Quantum theory of light propagation: Linear medium

PHYSICAL REVIEW A VOLUME 35, NUMBER 11 JUNE 1, 1987

Quantum theory of light propagation: Linear medium

I. AbramCentre National d'Etudes des Telecommunications, 196Avenue Henri Ravera, 92220 Bagneux, France

(Received 22 December 1986)

We have developed a quantum-mechanical formalism which permits the treatment of light propa-

gation within the conceptual framework of quantum optics. The formalism rests on the calculation

of the momentum operator for the radiation field, and yields directly a description for the spatial

progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment

for refraction and reflection by applying our formalism to propagation through a linear dielectric.

The fidelity with which this formalism reproduces all results known from classical optics demon-

strates its validity.

I. INTRODUCTION

Within the conventional formulation of the quantumtheory of light, traveling-wave phenomena are describedthrough the Hamiltonian of the electromagnetic field, andparticularly through the temporal evolution of the spatialmodes of the field. The modal Hamiltonian formalism isquite well suited for the description of radiative emissioninto the vacuum field When the Hamiltonian includes aterm coupling an emitter to an infinity of modes, energycan be dissipated into the radiation field. On the otherhand, the description of nonlinear wave interactions, inwhich two (or more) waves interact with each otherthrough the nonlinear polarization of a material, is quitecumbersome. The traveling waves have to be analyzedeach into an infinity of modes, and a nonlinear polariza-tion term which couples the individual modes is intro-duced in the Hamiltonian. Thus, the transfer of energybetween the two waves is described as a dissipative processfrom one infinite manifold of modes into another. Thiscomplicated quantum-mechanical procedure contrastssharply with the simplicity of the classical description ofnonlinear optical phenomena: the spatial differentialequations can be solved directly; energy is then transferredbetween the two waves as they advance together in thenonlinear material.

When the procedure used for nonlinear traveling-wavephenomena is applied to linear propagation throughmatter, the modal Hamiltonian formalism is seen to fail.Inclusion of the linear polarization term in the Hamiltoni-an

f (E +H +4rrXE )dv1

8~ v

(in standard notation) couples each mode to itself througha term quadratic in the electric field. This term leads toan increase in the energy of all the modes and thus givesrise to a renormalization of their frequencies. This resultis, of course, incorrect since the frequency of a light beamtraversing a dielectric is not changed. The linear polariza-tion of the dielectric gives rise to refraction which affectsonly the spatial characteristics of light by renormalizingits wave vector and its phase velocity, as is known from

classical optics. However, the modal Hamiltonian ad-dresses only the temporal evolution of the free-space nor-mal modes which constitute a fixed set of solutions of thespatial wave equation (with no polarization), and for thisreason it cannot account for spatial changes. This failureof the modal Hamiltonian in describing propagation in alinear dielectric indicates that its applicability to thedescription of nonlinear propagation must also have manylimitations.

The problem of propagation through a linear non-dispersive dielectric, one of the most basic problems ofclassical optics, has not been treated yet in quantum op-tics, to our knowledge. A theory of phenomenologicalquantum electrodynamics in refractive media wasdeveloped quite early, but has not be applied to quantumoptics. In quantum optics, on the other hand, a procedurehas been devised to circumvent the problem of a dielec-tric, by redefining the spatial modes in the cavity ofquantization; ' the dielectric function of the medium isintroduced in the Helmholtz equation for the field in thecavity, and thus a new spatial periodicity (wave vector) isobtained for the modes, different from its free-spacevalue. It is these new modes that are used in constructingtraveling wave packets to describe optical phenomenaoccurring in the medium, including nonlinear wave in-teractions. Another procedure, based on the space-timeanalogy for steady-state propagation, has been devised todescribe nonlinear propagation. By heuristically convert-ing spatial progression into temporal evolution (throughdivision by the speed of light) this procedure permits us toaddress nonlinear optical phenomena in a manner analo-gous to their corresponding classical treatment. Formalspace-time analogies have also been pointed out in the dif-ferential equations for the propagation of short lightpulses.

More recently, the problem of the quantum-mechanicaltreatment of light propagation has received renewed in-terest in connection with the description of nonclassicalstates of the radiation field, such as the "squeezed"states. Work focused mainly on the description of dif-fraction by applying the Green's function for the classicaldiffraction problem directly to the quantum-mechanicalfield operators.

35 4661 1987 The American Physical Society

4662 I. ABRAM 35

II. A REMINDER OF CLASSICALELECTRODYNAMICS

Propagation of the electromagnetic field is describedthrough the Maxwell equations,

1 BDVxH=-c Bt

(2.1a)

Clearly, it is necessary to extend the traditional theoryof quantum optics to describe propagative phenomena bydeveloping a quantum-mechanical formalism that cantreat traveling-wave phenomena without invoking themodal Hamiltonian, and can treat both linear and non-linear propagation on the same footing. In developingthis formalism, we may be guided by classical linear op-tics: Introduction of a dielectric in the path of a lightbeam does not change the total energy; the incident energyis simply redistributed into the transmitted and reflectedwaves. The quantum-mechanical Hamiltonian must thusremain unchanged. On the other hand, both the local en-

ergy density and the momentum of the wave (its wavevector) change because of the polarization induced in thedielectric. Clearly, the momentum is the concept that isappropriate for the description of refraction, and moregenerally of propagative phenomena within the frame-work of quantum optics: Since quantum mechanicallyspace and momentum are conjugate variables, the momen-tum operator of the electromagnetic field permits the cal-culation of its spatial progression, just like the Hamiltoni-an permits the calculation of its temporal evolution. Useof the momentum operator to describe the spatial progres-sion of a wave was first proposed by Shen. However, thisis the first paper where this proposal is thoroughly exam-ined.

