

(3.235)

where Hid) = ^^{d)-l],

(3.236)

Q(d) = l^Qfek,VT,0'ko^Y

(3.237)

with Uko

OL, 6~ko OZ

This provides the possibility of expanding the differential operator H(d): CX)

Hid) = J2 d"n^"\

(3.239)

n=0 7(")

where the differential operators H are just defined by the formal expression. The dimensionless amplitude ^ ( | , r, 0) has the expansion V(l T,e) = Y, 0"V^"\l T).

(3.240)

n=0

It is evident that the terms satisfy the following equations: i^W'\l dr

r)=Y^ n''-"'W''\l r),

« - 0,1,2,....

(3.241)

In the article by Deutsch and Garrison [1991b], the discussion of the classical equation of motion is completed by considering the initial-value problem. We rewrite eq. (3.226) as "Pix, 0) = — ^ f ^

K,(q)fM 0) e'»- d'q,

(3.242)

4, § 3]

Macroscopic theories and their applications

371

where the function Kx{q) is defined by

with ex{q) = ex{q + h\

(3.244)

Re-expressing eq. (3.242) in terms of the scaled variables gives l P ( | , r = O;0)= — ^ 3 ^ / Y. ^^^^

•'

jf,(i;, 0)7,(1/) e'-'-^d^I/.

(3.245)

A =1.2

The scaled kernel function is ifA(i;,e)=^#%,

(3.246)

where w(iy, 0) = yJ\ + e\2r]-_ + r]l) + d^r]l

(3.247)

Considering the expansion oo

Kx{n,e)-Y.^''K';:\t,),

(3.248)

«=0

we obtain the initial expansion of the envelope fields:

^'"*^^^= ( 2 ^ /

^

KT{n)f,{n)e'''d?t,.

(3.249)

A=l,2

Deutsch and Garrison [1991b] claim that the preceding arguments can be used to identify the subspace of the photon Fock space consisting of the paraxial states of the field. They resort to the space S, the infinitely differentiable functions that decrease, when |i/| -^ oo, faster than any power of |i/| ^

372

Quantum description of light propagation in dielectric media

[4, § 3

Let us recall the standard plane-wave creation and annihilation operators al(k), ax(k), and the wave-packet creation and annihilation operators

^h^=

f Yl ^x(k)al(k)d'k ^

(3.250)

X=\,2

and its conjugate. We now define

"^

A=l,2

where cl{q) = aliq + ko)

(3.252)

are the creation operators corresponding to the envelope field. For the subsequent analysis, a unitary operator T(d) is of interest, f(0) I/'; m) = \f(dy, m),

(3.253)

such that f(a)|0) = |0),

f(0)ct[/]7't(0) = c t [ y v / ( ^ , | ) ]

(3.254)

where (cf. eq. 3.221) /A, • A,„(?T 1, ^z I, • • • , ?Tm, ^zm, 6)

(3.255) /A,-A4 0 '

g2'--'

0 ' 02 ; •

In the description of the dynamics using the Schrodinger picture, the paraxial approximation means mainly the evolution of the initial state \(p{t,0)), ^^\
(3.256)

where m,e))\,^_, = m\0'{t = O)).

(3.257)

Defining the state \cp'{t,e)) = Pie)\0(t,d)),

(3.258)

4, § 3]

Macroscopic theories and their applications

373

for all times, we can rewrite eq. (3.256) in the form

I \0\t, 0)) = -'-H'{d) \^\t,6)),

(3.259)

where H'{d) = f\d)Hf{e\

(3.260)

Using the expansion

H'{d) = Y^ 0"'H'^"'\

(3.261)

we may expand eq. (3.259) into a set of coupled equations for the coefficients of the series oc

|0'(^, 6)) = Y^ e'" \0'^"'\t)).

(3.262)

w=0

Describing the dynamics in the Heisenberg picture, we should generalize the relations (3.260) and (3.261) to an arbitrary operator M(t), M\t,d) = f\e)M{t)f{e\

(3.263)

m=0

We can then rewrite the equation of motion _ M ( 0 = -^[M(/),^(/)]

(3.265)

in the form -M'{t,6)

= -'-[M\t,e\H'{t,

0)1

(3.266)

We may expand this equation into coupled equations similarly as eq. (3.259). Since f(l)=i,

M'{t,\) = M{t\

(3.267)

relation (3.264) simplifies for 0 = 1, or any operator M{t) can be expressed as CXD

M{t) = Y^M^"'\t).

(3.268)

In both pictures, definition (3.233) can be used whenever it is advantageous.

374

Quantum description

of light propagation

in dielectric media

[4, § 3

In accordance with Deutsch and Garrison [1991b], we introduce the operator ^f{x) by relation (3.226), with »F(jc, / = 0,0) »-^ ^{x) on the left-hand side and fxin^ S) "-^ Q ( ^ ) on the right-hand side. This operator can be expanded as oo

Wix) = Y^ W^"\xl

(3.269)

n=0

where •^^^^W = 7

^

/ E

^

^

-^ A=l,2

^1"^(^) ^A(^) e'^ ^ d^iy.

(3.270)

Using this expansion, we can compute the commutators between fields of different orders:

'^'^ '

•'

A =1,2

(3.271) As expected, the nth-order commutator can be expressed as n

Wi(x), W^(x')

E

[^^""\^), ^y "^^(JcO] .

(3.272)

m=0

The equal-time commutation relations are preserved by the dynamics in each order of the approximation scheme.

dt

Wi{xj\WJ{x\t)t^

= b.

(3.273)

In zeroth order, the theory yields a quantized analogue of the classical paraxial wave equation, and formally resembles a nonrelativistic many-particle theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same truenumber operator. In addition, it is shown that the 0{d^) difference between them vanishes for experiments described by stationary coherent states. 3.4. Optical nonlinearity and renormalization Abram and Cohen [1994] mainly applied a traveling-wave formulation of the theory of quantum optics to the description of the self-phase modulation

4, § 3]

Macroscopic theories and their applications

375

of a short coherent pulse of Hght. They seem to have been the first to use renormaUzation (Itzykson and Zuber [1980], Zinn-Justin [1989]). The renormalized theory successfully describes the nonlinear chirp of the propagating pulse, and permits the calculation of the squeezing characteristics of self-phase modulation. The description of the propagation of a short coherent light pulse inside a medium with an intensity-dependent refractive index (Kerr effect) has become relevant to optical-fiber communications, all-optical switching, and optical logic gates (Agrawal [1989]). The neglect of dispersion and of Raman and Brillouin scattering leads to the description of self-phase modulation. In classical theory it is derived that, in the course of propagation, the pulse becomes chirped (i.e., different parts of the pulse acquire different central frequencies), which also influences its spectrum. Abram and Cohen [1994] have pointed out many difficulties in the investigation of the quantum noise properties of a light pulse undergoing self-phase modulation. The traditional cavity-based formalism truncates the mutual interaction among the spatial modes to the selfcoupling of a single mode (or only a few modes) and cannot give a reasonable approximation to the frequency spectrum produced by self-phase modulation. In spite of the difficulties, papers based on a single-mode field description have indicated that the self-phase modulation can produce squeezed light (Kitagawa and Yamamoto [1986], Shirasaki, Haus and Liu Wong [1989], Shirasaki and Haus [1990], Wright [1990], Blow, Loudon and Phoenix [1991]), and others have treated squeezing in solitons (Drummond and Carter [1987], Shelby, Drummond and Carter [1990], Lai and Haus [1989]), an effect that has been verified experimentally (Rosenbluh and Shelby [1991]). Blow, Loudon and Phoenix [1991] have shown the divergence of nonlinear phase shift which Abram and Cohen [1994] treat through the process of renormalization. Let us review the basic features of the quantization of the electromagnetic field in a Kerr medium and discuss the relevance of the renormalization procedure to the treatment of divergences of effective medium theories. We consider a transparent, homogeneous isotropic and dispersionless dielectric medium that exhibits a nonlinear refi-active index. We examine a situation similar to that in Abram and Cohen [1991]. The Hamiltonian for the electromagnetic field in a Kerr medium is ^ ( 0 = J {^2 [^'(^'0 + ^E\z,O]

+ lxE\z,O}

dz,

(3.274)

where the integration along the direction of propagation, z, is written explicitly, while the integration over the transverse directions x and y will be implicit. We use the Heaviside-Lorentz units for the electromagnetic field without passing

Quantum description of light propagation in dielectric media

376

[4,

over to atomic units, i.e., h = c = I, and x is the nonlinear (third-order) optical susceptibility. From the perspective of substitution of eq. (3.22), the Hamiltonian (3.274) can be written as (Hillery and Mlodinow [1984]) H(t) =

/{

B^{z,t)+-b\z,t)

-^£)\z,/)|dz,

(3.275)

where the displacement field has been defined as b(z,t) = eE(z,t) + xE\zJ).

(3.276)

The canonical equal-time commutators are (cf. eq. 3.25) [A(zutXb(z2,t)]

= -ihcd{z, -Z2) i,

(3.277)

[B{zut)Mz2.t)]

= -ihcd'iz, -Z2) I

(3.278)

It is convenient to adopt a slowly-varying-operator picture in which the zerothorder dynamics of the field governed by the linear-medium Hamiltonian are already taken into account exactly, while the optical nonlinearity can be treated within the framework of perturbation theory. In such a picture, a field operator Q(z, t) evolving inside a nonlinear medium is related to the corresponding linearmedium operator go(^, 0 by Q{zj)=U-\t)Q^{z,t)U{t).

(3.279)

The unitary transformation Lf{t) is given by

^(0 = Texp

4/>'

(r)dr

(3.280)

where T = T denotes the time ordering and

m) = -^,jbtiz,t)
(3.281)

is the interaction Hamiltonian. The full nonlinear Hamiltonian (3.275) in which the exact displacement and magnetic induction field operators D(z,t) and B(z, t) are replaced by the corresponding zeroth-order (linear-medium) operators Z)o(z, t) and Bo{z, t) can be written as H{b^X)

= m ) + H,{t\

(3.282)

where

Ho{t)= \ j \Bl{z,t)+-^^^^^

(3.283)

is the linear-medium Hamiltonian. Following the traditional modal approach to relation (3.279), Kitagawa and Yamamoto [1986] developed a singlemode treatment of the self-phase modulation. Clearly, such an investigation

4, § 3]

Macroscopic theories and their applications

317

is valid only inside an optical cavity with a sparse mode structure. In this situation, the time evolution cannot be interpreted as space progression. In developing a traveling-wave theory for self-phase modulation, Blow, Loudon and Phoenix [1991] obtained a solution similar to the single-mode solution. They encountered a nonintegrable singularity upon normal ordering, a phenomenon known in quantum-field theory as an "ultraviolef divergence. To avoid an infinite nonlinear phase shift due to this simple description of the Kerr interaction. Blow, Loudon and Phoenix [1991, 1992] introduced a finite response time for the nonlinear medium as regularization in the Heisenberg picture. Alternatively, Haus and Kartner [1992] considered group-velocity dispersion for pulse propagation in the medium as a regularization. At any rate, regularization is a subsequent sophistication of the simple model known from classical theory. The response time as well as the group-velocity dispersion are necessary ingredients of a complete description of the propagation of an electromagnetic excitation in fibers. But they have no influence on the effects associated with the vacuum fluctuations under study. A systematic way of dealing with the vacuum fluctuations in quantum field theory is the procedure of renormalization (Itzykson and Zuber [1980], Zinn-Justin [1989]). In order to obtain finite results, the procedure of renormalization redefines all the quantities that enter the Hamiltonian. The renormalization point of view is that the new Hamiltonian is the only one we have access to. It contains the observable consequences of the theory, and the parameters are the ones we obtain from experiments. The bare quantities are only auxiliary parameters that should be eliminated exactly from the description (Stenholm [2000]). The re-defined (renormalized) quantities are able to incorporate the (infinite) effects of the vacuum fluctuations. We will provide the definitions of broad-band electromagnetic field operators and treat the propagation of light in a linear medium. The normal ordering is considered as the simplest renormalization, e.g., in the case of the effective linear Hamiltonian (3.283). To this end, the Hamiltonian Ho{t) is to be written in terms of the creation and annihilation operators. The normal ordering allows us to subtract the vacuumfield energy up to the first order from the effective Kerr Hamiltonian (3.275). However, when this Hamiltonian is used to describe propagation disregarding the richness of the quantum-field theory, the normal ordering gives rise to additional divergences that can be attributed to the participation of the vacuum fields. Upon renormalization, involving also the refractive index, the divergences are removed. As the Kerr nonlinearity involves the fourth power of the derivative §-fA(z, t), it cannot in general be renormalized to all orders with a finite number

378

Quantum description of light propagation in dielectric media

[4, §3

of corrections. Inspired by nonlinear optics, the slowly-varying-amplitude approximation decouples counterpropagating waves and renormalization to all orders becomes possible. All types of optical nonlinearity x^"^ give rise to divergences which require renormalization. In the treatment of parametric downconversion (Abram and Cohen [1991]), the problem of divergences and the need of renormalization were not formulated. Abram and Cohen [1994] defined the broad-band electromagnetic field operators, and treated the propagation of light in a nonlinear medium. In the absence of optical nonlinearities, / == 0 and the linear-medium displacement field has the usual proportionality relationship with the electric field: (3.284)

Do(z,t) = eEo(z,ty

The magnetic field and the displacement field in the linear medium obey the equal-time commutation relation Bo(zut),Do(z2,t)

(3.285)

= -\hcd\z\ -zi) i.

The operators V^ (cf eqs. 3.70, 3.71) from Abram and Cohen [1991] reappear as the operators

Mz,t)

1

V^(z,tl

1

^-(z,t)

Vo(z,t).

(3.286)

For V^±(z, t), the equations of motion in the Heisenberg picture may be calculated by use of the commutator (3.285),

l^-(^''^4

//o,t/;±(z,0

d^

(3.287)

where u = c/^/e is the speed of light inside a dielectric with refractive index e. Their solutions are t/;+(z,0

xjj+iz - ut, 0),

(3.288)

t/;_(z,0

V^_(z + t;^0).

(3.289)

The equal-time commutators of the copropagating field operators can be obtained from the definition and the commutator (3.285) as t/;+(zi,0,V^+(z2,0

= -ifiud\z\ -zi) i.

(3.290)

V^-(zi,0,V^-(z2,0

= ihud'{z\ -Z2) 1.

(3.291)

4, § 3]

379

Macroscopic theories and their applications

For the fields, the corresponding operators commute with each other, (3.292)

0.

\l}+{Zxj\\l)-{Z2j)

The operators (3.286) permit us to express the hnear-medium Hamiltonian (3.283) as

m) = \j\l)l{z,t)-^xi)l{zj)

(3.293)

dz.

thus separating it into a sum of two mutually commuting partial operators, one for each direction of propagation. In the homogeneous medium it is possible to separate the electromagnetic field operators V^±(z,0 into positive- and negative-frequency parts. (3.294)

i/;±(z,o = 0±(z,O + 0±(^,O, defined as h{zj)=\

\p±{z,t)±

i

r v^±(z^o,^,

(3.295)

z -z' The operators 0^(z, t) and 0±(z, t) can be considered as creation and annihilation operators, respectively, for a right- (or left-) moving electromagnetic excitation which at time t is at point z. The equal-time commutators of 0±(z, 0 are somewhat complicated, 0±(Zi,O,0±(Z2,O

hv d

1 Z\

-Zi

=F '\7lb(z\ -zi) 1, (3.296)

0+(Zi,O,0!(Z2,O = 0, where V refers to the familiar generalized fianction V-_. Nevertheless, an important simplification results when only unidirectional propagation is considered. Upon introducing the operators

b',\z,t)0+(z,O-0-(z,O ,

B'-\z,t)-

it

D':'{z,t) B';\z,t)

,

(3.297) (3.298)

[4, §3

Quantum description of light propagation in dielectric media

380

and considering the relations Do(z,t)

xl)^{z,t) + xl).{zj)

Bo(z,t) =

V2

xp^iz, t) - ip^z, t) (3.299)

and relation (3.294), we verify that Z)o(z, 0 = D^^\z. t) + b^^\z, t\

Bo(z, t) = B^^\z, t) + B^,\z, t).

(3.300)

Using the new operators, the equal-time commutation relation (3.285) can suitably be modified as B^^\zut\b',\z2.t) = --hcd\zi-Z2)l.

(3.301)

For a right-moving electromagnetic excitation, we observe that 0^(^)(z,,O = V2B[\l^(zutl

0.(^)(22,O = )J^b^oil)(^2,tl

(3.302)

where the subscript (+) refers to A: > 0. Using eq. (3.302), we obtain that

0+(zi,o,0!(z2,o

= -ihuS\z\ -zi) 1(+ (+)•

(3.303)

(+)

Similarly, for a left-moving electromagnetic excitation, we note that 0_(_)(zi,O = -y2^|^!^(zi,0,

0-(-)fc,O = ]J-^&oU^2,t%

(3.304)

where the subscript (-) refers to A: < 0. From this, (3.305) = i^t;^'(zi -Z2) i (-)• The electromagnetic creation and annihilation operators allow us to speak of the normal order, for instance, when we write the Hamiltonian (3.293) in the form 0_(zi,O,0!(z2,O

J(-)

Ho(t) = [ te(z, 0 Mz, t) + 0!(z, 0 0_(z, t)] dz.

(3.306)

We can define annihilation and creation wave-packet photon operators F^{z,t) = fF(z-z)Mz,t)dz, F|(Z,0

= /F*(z-z)0|(z,Odz,

respectively, with F(z) a complex function: 1 F(z) = exp(L^) F(z),

(3.307) (3.308)

(3.309)

where uK is the central (carrier) fi-equency and F(z) is the wave-packet envelope fianction peaked at z = 0 and A: = 0.

4, § 3]

Macroscopic theories and their applications

381

Under the usual assumption of narrow bandwidth and ihv I F\z)F\z)dz

= \,

(3.310)

where F' denotes the spatial derivative, we obtain that the operators F+ and F} follow the boson commutation relation [F4zJXFl(z,t)\=l

(3.311)

Let us remark that the commutation relation (3.311) is relation (A5) of Milbum, Walls and Levenson [1984], where the formalism of counterdirectional coupling was derived or rather this pitfall was underestimated. Now we can consider a coherent pulse whose shape is described by pF(z) with a scaling factor p. A coherent state appropriate to pF is defined as |pF)=exp[p(F,^-F,)]|0).

(3.312)

It satisfies the "single-mode" eigenvalue equation F^(z,t)\pF)=p\pF)

(3.313)

and, at the same time, it obeys the approximate quantum-field eigenvalue equation 0+(z,O \pF) = h\^pF\z-z)Qxp[-iK{z-z)]

\pF).

(3.314)

The approximation made in the derivation of eq. (3.314) has kindled the interest in the Glauber factorization conditions and the theory of coherence (see also Ledinegg [1966]). When we examine right-moving pulses, we can introduce a moving-frame coordinate, r] = z-ut,

(3.315)

and simplify relation (3.288), Mz,t)

= 0(ri,O) = m ,

(3.316)

dropping the subscript + whenever we use the coordinate t] explicitly. Similarly, the commutation relation (3.303) can be modified. The right-moving narrowbandwidth wavepacket operators (3.307) and (3.308) can now be written as F(fj) = jF(r]-f])mdr], where f] = z-ut,

(3.317)

and

F\r]) = j F\r]-m\r])dr]. In the moving-frame representation F{r], t) ^ F(rj, 0).

(3.318)

382

Quantum description of light propagation in dielectric media

[4, § 3

It is feasible to find a connection with the approaches leading to narrow-band field operators (a) appearing in papers by Shirasaki and Haus [1990], Drummond [1990] and Blow, Loudon, Phoenix and Sheperd [1990], and used in papers (Blow, Loudon and Phoenix [1991], Shirasaki and Haus [1990]). An important feature of these operators is that their commutator is a 6 fiinction ^>to(^l),«l,fe) = hukod(zi-Z2)l

(3.319)

Under the same narrow-bandwidth condition, the commutator of the AbramCohen operators, which is a delta-function derivative, can be approximated by d\z,-Z2)^-ikod(z,-Z2).

(3.320)

Abram and Cohen [1994] have analyzed the approximations that enter the quantum treatment of propagation in a Kerr medium and outlined the corresponding renormalization procedure. The slowly-varying-amplitude approximation according to Abram and Cohen [1991] is used by Abram and Cohen [1994]. According to eq. (3.299), the interaction Hamiltonian (3.281) can be written as

HM =16e2 -lL j\l)+{z - vt) + V^_(z + vt) dz.

(3.321)

The exact Hamiltonian (3.275) may be written up to the first order in x as H(t) = Ho(t) + H^s^(t) + H,s-(t) + d(x'l

(3.322)

where ^1S±(0 - - ^ ^ 2 / V^i(^ ^ ^^)d^

(^-^2^)

are the parts of the Hamiltonian (3.321) that commute with HQ, and the operator O(x^) involves all terms with x", « = 2,3,4,.... The authors do not content themselves with a partial formulation of substitution (3.315) and, appropriately, they complete the formulation with a conservation of the time t or of the spatial coordinate z. So we see that the application of the substitution (3.315) resulting in the solution (3.316) is incomplete. In this case, the transformation leaves the time coordinate

4, § 3]

Macroscopic theories and their applications

383

unchanged. In view of the approximation (3.322), the equation of motion of a right-moving field operator can be written in the interaction picture as

d^

i

(3.324)

His-„^iV,t)

This first-order approximation to the equation of motion can be solved formally using the corresponding time-evolution operator (cf eq. 3.280)

I7s40 = ^exp

-ll>

(r)dr

(3.325)

The classical slowly-varying-amplitude approximation has its quantum counterpart on a double assumption: (1) the initial state of the field is a narrowbandwidth state, and (2) the nonlinearity is weak enough so that the fiill nonlinear Hamiltonian (3.321) may be approximated by its first-order stationary component ^ i s , H,s=H,s^+His-

(3.326)

Therefore, ^is+ will be referred to as the slowly-varying-amplitude Hamiltonian. Now we turn to the renormalization. In the framework of the rotating-wave approximation, we obtain that

where (3.328) Upon normal ordering, the perturbative Hamiltonian (3.322) for the electromagnetic field in a Kerr medium can be written as ^(0= /0^(z,O^(z,Odz-y f0\z,t)^\zj)kz,t)0(zj)dz ^ -^

-hKZ I

(3.329)

0\z,t)0{zj)dz,

where K = 3;^/(4e^). We give the ftinction Z only asymptotically (cf. Abram and Cohen [1994]): Z

^

-^A^

(3.330)

A ^ oo ZJt

where A is a high-frequency cut-off. Whereas the first two terms in equation (3.329) are familiar, the third term, which is divergent, arises in the normal ordering procedure. For A fixed, this last term vanishes if ^ ^ 0.

384

Quantum description of light propagation in dielectric media

[4, § 3

In the renormalization procedure, a formal series (in h) of "counterterms" is added to the Hamiltonian in order to remove the divergences that arise upon normally ordering the results of calculation (Itzykson and Zuber [1980]). The Hamiltonian itself exemplifies that it is not sufficient for removing divergences, but at the same time renormalized parameters and renormalized field operators are introduced. In particular, a renormalized Kerr Hamiltonian //R may be defined by introducing a counterterm of order h as H^{t) = H{t) + 2hKZ / 0^(z,O0(z,Ociz.

(3.331)

• / * ' ( - - ,

The third term in eq. (3.329) changes sign, and for A ^ oo it is an infinite change in the inverse of the refractive index. The renormalized field operators 0 R ( Z , O = \ / 1 + /^Z0(Z,O,

0^(Z,O =

Vl+/cZ^t(z,0

(3.332)

are further quantities which, or at least whose Hermitian parts, etc., would relate to an experiment. Such a relationship is no more required from the bare quantities. At the same time, a renormalized refractive index is defined

The renormalized Kerr Hamiltonian can be written in terms of the renormalized field operators as ^ R ( 0 ^ ^OR ( 0 + H\ S+. R (0,

(3.334)

with

^OR(0 = J 4>i(z,t)M^,t)dz

(3.335)

OC

^IS.,R(0

=Y,h^'K-^'^'H\il,,(tl ./ = o

(3.336)

where

(3.337) where KH[llj^(t) is the "usual" Kerr term and h^K^'^^HlUj^it) is the yth quantum correction.

4, § 4]

Microscopic theories

385

Abram and Cohen [1994] have calculated the quantum noise properties of a coherent pulse undergoing self-phase modulation in the course of its propagation by eliminating the vacuum divergences through the renormalization procedure. The one-point averages were first determined. Two-point correlation fimctions were examined too. The result is similar to that obtained by linearizing the self-phase-modulation exponential operator exp(iy^]^Qfl'Ao) around the mean field (Shirasaki and Haus [1990]).

§ 4. Microscopic theories 4.1. Method of continua of harmonic oscillators 4.1.1. Dispersive lossy homogeneous linear dielectric Huttner and Barnett [1992a] started from the observation that the macroscopic approach to the theory of the electromagnetic field in a medium is a quantization scheme that does accept dispersion, but does not accept losses. Thus it does not deal with a fundamental property of the susceptibility, the Kramers-Kronig relations. Losses in quantum mechanics are treated by coupling to a reservoir, and thus a quantization scheme to describe the losses must introduce the medium explicitly. Huttner and Barnett [1992a] use the model of Hopfield [1958] and Fano [1956], having first treated the quantization of light in a purely dispersive dielectric (Huttner, Baumberg and Barnett [1991]) using a simple version of this model (Kittel [1987]). Their analysis is restricted to a one-dimensional model and to transverse electromagnetic fields. After introducing the Lagrangian densities, they discuss the effect of choosing the type of coupling between light and matter on the definition of the conjugate variables for the components of the vector potential. The matter is not quite identifiable with the reservoir, but there is a chain of couplings: the radiation is coupled to the matter (this is a field again) and the matter is coupled to the reservoir (this is a field of the dimension of the matter field increased by unity). Diagonalization is performed via the Fano technique (Fano [1961], Barnett and Radmore [1988], cf Rosenau da Costa, Caldeira, Dutra and Westfahl Jr [2000]). Huttner and Barnett [1992a] work, as usual, with fields in reciprocal space. Only in the very beginning do they consider radiation and matter in direct space and the reservoir in the Cartesian product of direct space and frequency space are considered. The description of the matter and reservoir is first diagonalized. This diagonalization gives rise to the (dressed) matter field B{k, w, t), whose operator exhibits the same dependence on the wavevector and the frequency as the

386

Quantum description of light propagation in dielectric media

[4, § 4

operator of the reservoir field. It is proven also that a coupling constant dependent on at least the fi*equency or the reservoir's "elementary" mode fijlfils the conditions for further diagonalization. This diagonalization gives rise to the field Cik, w, t) for polaritons. The operator of this field shares the dependence on the wavevector and the frequency with the operator of the reservoir field. In contrast with the free-field theory (the theory of the electromagnetic field in a vacuum), a macroscopic field emerges here whose operator depends also on the frequency. The vector potential depends on the spatial coordinate and time as usual, and it has the form of an integral of the vector potential for a unit density of polaritons with the wavevector k and the frequency o) multiplied by the polariton operator C{k, €0, t). The appropriate relation contains the complex relative permittivity of the medium e{o)) as a linear transform of the coupling constant g{o)) between the light and the dressed matter field B{k, co, t) (cf eq. 4.55). The complex relative permittivity e{o)) ftilfils the Kramers-Kronig relations. Taking into account the frequency decompositions of the fields E{x, t) and B{x,t) (see Huttner and Barnett [1992b]), one can introduce, in an "almost" conventional manner, the positive and negative propagating components. These differ almost negligibly due to the imaginary part of the refractive index (see eq. 3.286 or 3.69), and are proportional to the fields c+{x,cx),t) and c_(x, 0), t), respectively. Respecting the frequency decomposition of the field D(x, t), the fields c±(x, (o, t) are used and the spatial Langevin force/(x, O), t) is introduced. Using these definitions, two Maxwell equations are transformed into two spatial Langevin equations. The equal-space commutation relations between the operators at the frequencies co and co' can also be derived. Huttner and Barnett [1992a] list the papers devoted to the phenomenological approach to quantization (Levenson, Shelby, Aspect, Reid and Walls [1985], Potasek and Yurke [1987], Caves and Crouch [1987], Lai and Haus [1989], Huttner, Serulnik and Ben-Aryeh [1990]). Let us note that Huttner and Barnett [1992b] in their introduction mention also the popular approach (Huttner, Serulnik and Ben-Aryeh [1990]) in which spatial progression equations are derived and quantization of the field is performed by imposing the equal-space commutation relations. In contrast with the macroscopic theories, this technique is not derived from a Lagrangian and has not been justified in terms of a canonical scheme. Huttner and Barnett [1992b] provide the derivation of such equal-space commutation relations for the case of a linear dielectric. The canonical scheme and losses cannot be easily unified, but this has been solved by Huttner and Barnett [1992b]. The one-dimensional model (Huttner and Barnett [1992a]) has been expanded to three dimensions.

4, § 4]

Microscopic theories

387

The Hamiltonian is first derived, then diagonalized, and the expansions of the field operators are transformed. The propagation of fight in the dielectric is analyzed, the field is expressed in terms of space-dependent amplitudes, and their spatial equations of evolution are obtained. Huttner and Bamett [1992b] started the canonical quantization from a Lagrangian density

where -E

B

£e™ = | ( ^ ' + V f / ) 2 - — ( V x ^ ) 2

(4.2)

is the electromagnetic part expressed in terms of the vector and scalar potentials A and U; C^,t = ^(X'-wlx')

(4.3)

is the polarization part, modeled by a harmonic oscillator field X of frequency (OQ (the polarization field);

-H

(Yf,-w'Y;;,)dw

(4.4)

0

is the reservoir part, comprising a field F^,, of the continua of harmonic oscillators of frequencies co, used to model the losses (reservoirs); and C,ni = -a(A'X+UV'X)-X-

[

u(a))Kdw

(4.5)

is the interaction part with coupling constants a and u(a)). The interaction between the light and the polarization field has the coupling constant a, and the interaction between the polarization field and other oscillator fields used to model the losses has the coupling constant U((JO). In general, a could be generalized to a tensor. The displacement field is defined by Z)(r, 0 = eoE(r, t) - aX{r, t).

(4.6)

As (j does not appear in the Lagrangian, U is not a proper dynamical variable, but it can be written in terms of the proper dynamical variable X. The former has

388

Quantum description of light propagation in dielectric media

[4, § 4

an integral expression and that is why we go to reciprocal space. For example, the electric field is written

J_

E(r,t) = j : ^

jE{k,t)e"'d'k.