In this series of papers we develop the formalism forthe quantum-mechanical treatment of the spatial progres-sion of electromagnetic waves in matter, through the cal-culation of the momentum operator of the electromagnet-ic field. In this paper, we examine propagation through alinear dielectric, while in the following paper we treatpropagation through a nonlinear medium. When appliedto a situation already treated in classical optics (such asrefraction or reflection from a dielectric surface), this for-malism reproduces all the known classical results. At thesame time, however, it permits the treatment of purelyquantum-mechanical propagative phenomena, such aspropagation of nonclassical states, or spontaneously ini-tiated stimulated emission. The paper is organized asfollows. In Sec. II we review some well-known resultsfrom classical electrodynamics, recasting them in a waythat permits their direct application to the quantum-mechanical description of light propagation. The nota-tion is introduced in Sec. III by examining quantum-mechanical propagation in the trivial case of plane wavespropagating in free space. In Sec. IV we establish thequantum-mechanical Maxwell equations. These equationsare solved for a linear dielectric in Sec. V, where wepresent a quantum-mechanical treatment of the refractiveindex, of propagation through a refractive medium, andof reflection on a dielectric surface. Finally, in Sec. VI wepresent a discussion of our results.

(2.1b)

T-8=07' D=O,

(2. lc)

(2.1d)

where D=E+4mP is the electric displacement, B is themagnetic induction, E and H are the electric and magnet-ic field strengths, respectively, P is the (linear or non-linear) polarization induced in the medium, and c is thespeed of light. We assume that there are no free chargesor currents, and that we are dealing with nonmagneticmaterials, so that 8=H. We use Gaussian unitsthroughout the paper.

For simplicity, we shall consider only the case of planewaves propagating along the z axis, with the electric fieldpolarized along x, and the magnetic field along y. Thisreduces the Maxwell equations into scalar differentialequations, the directions of all vectors being implicit. Weshall further assume that light is propagating in lineardielectric, where the induced polarization is at all timesproportional to the incident electric field,

P =gE, (2.2a)

where X is the (linear) susceptibility of the material, whichwe assume for simplicity to be a scalar (neglecting its ten-sorial properties), independent of frequency (no disper-sion). It is convenient to define also the dielectric func-tion e of the material

e= 1+4~7

and the refractive index n

(2.2b)

(2.2c)

The Maxwell equations can be rearranged to give theelectromagnetic wave equation, which is usually the start-ing point for the classical treatment of (linear or non-linear) light propagation. Alternatively, they may be rear-ranged to give energy and momentum flow equations,which can also be used as the starting points for treatingpropagation. This latter approach to the propagation ofthe electromagnetic field will be reviewed in this section.Its advantage over the wave equation is that it deals withobservables (such as energy and momentum) whosequantum-mechanical equivalents can describe temporalevolution and spatial progression in the quantized elec-tromagnetic field. However, the energy and momentumfunctions have to be chosen appropriately so that their usein the quantum-mechanical description of propagationgives the right results: When a light beam traverses anonabsorbing dielectric, the number of photons containedin the beam, their frequency, and the total energy of thebeam must remain unchanged, while its wave vector andthe total momentum must increase proportionally to therefractive index.

The material polarization P is a key concept in classicalelectromagnetism, in that it can store part of the elec-tromagnetic energy: The energy density u within thevolume of a dielectric is greater than that of the free field,and this increase is attributed to the deformation of thematerial

35 QUANTUM THEORY OF LIGHT PROPAGATION: LINEAR MEDIUM 4663

1 (E'+H'+4~PE) .8m.

(2.3)aU =0at

(2.8a)

In quantum optics, however, the bookkeeping is different.The energy is carried by the photons and by the materialexcitations. Far from all resonances, the material under-goes transitions only to virtual states which generally liveless than 10 ' sec for a transparent material in the opti-cal region. Thus, for time scales of experimental interest,energy is carried only by the field. This implies that foran experiment that is not fast enough to resolve the pro-cesses associated with the virtual excitations, the increasein energy density in a transparent dielectric is regarded asan increase in the photon density. This partition betweenelectromagnetic and mechanical energies suggests theMinkowski definition for the momentum density'

g= D&B1

4mc(2.4)

(which includes both the momentum of the field and themomentum associated with the material polarization) asthe proper momentum function for quantum optics. Theusual (Abraham —von Laue) definition for the momentumdensity

g'= E&(H1

4vrc(2.5)

S= EXH.4~

(2.6)

The energy flux is conserved when a light beam goesthrough a dielectric; thus, an energy function proportionalto S would be the proper quantum-mechanical Hamiltoni-an.

For the classical field, and in the absence of an externaldriving agent, the energy-flow equation may be written as

—v-s= a E +H +Ea (2.7a)at 8m at

refers to the momentum that can be attributed to the fieldalone, and is independent of the medium in which lightpropagates. Its use would require an explicit treatment ofthe momentum associated with the virtual excitations ofthe material. For this reason, we shall adopt the defini-tion (2.4), even though its use in classical electrodynamicsis subject to controversy on thermodynamic grounds. Wemay also define the Poynting vector S, which gives a mea-sure of the energy flux as

aGat

whereu dv

v

(2.8b)

(2.9a)

is the total energy in the closed volume V, while

gdvv

(2.9b)

is the total momentum. To obtain the actual values of Uand G, however, we have to consider an open finitevolume through which flow can occur. For one-dimensional propagation Eqs. (2.7) reduce to scalar form

as auaz=at '

ag auat=az '

where we have used T =u, in Eq. (2.10b).

(2.10a)

(2.10b)

L't'=U

(2.1 1)

and is therefore independent of the medium. The energyand momentum carried by this pulse may be calculatedthrough Eqs. (2.10), by considering a surface A placeddownstream from the pulse. Since the electric and mag-netic fields vary as z ut, the total en—ergy and momentumcrossing the surface may be obtained by integrating Eqs.(2.10) over the time and over space, giving

SALC

SV (2.12a)

Equations (2.10) are usually integrated over a fixedvolume, in order to relate the energy flux to the change in

energy content in the volume. Here, however, in order tomake contact with the energy and momentum calculationsfor the quantized electromagnetic field, we shall solveEqs. (2.10) for the case of a finite light pulse, rather thanfor a fixed volume. For simplicity, we may consider a"square" pulse of length L (in free space), consisting of aplane wave packet. When the pulse enters a dielectric ofindex n, its velocity changes to v =c/n while all thewavelengths that compose it become A, '=A, /n. Thus, apulse of length L in free space shortens to L'=Lln.However, the duration of the pulse remains constant,

while the momentum-flow equation is

uV(2.12b)

g=V-Tat

(2.7b)

where T is the unsymmetrical Maxwell stress tensor, 'whose elements are

T p= [E Dp+H Bp ——,(E D+B.H)5 p]4m

(2.7c)

with a,P=x,y, z. For a field that decays at infinity, Eqs.(2.7) give energy and momentum conservation upon in-tegration over a volume totally enclosing the field,

where V is the volume occupied by the pulse in free space.For a general state of the radiation field, which is a super-position of forward- and backward-going waves (varyingas z ut and z+ ut, respectivel—y) Eqs. (2.12) become

U= —(S+ —S ),Vc

G= —(u+ —u ),V

c

(2.13a)

(2.13b)

where the + ( —) index, the energy density and flux dueto, the forward (backward) waves alone. Clearly, Eqs.