(4.7)

We shall underline the newly introduced quantities in order to differentiate between quantities in real and in reciprocal space. Let us recall that E*(k,t) = E_{-k,t). It comprises both the annihilation and creation operators, see below. The total Lagrangian can be written in the form ^

/

V^em "•" ^mat + £res + £int j ^ ^ '

(4.8)

where the prime means that the integration is restricted to half reciprocal space, and the Lagrangian densities become

/•OC

io

(4.9)

£int = - «

-L As usual in quantum optics, we choose the Coulomb gauge, k • A(k, t) = 0, so that the vector potential ^ is a purely transverse field. The scalar potential in reciprocal space can be obtained as

where K is a unit vector in the direction of k. The polarization field X and other oscillator fields Y(,j (the matter fields) are decomposed into transverse and longitudinal parts. For example, X can be written as X(k,t)=X^(k,t)+xHk,t)K, (4.11) and Y^^^ can be expressed similarly. The total Lagrangian can then be written as the sum of two independent parts. The transverse part, containing only transverse fields, is L^ = / ' {^L + £ i . + £ris + ^,n,) d^A,

(4.12)

4, § 4]

Microscopic theories

389

where

4-i, = eo(Up-cV|4|'), 4 t s = p / (lZ;lp-o>^|ZiP)d(y,

(4.13)

-'0

£^ = - aA'X

+/

ir^

i;(a;)X^*-y,,da;

+ C.C.

JO

The longitudinal part, containing only longitudinal fields, is also given by Huttner and Barnett [1992b]. It can be derived that /) is a purely transverse field. For convenience, one can restrict oneself to transverse components of other fields and omit the superscript ^. Unit polarization vectors ex{k), X= 1,2, are introduced, which are orthogonal to k and to one another, and the transverse fields are decomposed along them to get

A{Kt)= Y,A\Kt)ex{k)

(4.14)

A=l,2

and similar expressions for the other fields. C can now be used to obtain the components of the conjugate variables for the fields: -eo£^ = ^ dA dC

=eo/,

(4.15)

P^ = - ^

= pf- - aA\

(4.16)

at^^

=pit-^(«)r.

(4.17)

A famous ambiguity is worth mentioning: The conjugate of 4 can be -CoE_ (with the coupling ^A • P), as well as -D (with the coupling £ • X_). Thus, the choice of gauge determines the type of coupling. The Hamiltonian for the transverse fields is // = !

(2iem+2i„a. + 2i,n.)d'A,

(4-18)

where H,^ = eo{\Ef + c^P\A\^)

(4.19)

390

Quantum description of light propagation in dielectric media

[4, § 4

is the electromagnetic energy density; k is defined by ^ = y/k^ + k^ with kc = cOc/c = y/a^/pc^Co;

do)

Ziimaf

Jo

P

r

r

(4.20) is the energy density of the matter fields, including the interaction between the polarization and the reservoirs; 0)1 = 0)1 + j ^ ^^^ do; is the renormalized frequency of the polarization field; and (4.21)

Wi„, = - ( 4 * - ^ + c.c.)

is the interaction energy between the electromagnetic field and the polarization. Part of the interaction energy with the matter, namely ^ | 4 p , has already been classified into eq. (4.19). Fields are quantized in a standard fashion (Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]) by postulating equal-time commutation relations between the variables and their conjugates.

A(k,tlE

(k\t)

(4.22)

Jlsu'6{k-k')l ^0

X\k,t),p''\k',t)

=

tlikJXQ'Ak'j)

ihd,^.d{k-k')\. ihdxx>d{k-k')6{o)-o)')\.

(4.23) (4.24)

with all quantized operators identified by a caret. As usual, one introduces the annihilation operators

a{X,k,t)

=

b{X,k,t)

=

\o(X,k,t)

=

_eo_ ~kct{k, t) - iE\k, t) 2h~kc (boX\k,t)+-p\k,t) P ~P~

-i(oYlik,t)

+

(4.25) (4.26)

^Qlik,t)

(4.27)

391

Microscopic theories

4, § 4 ]

From the equal-time commutation relations for the fields (4.22)-(4.24), one obtains the equal-time commutation relations for the creation and annihilation operators:

(4.28) ko{KK t\ bla',k\

t)] = 6,,.8(0) - co')6(k

-k')l

The normally ordered Hamiltonian for the transverse fields is (4.29)

^ ( 0 = 4 m ( 0 + ^mat(0 + 4 n t ( 0 ,

where

4m(0== / Yl "^

(4.30)

^^ca\k,kJ)a{X,kj)d^k,

A=l,2

ffmai(t)= / ^

lh(bob\X,k,t)b(X,k,t)+

/

h(x)bl{^,k,t)b,o(X,k,t)dw

A=l,2

[b(X,k,t)bla,k,t) +b\x,-k,t)bia,kj)^R.c.'\

(4.31) dco } (Tk,

HUt) = il f Yl Mk)[a{KKt)b\KKt) *^

(4.32)

A=l,2

+ a\X, -k, t) b\X, k, t) + H.c] d^k, where

Via,)

u{o)) jlo (Wo

AW^J^.

and the k integration has been restored to the fiall reciprocal space. It is worth mentioning that the Maxwell-Lorentz equations can be derived from the Hamiltonian. It is important that the matter can be formally decoupled from the reservoir by the Fano technique, obtaining a dressed matter field. Following Fano [1961],

392

Quantum description of light propagation in dielectric media

[4, §4

the polarization-reservoir Hamiltonian can be diagonalized. The dressed matter creation and annihilation operators B\X, k, co, t) and B{X, k, o), t) are introduced, respectively, which satisfy the usual equal-time commutation relations. B{X, k, 0), t\ B\X\

k',0)', o] = ^kk' ^{k - k')6{(o

(4.33)

-(D')\,

B(X, k, (JO, t) = a^io)) b(X, k, t) + ji^iw) b\X, -k, t) , (4.34)

+ / \ai((o,w')b,,AX,kj)-^P\{w,a)')bl,(X,-k,t)\ do;' Jo ^ ^ The coefficients ao(co), fio(co), a\(co,aj^) and li\{w,co') are defined in such a way that the polarization-reservoir Hamiltonian ffmaxiO is diagonal in the operators B^X, k, co, t) and ^(A, /r, co, t). It is interesting that the diagonalization is performed once for the polarization-reservoir Hamiltonian and once for the total Hamiltonian. From relation (4.34) it can be seen that the diagonalization is performed independently for every pair of counterpropagating modes of the polarization field (these "modes" are only formally similar to those of the electromagnetic field) and that it is performed using a Bogoliubov transformation. The dressed matter-field annihilation operator is an "eigenoperator" (Barnett and Radmore [1988]), = ha)B(X,k,co,t).

B(X,k,co,t\Hm,r(t)

(4.35)

The coefficients of the Bogoliubov transformation are calculated by a Fanotype technique, cto(a)) = l3o{w) = aiico.w')

V{w) co^ - ajQzicoY V(co) co^ - cb^zicoY

(O + O^o

2 CO-

Cbo

= 6(0)-w')^

(4.37) V(cjo)

( — CO-

l3,(co,w') =

(4.36)

CO'

iOj co^ - cb^zico)

V{co^) V(co) co + co') CD- - cbh{coy

(4.38) (4.39)

where z(co) is defined by z(co) = 1

1 lim 2cbo

/

V(co^) CO'

dco'

(4.40)

C0-\-l8

From the study, V(co) = \V(co)\^ independent of the frequency is excluded by the assumption that the analytical continuation of V(co) to negative frequencies is an odd fiinction.

4, §4]

393

Microscopic theories

As usual with substitutions, we are also interested in the inverse transformation. It is given by the relations

(4.41)

ao (CO) B(?i, k, CO, t) - Poio)) B\X, -k, co,d(X>, t) Jo

a\{w',(D)B{X,k,w\t)-P\{(jo',(jo)B\X,-k,(D\t)\

dw'. (4.42)

Jo The conditions /•DC

/ = /

[|ao(a;)|' - |A)(^)|'] do; = 1,

(4.43)

/•OC

I{a),(D')= /

[a\{v,(x))ax{v,o)')-l]x{v,(D)P;{v,(ji)')\

dv = d{co-w')

(4.44)

for the coefficients of the Bogoliubov transformation look familiar. Relation (4.31) becomes

H^^x{t)=y^ 11 . , J Jo X=\2

hcoB\Kk,co,t)B{X,k,co,t)do)d?k.

(4.45)

It has been shown that the diagonalization cannot be performed on the common assumption of white noise (Markov-type coupling). We note that free charges and a conducting medium are beyond the scope of Huttner and Barnett [1992b]. The diagonalization of the total Hamiltonian is formally very similar to the diagonalization of its matter part. The expression for the total Hamiltonian H{t) has the same form as the expression of the matter-reservoir Hamiltonian HmaM in terms of the initial creation and annihilation operators, when the parameters a>o and

V((JO)

in relation (4.31) are replaced by kc and yS-^(a))^

with

^(a>) = i^/a)Q[ao(cJO) + ft(w)], and the operators b and ba, are replaced by the operators a and B(a)). The creation and annihilation operators C\?i,k, co, t) and C(A, k, CO, t) are considered: C(A, k, CO, t) = ao(k, CO) a(X, k, t) +ft^{k,co) a\X, ~k, t)

f Jo

a\ (k, CO, co') B{X, k, co' ,t) +ft\(k, co, co') B\X, -k,

dcx)', CJJ' , t) (4.46)

[4, §4

Quantum description of light propagation in dielectric media

394

where the coefficients ao(k, w), ^{k, w), a\(k, a), o)') and ^i(A:, o), o)') appear in slightly modified form, \ 0)}

OoCA:, a;) = an

(ji)-\-kc

t{^)

2

I e*(ft;)co2-Fc2'

kc \

^

/ ^c l 0) - kc

kc \ a,ik,w,co')

2

= 6(0)-0)')+

P\ (k, W, 0)')

(4.47)

(4.48)

e'^(co)co^-k^c^' ^^' ^ ^*^'^'^ 2 \oj-a)'-iOj

0)1 ( K{0)') \ 2 \o) + o)')

U(o) e''(a))a)^-k^c^

t{0)) e''{o))o)^-k^c^'

,

(4.49) (4.50)

as the normalization condition j ^ [\t{o))\^/o)) do; = 1 has been used and the complex relative permittivity e{o)) is introduced: '^2 ^^ \t{0)'f 0)^ e(o)) = 1 + —^ lim , ^ , . ^ , da;^ 20) e^+O y _ ^ (O)^ - 0)-

(4.51)

18)0)'

Relation (4.29) (-4.32) becomes ^ ( 0 = y2

I I

^^C\X, k, 0), t) C(A, k, 0), t) do; d^k.

(4.52)

A=l,2

The operators C(A, k, o), t) and C^(A, k, o), t) also satisfy the usual commutation relations, C(A, k, 0), t\ C\X', k\ o)\ 01 = ^U' ^{k - k')d{oj

-o)')\,

(4.53)

and they have a harmonic time dependence (4.54)

C(A, A, CO, 0 = C(A, ^, a;, 0) e-*^'^^ The vector potential is now given by A{r,t)

1 (2;r)3^2

A=l,2

ho)l

/ / 26^

r(co) -C(A,)t,a;,0)e O)^e(o))- k^c-

-i((0/-A-r)

+ H.C. do)d^k.

(4.55) Huttner and Barnett [1992b] restrict themselves to a one-dimensional case when justifying the temporal mode approach to the propagation in the dielectric.

4, § 4]

Microscopic theories

395

The complex refractive index n(a)) is introduced as the square root of the relative permittivity e(co), with a positive real part r]((i)). The vector potential is considered in the simpler form 1

A(x, t) = —== /

V4jr Jo

f^

A{o)) [c+(jc, a;, 0) e'"'' + C_(JC, O;, 0) e'^"' + H.c.l dco,

(4.56) with a normalization factor

^ ^

Y 6o5cft;|«(a;)P'


(4.58)

with Kico) = " ^ ^ ,

(4.59)

Mco) ^ t*(c^) | K ^ ) | |C(a;)| n(co) ' C(k,a),t)

^

(4.60)

^C{X,k,co,tl

(4.61)

the complex wave number, a phase factor and a polariton operator, respectively. Since the magnetic field can be expressed similarly as the vector potential, the spatial quantum Langevin equations of progression can be obtained: d

_ c±(x,CO,t) = ±iK(w)c±(x,CO,0±

y/2lm{K{w)}f(x,co,t),

(4.62)

(JX

where fix, 0), t) = --=e'^(^'^) / C(k, CO, t)e'^^ dk v2jr J-oo

(4.63)

is the Langevin noise operator, which enters a rather similar expression for the electric displacement operator. Let us note that Im{A^(co)} > 0. Equations (4.62) have been obtained from the Maxwell equations for monochromatic fields.

396

Quantum description of light propagation in dielectric media

[4,

Huttner and Barnett [1992b] remind of the simple commutation relations f{x,o),t\f\x\a)\t)

d{x-x')d{(i)-o)')\,

(4.64)

of the equal-space commutation relations \c±{x,(j),t),c^j^{x,o)' ,t)\ = d{(ji)-(o')\,

\c±{x,(jo,t),c\^{x,(j)\t)\

=0, (4.65) and, finally, that the temporal mode operator c+(x, o), t) commutes with all the Langevin operators/(x^ o)', t) and/^(x^ o)', t) for all x' > x, while the operator c_(x, 0), t) commutes with all the Langevin operators/(x^ co', t) and/^(x^ w', t) for all x' < X. Wubs and Suttorp [2001] have solved the initial-value problem for the damped-polariton model formulated by Huttner and Barnett [1992a,b] and have found that for long times all field operators can be expressed in terms of the initial reservoir operators. They have investigated the transient dynamics of the spontaneous-emission rate of a guest atom in an absorbing medium. Hillery and Drummond [2001] have studied the scattering of the quantized electromagnetic field from a linear dispersive dielectric in the limit of "thin" absorption lines. The field is represented by means of the dual vector potential. Input-output relations are unitary and no additional quantum-noise terms are required. Equations specialized to the case of a dielectric layer with a uniform density of oscillators are usual expressions. 4.1.2. Correlation of ground-state fluctuations The quantization of the radiation imbedded in a dielectric with a space-dependent refractive index has been expounded in the book by Vogel and Welsch [1994]. A canonical quantization scheme for radiation fields in linear dielectrics with a space-dependent refractive index has been developed by Knoll, Vogel and Welsch [1987] and later by Glauber and Lewenstein [1991]. For application see, for example. Knoll, Vogel and Welsch [1986, 1990, 1991], Knoll and Welsch [1992] and a related work Knoll and Leonhardt [1992]. Gruner and Welsch [1995] have contributed to the stream of papers aiming at a description of quantum properties of dispersive and lossy dielectrics including the vacuum fluctuations, i.e.,fluctuationsof the radiation field in the ground state of the coupled light-matter system. They study it in terms of a symmetrized correlation fianction. They try to expound and supplement the paper by Huttner and Barnett [1992b] from the point of view of the quantization of the phenomenological Maxwell theory.

4, § 4]

Microscopic theories

397

First, the quantization of radiation in a dispersive and lossy dielectric is performed. This begins with the classical Maxwell equations (3.121), with the constitutive relations (3.122) and a constitutive relation comprising an integral term. ^ ( ^ 0 = ^0 E{r,t)^

/I

xir)E(r,t-T)dT

(4.66)

0

Jo Everything is transformed into Fourier space, where relation (4.66) becomes b(r, 0)) = 6o6(w) E{r, w), (4.67) and the Helmholtz equation is presented. The Huttner-Bamett quantization scheme is introduced with a diagonalized Hamiltonian (4.52). The effect of the medium is entirely determined by the complex permittivity e{co). It still has no tensorial character. Let us refer to original relations (4.3), (4.5), (4.6) and (4.7) after Huttner and Barnett [1992b]. In these relations, one should make use of the identity ^ r ( w ) = (o^/lm{e(a))}. Frequency-dependent field operators are introduced in the three-dimensional case. Not only the equal-time commutation relations, but even the most general ones are presented. The vector-field operators a(r,co,t) and f(r,a),t) are introduced, the vector a(r, co, t) being a generalization of the component c(x, co, t) from (Huttner and Barnett [1992b]). Using the operators a(r,co,t) a n d / ( r , co,/), an analogue of relation (5.21) of Huttner and Barnett [1992b] (cf eq. 4.56) is written. Then, analogues of their frequency decompositions of the vector-potential operators, electric-fieldstrength operators, etc. are presented. The operator constitutive equation (in Fourier space) b(r, w, t) = eoe(a>) k{r, w, t) - eoT(aj)f{r, (o, t\

(4.68)

where eo^(co) = y ^eo Im{e(a;)},

(4.69)

differs from the classical equation (4.67) by an additional term. On substituting into the phenomenological Maxwell equations, the partial differential equation

398

Quantum description of light propagation in dielectric media

[4, § 4

for the operatorfl(r,co, t) is obtained, which is a Helmholtz equation with a righthand side. In the three-dimensional case, there exists no decomposition into firstorder equations. The canonical commutation relations are Ai{rJ\Ej{r'A = --d^(Ar) 1, J

(4.70)

Co

with the abbreviation Ar = r-r'.

(4.71)

A test of the consistency of the theory in the limit e((jo) —> 1 has been accomplished. Let us recall the usual annihilation and creation operators entering the expansion for A(r,t), i.e. a(X,k,t) and a^(X,k,t), respectively; they satisfy the commutation relations [aa,k,tXa^(X\k\t)]=d^;,^d(k-k')l

(4.72)

The operators a(r, w, t) and / ( r , w, t) derived from C(A, Ar, (o, t) are not independent operators. Compare Huttner and Bamett [1992b] who in the onedimensional case introduce forward- and backward-propagating fields and show that such a definition ensures the causal (one-sided) independence of the respective temporal mode operators of the operator / ( r , co, t). In the threedimensional case, there exists no generalization of relation (4.58) and no equation for such quantities. The theory is applied to the determination of the correlation of the groundstate fluctuations of the electric field strength. A symmetric correlation ftinction of the electric field strength is considered: Kmn{Ar, r) = \ (0| [^^(r, t + T)E,{r + Ar, t) + E,{r + Ar, t)E,,{r, t + r)] |0). (4.73) The influence of absorption, phase and group velocities and group-velocity dispersion on the dynamics of the field fluctuations within a frequency interval have been studied. The absorption causes a spatial decay of the correlation of the field fluctuations. The light cone of strong correlation, which in empty space is determined by the speed of light in vacuum, is now given by the group velocity in the medium, provided that the spatial distance is not too large. With increasing distance, the dispersion of the group velocity should be taken into account. 4.13. Green-function approach If one allows a frequency-dependent complex permittivity that is consistent with the Kramers-Kronig relations and introduces a random operator noise source

4, § 4]

Microscopic theories

399

associated with the absorption of radiation, the classical Maxwell equations can be considered as quantum operator equations. Their solution based on a Green-function expansion of the vector-potential operator seems to be a natural generalization of the mode expansion applicable to source-free radiation in nearly lossless dielectrics. (i) Dispersive lossy linear inhomogeneous dielectric. Gruner and Welsch [1996a] have expounded a quantization scheme which starts from the phenomenological Maxwell equations instead of the Lagrangian densities, and is consistent with the Kramers-Kronig relations and the familiar (equaltime) canonical commutation relations for the vector-potential and electricfield-strength operators. This is realized for homogeneous and inhomogeneous (especially multilayered) dielectrics. In the phenomenological classical Maxwell theory, the equations comprise e(a;), the frequency-dependent complex relative permittivity introduced phenomenologically. This function has an analytical continuation in the upper complex half-plane, e(0), which satisfies the relation 6(-0*) = e*(^). The real and imaginary parts of the relative permittivity satisfy the well-known KramersKronig relations: Re{e(a,)} - 1 = I v . p . H

'-^^^^^

lm{e(co)} = -iv.p. r

da,',

M^(^!^:)hl do.;

(4.74) (4.75)

The quantization scheme is based on the Helmholtz equation with the source term AA{r, 0), t) + K^{a))A{r, w, t) =j^{r, w, t),

(4.76)

where A(r, w, t) = A(r, o), 0) e'^'^^ with A(r, co, 0) the "Fourier transform" of the (known) operator-valued vector potential A{r, t), andy^(r, w, t) =jjj', (o, 0) e'^'^^ with 7j^(r, w, 0) the "Fourier transform" of the operator-valued noise current. In fact, from the exposition it can be seen that the vector-potential operator is introduced by the relation A(r,0)=

/

A(r, w,0)da)^H.c.,

(4.77)

where quantum-mechanically also the frequency-dependent operators can be time dependent.

Quantum description of light propagation in dielectric media

400

[4, §4

When \m{e{Ci))} > 0, a (hypothetical) addition of a nontrivial solution of the homogeneous wave equation would violate the boundary condition at infinity. Hence, the operator A{r, co, t) is uniquely determined by a linear transformation of the source operator/^^(r, (0,0• This operator can be chosen in the form (cf. Gruner and Welsch [1995]) CO

(4.78)

j^(r, 0), t) = Hco) —f(r, CO, t\

with T{cji)) given in eq. (4.69). The Hamiltonian H{t) is diagonal in the operators / ( r , co, t). H{t) =

hcof\r, CO, t) • / ( r , CD, t) dco dV,

(4.79)

and these operators have the usual properties \fi{r,coj)J^{r\co',t)\^

=

\fi{r,co,t)Jj{r\co'j)]

=

d^{r-r')d{cj)-co')\, \f^{r,co,t)/f^{r',co',t) = 0.

(4.80) (4.81)

From the foregoing considerations it follows that (when all appropriate conditions are fulfilled) the operator of the vector potential can be defined by the relation A{rj)=

/

j

G(r,r\co)}\{r\co,t)d^/dco^H.c,

(4.82)

where the Green function G(r,r\co) satisfies AG(r, / , CO) + K\CO) G(r, r\co) = d(r - /)

(4.83)

and the boundary condition that it vanishes at infinity. Another required property is E(r,0) = -A(r,0)

(4.84)

and the canonical field commutation relations

i,(^0),£;(/,0)l

=--S;^(r-r')l

(4.85)

Relations (4.85) must be verified by straightforward calculation. For the sake of clarity, Gruner and Welsch [1996a] illustrate this procedure in linearly polarized

4, §4]

Microscopic theories

401

radiation propagating in the x-direction. Relations (4.85) are replaced by the relation (4.86)

A(x,0),E(x,0)

where A is the normalization area perpendicular to the x-direction. It is shown that when losses in the dielectric may be disregarded, lm{e((j))} -^ 0, the concept of quantization through the mode expansion can be recognized. The operators f(x, 0), t) are replaced by the operators a±(x, co, t), which satisfy the commutation relations a±{x, CO, t), al_(x, (JO\ t) exp

\m{n{(x))} — \x - x' 6{co-co')\, c

(4.87)

CO

a±{x,co,t),ciL{x ,CD'J)\ = 2lm{n(co)}— expH=i^(<^) ^(x-\-x^) ^ J c I ^ lm{n(co)} — \x-x^ X exp c sm[RQ{n(co)}'-^\x-x'\] "" Rc{n(co)}^-f X

(4.88)

6[±{x-x)]dico-co')l,

where d(x) is the Heaviside function. These operators become independent of x in the limit lm{n(co)}f\x - x ' | -^ 0. As the commutation relation (4.86) is in obvious contradiction with the macroscopic approach, it is important that Gruner and Welsch [1996a] have derived the relation AAa)(x,0%EAa)(x\0)

ih 8(x-x)l, AeR(co,)eo

(4.89)

where cOc is the center frequency for suitably defined operators, A/^a)(x,0), The theory further reveals that the weak absorption gives rise to spacedependent mode operators that spatially progress according to quantum Langevin equations in direct space. As could be expected, the operators a±(x,co,t), representing the forward and backward propagating fields, are governed by quantum Langevin equations. In other words, the operators a±(x, co,t) progress in space. As an example of inhomogeneous structure, two bulk dielectrics with a common interface are considered. The problem of determining a classical Green function reappears. The verification of the commutation relation (4.86) is performed by straightforward calculation, more complicated this time. A general

EA(O(X, 0).

402

Quantum description of light propagation in dielectric media

[4, § 4

proof of this relation is not presented, causality reasons are only pointed out. There exists a straightforward generalization of the quantization method based on a mode expansion (Khosravi and Loudon [1991, 1992], Agarwal [1975]). (ii) Dispersive lossy nonlinear inhomogeneous dielectric. Emphasizing the important differences from the linear model, the Lagrangian and Hamiltonian for the nonlinear dielectric are introduced by Schmidt, Jeffers, Barnett, Knoll and Welsch [1998]. The Lagrangian density (4.1) is denoted by C\{r), and this relation with C replaced by C\{r) has been utilized in a more general Lagrangian density: C{r)-^C^{r) + C,^{r\

(4.90)

where moreover C^x{r)=f[X{r)l

(4.91)

While in the linear case it is sufficient to quantize only the transverse fields, in the nonlinear case such a procedure would result in a loss of generality. The result of substitution from relations (4.30)-(4.32) into (4.29), which we have denoted by H, is denoted here by H^^. The normally ordered Hamiltonian for the longitudinal fields is

Hl\t) = J(ficoob\lkj)klkj)^ h j j V{o)) \^b\l-Kt) 2 ./ ./o

J + klkj)]

hajblXlkj)bM^k,t)d€o)d'k \bl{lKt)-^h,{l-Kt)\

da;d^/r,

(4.92) where the components b{\\,k,t), b^o{\\,-k,t) must be defined appropriately (see Schmidt, Jeffers, Barnett, Knoll and Welsch [1998]) for b\^{k\ b^\{k,o))). The total Hamiltonian can be written as H(t) = HM + HM,

(4.93)

where the nonlinear interaction term ^ni(0 is given by Hni(t) = -jf[Xir,t)]d'r,

(4.94)

and the Hamiltonian ff\(t) that governs the linear dynamics can be written as ^ , ( 0 = A" (0 + ^1^(0-

(4.95)

In general, Hn\{t) couples the transverse and longitudinal fields, cf, the relation (4.11).

4, § 4]

Microscopic theories

403

Schmidt, Jeffers, Barnett, Knoll and Welsch [1998] have derived evolution equations for the field operators, showing that additional noise sources appear in the nonlinear terms. Linear relationships between quantum (operator-valued) fields are introduced following Huttner and Barnett [1992b] as well as Gruner and Welsch [1995]. The relations hold for all times and for both linear and nonlinear cases. We now add the following representations of the matter fields. The longitudinal matter field X^r, t) can be expressed in terms of the field/"(r, co, t) as

'•''-{ik[^"'

(4.96)

and the transverse matter field X^{r,t) / ( r , 0), t) as

can be expressed in terms of the field

*"<•••'>" l / v i T r /

X^{r,t)=

/ Jo

K<<>')-fi;(»')]/"(MO,<)'lo' + H,c.,

X (r,ft>,Oda; + H.c.,

(4.97)

where X

{r,o),t)=-

c -\a)[e{w)- \]A{r,o),t)+

h W — Im{e(co)}/(r, (0,0

(4.98) with^(r, 0), t) connected with the field/(r, co, t) as the solution of eq. (4.76) and by the explicit relation (4.78). The vector-potential field has the representation A{r,t)^

/

^(r,co,Oda; + H.c..

(4.99)

Relating the validity of expressions (4.99), (4.96) and (4.97) to a time evolution like eq. (4.54), we might suspect that it will not survive the change to the nonlinear case. This change is reflected in the equations of motion for the basic fields and the vector-potential field in the Heisenberg picture, i ^ | / l l ( r , CD, t) = [/ll(r, CO, 0 , ^ ( 0 ] - hojfhr, co, t) + [fHr, w, 0 , 4 i ( 0 ] , (4.100) i ^ - / ( i - , (o, t) = [/(/•, w, t),H{t)] = hwf(r, w , 0 + [/(^ 0), t),HM],

(4.101)

ih—^(r,(jo,t)=

(4.102)

[2(r,a),t),ffit)]

= ha)A{r,(i),t)-^ [k{r,a),t),Hn\(t)]-

404

Quantum description of light propagation in dielectric media

[4, § 4

Among the relations which do hold in both linear and nonlinear cases is the nonhomogeneous Helmholtz equation:

A^(r,a),t) + K\co)Air,(o,t)=

%J—lm{e(a))}f{r,a),t). c^ y Jteo

(4.103)

Considering the notation K^{w) = c~^co^e((jj) (cf. eq. 4.59), we can see that K\a))A(r,

(O, t) = K\co\)A{r,

co, t\

(4.104)

where 1 is the identity superoperator and relation (4.102) implies that w\ = \^^+^-H,,{tr,

(4.105)

where for the sake of clarity we have written ^ to the right of 1, and the action of ^ni(0^ 01^ ^^ operator 0{t) is defined by

H,x{trO{t)

Hn\{t\0{t)

(4.106)

Relation (4.105) can be written in the form

(4.107) where the elimination of the field X{r, t) using relations (4.96) and (4.98) indicates new noise sources. All of the fields/"(r, a;, t),f{r, w, t) and A{r, a), t) obey the nonlinear dynamics. By integration of eq. (4.107) over w, an equation adequate for both linear and nonlinear cases is obtained. The wealth of operatorvalued fields facilitates the expression of dispersion and absorption in the nonlinear medium. The basic equations are applied to the one-dimensional case and propagation equations for the slowly varying field amplitudes of pulse-like radiation are derived. The scheme is related to the familiar model of classical susceptibilities, and is applied to the problem of propagation of quantized radiation in a dispersive and lossy Kerr medium. In the linear theory it is possible to separate the two transverse polarization directions from each other and from the longitudinal direction. As has already been stated, this is not possible for nonlinear media. In practice, in a single-mode optical fiber, only one transverse

4, § 4]

Microscopic theories

405

polarization direction will be excited. Then the total Hamiltonian (4.93) can be reduced to a one-dimensional single-polarization form. Let us consider the propagation in the x-direction of plane waves polarized in the j;-direction. The one-dimensionality of the problem permits one to decompose the field A{x, w, t) into components A+(x, w, t) and A^{x, co, t), respectively, propagating in the positive and negative x-directions, A(x, 0), t) = A+{x, CO, t)-\-A.(x, 0), t), where A±(x,w,t)

(4.108)

are the solutions of spatial equations of progression:

—A±(x, (o, t) = ±iK(co)k:t(x, CO, t)^\MJ (jx y

^"^^^^^^^ /(;c, CO, t\ e{co)

(4.109)

with a normalization factor M = y/h/AjreoAc^. Similarly as from relations (4.103) and (4.107), one can arrive from relation (4.109) at the relation d ^

/^9

1.