4664 I. ABRAM 35

(2.12) and (2.13) are independent of the medium in whichthe pulse propagates, and permit us to keep track of theenergy and the momentum of the pulse as it traverse dif-ferent media.

One of the basic results of classical optics is that the en-

ergy flux is conserved when an electromagnetic wave en-counters a dielectric. Thus, for normal incidence on asharp vacuum-dielectric interface, we have the well-known relationship among the incident ( I), reflected ( R ),and transmitted ( T) waves

SI ——Sg +ST (2.14a)

and, in particular,2

,n —1

S~ ——' SrI, n+1

4nST —— 2Sr

(n +1)while the electric fields follow

(2.14b)

(2.14c)

n —1

n+1 (2.14d)

2ET—n+1 (2.14e)

since the electric field in each medium is related to the en-

ergy flux through

S =nE (2.14fl

If the refractive index at the surface changes graduallyfrom 1 to n so that there are no reflections, or if thedielectric is entered through an "antireflective" coating,then the flux in the dielectric equals the flux in free space,

ST ——SI . (2.15a)

Equation (2.15a) together with Eq. (2.12a) imply that theenergy content of a pulse is not affected by the medium inwhich the pulse is traveling, as long as there are no reflec-tion losses. In the absence of reflections, the electric andmagnetic fields of the transmitted wave in the dielectricare related to the corresponding incident fields (in freespace) by

Er EIIv'n—HT ~nHI .

(2.15b)

(2.15c)

which implies a similar increase for the total momentumof the pulse

GT ——nGg . (2.15e)

That is, the momentum carried by the pulse, defined ac-cording to Eqs. (2.4) and (2.9b), is always proportional tothe wave vector of its photons, which in turn is propor-tional to the refractive index.

The energy density inside the dielectric is therefore in-creased with respect to the free-space energy density by afactor of n,

(2.15d)

We are now in a position to apply the above discussionto the quantized electromagnetic field. For this, we notethat the periodic boundary conditions used in quantumoptics for the cavity of quantization are equivalent to con-sidering the cavity as an open volume through which flowoccurs cyclically: The flux exiting on the right reenterson the left. Thus, a propagating field may be consideredas a "square" pulse that fills exactly the volume of quanti-zation at r=o Pr.opagation causes the energy (and themomentum) of the square pulse to flow through the boun-dary of the cavity, while the periodic boundary conditionsreplenish the cavity through the opposite boundary. Theenergy and momentum of the cavity of quantization cor-respond to the energy and momentum of that pulse. Theyare therefore given by Eqs. (2.13) (with V being now thevolume of the cavity of quantization) and exhibit theproper behavior for the description of temporal evolutionand spatial progression: The energy of the cavity is in-dependent of the medium it contains, while its momentumis directly proportional to the wave vector of the excitedfield modes, and depends on the medium. Equations(2.13) will therefore be used as the starting point for thequantum-mechanical treatment of light propagation.Their use to describe quantum-mechanical light propaga-tion through a dielectric implies that the optical pathlength of the cavity of quantization (i.e., the number ofwavelengths it contains for each mode) remains constantupon introduction of a dielectric. This insures that theperiodic-boundary conditions of the cavity are the sameirrespective of whether the cavity is empty or contains adielectric, and establishes a correspondence between thefree-field and refracted-wave modes of the cavity. At thesame time, it means that the physical length of the cavityof quantization shortens in the presence of a dielectric, sothat energy may be conserved. This result may seemparadoxical when thinking of the cavity as a finite box offixed size and set boundary conditions. However, itshould be remembered that the periodic-boundary cavityis simply a device for treating the propagating modes ininfinite space where energy conservation (as light traversesdifferent media) is insured by Eq. (2.8a).

We note that in this treatment of quantum optics ener-

gy is conserved while momentum is not conserved whenentering a dielectric, a problem that exists also in the con-ventional treatment of classical optics. This apparentdiscrepancy arises from the phenomenological treatmentof the material response through the polarization (or sus-ceptibility) rather than through the explicit treatment ofthe transitions to virtual states undergone by the atomswhen the dielectric is traversed by a light beam. Thus,processes associated with these transitions, such as therecoil of the atoms and the resulting Doppler shift of thefield, are neglected. However, we shall retain this level ofapproximation in our formalism, since its assumptions arecompatible with the times scales and experimental condi-tions of quantum optics.

III. QUANTUM PROPAGATION IN FREE SPACE

Following any classic textbook, ' we can quantize the

electromagnetic field in a box large compared with experi-


T

A(z, t) =c gVcuj

' 1/2iru t ik—z+iP

aJe

where a J,aJ are the creation, annihilation operators for aphoton in the jth mode of wave vector kJ (withk J= —k~) and frequency co~=c

~kJ ~. To simplify no-

tation, in all equations we use %=1, and omit unit vectors(x, y, and z) since the directions of all vector operatorsare fixed. The a J,QJ operators follow Bose commutationrelations. It is convenient to rearrange Eq. (3.1a) in amanner that is familiar to solid-state physics, by groupingtogether the creation and the annihilation operators oftwo counterpropagating waves,

A(z, t)=c gVcoj

1/23Q) '3 ~ —3' t, —3k Z'a Je '+a Je ' 'e

(3.1b)

where we have chosen the initial phase /=0 for simplici-ty. In this paper we shall not treat P explicitly, sincelinear-propagation phenomena do not depend on the ini-tial phase of the field. Thus, in all our results, a nonzeroinitial phase can be treated simply by considering the tem-poral evolution of the relevant operators, and translatingthe time origin appropriately. The individual j com-ponents thus defined correspond to the complex "coeffi-cients" of the spatial Fourier expansion of A. The electricand magnetic field operators may then be obtained as

mental dimensions under periodic-boundary conditions.For simplicity, we look only at propagation along the +zaxis and consider only the set I j I of longitudinal modeshaving the lowest-order transverse structure (e.g. , TEMOOor plane waves), with the electric field polarized along x,and the magnetic field along y. The electromagnetic vec-tor potential operator 2 is usually written as

—3' 8+3k.zbJ ——aJe (3.3b)

We note that the individual components eJ and Aj of theelectric and magnetic fields are not Hermitian,

J —J

hj ——h

(3.4a)

(3.4b)

and thus do not constitute observable quantities. They aresimply a convenience to facilitate bookkeeping when per-forming a spatial integration. The electric field of the jthmode on the other hand, is an operator of the form

1/22&Mj

VE = —iJ (b, b~)—, (3.5)

which is Hermitian, and thus constitutes an observable.Similarly, this applies for the magnetic field operator Hj.