H„,it)xy^(x,co,t)Ti^fJ^^^^^^fix,OJ,t). (4.110)

In analogy with eq. (4.99), the operators A±. {x,t) can be introduced as -M)

A^(x,t)=

'"" -

I

A±(x,co,t)dco.

(4.111)

Integrating eq. (4.111) over co yields an equation appropriate for both linear and nonlinear cases. Adequately to the derived equations which we consider to be mere approximations in the nonlinear case, Schmidt, Jefifers, Barnett, Knoll and Welsch [1998] study the narrow-bandwidth field components and narrow-bandwidth pulses. The theory has been applied to narrow-bandwidth pulses propagating in a dielectric with a Kerr-like nonlinearity. (Hi) Elaboration of the linear theory. Dung, Knoll and Welsch [1998] have developed a three-dimensional quantization presented in part by Gruner and Welsch [1996a] concerning a dispersive and absorbing inhomogeneous dielectric medium. The approach starts directly from the Maxwell equations in the frequency domain for the macroscopic electromagnetic field. It is shown that the classical Maxwell equations together with the constitutive relations, except relation (4.66), can be transferred to quantum theory. In considering the charge

406

Quantum description of light propagation in dielectric media

[4, § 4

and current densities, one concentrates on the noise-charge and noise-current densities. An operator-valued noise-charge density p(r, w, t) and an operatorvalued noise-current density j{r, co, t) are introduced, related to the operatorvalued noise polarization P{r, (o, t): p{r, CO, 0 = - V • P{r, 0), t\

(4.112)

y(r, CO, t) = -ia)P(r, co, t).

(4.113)

It follows from relations (4.112) and (4.113) that p(r,a),t) md](r,co,t) the continuity equation V j(r, (O, t) = ico^(r, co, 0-

fulfil

(4.114)

The source term j{r, co, t) is related to a bosonic vector field / ( r , co, t) by a relation like (4.78). The commutation relation (4.81) remains valid, and relation (4.80) must be modified to the form fi{r,CO,t),fj^{r',co',t)\ = dij8(r-/)d(co-co')

I

(4.115)

It is pointed out that the current density j{r, co, t) is not transverse, because the whole electromagnetic field is considered. Hence, the vector field f{r, co, t) assumed here is not transverse either, and the spatial 6 fiinction in the relation (4.115) is an ordinary d function instead of a transverse d fiinction. Relation (4.77) is an integral representation of the vector-potential operator. Dung, Knoll and Welsch [1998] start from the partial differential equation C

C02

C

C

V X V XE(r,co,t)-—^e(r,co)E(r,co,t)

= icofi^jir,co,t),

(4.116)

whose solution can be represented as (different notations will be used here and below)

) I/ G{r,s,co)j{s,co,t)<\ G{r,s, E{r,co,t) ='\coiM)

s,

(4.117)

where G{r, s, co) is the tensor-valued Green fiinction of the classical problem. It satisfies co^

V , V , - l M , + -ye(r,co)

G(r,s,co)^8(r-s)l

(4.118)

4, § 4]

Microscopic theories

407

together with appropriate boundary conditions. Dung, Knoll and Welsch [1998] have derived commutation relations 'Urj)Mr\t)] = ^ Y , e , „ ~ m,7

'"

H

^Gij(r,r\ w)dw,

(4.119)

-^-^

where ekmj is the Levi-Civita tensor, Gij{r, r^ (o) = a • G(r, r\ co) • Cj,

(4.120)

and [£K^O,^K^',0] - 6 = [Ur,t\B,{r'j)\

.

(4.121)

In the sense of the Helmholtz theorem there exists a unique decomposition of the electric field E(r, co, t) into a transverse part E (r, co, 0 and a loncll gitudinal part £" {r,aj,t), i.e., the Coulomb gauge can be introduced, where E (r, CO, t) = i(joA(r, co, t) and E (r, a>, t) = -Vqp{r, co, t). In the Coulomb gauge, the vector and scalar potentials A{r, co, t) and (p{r, co, t), respectively, are related to the electric field as Ai{r,CD,t) = — / dlr{r-s)Ei{s,Ci),t)dh,

(4.122)

ILL/ J

—-^(r,co,t)

= - I dl(r-s)l:j{s,a),t)dh,

(4.123)

where 6^- and djj are the components of the transverse and longitudinal tensorvalued d fiinctions d^(r)

= d(r)l + VS/(4jT\r\y\

S\\(r) = -VV(4jr|r|)-*.

(4.124) (4.125)

It is recalled that A(r, t) and eoA{r, t) are canonically conjugated field variables. In contrast, the complexity of the commutation relation (4.119) suggests that the "canonical" commutators are not as simple as one would expect from the definition. The commutation relation between the vector potential and the scalar potential is as complicated, when one and only one of these quantities is

Quantum description of light propagation in dielectric media

408

[4, §4

differentiated with respect to the time or comprises such a derivative. The simple commutation relations are Ai(r,t)Jj(r\t)

(4.126)

= 0 = Mr,t)Ji{r\t)

[q)(r, t\ qp(r\ t)] =0= \qp(r, t)Ji(r\

t)

(4.127)

Then, the theory is applied to the bulk dielectric such that the dielectric function can be assumed to be independent of space: e(r, (o) = e(oj) for all r. In this case, the solution of eq. (4.118) that satisfies the boundary condition at infinity is (cf Tomas [1995]) G(r,r\a))=

[V,Vr+K\a))l]K-\aj)g{\r~/\,aj),

(4.128)

where g(r, CO) =

Qxp[iK(a)) r] 4jtr

(4.129)

Relation (4.119) can be simplified: \Ei(r,tlMr\t)

— ei/„„-—d(r-/)U Co dx„,

(4.130)

and the "canonical" commutator corresponds to the definition,

J

ih eo

(4.131)

= 0.

(4.132)

Moreover, \qp{r,t)Jj(r\t)

The commutation relations presented are in the equal-time Heisenberg picture, and therefore it is emphasized that they are conserved. To make contact with the earlier work. Dung, Knoll and Welsch [1998] define the vectors (4.133) fhr,coj)

= f

dHr-s)f{s,a),t)dh.

(4.134)

4, § 4]

Microscopic theories

409

The commutation relations (4.115) and (4.81) imply that

't^%,co,t),{f^^^%',aj',t))'

(5,^""(/--/-')<3(w-w')i,

f,^^%,co,t),f.^^%\oj',t))

(4.135) = 0.

(4.136)

The representation of the transverse vector potential simplifies to k';o},t) = ^k>fg(\r-r'\,w)]^{r',(o,t)d'r'.

(4.137)

It can be derived that the scalar potential operator ^^r,co,t)=---^^ f^^^dh, 4jteoe(a))

(4.138)

where p(r, co, t) = (ico)~^ V j (r, w, t). Another application is the quantization of the electromagnetic field in an inhomogeneous medium that consists of two bulk dielectrics with a common interface. The determination of the tensor-valued Green function for threedimensional configuration of dielectric bodies is in general a very involved problem. Dung, Knoll and Welsch [1998] return to the simple configuration mentioned by Gruner and Welsch [1996a]. The reader is referred to (Tomas [1995]) for the classical treatment of multilayer structures. It is shown that for the configuration under study, the commutation relations (4.130)-(4.132) hold. The necessity of a new calculation of the quantum electrodynamical commutation relations for a new three-dimensional configuration (cf Dung, Knoll and Welsch [1998]) is not absolute. Scheel, Knoll and Welsch [1998] have proven that the fiindamental equal-time commutation relations of quantum electrodynamics are preserved for an arbitrarily space-dependent Kramers-Kronig dielectric ftinction. Let us recall that the complex-valued dielectric function 6(r, of) depends on frequency and space: e(r,a;)^l

if

co-^ oo.

(4.139)

It is assumed that the real part (responsible for dispersion) and the imaginary part (responsible for absorption) are related to each other according to the

410

Quantum description of light propagation in dielectric media

[4, § 4

Kramers-Kronig relations, because of causality. This also implies that e(i*, oj) is a holomorphic function in the upper complex half-plane of frequency ~—6(r, (o) = Q,

Im ft; > 0.

(4.140)

aft;*

Scheel, Knoll and Welsch [1998] study relation (4.119). By comparison of the right-hand sides of this relation and (4.130), they arrive at the following identity to be proved: -^G(r,r\co)dco x Vr' = -ijnd(r-r)

x W-

(4.141)

oo ^

Here the left arrow means that the operators d/dx',^, in the expansion of the nabla operator with this upper limit will first be written on the right-hand side as

d/dxl. Based on the partial differential equation (4.118) for the tensor-valued Green fianction, an integral equation will be presented in what follows. The partial differential equation and the boundary condition at infinity determine the Green ftmction uniquely. By comparison of relation (4.117) with a constitutive relation, we could derive that ijHoCoGijir, s, (o) are holomorphic functions of co in the upper complex half-plane, i.e., - - ^ [coGkjir, 5, ft;)] = 0 ,

Im ft; > 0,

(4.142)

with (oGkj(r,s, CO) ^ 0

if

|ft;| -^ oo.

(4.143)

Second derivation of the Cauchy-Riemann equations (4.142) consists in the application of 9/9ft;* to relation (4.118). The left-hand side of relation (4.142) is then the unique solution of the homogeneous problem. Knoll and Leonhardt [1992] calculate the time-dependent Green ftanction. Scheel, Knoll and Welsch [1998] have derived the relation i!J^a)Gij(r,s,co) = bijir,s,co)=

f

e'^'^^Ay(r,5, r)dr,

(4.144)

where Dij(r,s,T) are components of the tensor-valued response function that causally relates the electric field E(r, t) to an external current yext('S', t - r), so that Dij{r,s, ^)=^J

^~'"''D,ir,s, w)dco.

(4.145)

From the theory of partial differential equations it is known (see, e.g., Garabedian [1964]) that there exists an equivalent formulation of the problem in

4, § 4]

Microscopic theories

411

terms of an integral equation. For eo(co) = J e(r, (o) d V / / d^r, an appropriately space-averaged reference relative permittivity, the integral equation for the tensor-valued Green function can be written as G(r, s,a))= I K{r, v, o)) • G{v, s, aj)d^v-^ G^^\r, s, co),

(4.146)

where G^yr,s, (o)=[l-

VrVsK-\s,

co)] gi\r - s], (o),

(4.147)

K(r,V, w) = [V,g(|r-1;|, co)] [V, In^^^^^ ^^j +

[K\v,co)-K^{co)]g{\r-v\,co)]l.

Here g(r, co) = go(r, co) is given by eq. (4.129), where K(co) = Ko(co), K\r,co)

= ^e(r,a>),

(4.149)

J-

K^(co) = —6o(a>).

(4.150)

It can be seen that the components of the kernel Kik(r,v,co) are holomorphic functions of co in the upper complex half-plane, with Kik(r,v,co)-^0

if

\co\ ^oo.

(4.151)

To prove the fundamental commutation relation (4.130), we first decompose the tensor-valued Green function into two parts: G(r,s,co) = Gi(r,s,co) + G2(r,s,co),

(4.152)

where G\ (r, s, co) satisfies the integral equation Gi = j K-Gxd^v + Gf\

(4.153)

G | V ^ , W ) =g{\r-slco)h

(4.154)

C72(r,5,co) =r(r,5,a>)Vv

(4.155)

with

In relation (4.155), F is the solution of the integral equation

r= j K-rd^v-^r^^\

(4.156)

with r^yr.s,

CO) = -V,[K-\s,

aj)g{\r-sl

co)].

(4.157)

Scheel, Knoll and Welsch [1998] derive that i^^o^^i and /.IQCO^F are the Fourier transforms of the response functions to the noise-current density and the noisecharge density, respectively.

412

Quantum description of light propagation in dielectric media

[4, § 4

Combining relations (4.152) and (4.155), and recalling that Vr' x Vr' = 0, we see that the left-hand side of relation (4.141) can be rewritten as f^

-

0)

-^G(r,r\w)da)x\/,^ J-oc

^

^

f^

=-

CO

-

— G,(r,/,a;)dw x Vr'J-oc

(4.158)

^

Thus, only the noise current response function 'm^(x)G\ contributes to the commutator (4.119). Muhiplying the integral equation (4.153) by the function ^ and integrating over co, we obtain, similarly as in the derivation of relation (4.86), from the holomorphic properties of the tensors K and (j)G\ that

f

%Gx{r,r\o))di(D ^

'\JT\d{r-r').

(4.159)

J -o

The outer product of this equation and the operator (-VrO can be taken, and together with relation (4.158) implies relation (4.141). In addition, it is shown that the scheme also applies to media with both absorption and amplification (in a bounded region of space). An extension of the quantization scheme to linear media with bounded regions of amplification is given, and the problem of anisotropic media is briefly addressed: there the permittivity is a symmetric complex tensor-valued ftinction of w, eij{r,CD) = eji{r,w).

(4.160)

Di Stefano, Savasta and Girlanda [2000] have developed a quantization scheme for the electromagnetic field in dispersive and lossy dielectrics with planar interface, including propagation in all the spatial directions, and considering both the transverse electric and transverse magnetic polarized fields. Di Stefano, Savasta and Girlanda [2001a] have presented a one-dimensional scheme for the electromagnetic field in arbitrary planar dispersing and absorbing dielectrics, taking into account their finite extent. They have derived that the complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the classical light modes, they have obtained a generalization of the method of modal expansion (e.g.. Knoll, Vogel and Welsch [1987]) to absorbing media. Di Stefano, Savasta and Girlanda [2001b] have based an electromagnetic field quantization scheme on a microscopic linear two-band model. They have derived for the first time a noise current operator for general anisotropic and/or

4, § 4]

Microscopic theories

413

spatially nonlocal media, which can be described only in terms of an appropriate frequency-dependent susceptibility. (iu) Modification of spontaneous emission by dielectric media. Dung, Knoll and Welsch [2000] have developed a formalism for studying spontaneous decay of an excited two-level atom in the presence of arbitrary dispersing and absorbing dielectric bodies. They have shown how the minimal-coupling Hamiltonian simplifies to a Hamiltonian in the dipole approximation. The formalism is based on a source-quantity representation of the electromagnetic field in terms of the tensor-valued Green function of the classical problem and appropriately chosen bosonic quantum fields. All relevant information about the bodies such as form and dispersion and absorption properties is contained in the tensor-valued Green function. This function is available for various configurations such as planarly, spherically, and cylindrically multilayered media (Chew [1995]). The theory has been applied to the spontaneous decay of a two-level atom placed at the center of a three-layer spherical microcavity, the wall being modeled by a Lorentz dielectric. The tensor-valued Green function of the configuration is known (Li, Kooi, Leong and Yeo [1994]). The calculations have been performed on the assumption of a dielectric with a single resonance. For simplicity, it has been assumed that the atom is positioned at the center of the cavity. Weak and strong couplings are studied, and in the study of the strong couplings both the normal-dispersion range and the anomalous-dispersion range associated with the band gap are considered. Whereas in the range of normal dispersion the cavity input-output coupling dominates the strength of the atom-field interaction, the most significant effect within the band gap is photon absorption by the wall material. Dung, Knoll and Welsch [2001 ] have studied nonclassical decay of an excited atom near a dispersing and absorbing microsphere of given complex permittivity that satisfies the Kramers-Kronig relations laying emphasis on a Drude-Lorentz permittivity. Among others, they have found a condition on which the decay becomes purely nonradiative. For a transparent dielectric, theoretical studies can take a traditional approach. Inoue and Hori [2001] have developed a formalism of quantization of electromagnetic fields including evanescent waves based on the detector-mode functional defined in terms of those for the widely used triplet modes. They have evaluated atomic and molecular radiation near a dielectric boundary surface. Matloob and Pooseh [2000] have discussed a fully quantum mechanical theory of the scattering of coherent light by a dissipative dispersive slab. Matloob and Falinejad [2001] have calculated the Casimir force between two dielectric slabs by using the notion of the radiation pressure associated with the quantum

414

Quantum description of light propagation in dielectric media

[4, § 5

electromagnetic vacuum. Specifically, they have used the fact that only the field correlation fiinctions are needed for the evaluation of vacuum radiation pressure on an interface. Matloob [2001] has postulated an electromagnetic field Lagrangian density at each point of space-time to be of an unfamiliar form comprising the noise current density. He has expressed the displacement D(r, t) merely in terms of the electric field E{r, t'), t' ^ t, without adding a noise polarization term. Leonhardt and Piwnicki [2001] have analysed the propagation of slow light in moving media in the case where the light is monochromatic in the laboratory frame. The extremely low group velocity is caused by the electromagnetically induced transparency of an atomic transition. § 5. Microscopic models as related to macroscopic concepts 5.7. Quantum optics in oscillator media A quantum-optical experimental setup may consist of active and passive devices: active devices to generate light with certain properties (e.g., nonclassical light), and passive ones to modify and distribute it. It is an interesting nontrivial problem to study how quantum-statistical properties of light are influenced by passive optical devices like mirrors, resonators, beamsplitters or filters. Knoll and Leonhardt [1992] have elaborated on the paper by Knoll, Vogel and Welsch [1987] where the medium is nondispersive and lossless, but they now intend to consider dispersion and losses. On introducing the Hamiltonian for the complete system, the Heisenberg equations of motion for field operators and medium (not source) quantities are derived. The complete system under consideration consists of the following subsystems: optical field, medium atoms and sources. The field is described by the electric-field-strength operator E(x,t) and the electromagnetic vectorpotential operator A(x,t) in the Coulomb gauge. The medium is modeled by damped harmonic oscillators {qa(t),pa(t)}, namely, oscillators coupled to reservoirs composed of bath oscillators {quB(t), J&//B(0}? whose energy quanta may be, for example, phonons. The medium oscillators are localized at x^i, and they all have the same mass m and elasticity (force) constant k. The bath oscillators are characterized by masses m^ and elasticity constants ks, and the coupling is expressed by the coupling constants o^. The atomic sources are described by a current operator j(x, /), of which the dynamics need not be specified, and e is the electron charge. For simplicity, a one-dimensional model is considered only.

4, § 5]

Microscopic models as related to macroscopic concepts

415

The Hamiltonian of the complete system is (5.1)

^ ( 0 = ^ R ( 0 + ^ M ( 0 + ^RS(0 + ^S(0,

where ^ R ( 0 is the Hamiltonian of the optical field, Hu{t) is that of the medium atoms, and^Rs(0 describes the interaction between the optical field and sources: ,. ,, , 2 / dA{xJ) E\x,t) + c' dx

^R(0 = J

dx.

(5.2)

n2

Hu{t) - 2 ^ i

—

+ -q,{t) (5.3)

B

L

(5.4)

^Rs(0 = - / Kx, t)A(x, t) Ax,

and ^ s ( 0 is the Hamiltonian of the atomic sources which is left unspecified. The usual commutation rules are i(x, 0, -eQE{x, t) = ifid(x

-x)\. (5.5)

The Heisenberg equations of motion for field operators, medium operators, and bath operators have been obtained. As a result of a Wigner-Weisskopf approximation for the interaction of medium oscillators with the bath operators, quantum Langevin equations have been obtained. By eliminating the medium quantities from equations for field operators and otherwise by a common procedure, a generalized wave equation for the vector potential is obtained. Using a Green function, this wave equation is solved. The decomposition of the timeordered quantum correlation functions into time-ordered correlation functions of the source operators and the free-field operators has been derived. The time-dependent Green function for a dielectric layer as the simplest optical device is calculated. The field behind the layer is discussed and represented by the negative-frequency part of the field, its expectation value and the normallyordered quadrature variance determined for the sake of squeezing analysis.

416

Quantum description of light propagation in dielectric media

[4, § ^

5.2. Problem of macroscopic averages 5.2.1. Conservative oscillator medium Dutra and Furuya [1997] have investigated a simple microscopic model for the interaction between an atom and radiation in a linear lossless medium. It is a guest two-level atom inside a single-mode cavity with a host medium composed of other two-level atoms that are approximated by harmonic oscillators. The intention is to show, in general, that ordinary quantum electrodynamics suffices, at least in principle, and that there is no need to quantize the phenomenological classical Maxwell equations. If a macroscopic description is possible, it should appear as an approximation to the fundamental microscopic theory under certain conditions. Such a "macroscopic" approximation is obtained and conditions for its validity are derived. All of the medium harmonic oscillators are represented by a collective harmonic oscillator, then two modes of a polariton field are defined. The microscopic average is regarded as filtering out higher spatial frequencies. The field that influences the guest atom is modified and the characteristics of the effect of such a microscopic field are calculated. A condition is pointed out under which the contribution of the atoms to the quantum noise appears only through a frequency-dependent dielectric constant. An effective description is obtained by leaving out the polariton mode which is approximately equal to the collective mode of the medium. Dutra and Furuya [1997] introduced a microscopic model for a material medium that they have adopted: A^ two-level atoms having the same resonance frequency WQ in a single-mode cavity with resonance frequency co. They consider a guest two-level atom with resonance frequency o)^ and strongly coupled to the field so that it will not be approximated by a harmonic oscillator. The operator of electric displacement field in the cavity is given by

b(x, t) = y

^

[a(t) + at(0] sin (^.v) ,

(5.6)

where L is the length of the cavity. It is assumed that co and L for the singlemode cavity are related as (D/C = JT/L. The operator of the polarization of the medium is given by A'

P(x, t) = Y^ dj \f)j{t) + ^/(O] d{x-xj\

(5.7)

4, §5]

Microscopic models as related to macroscopic concepts

417

where (5.8) 2womo^J are electric dipole moments. In eq. (5.8), mo is an effective mass, q, are effective charges, and products dj[bj{t) + ^J(0] are the electric dipole moment operators of the atoms of the medium that are located at Xj. The operators a{t), a\t), bj(t) and bj(t) satisfy the bosonic commutation relations, in particular dj =

\bj(tlb]^(t)\ = d-.l.

(5.9)

= 0,

bj(t),br{t)

and a(t),a\t) commute with bj(t),b^{t). The Hamiltonian is given by the relation H(t) = ha)a\t)a(t) + Picooy^bUt)bj(t)

/

b(x,t)P(x,t)dx (5.10)

+ ^o,(t)

+ hQ [a{t) + a\t)] [a(t) + a\t)] ,

where a-(t) and a(t) are the pseudo-spin operators, and Q

-d^l—rrSin CoLfi

(5.11)

\ c

is the Rabi frequency of the guest atom located at x^ whose electric dipole moment is d and the effective mass Wg. We notice that 1 /• ~ j b{x,t)P{x,t)dx

^ = hJ28j[^iO

+ ciHt)][bj(t) + b]{t)],

(5.12)

where como

Sj

Hh)-

(5.13)

In simplifying the Hamiltonian, Dutra and Furuya [1997] denote N

G=

H^h

V

(5.14)

the coupling constant between the field and the collective harmonic oscillator described by the annihilation operator

m = -^Y.^Mt).

(5.15)

7=1

In the Hamiltonian (5.10), the self-energy terms (Cohen-Tannoudji, DupontRoc and Grynberg [1989]) have been neglected. This resuhs in the condition

418

Quantum description of light propagation in dielectric media

[4, § 5

4G^ ^ a^cao. Further, the original problem is reduced to the case of a single atom coupled to two polariton modes. The frequencies of these modes are denoted by ^1 and Qj so that Q\ :^ coJl - 4 ^ , co and Q2 -^ <^o when co ^ 0, oc, respectively, with G' = J^G.

(5.16)

In other words, Q\ < Q2 for (x) < (JOO, Q\ > Q2 fox o) > co^. The dressed operators are denoted by Ck{t) and c\{t), k = 1,2, and the operators a(t) and B(t) are expressed in their terms (Chizhov, Nazmitdinov and Shumovsky [1991]). The problem of the extra quantum noise introduced by the atoms of the medium is discussed. When the atoms of the medium are coupled only weakly to the field, i.e., G\ co ^ ^ , where A[a(t) + a\t)] permittivity: e r ^ 1+4—^.

= a(t) + a\t) - {a(t) + ci\t))l,

(5.17) and er is the relative

(5.18)

From relation (5.17) the variance of the electric displacement field D(x,t) (cf. Glauber and Lewenstein [1991]) can be calculated. Adopting a continuous distribution of atoms in the medium instead of the realistic discrete one implies a greater variance (Rosewarne [1991]). Let us address the problem of macroscopic averages. The macroscopic theories of quantum electrodynamics in nonlinear media have often, by "definition", avoided discussing the macroscopic averaging procedure. The quantummechanical averaging advocated by Schram [1960] removes the quantum fluctuations from the macroscopic theory. The problem of what macroscopic averaging procedure to use had evaded solution for many years. Lorentz, in the early twentieth century, was the first to attempt such a derivation (cf. the chapters by de Groot [1969] and van Kranendonk and Sipe [1977] in previous volumes of Progress in Optics). Robinson [1971, 1973] proposed a different kind of macroscopic average. He regards a macroscopic description as a description where spatial Fourier components of the field variables are irrelevant above some limiting spatial frequency ko. Dutra and Furuya [1997] consider Fourier components with spatial frequencies above oj/c irrelevant in a macroscopic

Microscopic models as related to macroscopic concepts

4, §5]

419

description. They arrive at the following expression for the operator of the macroscopic polarization: P(x,t) = -2G^

IJieo_

\B(t)-^B\t)\

(5.19)

sin(^-x^

(joLmo

The macroscopic "average" does not change the electric-displacement-field operator b(x, t), and the macroscopic electric-field-strength operator is given by 1 E{x, t) = (5.20) D(x,t)-P(x,t) eo The calculation of the variance of a quantity typical of the operator E(x, t) yields a larger value than e~^^^, which agrees with Rosewame's [1991] result. Thus, it has been shown that the contribution from the atoms to the quantum noise of the field does not restrict itself to inclusion of the dielectric constant. We will now report a suitable macroscopic theory of electrodynamics in a material medium which does not suffer from the problems which are discussed here. It is shown that under certain conditions a macroscopic description incorporating the frequency dependence of the relative permittivity provides a good approximation. In this domain, Milonni's [1995] result has been recovered. The guest atom is not affected by the polariton mode if the frequency of the atom is far from O2. Analysis of the probability of this mode inducing transitions shows that this probability is negligible when ^2^0

\Q^-o)'\~-

ft>2)2

7«l^2-

+A0)QWG-

(5.21)

0)^

In the regime described by relation (5.21), if one leaves out the polariton mode described by C2{t) and ^2(0 and preforms macroscopic averaging, the resulting macroscopic Hamiltonian is / r ^ a c ( 0 = ^ ^ l ^ | ( 0 ^ l ( 0 + ^ ^ r ( 0 - - ^ m a c ( X a , 0 [d{t) + d\t)\

,

(5.22)

with the macroscopic displacement field ^macv-^? 0

\hQ\eQe,JTr Ly

c,(0 + c (0 sm

er—X

(5.23)

where

y-do;(^'^)

(5.24)

By relation (5.24), y is the ratio between the speed of light in the vacuum and the group velocity in the medium. The macroscopic polarization Pmac(^, 0 is given

420

Quantum description of light propagation in dielectric media

[4, § 5

by relation (5.19), simplified by leaving out its C2-cl polariton component. Then, the macroscopic electric field is obtained ft-om the relation 1 ^macv-^? 0

^macl-^? * j

^macv-^5 * j

(5.25)

^0

It is stated that the results of Dutra and Furuya [1997] for the macroscopic fields coincide with those derived by Milonni [1995] for the case of one and more modes. 5.2.2. Kramers-Kronig dielectric Dutra and Furuya [1998a] have pointed out that the Huttner-Bamett model at the stage after the diagonalization of the Hamiltonian for the polarization field and reservoirs can operate with a larger class of dielectric fianctions than that admitted by the original microscopic model. At this stage, the relative permittivity is expressed dependent on the dimensionless coupling function ^(o;) as introduced in eq. (4.51). For example, the permittivity obtained in the Lorentz oscillator model (Klingshirn [1995]) can be recovered by adopting C(w) = —.—z-r—z

1=.

w^ -
(5.26)

yjn

where K is the frequency-independent absorption rate. The relative permittivity for the original Huttner-Barnett microscopic model has the form e(o;) = 1

—^.

,

(5.27)

where F ( a ; ) - lim

r

l^^^'^l'

dco^

(5.28)

f -^ +0 7_oc (^ - CO - If

It is indicated that, in the Lorentz oscillator model, eq. (5.27) yields the solution F(cy) = i ^ ,

(5.29)

but the integral equation (5.28) with this left-hand side (= replaced by =) cannot be solved to yield a coupling ftinction V{o)). This is the main difficulty, because from the relation t(a» = i^^;^

" ^

(5.30)

4, § 5]

Microscopic models as related to macroscopic concepts

421

we obtain that |K(a»|^ = ^

(5.31)

or we could determine v{a)) = p\ — V(a)) = ± p ^ .

(5.32)

y/jT

V (O

This presents a restriction to the Huttner-Barnett microscopic model. 5.2.3. Dissipative oscillator medium Let us recall that in the microscopic model, the electromagnetic-field operators are given by integrals both over k and co. Huttner and Barnett [1992a] themselves say that they lose the relationship between the frequency and the wavevector k. This observation relates to the macroscopic theories as well: the Dirac delta fianction suitable for the expression of such a relationship is never replaced by another (generalized) function. Quantities such as (4.47)-(4.50) are formulated dependent on the relative permittivity e{(x)). Dutra and Furuya [1998b] suggest a simplified expression: e(a;)=l + ^ l i m / " / ^ ^ ' ^ . da;^ 2a; f -^ +0 J_^ oj' - (D-\£

(5.33)

where (Dutra and Furuya [1998b]) S(co) = cboco

l^^^jf

(5.34)

with F(a)) defined by relation (5.28). They try to calculate the relative permittivity for the Huttner-Barnett microscopic model by means of classical electrodynamics. The Huttner-Barnett approach is applied to the particular case where the coupling strength is a slowly varying function of frequency. In continuation of Dutra and Furuya [1997], a modified version of the simple model takes account of absorption. The inclusion of losses necessarily introduces a continuum of modes in the model. The consequences are minimized by adopting the standard elimination of the reservoir. The interaction between the radiation field and the medium is described by a dipole-coupling Hamikonian, where the canonically conjugated field is

422

Quantum description of light propagation in dielectric media

[4, § 5

the displacement field, instead of a minimal-coupling Hamiltonian, where the canonically conjugated field is the electric field. For simplicity, a Lorentzian shape is assumed for |F(a>)p, given by the relation V((0)=

^^—-ri/^,

(5.35)

where o^o > ^ > ^. The Hamiltonian incorporating absorption is assumed to be N

H(t) = h(D^a\t) a(t) + hcoo ^

b^.(t) bj(t)

7=1 1

--

/•

poo

I b{x,t)P{x,t)dx-^h

^J

N >

/

y^QWJ{Qj)Wj{Qj)dLQ

Jo

j=,

(5.36) where Wj{Q,t), Wj{Qj) are reservoir creation and annihilation operators that commute with every other operator, but the commutation relation holds: \Wj{Qj\

Wj\Q',t)\

= djj>d{Q-Q') i.