Another way of rearranging the electric and magneticfield operators is to distinguish a positive- and anegative-frequency part. The positive-frequency part ofthe electric field operator corresponds to the sum of allthe annihilation-operator terms in Eq. (3.2a), while thenegative-frequency part corresponds to the sum of all thecreation-operator terms.

The implicit spatial and temporal dependence of theb J,bJ operators constitutes the propagation of the elec-tromagnetic field Thi.s propagation (spatial progressionand temporal evolution) can be recalculated quantummechanically by evaluating the tota1 momentum and ener-gy (Hamiltonian) operators for the electromagnetic field,and setting up the corresponding differential equations.The calculation of these operators can be done by usingEqs. (3.2), which permit us to convert all classical quanti-ties associated with the electromagnetic field to the corre-sponding quantum-mechanical operators, simply by re-placing the electric and magnetic fields in a classical ex-pression by Eqs. (3.2).

Thus, the energy-density operator can be written as

u = (E +H )= g(e~et+h~hi) .8~ 8~ j,

(3.6)E(z, t) = ———A = g eJc Bt

' 1/2

(b J bJ ) (3.2—a)i—and

H(z t)= A = gh,J

1/2

V(b J +b

~).(3.2b).= g —lS)

Ju= gu, ,

J(3.7a)

Throughout the paper, the caret distinguishes a Hilbert-space operator from the corresponding classical quantity.In Eqs. (3.2) s& ——+1 is the sign of j, and the t and zdependence is understood to be implicit in the b J,bJoperators. That is,

where

uJ = (eJe J+hih ~)8~

This operator can be used in the calculation of the totalmomentum of the field, through the quantum-mechanicalequivalent of Eq. (2.13b). However, since this latter equa-tion results from integration over space of the spatialderivative of u, the cross terms with l& —j may be elim-inated in Eq. (3.6). The reason is that these terms are sub-

i(k +k()zject to spatial oscillations of the form e ' ', and thuscancel out upon spatial integration, while the terms withI = —j which do not oscillate, survive. Equation (3.6) canthen be written as

3' t —3k.Zb J=Q Je (3.3a) (b j~b)+b ~ b~ +1) . . . (3.7b)

4666 I. ABRAM 35

Thus, the energy-density operator is given by

The total momentum G then is

(3.7c)to give

b)(t) =b)e

(3.15a)

(3.15b)

G= —(u+ —u )= gk)b&bj .C

(3.8)

Gg= i-az

' (3.9)

while the spatial derivative of any operator Q is given bya Heisenberg-like equation involving the momentum,

= —i[G,Q] .az

(3.10)

Equation (3.10) is a differential equation that permits thecalculation of the spatial dependence of the field opera-tors, once the momentum operator is known. It thus givesthe spatial progression of an electromagnetic wave, justlike the ordinary Heisenberg equation gives its temporalevolution.

For free-space propagation of plane waves, the momen-tum operator of Eq. (3.8) gives the differential equation(e.g. , for the operator b) ),

i [G,b ]—=ik)b& . (3.1 1)a,

' =-

This is the operator that describes the spatial characteris-tics of the electromagnetic field: As is known from ele-mentary quantum mechanics, the momentum operatorgives the spatial derivative of any wave function to whichit is applied,

This concludes the quantum-mechanical calculation oflight propagation in free space. The results are, of course,trivial, as they simply reproduce the spatial progressionand temporal evolution given in the definition of the bJoperators [Eq. (3.3)]. They demonstrate, however, theproperties of the different quantum-mechanical operatorspertaining to the electromagnetic field that are used inthis paper.

IV. THE QUANTUM-MECHANICAL MAXWELLEQUATIONS

The individual components of the rearranged electricand magnetic field operators defined in Eq. (3.2),

1/22&COj

ej= —lV

(b) b )»—- (4.1a)

Aj = —1SJ.

]. /2

(b, +b, ), (4. lb)

have a structure similar to that of the momentum and po-sition operators (respectively) for the harmonic oscillator,however, with a very important difference: Eqs. (4.1) arelinear combinations of a creation and an annihilationoperator of opposite indices (+j). This gives rise to amodified operator algebra (with respect to that of the har-monic oscillator) in that it is the e) and h) operators ofopposite index that give a nonzero commutator,

Integration of Eq. (3.11) gives the spatial progression ofthe annihilation operator for the jth plane wave as [e, ,e, ]=[h, , h, ]=0, (4.2a)

ik.zb)(z) =b)e (3.12)

[e),ht] =s (4.2b)

S= EH= g e)h4~ 4~ J

giving

(3.13a)

where the phase at z=0 is chosen as /=0. The spatialprogression of all other operators may be calculated in asimilar fashion.

Similarly, we may calculate the Poynting vector opera-tor as

where 6J l is the Kronecker 6 symbol. The structure ofEqs. (4.1) implies that e) and hj follow equations ofmotion similar to those of the harmonic oscillator. Weshall show in this section that these harmonic-oscillator-like equations of motion are the quantum-mechanicalequivalent of the Maxwell equations (2.1a) and (2.1b),which for one-dimensional propagation through a linearisotropic medium can be written as

(3.13b)aa & aEBz c c3t

(4.3a)

The total energy of the free field inside the cavity ofquantization is thus

aE 1 aaBZ C Bt

(4.3b)

~=U= —(S+ —S )= +co~b)b, . .C

(3.14)

This operator corresponds to the Hamiltonian of the freefield, and describes the oscillatory temporal evolution ofeach mode through the Heisenberg equation, for example,

BeJ

at' =i[~,e)]=i ([S,e)]+—[S,—e)] ), (4.4a)

where the + ( —) indices of the commutators refer to the

The temporal derivatives of eJ and hj can be calculatedthrough the Hamiltonian (2.13a),


forward- (backward-) going waves as in the definition ofA . Evaluating the commutator of ej with the Poyntingvector operator S,

S= g e)h4~(4.4b)

we have

Be&=ice)s)I(e))+ (e—)) I =ice)s)h, (4.4c)

and similarly

Bhj = l co)s) I (h) )+ —(h) ) )= l co)s)8~

where, according to the definition (4.1),1/2

(4.4d)

(e) )+ = i— 5& if j&0 (4.5a)

27TCO)

V

1/2

b if j(0, (4.5b)

and

and so on, of (e)), (h))+, and (h)) . Equations (4.4)

show that ej and hz are each other's derivative, and aretime harmonic with frequency co~, both in free space andinside matter, since the Poynting vector operator has thesame form (4.4b) irrespective of the medium in whichlight is propagating.

The spatial derivatives are given through the momen-tum operator (2.13b). To evaluate them, we first calcula-

tion the commutators of ej and hj with the energy-density operator inside a dielectric

sJ.S) ~[u, e ]= g [(cele ~+bib ~),e)]= h) (4.6a)

operators were defined by solving the Maxwell equationsin free space. However, Eqs. (4.7) indicate that this con-clusion is much more general, and applies also to propaga-tion through matter. That is, although the operators e~

and h& are initially defined for the free field, their equa-tions of motion under the Hamiltonian and momentumoperators of the form (2.13) are valid also inside a material system, since they reproduce the Maxwell equations. Itis therefore possible to describe the temporal evolutionand spatial progression of the electromagnetic field insidea material within the framework of the quantum-mechanical equations of motion. Alternatively, theMaxwell equations inside the material may be solvedquantum mechanically through the same algebraic tech-niques of second quantization that permit the solution ofthe equations of motion for Bose operators.

JTo this end we note that the commutators of e and h ~

Jwith the energy-density operator (4.6) have the same formas the equations of motion for the momentum and posi-tion operators under the Hamiltonian of a harmonic oscil-lator with a change in its mass, from m to m/e. Thesolution to this problem is well known: Diagonalizationof the Hamiltonian yields the renormalization of the oscil-lation frequency due to the mass change. The samemethod can be applied to the solution of the Maxwellequations (4.7). Diagonalization of the energy-densityoperator should give the renormalization of the wave vec-tor due to refraction.

V. REFRACTION AND REFLECTION

We consider the free-space electromagnetic field quan-tized inside a large volume under periodic-boundary con-ditions. If the cavity of quantization is filled with a lineardielectric material, the electromagnetic energy-densityoperator is given by Eqs. (2.3), (2.2), and (3.2) as

[u, h). ]= ee) .V

The spatial derivatives are then given by

(4.6b) u = (E +H +4~XE )8m.

8~ g (e)e )+h)h )+ vrXe)e ))

and

i [G,e ]= —is) I (h—) )+ —(h) )Bz '' 'c

= ———h ~cBt'

Bh~ Cc)J.Ei [G,h) ]=— is) I (e) )+ —(—e) )

(4.7a)

where, as discussed in Sec. III, the cross terms are elim-inated when dealing with the field inside a large volume.Substituting e~ and h~ in terms of the free-field Boseoperators, the jth component of the energy-density opera-tor can be written as

u) = Ib)tb)+b )b ) 2vrX(b) b)—)(b ) b)—) I . —

(5.2)E' B

ec Bt ' (4.7b)

Clearly, Eqs. (4.7) have a form identical to that of theclassical Maxwell equations (4.3), indicating that theMaxwell equations are implicit in the structure of theoperators (4.1), and correspond simply to their equationsof motion. This conclusion is, of course, trivial when ap-plied to the free field, since the electric and magnetic field

Thus, inside a dielectric, the energy-density operator u isnot diagonal when expressed in terms of the free-fieldoperators. This implies that the free-field states are notmomentum eigenstates in the dielectric, since the linearpolarization introduces a coupling between the free-field+j waves.

The energy-density operator u may be diagonalizedthrough a Bogolyubov transformation which permits us to

4668 I. ABRAM 35

express Eq. (5.2) in terms of refracted-wave operators.This transformation consists of the application of a uni-tary operator of the form

(5.3a) E(z, t)= g i—

electric and magnetic field operators inside the dielectriccan be obtained by inserting Eqs. (5.5) into Eqs. (3.2) as

i /2

(8 J—8,) (5.10a)

nV

where y is a real number, while R is an anti-Hermitianoperator of the form

R = g(b&b) bib—+~ )= .g (B&BJ . Bt8—' )) .J J

H(z, t) = g —is~J

1/22&n &j

V(8,+8, ) . (5.10b)

(5.3b)

1t relates the refracted-wave creation (annihilation) opera-

tors 8 /(B. ) to the corresponding free-field operatorsJ J

through

Similarly, the Poynting vector operator inside the dielec-tric is given by

(5.1 1)

and

8 t=e r~b e&'~=(coshy)b J—(sinhy')b

8 =e r~b er~=(coshy)bj —(sinhy)bJ J

b . =er 8 e r =(coshy)B~+(sinhy)B

bz er Bze r =——(coshy)B~+(sinhy)B

(5.4a)

(5.4b)

(5.5a)

(5.5b)

so that the Harniltonian can be calculated through Eq.(2.13a) as

cW= g co)B ~8~.J

(5.12)

B)(t)=8)e (5.13)

Thus, the temporal evolution of the refracted wave isgiven by

y = —ln(1+4m. Y) = —, inn1 1

4(5.6)

to obtain

2V(5.7)

where n is the refractive index. The eigenvalues of theenergy-density inside the dielectric are thus increased by afactor of n with respect to the corresponding values infree space, as in the classical case (2.15d). The momen-tum operator is then given by

6= (u+ —u )= QKJB j~BJC

(5.8)

with KJ =nkJ. The spatial progression of the refractedwave is given by Eqs. (3.10) and (5.8) as

iK.zBi(z) =BJe (5.9)

as expected from classical considerations. We note that inthis formalism the exact form of the refractive index as asquare root arises from the nonperturbative treatment ofthe antiresonant terms in Eq. (5.2). These terms have theform b Jb J+bJb J, oscillate rapidly at a frequency of2coJ. , and do not conserve energy to first order. However,if these terms are neglected, the refractive index becomesn =1+2vrX [by inspection of Eq. (5.2)], which is the re-sult of first-order perturbation theory, and is valid forsmall g.

The unitary operator (5.3) that diagonalizes the energydensity relates all refracted-wave operators to their free-field equivalents, through Eqs. (5.4) and (5.5). Thus, the

Substituting Eqs. (5.5) into Eq. (5.2) we can eliminate theoff-diagonal terms (the cross products of + j and —joperators) for

Equations (5.8) and (5.12) show that both the Hamiltonianand the momentum operator are diagonal in therefracted-wave basis set. The refracted-wave Hamiltonianretains the same eigenvalues as the free-field Hamiltonian,whereas the eigenvalues of the refracted-wave momentumoperator are renormalized by a factor of n with respect tothe corresponding free-field eigenvalues. This is, ofcourse, in accord with the well-known result of classicaloptics, that inside a linear dielectric the frequency of atime-harmonic electromagnetic wave retains its free-spacevalue, whereas its wave vector is changed by the refractiveindex. Thus, Eqs. (5.8) and (5.12) describe completely thepropagation of the electromagnetic field inside a dielec-tric, yielding the results expected from classical optics.

One of the basic problems of classical optics is thetreatment of a vacuum-dielectric interface. This problemhas been addressed through the conventional formulationof quantum optics, by first redefining the radiationmodes so that the new modes consist each of three seg-ments, representing the incident, reflected, and transmit-ted (possibly evanescent) waves; the new modes are thenquantized in a manner analogous to that of the free-spacemodes. However, the mode redefinition requires the ful/classical solution of the wave equation at the interface, im-plying that the spatial progression of the light waves in-side the dielectric is still addressed within a classicalframework. We shall show that the relationship betweenthe free-field and refracted-wave operators permits arigorous quantum-mechanical treatment of the vacuum-dielectric interface. We note, however, that since in thispaper we consider only one-dimensional propagation (andconsequently normal incidence on the interface) we do notget evanescent wave solutions except, of course, when therefractive index is purely imaginary as in the polaritonproblem.

We now consider that the volume of quantization of the


electromagnetic field consists of two half-spaces separatedby a plane interface at z=O, perpendicular to the directionof propagation of the plane-wave modes of the field. The—z half-space is empty (free space) so that the free-fieldenergy and momentum operators (3.8 and 14) are applic-able to it, while the + z half-space is filled with a lineardielectric, so that the refracted-wave energy and momen-tum operators describe propagation in it. We consider aforward-going wave packet of finite extent, so that it is in-itially totally within the empty half-space, and can thus bedescribed completely in terms of free-field operators.

If the interface is such that the refractive index variesgradually between 1 and n, so that reflections are elim-inated, then the forward-going wave packet enters thedielectric with no loss. The smooth variation of the re-fractive index can be formalized as an "adiabatic switch-ing" of the interaction term proportional to X in Eqs. (5.1)and (5.2), so that when the wave packet enters the dielec-tric all free-field eigenstates in the —z half-space go overto the corresponding refracted-wave eigenstates in the+ z half-space. Equivalently, the free-field creation (an-

nihilation) operators for each mode go over to the corre-sponding refracted-wave operators. We may thus consid-er the behavior of each mode across the interface separate-ly. In particular we examine the case in which each modein the empty half-space is in a coherent state (in the P =0quadrature). To simplify notation in what follows we ex-amine only the +j modes. When the wave packet entersthe refractive half-space, it will still consist of a superpo-sition of modes excited to a coherent state,

ET (——a iE ia)=2a1/2

27TCOJ

nV(5.15a)

and in free space as

E,=(a~

E~

a) =2a1/2

27TEOJ.

V(5.15b)

where the subscripts I and T refer to the incident andtransmitted fields. Equations (5.15) imply that

ET=EI/v n (5.16a)

as in the classical equations (2.15b). Similarly, for themagnetic field Eq. (5.14) gives

HT =H, v'n (5.16b)

identical to the classical Eqs. (2.15c). It is easy to verifythat the mean number of photons in the mode, the energy,and the energy flux do not change, while the energy densi-ty and momentum are multiplied by the refractive index.

Ordinarily, a sharp vacuum-dielectric interface entailsan abrupt discontinuity in refractive index from 1 to nacross the interface. When the wave packet reaches theinterface, its jth component is reflected into the —j free-field wave and/or is transmitted into the +j refractedwave. The behavior of the wave packet due to the suddenchange in the momentum operator at z=O can be treatedwithin the "sudden approximation": The free-field wavesget projected onto the refracted waves. We may calculatethis projection through Eqs. (5.4), which can be rewrittenas

(5.14) b = B.+ b2 n ~p n 1

n+1 ' n+1 (5.17a)

where i0) is the free-space vacuum in the —z half-space,and the refracted-wave vacuum in the + z half-space.Such states have quasiclassical properties in that theypresent a nonzero expectation value for the electric andmagnetic field operators; in particular, in the refractivemedium their values may be obtained through Eqs. (5.10)as

b= B+ bn ~ n —1

n+1 ' n+1 (5.17b)

where we have used the explicit expressions for coshy andsinhy through Eq. (5.6). When a coherent wave packetreaches the interface, it gets split into a transmitted and areflected coherent wave packet,

—ia(bi+b ). . . ia([(2V n—)/(n+1)](B +B )+[(n —1)/(n+1)](b .+b )]J 1 J J 0) (5.18)

Clearly, the corresponding expectation values of the elec-tric field satisfy

T 2n —1 NI,n+ I

(5.20b)

2E EIn+1n —1

n+1

(5.19a)

(5.19b)

4nNT —— 2N

(n +1) (5.20a)

(where the subscript R denotes the reflected wave), whilethe mean photon numbers (or energies) verify

giving transmission and reflection coefficients for thefield and for the energy identical to the correspondingclassical coefficients (2.14).

Equations (5.17) imply that the negative-frequency partof the incident electric field wave projects onto thenegative-frequency part of the transmitted wave and ontothe positiUe-frequency part of the reflected wave. Conse-quently, as in Eq. (5.18), both the negative- and thepositive-frequency parts of the electric field operator arenecessary to describe reflection within this formalism,

4670 I. ABRAM 35

even though the positive-frequency part (which corre-sponds to the annihilation operators) may give zero con-tribution to the description of the incident wave in termsof photons.

A single photon, that is, a wave packet created by a sin-gle excitation of the electric field (e.g. , dipole emission ofone quantum) in the jth free-field wave,

(bj+b j) ~0) =b j ~

0),will give at the interface

(5.21a)

(Bj+Bj)+ (b —j+b —j)n+1 ' ' n+1

Bj. ~o)~ b j ~0) (5.21b)n+1 ' n+1that is, it will be transmitted with a probability given bythe classical energy-transmission coefficient, or will be re-flected with a probability given by the classical energy-reflection coefficient.

Similarly, a wave packet created by a double excitationof the electric field (e.g. , formally by a term quadratic inthe electric field),

l Nj ) retr=e 1Nj ) tree

~Nj )tree=e

l Nj )retr i

(S.23a)

(5.23b)

frequency part of the field is considered, while both in theclassical and in the quantum versions of the Maxwellequations, propagation and boundary conditions are for-mulated for both the positive and the negative parts of thefields simultaneously.

Before closing this section we shall point out the closerelationship that exists between the classical phenomenonof refraction, and the recent considerations on quantummechanical "squeezing, " that is, on the creation of statesof the electromagnetic field which exhibit a quantum-mechanical noise level (in one quadrature) below that ofthe vacuum fluctuations. The unitary transformation(5.3) that diagonalizes the energy-density operator inside arefractive material is precisely the "two-mode squeezeoperator" introduced for describing a class of two-photoncoherent states that exhibit nondegenerate squeezing. "This transformation relates free-space waves to refractedwaves through Eqs. (5.4) and (5.5), and this means thatfree-space number eigenstates (photons) are related to re-fracted wave eigenstates through

(b +b )~

0) =b,'b,'~

0)+ ~0),may split in three different ways at the interface,

. 2

(n+1) n+1

(5.22a)

(S.22b)

where N~ is a positive integer or zero. In other words, re-fracted photon states are two-mode squeezed states of thecorresponding free-space photons, and vice versa. In par-ticular, the refracted-wave vacuum (Nj =0), when ex-pressed in terms of free-space eigenstates, consists of a su-perposition of all two-mode number eigenstates whereboth modes are equally excited,

that is, both photons may be reflected or both transmitted,or one photon may be reflected and one transmitted, eachprocess occurring with the probability of the correspond-ing product of the classical reflection and transmissioncoefficients. A more thorough examination of two-photon (nonlinear) processes will be given elsewhere.

These results demonstrate that the diagonalization ofthe energy-density operator for a linear medium isequivalent to the solution of the quantum-mechanicalMaxwell equations, and permits a quantum-mechanicaldescription of linear propagative phenomena, reproducingthe results of classical optics. In particular, our formal-ism gives a quantum-mechanical theory of the refractiveindex as the parameter that renormalizes the momentumeigenvalues in the medium and describes the spatial pro-gression of the refracted waves. The quantum-mechanicaldescription of propagation through the momentum opera-tor permits us also to address the problem of a mediuminterface through the standard techniques of quantum-perturbation theory, and gives a successful quantum-mechanical description of reflection and transmission ofthe electromagnetic field at an interface. Within thisquantum-mechanical formalism, the entity reflected ortransmitted at an interface is not the photon, but rather theelectric and magnetic fields that create it, as in classicaloptics. This seemingly paradoxical result is due to thefact that in the conventional picture of the photon as aneigenstate of the number operator, only the negative-

g tanh y ~

Nj. ,N j)f„„,cosh/ ~ o

(5.24)

and

ai, ——— (b j b,)—J

a~i —— (b j+b j),v'2

(5.25a)

(5.25b)

and has thus a structure identical to that of the output ofan ideal two-photon device. However, Eq. (5.24) has nophysical significance in the time scales of quantum optics,since the free-field photons that appear in it are producedby energy nonconserving terms, and constitute thereforevirtual states. In other words, inside a dielectric there isno experiment that can detect free-space photons and thusthere is no way to investigate the consequences of Eqs.(5.23) and (5.24) regarding squeezing. To clarify thispoint, we look at the quantum noise characteristics of therefracted-wave vacuum as measured in a hypothetical ex-periment that mixes the + j and —j modes with a localoscillator on the surface of a photodetector in the dielec-tric (notwithstanding the geometrical restrictions of one-dimensional propagation) and detects the two quadraturesof the resulting photocurrent. If free-space photons weredetectable inside the dielectric, then the two-modequadrature-phase amplitude operators relevant to such anexperiment would be


whose variances for the refracted-wave vacuum are

1rerr&0

I~rztj

I0&ret'r=

2fl(5.26a)

rerr&0I~~2j

I0&ret'r= & 22

(5.26b)

where n is the refractive index. Similar variances wouldbe obtained for refracted coherent states, indicating that ifthe free-field could be measured inside a dielectric thenoise that is in phase with the signal would exhibit, fluc-tuations below those of free-space vacuum (b, = —, ), whilethe noise level in quadrature with the signal would behigher than that of the vacuum fluctuations. However,inside a dielectric we can only detect refracted waveswhose electric field operators are given by Eq. (5.10).Since a photodetector is a photon number or energymeasuring device, Eq. (2.14f) implies that the relevantquadrature-phase amplitude operators inside a refractivemedium are

(5.27a)

(5.27b)

VI. CONCLUSIONS

The conventional theory of quantum optics rests on aset of basic assumptions, such as the following.

(1) The field is quantized as a set of quasidiscretemodes inside an empty (vacuum) cavity with periodicboundary conditions.

(2) The total energy of the field (its Hamiltonian) isgiven by the integral of the electromagnetic energy densityover the physical volume of the cavity in assumption (1).

(3) The dynamical behavior of the field may be calcu-lated by applying the Hamiltonian obtained in (2) to theproper superposition of modes, corresponding to the ini-tial conditions of a given problem.

These assumptions permit a successful treatment of theelectromagnetic field in vacuum, as it interacts with ma-terial points representing emitters, absorbers, or detectors.At the same time, however, these same assumptions are atthe root of the shortcomings of conventional quantum op-tics. In particular, it is impossible to treat directly thespatial progression of a wave under these assumptions,since the Hamiltonian formalism in (3) can only deal withthe time dimension. The calculation of spatial progres-sion is avoided by fixing the spatial characteristics of themodes defined in (1) to the proper form, and then calcu-lating only the temporal evolution of a wave packet com-posed of these modes, traveling like a train on predeter-mined tracks. This is in contrast to the treatment of

which give identical variances for both quadratures,

"r.&0I ~&lj I0&refr refr&0I ~&2, I0&refr 2

meaning that squeezing due to refraction is not observ-able.

propagation in classical optics, where spatial differentialequations can be set up to calculate explicitly the spatialprogression of the field, including the determination of itsmode structure. The difficulties associated with spatialcalculations in quantum optics are even more pronouncedwhen examining propagation through bulk matter, sincein that case its basic assumptions break down. In particu-lar, there is no prescription on how the periodic-boundaryconditions in (1) get modified upon introduction of amedium, while at the same time assumption (2) gives achange in total energy of a traveling wave when a polariz-able material is present, a result that is at variance withthe actual behavior of the electromagnetic field.

These difficulties may be overcome by qualifying or re-formulating these assumptions, and thus extending thescope of quantum optics to include the description of pro-pagative phenomena inside a medium. The new assump-tions are the following.

(1') The periodic-boundary cavity constitutes an openvolume permitting flow through its boundaries, and is adevice for dealing with traveling waves in infinite space(where the modes are standing waves vanishing at infini-ty). We may postulate that, upon introduction of a linearmedium in infinite space, the dimensions of the periodic-boundary cavity get modified so that the number of wave-lengths it contains remains unchanged. This insures thatperiodic-boundary conditions still hold, and establishesthe right correspondence between the free-space andrefracted-wave modes of infinite space.

(2') Since quantization is done in a "flow cavity, " thetotal energy of the field corresponds to the energy flux.The same holds for the total momentum of the field, themomentum flux being given by the energy density.

(3') The temporal evolution of the field is given by theHamiltonian (2 ), while its spatial progression can be cal-culated through the momentum operator given in (2').This permits us to treat on the same footing both the tem-poral and the spatial coordinates of an electromagneticwave.

The simultaneous use of the Hamiltonian and themomentum operators defined according to (2') yieldsspatial-temporal equations of motion for the electric andmagnetic fields, having a form identical to that of theclassical Maxwell equations. This confirms the validity ofassumptions (1')—(3') and implies that the formalismbased on them is not subject to the limitations of conven-tional quantum optics, but can also treat medium-dependent propagative phenomena quantum mechanical-ly. Thus its use should permit a rigorous description ofquantum propagative phenomena, as for example spon-taneously initiated (noise-driven) stimulated processes,which are usually treated through classical propagationtheory. In classical theory initiation is produced by a sta-tistical noise field, which in essence simulates quantum-mechanical uncertainty.

For propagation through a linearly polarizable medium,the quantum-mechanical Maxwell equations can be solvedexactly through second quantization operatorial tech-niques. In particular, diagonalization of the energy densi-ty (or momentum) operator inside a refractive mediumgives directly the refractive index, the renormalization of

4672 I. ABRAM 35

the wave vectors (i.e., the momentum eigenvalues) and thecorresponding change in the spatial structure of the fieldmodes. This treatment can readily describe all refractivephenomena, such as reflection and transmission at an in-terface, as well as the propagation (i.e., spatial progressionand temporal evolution) of an electromagnetic wave in arefractive medium. In the diagonalization of the momen-tum (energy-density) operator, all the terms resulting fromthe different products of positive- and negative-frequencyparts of the field have to be taken into account to describerefraction: In the expansion of the refractive index as apower series of the susceptibility, only the first-order termis due to the energy-conserving (nonoscillating) fieldterms. A11 higher-order terms result from rapidly oscillat-ing, energy-nonconserving products of positive- or ofnegative-frequency-field terms.

To obtain a correct description of propagative phenom-ena, the contributions of both the positive- and the

negative-frequency parts of the electromagnetic field haveto be treated on the same footing, even if one of the twoparts may be redundant in some situations. In particular,in the description of the field in terms of photons, thepositive-frequency part is usually dropped, since it corre-sponds to negative-energy photons, or equivalently to pho-ton annihilation in the vacuum. However, in the descrip-tion of reflection and transmission of the field at an inter-face, if the positive-frequency part of the incident wave isneglected, the reflection of photons from a dielectric sur-face cannot be described.

The fidelity with which this quantum-mechanical treat-ment of linear propagation reproduces all the resultsknown from the classical treatment of refraction demon-strates the validity of our approach, based on the momen-tum operator. In the next paper in this series we shall usethe momentum-operator approach to treat propagationthrough a nonlinearly polarizable medium.

tR. Loudon, The Quantum Theory of Light, (Oxford UniversityPress, New York, 1983); W. H. Louisel, Quantum StatisticalProperties of Radiation (Wiley, New York, 1973).

2This viewpoint is used, for example, in C. K. Hong and L.Mandel, Phys. Rev. A 31, 2409 (1985).

J. M. Jauch and K. M. Watson, Phys. Rev. 74, 950 (1948).~Y. R. Shen, Phys. Rev. 155, 921 (1967); in Quantum Optics,

edited by R. J. Glauber (Academic, New York, 1969), p. 473.5C. K. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).S. A. Akhmanov, A. P. Sukhurukov, and A. S. Chirkin, Zh.

Eksp. Teor. Fiz. 55, 1430 (1968) [Sov. Phys. —JETP 28, 748(1969)].

7D. Stoler, Phys. Rev. D 1, 3217 (1980); 4, 1925 (1971); H. P.Yuen, Phys. Rev. A 13, 2226 (1976).

H. P. Yuen and J. Shapiro, IEEE Trans. Info. Theory IT-24,657 (1978); IT-26, 78 (1980); H. P. Yuen, J. Shapiro, and J. A.Machado Mata, ibid. IT-25, 179 (1979).

I. Abram and R. Raj (unpublished).J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, NewYork, 1975), Sec. 6.9; J. A. Stratton, Electromagnetic Theory(McGraw-Hill, New York, 1941); R. Becker, ElectromagneticFields and Interactions (Dover, New York, 1982).

"C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068(1985); 31, 3093 (1985).

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Quantum theory of light propagation: Linear medium