(5.37)

Upon substitution of relations (5.6) and (5.7) into the Hamiltonian (5.36) and introducing appropriate collective operators, the total Hamiltonian becomes a sum of two uncoupled Hamiltonians H(t) = Hi(t)^H2(t).

(5.38)

The second Hamiltonian H2(0 describes A^^ - 1 damped collective excitations of the medium to which the field does not couple. The field and the single damped collective excitations of the medium, to which the field couples, are described by the Hamiltonian H\(t) alone. This Hamiltonian is given by Hi (t) = HUt) + ^mat(0 + ^int(0, (5-39) where the Hamiltonian of the field, that of medium and their interaction Hamiltonian are expressed as follows HUt) = hw,a\t)a(t), H^!,tit) = fi(t}oBHt)B{t) + f' / +h I

\V{Q)B\t)Y{Q,t)+

HUt) = hG[a\t) + a{t)][B{t) + B\t)\,

(5.40) QY\Q,t)Y(Q,t)dQ V*(Q)Y\Q,t)B{t)\

dQ, (5.42)

4, § 5]

Microscopic models as related to macroscopic concepts

423

respectively. The collective annihilation operators B{t) and Y{Q, t) are given by N

B(t) =

Y,0jbj(tl 7=1

'"'

(5.43) N

where (frj = gj/G and gj and G are given by eqs. (5.13) and (5.14), respectively. Dutra and Furuya [1998b] have a (conventional) strictly microscopic model, where the medium is not continuous, but discrete. They admit the practicality of the macroscopic average of the physical quantities. Following Robinson [1971, 1973], they understand the macroscopic averaging as filtering out of higher spatial frequencies. Considering a classical Hamiltonian which is a modification of relation (5.39), they derive a relative permittivity. A further topic in the article by Dutra and Furuya [1998b] is essentially relation (4.68) due to Gruner and Welsch [1995]. In particular, it is shown that also in the case of the simple microscopic model of the medium used by Dutra and Furuya [1998b], it is possible to first diagonalize HmatiO a^d then the total Hamiltonian (5.39). The diagonal form of the Hamiltonian ffmaM is achieved in terms of continuous operators B(v,t), /•OC

B(v,t) = a{v)B(t)+

/

l3(y,Q)Y(Q,t)dQ,

(5.44)

where a(v) and ^(v, Q) are coefficients like (4.36) and (4.38). The diagonal form of the total Hamiltonian (5.39) is achieved in terms of continuous operators h3i((0, v)^(v, 0 + ^ ( 0 ; , v)^^(v,0 dv. '(5.45) where a\((o), a2((x)), l5[((o,v), ft(a;, v) are coefficients like (4.47)-(4.50). Suitable operators, namely those of the electric displacement field and of the macroscopic electric field strength, are defined such that

A(a),t) = ai(a))a(t) + a2((o)a\t)-\- / Jo

eoe((i))

/ (^^QL

2G'a*(a;)yi(co,0dwsinf—jcj+H.-

JO

(5.46) The difference between this and relation (4.68) arises because Dutra and Furuya have only a single mode of the field, use a dipole-coupling Hamiltonian instead

424

Quantum description of light propagation in dielectric media

[4, § 6

of a minimal-coupling Hamiltonian, and have defined their field operators in terms of different quadratures of the annihilation and creation operators. Ruostekoski [2000] has theoretically studied the optical properties of a FermiDirac gas in the presence of a superfluid state. He also considered diffraction of atoms by means of light-stimulated transitions of photons between two intersecting laser beams. Optical properties could possibly signal the presence of the superfluid state and determine the value of the Bardeen-Cooper-Schrieffer parameter in dilute atomic Fermi-Dirac gases. § 6. Conclusions In this chapter we have mainly reviewed canonical quantum descriptions of light propagation in a nonlinear dispersionless dielectric medium and in linear and nonlinear dispersive dielectric media. These descriptions have regularly been simplified by a transition to one-dimensional propagation, which has been illustrated also by some original simple descriptions. Besides this we have reported criticisms of the description of light propagation in a nonlinear medium using a spatial variable instead of and similarly as the quantum-mechanical time parameter. We have adopted the standpoint that macroscopic quantum electrodynamics arises both through reduction of a microscopic description and through immediate application of a quantization scheme to macroscopic fields. The origins of immediate macroscopic theories are connected with the possibility of determining the electrical displacement field as the canonical momentum (up to the sign) to the vector potential. One may also consider as a member of this class the possibility of starting from the assumption that the canonical momentum is the magnetic induction field with the dual vector potential whose curl is the electric displacement field. Due to the relatively large number of various fields and their components, and the relative complexity of commutators, discrete and continuous expansions in terms of annihilation and creation operators must be presented. In the papers reviewed here, the ground (vacuum) state of the electromagnetic field has mostly been assumed. The Heisenberg picture of time evolution has been preferred. In one of these papers the opinion has been formulated that the quantum description of sources is to be treated separately, but a detailed treatment cannot be found there. As a criterion for the correctness of the theories, equivalence between the Heisenberg equations of motion for the electric displacement and magnetic induction fields and the Maxwell equations is required.

4]

Acknowledgments

425

The role of the momentum operator has been analyzed: its limited domain of application, one-dimensional propagation, is still appealing for its simplicity. The integral expression for the momentum operator has been given. It has been stressed that the use of this operator as the generator of spatial progression depends on the knowledge of appropriate equal-space commutators between field operators. Results on the connection with the slowlyvarying-envelope approximation have been presented, including some important applications. The possibility has been examined of determining a phenomenological Hamiltonian, and the quantization scheme with dual potential has been expounded and appropriately modified, so that the real relative permittivity of the lossless medium is expressed at least locally (in terms of a quadratic Taylor polynomial). In simplifying to the one-dimensional case, an application to a nonlinear dispersive (Kerr-like) medium and quantum solitons has been presented. Various notions of the slowly-varying-envelope approximation and the quantum-paraxial approximation have been presented, along with applications of these concepts in both linear and nonlinear cases. A new application to a nonlinear dispersionless (Kerr-like) medium has been concerned with the propagation of arbitrary pulses. The need for renormalization has been declared even in this, one-dimensional case. Derivation of a macroscopic description from a microscopic model has been performed; it is not possible - in the prevailing Heisenberg picture - to eliminate the matter fields completely. The argument has been provided not only with the description of the dispersion up to any order of accuracy, but also with losses, as follows from the Kramers-Kronig relations. The exposition of this microscopic model has been delivered from the viewpoint of the macroscopic variables and the operator-valued noise current, which is a form of matter field. A nonlinear modification of this description of the linear dispersive absorbing medium has been performed for the Kerr medium. The application to a two-level atom positioned in the center of a spherical cavity has been mentioned. A different notion of the macroscopic fields has been mentioned as well: these are not comprised in a microscopic model in advance, but approximately equal operators of measurable modes of the dressed matter fields. Acknowledgments This work under project number LN00A015 was supported by the Ministry of Education of the Czech Republic.

426

Quantum description of light propagation in dielectric media

[4

References Abraham, M., 1909, Rend. Circ. Mat. Palermo 28, 1. Abram, I., 1987, Phys. Rev. A 35, 4661. Abram, I., and E. Cohen, 1991, Phys. Rev. A 44, 500. Abram, I., and E. Cohen, 1994, J. Mod. Opt. 41, 847. Agarwal, G.S., 1975, Phys. Rev A 11, 230. Agrawal, G.P., 1989, Nonlinear Fiber Optics (Academic Press, New York). Aiello, A., 2000, Phys. Rev A 62, 063813. Artoni, M., and R. Loudon, 1997, Phys. Rev A 55, 1347. Artoni, M., and R. Loudon, 1998, Phys. Rev A 57, 622. Artoni, M., and R. Loudon, 1999, Phys. Rev A 59, 2279. Babiker, M., and R. Loudon, 1983, Philos. Trans. R. Soc. London Ser. A 385, 439. Ban, M., 1994, Phys. Rev A 49, 5078. Bamett, S.M., C.R. Gilson, B. Huttner and N. Imoto, 1996, Phys. Rev Lett. 77, 1739. Bamett, S.M., J.R. Jeffers, A. Gatti and R. Loudon, 1998, Phys. Rev A 57, 2134. Bamett, S.M., R. Matloob and R. Loudon, 1995, J. Mod. Opt. 42, 1165. Bamett, S.M., and PM. Radmore, 1988, Opt. Commun. 68, 364. Bechler, A., 1999, J. Mod. Opt. 46, 901. Ben-Aryeh, Y, 1999, J. Opt. B 1, 234. Ben-Aryeh, Y, A. Luks and V Pefinova, 1992, Phys. Lett. A 165, 19. Ben-Aryeh, Y, and S. Serulnik, 1991, Phys. Lett. A 155, 473. Bialynicka-Bimla, Z., and I. Bialynicki-Birula, 1987, J. Opt. Soc. Am. 4, 1621. Bialynicki-Birula, L, 1996a, in: Progress in Optics, Vol. 36, ed. E. Wolf (Elsevier, Amsterdam) p. 245. Bialynicki-Birula, I., 1996b, in: Coherence and Quantum Optics VII, eds J.H. Eberly, L. Mandel and E. Wolf (Plenum, New York) p. 313. Bialynicki-Birula, I., 1998, Phys. Rev Lett. 80, 5247. Bialynicki-Birula, I., 2000, Opt. Commun. 179, 237. Bjorken, J.D., and S.D. Drell, 1965, Relativistic Quantum Fields (McGraw-Hill, New York). Bleany, B.I., and B. Bleany, 1985, Electricity and Magnetism (Oxford University Press, Oxford). Bloembergen, N., 1965, Nonlinear Optics (Benjamin, New York). Blow, K.J., R. Loudon and S.J.D. Phoenix, 1991, J. Opt. Soc. Am. B 8, 1750. Blow, K.J., R. Loudon and S.LD. Phoenix, 1992, Phys. Rev A 45, 8064. Blow, K.J., R. Loudon, S.J.D. Phoenix and T.J. Sheperd, 1990, Phys. Rev A 42, 4102. Boivin, L., EX. Kartner and H.A. Haus, 1994, Phys. Rev Lett. 73, 240. Bom, M., and L. Infeld, 1934, Proc. R. Soc. London A 147, 522. Bom, M., and L. Infeld, 1935, Proc. R. Soc. London A 150, 141. Boyd, R.W., 1999, J. Mod. Opt. 46, 367. Boyd, R.W., 2000, J. Mod. Opt. 47, 1147. Brown, S.A., and B.J. Dalton, 2001a, J. Mod. Opt. 48, 597. Brown, S.A., and B.J. Dalton, 2001b, J. Mod. Opt. 48, 639. Brun, TA., and S.M. Bamett, 1998, J. Mod. Opt. 45, 777. Cahill, K.E., and R.J. Glauber, 1999, Phys. Rev A 59, 1538. Carmichael, H.J., 1987, J. Opt. Soc. Am. B 4, 1565. Carter, S.J., PD. Drummond, M.D. Reid and R.M. Shelby, 1987, Phys. Rev Lett. 58, 1841. Caves, CM., and D.D. Crouch, 1987, J. Opt. Soc. Am. B 4, 1535. Caves, CM., and B.L. Schumaker, 1985, Phys. Rev A 31, 3068. Chew, W C , 1995, Waves and Fields in Inhomogeneous Media (IEEE Press, New York).

4]

References

427

Chizhov, A.V, R.G. Nazmitdinov and A.S. Shumovsky, 1991, Quantum Opt. 3, 1. Cohen-Tannoudji, C , J. Dupont-Roc and G. Grynberg, 1989, Photons and Atoms: Introduction to Quantum Electrodynamics (Wiley, New York). Collett, M.J., and C.W. Gardiner, 1984, Phys. Rev. A 30, 1386. Crouch, D.D., 1988, Phys. Rev. A 38, 508. Dalton, B.J., S.M. Bamett and PL. Knight, 1999a, J. Mod. Opt. 46, 1107. Dalton, B.J., S.M. Bamett and PL. Knight, 1999b, J. Mod. Opt. 46, 1315. Dalton, B.J., S.M. Bamett and PL. Knight, 1999c, J. Mod. Opt. 46, 1495. Dalton, B.J., S.M. Bamett and PL. Knight, 1999d, J. Mod. Opt. 46, 1559. Dalton, B.J., E.S. Guerra and PL. Knight, 1996, Phys. Rev A 54, 2292. de Groot, S.R., 1969, in: Studies in Statistical Mechanics, Vol. 4, ed. G.E.U.J. de Boer (North-Holland, Amsterdam) ch. 1, p. 11. Dechoum, K., T.W. Marshall and E. Santos, 2000, J. Mod. Opt. 47, 1273. Deutsch, LH., and J.C. Garrison, 1991a, Opt. Commun. 86, 311. Deutsch, LH., and J.C. Garrison, 1991b, Phys. Rev A 43, 2498. Di Stefano, O., S. Savasta and R. Girlanda, 1999, Phys. Rev. A 60, 1614. Di Stefano, O., S. Savasta and R. Girlanda, 2000, Phys. Rev. A 61, 023803. Di Stefano, O., S. Savasta and R. Girlanda, 2001a, L Mod. Opt. 48, 67. Di Stefano, O., S. Savasta and R. Girlanda, 2001b, J. Opt. B 3, 288. Dirac, PA.M., 1927, Proc. R. Soc. London A 114, 243. Dirac, P.A.M., 1964, Lectures on Quantum Mechanics (Belfer Graduate School of Science, New York). Drummond, P D , 1990, Phys. Rev A 42, 6845. Drummond, P.D., 1994, in: Modem Nonlinear Optics, Part 3, eds M.V Evans and S. Kielich (Wiley, New York) p. 379. Drummond, PD., and S.J. Carter, 1987, J. Opt. Soc. Am. B 4, 1565. Dmmmond, P D , S.J. Carter and R.M. Shelby, 1989, Opt. Lett. 14, 373. Dmmmond, P D , and M. Hillery, 1999, Phys. Rev A 59, 691. Dung, H.-T., L. Knoll and D.-G. Welsch, 1998, Phys. Rev A 57, 3931. Dung, H.-T, L. Knoll and D.-G. Welsch, 2000, Phys. Rev A 62, 053804. Dung, H.-T, L. Knoll and D.-G. Welsch, 2001, Phys. Rev A 64, 013804. Dutra, S.M., and K. Fumya, 1997, Phys. Rev A 55, 3832. Dutra, S.M., and K. Fumya, 1998a, Europhys. Lett. 43, 13. Dutra, S.M., and K. Fumya, 1998b, Phys. Rev A 57, 3050. Dutra, S.M., and G. Nienhuis, 2000, Phys. Rev A 62, 063805. Fano, U, 1956, Phys. Rev. 103, 1202. Fano, U, 1961, Phys. Rev. 124, 1866. Fox, A.G., and T Li, 1961, Bell Sys. Tech. J. 40, 453. Friberg, S.R., S. Machida and Y Yamamoto, 1992, Phys. Rev Lett. 69, 3165. Garabedian, PA., 1964, Partial Differential Equations (Chelsea, New York) ch. 10. Gardiner, C.W, and M.J. Collett, 1985, Phys. Rev A 31, 3761. Gea-Banacloche, J., N. Lu, L.M. Pedrotti, S. Prasad, M.O. Scully and K. Wodkiewicz, 1990a, Phys. Rev. A 41, 369. Gea-Banacloche, J., N. Lu, L.M. Pedrotti, S. Prasad, M.O. Scully and K. Wodkiewicz, 1990b, Phys. Rev. A 41, 381. Glauber, R.J., and M. Lewenstein, 1989, in: Squeezed and Nonclassical Light, eds P. Tombesi and E.R. Pike (Plenum, New York) p. 203. Glauber, R.J., and M. Lewenstein, 1991, Phys. Rev. A 43, 467. Graham, R., and H. Haken, 1968, Z. Phys. 213, 420.

428

Quantum description of light propagation in dielectric media

[4

Gruner, T, and D.-G. Welsch, 1995, Phys. Rev. A 51, 3246. Gruner, T., and D.-G. Welsch, 1996a, Phys. Rev. A 53, 1818. Gruner, T., and D.-G. Welsch, 1996b, Phys. Rev. A 54, 1661. Haus, H.A., and EX. Kartner, 1992, Phys. Rev. A 46, Rl 175. Haus, H.A., and Y. Lai, 1990, J. Opt. Soc. Am. B 7, 386. Haus, H.A., K. Watanabe and Y. Yamamoto, 1989, J. Opt. Soc. Am. B 6, 1138. Hawton, M., 1999, Phys. Rev. A 59, 3223. Healey, W.P., 1982, Nonrelativistic Quantum Electrodynamics (Academic Press, London). Hillery, M., and PD. Drummond, 2001, Phys. Rev. A 64, 013815. Hillery, M., and L.D. Mlodinow, 1984, Phys. Rev. A 30, 1860. Hillery, M., and L.D. Mlodinow, 1997, Phys. Rev. A 55, 678. Hinds, E.A., 1990, in: Advances in Atomic, Molecular and Optical Physics, Vol. 28, eds D. Bates and B. Bederson (Academic Press, New York) p. 237. Ho, K.C., PT. Leung, A. Maassen van den Brink and K. Young, 1998, Phys. Rev. E 58, 2965. Ho, S.-T., and P Kumar, 1993, J. Opt. Soc. Am. B 10, 1620. Hopfield, J.J., 1958, Phys. Rev. 112, 1555. Hradil, Z., 1996, Phys. Rev. A 53, 3687. Huttner, B., and S.M. Barnett, 1992a, Europhys. Lett. 18, 487. Huttner, B., and S.M. Barnett, 1992b, Phys. Rev. A 46, 4306. Huttner, B., J.J. Baumberg and S.M. Barnett, 1991, Europhys. Lett. 16, 177. Huttner, B., S. Serulnik and Y Ben-Aryeh, 1990, Phys. Rev. A 42, 5594. Imoto, N., 1989, in: Proc. 3rd Int. Symp. on Foundations of Quantum Mechanics (Tokyo), eds S. Kobayashi, H. Egawa, Y Murayama and S. Nomura (Physical Society of Tokyo, Tokyo) p. 306. Inagaki, T, 1998, Phys. Rev. A 57, 2204. Inoue, T, and H. Hori, 2001, Phys. Rev. A 63, 063805. Itzykson, C , and J.B. Zuber, 1980, Quantum Field Theory (McGraw-Hill, New York). Jauch, J.M., and K.M. Watson, 1948, Phys. Rev. 74, 950. Javanainen, J., J. Ruostekoski, B. Vestergaard and M.R. Francis, 1999, Phys. Rev. A 1959, 64. Jeffers, J.R., and S.M. Barnett, 1994, J. Mod. Opt. 41, 1121. JefiFers, J.R., S.M. Barnett, R. Loudon, R. Matloob and M. Artoni, 1996, Opt. Commun. 131, 66. Jeffers, J.R., N. Imoto and R. Loudon, 1993, Phys. Rev. A 47, 3346. Kennedy, TA.B., and E.M. Wright, 1988, Phys. Rev. A 38, 212. Khosravi, H., and R. Loudon, 1991, Proc. R. Soc. London Ser. A 433, 337. Khosravi, H., and R. Loudon, 1992, Proc. R. Soc. London Ser. A 436, 373. Kitagawa, M., and Y Yamamoto, 1986, Phys. Rev. A 34, 3974. Kittel, C , 1987, Quantum Theory of Solids, 2nd Ed. (Wiley, New York). Klingshim, C.F., 1995, Semiconductor Optics (Springer, Berlin) p. 57. Knoll, L., and U. Leonhardt, 1992, J. Mod. Opt. 39, 1253. Knoll, L., S. Scheel, E. Schmidt, D.-G. Welsch and A.V Chizhov, 1999, Phys. Rev. A 59, 4716. Knoll, L., S. Scheel and D.-G. Welsch, 2001, in: Coherence and Statistics of Photons and Atoms, ed. J. Pefina (Wiley, New York) p. 1. Knoll, L., W Vogel and D.-G. Welsch, 1986, J. Opt. Soc. Am. B 3, 1315. Knoll, L., W Vogel and D.-G. Welsch, 1987, Phys. Rev. A 36, 3803. Knoll, L., W Vogel and D.-G. Welsch, 1990, Phys. Rev A 42, 503. Knoll, L., W Vogel and D.-G. Welsch, 1991, Phys. Rev. A 43, 543. Knoll, L., and D.-G. Welsch, 1992, Prog. Quantum Electron. 16, 135. Kobe, D.H., 1999, Phys. Lett. A 253, 7. Kolobov, M.I., 1999, Rev. Mod. Phys. 71, 1539.

4]

References

429

Korolkova, N., R. Loudon, G. Gardavsky, M.W. Hamilton and G. Leuchs, 2001, J. Mod. Opt. 48, 1339. Kubo, R., 1962, J. Phys. Soc. Jpn. 17, 1100. Lai, Y, and H.A. Haus, 1989, Phys. Rev. A 40, 844. Landau, L.D., and E.M. Lifshitz, 1960, Electrodynamics of Continuous Media (Pergamon Press, Oxford). Lang, R., and M.O. Scully, 1973, Opt. Commun. 9, 331. Lang, R., M.O. Scully and W.E. Lamb Jr, 1973, Phys. Rev. A 7, 1788. Lax, M., W.H. Louisell and W.B. McKnight, 1974, Phys. Rev A 11, 1365. Ledinegg, E., 1966, Zeit. Phys. 191, 177. Leonhardt, U, 2000, Phys. Rev A 62, 012111. Leonhardt, U, and P Piwnicki, 2001, J. Mod. Opt. 48, 977. Levenson, M.D., R.M. Shelby, A. Aspect, M.D. Reid and D.R Walls, 1985, Phys. Rev A 32, 1550. Ley, M., and R. Loudon, 1987, J. Mod. Opt. 34, 227. Li, L.W., PS. Kooi, M.S. Leong and T.S. Yeo, 1994, TEE Trans. Microwave Theory Tech. 42, 2302. Loudon, R., 1963, Proc. Phys. Soc. 83, 393. Loudon, R., 1998, Phys. Rev A 57, 4904. Loudon, R., and PL. Knight, 1987, J. Mod. Opt. 34, 709. Luis, A., and J. Peiina, 1996, Quantum Semiclass. Opt. 8, 39. Mandel, L., 1964, Phys. Rev 136, B1221. Mandel, L., and E. Wolf, 1995, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge). Matloob, R., 1999a, Phys. Rev A 60, 50. Matloob, R., 1999b, Phys. Rev A 60, 3421. Matloob, R., 2001, Opt. Commun. 192, 287. Matloob, R., and H. Falinejad, 2001, Phys. Rev A 64, 042102. Matloob, R., A. Keshavarz and D. Sedighi, 1999, Phys. Rev A 60, 3410. Matloob, R., R. Loudon, M. Artoni, S.M. Bamett and J.R. Jeffers, 1997, Phys. Rev A 55, 1623. Matloob, R., R. Loudon, S.M. Bamett and J.R. Jeffers, 1995, Phys. Rev A 52, 4823. Matloob, R., and G. Pooseh, 2000, Opt. Commun. 181, 109. McDonald, K.T., 2001, Am. J. Phys. 69, 607. Mendon^a, J.T., A.M. Martins and A. Guerreiro, 2000, Phys. Rev E 62, 2989. Milbum, G.J., and D.R Walls, 1981, Opt. Commun. 39, 40. Milbum, G.J., D.F. Walls and M.D. Levenson, 1984, J. Opt. Soc. Am. B 1, 390. Milonni, PW, 1995, J. Mod. Opt. 42, 1991. Misner, C.W, K.C. Thome and J.A. Wheeler, 1970, Gravitation (Freeman, San Francisco, CA). Mrowczynski, S., and B. Muller, 1994, Phys. Rev D 50, 7542. Nilsson, O., Y Yamamoto and S. Machida, 1986, IEEE J. Quantum Electr. QE-22, 2043. Peierls, R., 1976, Proc. R. Soc. A 347, 475. Peierls, R., 1985, in: Highlights of Condensed Matter Theory, eds F. Bassani, F. Fumi and M.P Tosi (North-Holland, Amsterdam). Perina, J., 1991, Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd Ed. (Kluwer, Dordrecht). Perina Jr, J., and J. Perina, 2000, in: Progress in Optics, Vol. 41, ed. E. Wolf (Elsevier, Amsterdam) p. 361. Pefinova, V, A. Luks, J. Kfepelka, C. Sibilia and M. Bertolotti, 1991, J. Mod. Opt. 38, 2429. Potasek, M.J., and B. Yurke, 1987, Phys. Rev A 35, 3974. Power, E.A., and S. Zienau, 1959, Philos. Trans. R. Soc. London Ser. A 251, 427. Robinson, F.N.H., 1971, Physica (Utrecht) 54, 329.

430

Quantum description of light propagation in dielectric media

[4

Robinson, F.N.H., 1973, Macroscopic Electromagnetism, 1st Ed. (Pergamon Press, Oxford). Roman, P., 1969, Introduction to Quantum Field Theory (Wiley, New York). Rosenau da Costa, M., A.O. Caldeira, S.M. Dutra and H. Westfahl Jr, 2000, Phys. Rev. A 61, 022107. Rosenbluh, M., and R.M. Shelby, 1991, Phys. Rev. Lett. 66, 153. Rosewame, D.M., 1991, Quantum Opt. 3, 193. Ruostekoski, J., 2000, Phys. Rev A 61, 033605. Scheel, S., L. Knoll and D.-G. Welsch, 1998, Phys. Rev A 58, 700. Scheel, S., L. Knoll, D.-G. Welsch and S.M. Bamett, 1999, Phys. Rev A 60, 1590. Schmidt, E., J.R. Jeffers, S.M. Bamett, L. Knoll and D.-G. Welsch, 1998, J. Mod. Opt. 45, 377. Schmidt, E., L. Knoll and D.-G. Welsch, 1996, Phys. Rev A 54, 843. Schram, K., 1960, Physica (Utrecht) 26, 1080. Schubert, M., and B. Wilhelmi, 1986, Nonlinear Optics and Quantum Electronics (Wiley, New York). Scully, M.O., and M.S. Zubairy, 1997, Quantum Optics (Cambridge University Press, Cambridge). Sczaniecki, L., 1983, Phys. Rev A 28, 3493. Serulnik, S., and Y Ben-Aryeh, 1991, Quantum Opt. 3, 63. Shelby, R.M., PD. Drummond and S.J. Carter, 1990, Phys. Rev A 2, 2966. Shen, Y.R., 1967, Phys. Rev 155, 921. Shen, Y.R., 1969, in: Quantum Optics, ed. R.J. Glauber (Academic Press, New York) p. 473. Shen, Y.R., 1984, The Principles of Nonlinear Optics (Wiley, New York). Shirasaki, M., and H.A. Haus, 1990, J. Opt. Soc. Am. B 7, 30. Shirasaki, M., H.A. Haus and D. Liu Wong, 1989, J. Opt. Soc. Am. B 6, 82. Slusher, R.E., P Grangier, A. LaPorta, B. Yurke and M.J. Potasek, 1987, Phys. Rev Lett. 59, 2566. Steinberg, S., 1985, in: Lie Methods in Optics, eds J. Sanchez Mondragon and K.B. Wolf (Springer, Berlin). Stenholm, S., 2000, Opt. Commun. 179, 247. Tarasov, VE., 2001, Phys. Lett. A 288, 173. Tip, A., 1998, Phys. Rev A 57, 4818. Tip, A., L. Knoll, S. Scheel and D.-G. Welsch, 2001, Phys. Rev A 63, 043806. Toda, M., S. Adachi and K. Ikeda, 1989, Prog. Theor. Phys. Suppl. 98, 323. Tomas, M.S., 1995, Phys. Rev A 51, 2545. Toren, M., and Y. Ben-Aryeh, 1994, Quantum Opt. 6, 425. Tucker, J., and D.E Walls, 1969, Phys. Rev 178, 2036. Ujihara, K., 1975, Phys. Rev A 12, 148. Ujihara, K., 1976, Jpn. J. Appl. Phys. 15, 1529. Ujihara, K., 1977, Phys. Rev A 16, 562. van Kranendonk, J., and J.E. Sipe, 1977, in: Progress in Optics, Vol. 15, ed. E. Wolf (North-Holland, Amsterdam) ch. 5, p. 245. Vogel, W, and D.-G. Welsch, 1994, Lectures on Quantum Optics (Akademie Verlag, New York). Walls, D.E, 1983, Nature (London) 306, 141. Watanabe, K., H. Nakano, A. Honold and Y Yamamoto, 1989, Phys. Rev Lett. 62, 2257. Woolley, R.G., 1971, Philos. Trans. R. Soc. London Ser. A 321, 557. Wright, E.M., 1990, J. Opt. Soc. Am. B 7, 1141. Wubs, M., and L.G. Suttorp, 2001, Phys. Rev A 63, 043809. Yablonovitch, E., and T. Gmitter, 1987, Phys. Rev Lett. 58, 2486. Yamamoto, Y, and N. Imoto, 1986, IEEE J. Quantum Electr. QE-22, 2032. Yang, C.N., 1967, Phys. Rev Lett. 19, 1312. Yang, C.N., 1968, Phys. Rev 168, 1920. Yariv, A., 1985, Optical Electronics, 3rd Ed. (Wiley, New York).

4]

References

431

Yariv, A., 1989, Quantum Electronics (Wiley, New York). Yariv, A., and P. Yeh, 1984, Optical Waves in Crystals (Wiley, New York). Yurke, B., 1984, Phys. Rev. A 29, 408. Yurke, B., 1985, Phys. Rev. A 32, 300. Yurke, B., P Grangier, R.E. Slusher and M.J. Potasek, 1987, Phys. Rev. A 35, 3586. Zalesny, J., 2001, Phys. Rev. E 63, 026603. Zinn-Justin, J., 1989, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford).