VI. Quantum optics. Quantization of the electromagnetic field. 2 = + K x (VI-1)

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1 VI. Quatum optics Quatizatio of the electromagetic field Historically it was foud that ay attempt of a classical descriptio of the motio electros i atoms failed. Oly quatum mechaics succeeded i describig pheomea as atomic spectra, electro diffractio, electrical coductio i crystals ad of cetral importace for photoics the iteractio of light with matter, icludig the operatio of lasers. I quatum mechaics operators replace the familiar variables of classical theory ad the state of the system is replaced by state vectors. This geeral procedure is ot limited to ay particular system, but accordig to our curret isight must be applied to all classical variables to provide the most correct ad most accurate descriptio of ature amog all curretly available physical theories. Accordig to this geeral procedure all observable quatities of physics icludig field variables describig wave pheomea must be accouted for by operators i the same way as we replaced coordiates, mometa or the eergy of a mechaical system by operators i the previous chapter. The quatizatio of fields, that is the descriptio of field variables by operators leads to quatum field theory, which i case of electromagetic fields is referred to as quatum electrodyamics. The applicatio of the laws of quatum electrodyamics to optical fields ad their iteractio with matter has bee termed quatum optics. Beyod the cosistet extesio of the laws of quatum physics from mechaical systems to field variables (last setece of the previous paragraph quatum optics does ot require ay ew postulates. Field quatizatio is also eforced by experimetal evidece. The spectrum of blacbody radiatio, the temporal evolutio of spotaeous emissio as well as the oise characteristics of laser radiatio could oly be explaied i the framewor of quatum optics. Quatum theory of the harmoic oscillator The modes of electromagetic radiatio i waveguides as well as i free space ca be treated as harmoic oscillators, therefore we address the quatum mechaical descriptio of a harmoic mechaical oscillator before proceedig to field quatizatio. The simplest harmoic oscillator is a mass attached to a sprig, which provides a restorig force Kx proportioal to the displacemet x of the mass from its equilibrium positio (Fig. VI-. Fig. VI- The Hamiltoia of this mechaical harmoic oscillator is give by H p = + K x (VI- m - 7 -

2 which with Hamilto s equatios of motio (V- ad (V- leads to the Newto equatio for the classical oscillator d x m = Kx (VI- dt with the well-ow solutio x = A si( ω t+ϕ (VI-3 where A is the amplitude, φ is the phase ad K ω= (VI-4 m is the (agular frequecy of the oscillatio. Expressig K i terms of m ad ω, we ca rewrite the Hamiltoia of the harmoic oscillator as H = (p + m ω x (VI-5 m The usual way of describig the harmoic oscillator quatum mechaically is to use the Schrödiger represetatio: replace p by ih / x i (VI-5 ad for obtaiig the eergy eigevalues of the oscillator solve h ψ m x E + ω ψ = m x ψ (VI-6 Secod quatizatio, creatio ad aihilatio operators The solutio of (VI-6 yields ot oly the eergy eigevalues but also the eigefuctios ψ i terms of Hermite-Gaussia polyomials. However, we ca also treat the harmoic oscillator i a more abstract way without the use of ay particular quatum mechaical represetatio. The followig treatmet is borrowed from Dirac. First, we itroduce the ew operators a= ( mωx ip mhω + (VI-7a a = ( mωx ip mhω (VI-7b P. A. M. Dirac, The Priciples of Quatum Mechaics, 4 th ed. Oxford,

3 The operator â is the Hermitia adjoit to a, sice by defiitio p ad x are Hermitia operators. The coefficiets occurrig i Eqs. (VI-7 have bee chose to simplify certai relatios which follow. The commutatio relatio of â ad â [ aa, ] = aa a a= i ( px xp = h (VI-8 follows (exercise from that of Eqs. (VI-7 yields p ad x give by (V-46 ad will be itesively used i the followig treatmet. Ivertig h x = ( a a m ω + (VI-9a mhω p = i ( a a (VI-9b ad substitutig ito (VI-5 leads to H = hω ( a a+ aa = h ω ( a a+ (VI-0 ad (VI-8 implies that [ a, H] = hω a; [ a, H] = ωa h (VI-a,b The expressio of the Hamiltoia i terms of â ad a is ofte referred to as secod quatizatio i textboos. To obtai the eergy eigevalues ad eigevectors of the harmoic oscillator we start out from a particular eergy eigevector E ad eigevalue E, apply the commutator of a ad H to this eigevector ad utilize (VI-a ad H E = E E to obtai (exercise H ( a E = ( E h ω( a E (VI- If E is a eigevector of H with a eigevalue E the E = a E is also a eigevector with a eigevalue E = E hω. By repeated operatio with a o E, we fid that a E Ĥ E hω is a eigevector of with the eigevalue. For a sufficietly large value of, the eigevalue appears to become egative. But are egative eigevalues

4 possible? To aswer this questio, let us calculate the expectatio value of eergy i the eergy eigestate E by usig (VI-0 E H E E a a E E E E E E E = hω + = hω + = E (VI-3 It follows from the defiitio of the scalar product (V-8 that the scalar product of a state vector ψ with itself caot be egative. Ideed, by expressig the idetity operator (V-4 i terms of the complete orthoormal set Î = a a the scalar product ca be reexpressed as a as I a a a 0 (VI-4 ψψ ψ ψ = ψ ψ = ψ As a cosequece, the eigevalues of the Hamiltoia of the harmoic oscillator ca ot be egative. Hece the reductio of the eigevalue upo the applicatio of a to E must be termiated. This is oly possible if there exists oe eigevector with the property ae 0 = 0 (VI-5 Because (VI-5 implies that we ca geerate o further eigevectors (with eve lower eergy by applyig â sice a E0 = 0. The eigevalue of this lowest-eergy state, that is the groud state of the harmoic oscillator ca be obtaied from (VI-3 by taig E ' = E0, E = E 0 ad, as a cosequece of (VI-5, E = 0. The result is E 0 = hω (VI-6 The effect of applyig â to E ca be derived by usig the procedure that leads to (VI- H( a E = ( E + h ω( a E (VI-7 that is E = a E is also a eigevector with a eigevalue E = E + hω. By repeated operatio with a o E, we fid that a E is a eigevector of Ĥ with the eigevalue E + hω. Startig out from the groud state E0 we ca thus geerate ew eigevectors ( a E0 = E (VI-8 with eigevalues

5 E = ω ( + h (VI-9 Are these all possible eigevalues ad vectors, or are there others? Repeated applicatio of a to ay arbitrary E must lead us to E 0 otherwise we would ed up with egative eigevalues. The eigevalue of E 0 must be ½ hω accordig to (VI-3. No matter which eigevector we start out from, we always ed up with the same set of eigevalues, give by (VI- 9. Cosequetly, these eigevalues must be uique. May there be differet vectors E0 obeyig (VI-5? If yes, we would have the case of degeeracy sice all these eigevectors must possess the eigevalue ½ hω. I the groud state ad hece all other eergy eigestates would be degeerate, there would have to be other operators commutig with, but idepedet of H whose eigevalues would suit for labellig the set of degeerate eigevectors uiquely (as i the case of the electro movig i the Coulomb potetial of the proto i the hydroge atom the eigevalues of the agular mometum operator are used to label the degeerate eigestates of idetical eergy. I the lac of such a operator i the case of the curret problem we coclude that the eergy eigevectors (VI-8 ad eigevalues (VI-9 are uique. Eq. (VI-9 reveals that the miimum eergy of the harmoic oscillator, reached i its groud state E 0, is ½ hω, ad ca oly be icreased i discrete steps, by the eergy quatum h ω or its iteger multiple (Fig. VI-. The operator a is called a destructio or aihilatio operator sice it destroys oe quatum of eergy. Its Hermitia cojugate, â is called a creatio operator sice it creates a quatum of eergy. At the oscillatio frequecies of mechaical oscillators hω is hardly measurable, hece the mechaical oscillators ca for all practical purposes be well described by classical physics. However, at optical frequecies h ω exceeds the wor fuctio of specific solids so that the eergy quatum, referred to as a photo, becomes easily detectable by utilizig the photoeffect, as we shall see later. Fig. VI- Are the eigevectors (VI-8 orthogoal? To aswer this questio, we calculate the scalar product of to differet eigevectors defied by (VI-8. By maig use of the idetity m m- m- - a( a = ( a a+ ( a a ( a = a ( a a + a ( a (VI-0a,b which ca be derived by repeated applicatio of the commutatio relatio (VI-8 (exercise, we fid m m E E = E a ( a E = E a ( a E (VI- m By repeated applicatio of this rule we obtai

6 m! 0 0 = 0 if > E a E m! m Em E = E0 ( a E0 = 0 if m< ( m!! E0 E0 =! if m= (VI- where we assumed that the groud state E 0 is ormalized, E E0. From (VI- we may coclude that the eergy eigestates (VI-8 are ideed orthogoal as expected for the eigestates of a Hermitia operator ad with the ormalizatio 0 = E ( = a E 0 (VI-3! form a complete orthoormal set, which we will use throughout the rest of this chapter. Number operator, umber states With the help of (VI-0a we fid a E = E (VI-4 a E = + E + (VI-5 from which it immediately follows that a a E = E (VI-6 that is the operator a a couts the umber of eergy quata i the eergy eigestates ad is therefore referred to as the umber operator. Because the eergy of the system i state E cosists of quata, this state is also referred to as a umber state. The matrix elemets of the creatio ad destructio operators i the from (VI-4 ad (VI-5 (ad utilizig the orthoormality of E E represetatio ca be writte dow immediately am = Em a E = δ m, (VI-7a ad

7 a m Em a E m, + = = + δ (VI-7b Furthermore, the matrix represetatio of the mometum ad positio operators read as (, +, m hω p = E p E = i + δ δ m m m m (VI-8 ad (, +, h x = E x E = + δ + δ mω m m m m (VI-9 Ucertaity products The operators of the mometum ad positio do ot commute, hece these quatities ca ot be measured simultaeously accurately. The ucertaity product ΔpΔx for the eergy eigestates of the harmoic oscillator ca be calculated by usig (exercise ( ( mhω Δ p = E p p E = (+ ad ( = E ( x x E = h Δx ( mω + (VI-30 (VI-3 where we utilized that as a cosequece of (VI-8 ad (VI-9 the expectatio values p = E p E = 0 ad x E x E = 0. This leads to the ucertaity product = ΔpΔx = ( + h (VI-3 Accordig to the Heiseberg ucertaity priciple ΔpΔx h (VI-33 We see that the ucertaity product i the groud state of the harmoic oscillator reaches the smallest possible value allowed by Heiseberg s ucertaity priciple. The message (VI-3 coveys is that eve i its state of lowest eergy the particle does ot come to rest, which would imply Δp = 0 ad Δx = 0. Istead, it oscillates with a residual eergy ½ hω aroud the mea values p = 0 ad x =

8 The above treatmet shows the power of the abstract Dirac formulatio of quatum mechaics. Without the use of ay particular represetatio of the Hilbert space of the abstract state vectors, we have bee able to derive all results relevat for physical measuremets by merely usig the respective operators of the physical measurables ad their commutatio relatios. We have ow developed the formalism required for quatum optics. The quatizatio of the radiatio field i a resoator, defiitio of the photo Let us cofie a plae optical wave propagatig i the z directio betwee two plae x-y surfaces of perfect coductivity. Such a plae-wave resoator, strictly speaig, would have to be bouded by two plae-parallel mirrors of ifiite cross sectio. This is, of course, ot feasible. However, at optical frequecies, the wavelegth is may orders of magitude smaller tha a reasoable-sized mirror (say radius r cm. Deviatio of the eclosed resoator beam from a plae wave ca be quatified by the divergece agle θ λ / r, which for visible light (λ 0.5 μm amouts to θ 50 microradias (Fig. VI- 3. Over a propagatio legth of L m this causes a icrease of the beam radius by merely 0.5%. Hece the eigemodes of such a plae-mirror resoator ca be well approximated by plae waves of appropriate frequecies (eigefrequecies of the resoator thas to the short wavelegth of optical radiatio. For quatizatio purposes, the plae-wave approximatio is also applicable to stable, Gaussia-beam resoators, as log as the radial variatio of the field is egligible withi oe wavelegth: F 00 / r << / λ, which implies that the logitudial field compoets are egligible i Eqs. (IV-4 ad (IV-43. Fig. VI-3 Sice the electric field E( r, t must fulfil E ( z = 0, t = E( z = L, t = 0, the fields iside this plae-wave resoator of volume V=LA with its axis aliged alog the z directio ca be expaded as Er (, t = pl, ( t El, ( z l ε, 0 (VI-34 Br (, t = μ ω q ( t B ( z l, 0 l l, l, (VI-35 where the (stadig-wave field distributios of the l th resoator mode with its electric field polarized alog e are give by

9 El e l Bl e e z V V ad, ( z = si z ;, ( z = z cos l (VI-36 π = ; = x, y L l l (VI-37 with l beig a positive iteger ad e deotig the uit vector poitig alog the directio. Here μ = ε c is the 0 / magetic permeability of vacuum a ωl = l / ε0μ0. The modes are orthogoal, that is after proper ormalizatio obey 0 E V V 3 l, Em, γd r=δl, mδ,γ 3, Bm, γd r=δ, mδ,γ B l l (VI-38 Eqs. (VI-34-(VI-37 represet the ormal mode expasio of the resoator. Substitutig (VI-34 ad (VI-35 ito the first ad secod Maxwell s equatios, (IV- ad (IV-, we obtai (exercise p dq,, = l l (VI-39 dt dp ω = l dt, l ql, (VI-40 which implies d q dt l, +ω q l l, = 0 (VI-4 Eq. (VI-4 idetifies ωl as the oscillatio frequecy of the l th mode. The total electromagetic eergy stored i the cavity H E B V μ 0 3 field = ε 0 + d r (VI

10 which is equivalet to the Hamiltoia of the system, ca be expressed i terms of the dyamical variables q l, t ( by substitutig (VI-34 ad (VI-35 ito (VI-4 ad usig (VI-38 p l, t ( ad l, (,, Hfield = p +ω q l l l (VI-43 Compariso of (VI-4 with (VI-5 reveals that the electromagetic field i the resoator behaves mathematically lie a esemble of idepedet harmoic oscillators. The dyamical variables p l, ( t ad q l, ( t costitute the caoically cojugate mometum ad positio variables, which ca be verified by derivig from Hamilto s equatios of motio H H q& p p& q (VI-4 l, = = l, ;,, p l = = ωl l, q 4 l l, The same equatios which we previously obtaied from Maxwell s equatios for ( ad (. p l, t q l, t We ca thus proceed with the quatizatio precisely i the same maer as we did i the case of the harmoic oscillator, by defiig the creatio ad aihilatio operators ( l, l l, l, a = ω q i p (VI-45 hω l a l, l l, ( = ω q + i p l, hω l (VI-46 with the commutator relatios l, m, γ = l, m, γ = l, m, γ = δl, mδ, γ (VI-47 [ a, a ] 0 ; [ a, a ] 0 ; [ a, a ] Ivertig (V-45 ad (VI-46 yields (, hω p i a a l l, = l, l (VI-48 ad

11 q ( a a h = + l, l, l, ωl (VI-49 The operators of the electric ad magetic fields of the resoator i the plae-wave approximatio ca ow be obtaied by substitutig (VI-48 ad (VI-49 ito (VI-34,(VI-35 l, (,, hω E= ie a a si z l, l l l l ε0v 0 (,, ( z h B= e e l a l + a l coslz ε Vω l (VI-50 (VI-5 The summatio must be exteded to the trasverse mode idices if more tha oe trasverse mode is oscillatig. The Hamilto operator of the field stored i the cavity ca also be expressed i terms o f â ad â by substitutig (VI-48 ad (VI-49 ito (VI-43 H a a field = h ω l l, l, + (VI-5 l, If the resoator also cotais a atomic system (e.g. gai medium i a laser iteractig with the modes of the resoator, the total Hamiltoia of the system ca be writte as H total = Hfield + Helectro = Hfield + Ĥ0 + Hit (VI-53 he state vector of the atom-field system ca be expaded i terms of the eigestates of H H = H + H : T total it field 0 Φ = c jm φjm ; jm 3 jm, φ =,,,...,,... um (VI-54 i where, the idex i comprises all mode idices (icludig the logitudial mode idex l, the trasverse mode idices if apply ad the polarizatio idex, the idex j comprises the photo umbers i all modes ad u is the mth eigestate of the Hamiltoia of the atom i the absece of fields, Ĥ 0. Applicatio of the field ad (uperturbed atomic Hamiltoia to this state vector results i m - 8 -

12 H field φ jm = hω i i + φjm i ; H 0 φ = E φ (VI-55 jm m jm idicatig that the field eergy stored i the ith mode of the resoator is equal to h ω i ( i +/, i.e. to the zero-field eergy plus a iteger multiple of the elemetary excitatio of this resoator mode, which has bee referred to as a photo. The Hamiltoia (VI-53 together with (V-66, (V-67, ad (V-68 provides a full quatum descriptio of light-matter iteractio i a optical resoator. Travellig-wave quatizatio Radiatio may iteract with matter outside a resoator. I this case, the field ca ot be expressed i terms harmoic oscillators ad the previous approach does ot wor. Rather, we expad the electromagetic field i terms of plae waves, which mathematically costitutes a 3-dimesioal Fourier expasio (similar to the D expasio used i Fourier optics, see. Chapter III. As it is more coveiet to hadle series rather tha itegral represetatios, we itroduce the so-called box ormalizatio. I this cocept, space is limited to a arbitrary but usually large volume surroudig the regio of iterest, for example a atomic system ad we assume that the field outside is a periodic repetitio of the field iside the volume. The use of a square box with sides of equal legth 0 x L ; 0 y L ; 0 z L (VI-56 maes it easy to impose this periodic boudary coditio. The procedure outlied above is the prescriptio of a three dimesioal Fourier series expasio. Followig this procedure we ca expad the vector potetial A(r,t ito a Fourier series, the compoets of which are solutio of A( rt, A ( rt, = 0 (VI-57 c t ad ca be formally regarded as harmoic oscillators of discrete eigefrequecies, which are defied by the periodic boudary coditios. Eq. (VI-58 is obtaied here by usig the Coulomb gauge A( r, t = 0 (VI-58 ad assumig a time idepedet scalar potetial φ= 0 (VI-59 Uder these circumstaces (IV-4b simplifies to (VI-57 i the absece of free currets ad we have B = A; E= A (VI-60 t implyig trasverse radiatio fields (a assumptio also implicit i our procedure for resoator mode quatizatio. The quatizatio of geeral the electromagetic fields (i.e. those havig field compoets poitig alog the wave vector, termed - 8 -

13 logitudial field compoets should be doe i the Loretz gauge, which is a rather complicated procedure treated i a few boos o quatum electrodyamics. For most problems i quatum optics, it is a good approximatio to quatize oly the trasverse field compoets obeyig (VI-58 ad (VI-60 ad treat the time-idepedet scalar potetial ad its resultat electric field (such as that bidig electros to the ucleus as a classical, uquatized field icorporated i Ĥ 0 of (VI-53. Quatizatio of the total field maes this force appear as a result of a exchage of virtual logitudial photos betwee these particles ad gives rise to small (but well measurable effects such as e.g. the Lamb shift, which ca ot be described by our simplified treatmet. The formal aalogy of the terms of the Fourier-series expasio of A(r,t to the descriptio of harmoic oscillators allows agai the itroductio of the creatio ad aihilatio operators, i terms of which the operator of the vector potetial of a trasverse radiatio field ca be expaded as 3 h (,, ε0v ω ir A= e a e a, e, where ir + (VI-6 π = ( exx + eyy + ezz; ω = c ;, L with x, y ad z beig itegers. The Hamiltoia taes the usual form,,, = (VI-6 H a a = hω + (VI-63 The creatio ad aihilatio operators commute accordig to (VI-45. From Eq. (VI-6 we ca derive the electric field ad magetic field operators by usig (VI-60: E = A= e +, h da ir da, ir, e e (VI-64 t, ε0v ω dt dt B = = i h ( i,,, a e r ε0v ω ir A e a, e (VI-65 The temporal derivatives of the creatio ad aihilatio operators ca be calculated by applyig the operator equatio of motio (V-45 ad usig the commutator relatios (VI- (exercise,, da = [ a, H ] = iω a dt ih, (VI-66 A. I. Ahiezer ad V. B. Berestetsii, Quatum Electrodyamics (Itersciece Publishers, 965 W. Heitler, The Quatum Theory of Radiatio 3 rd Ed. (Oxford, D. Marcuse, Priciples of Quatum Electroics, Academic Press, Ic.,

14 da, = [ a,, H] = iω a, dt ih The substitutio of these expressios ito (VI-64 yields hω ( ir ir,, a, e (VI-67 ε0v E = i e a e Equatios (VI-65 ad (VI-67 specify the field operators i the Schrödiger picture, where the state vectors evolve ad the operators are froze i time. I quatum optics, it is ofte more coveiet to wor i the Heiseberg picture, where the state vectors are idepedet of time, but the operators evolve accordig to (VI-66. Eqs. (VI-66 ca be readily itegrated to yield i the Heiseberg picture i ( a t = a e ω,, i a ( t a e ω, =, t t (VI-68 Substitutio of these time depedet operators ito (VI-64 ad (VI-65 leads to the field operators i the Heiseberg picture E ( t = e a e + a e, π π hω i( r ω ( t+ i r ω t+,,, ε0v (VI-69 π π h i( r ω + ( ω + t i r t B ( t = e +, a,, e a e, ε0v ω 70 (VI- Spotaeous atomic trasitios I the framewor of the semiclassical theory of light-matter iteractios we have bee able to calculate the trasitio rate of atomic trasitios iduced by the field. However the rate of spotaeous trasitios had to be itroduced pheomeologically. With the field quatized, the rate of spotaeous emissio ca ow be readily calculated. Let us assume that a plae wave propagatig alog with a polarizatio iteracts with atoms prepared i a excited state u with eergy E resoatly to iduce a trasitio to a lower state u with eergy E such that E = ω hω. E h

15 U Fig. VI-4 As Fermi s golde rule for the trasitio rate π W H E h iitial fial = fi δ( fial E iitial (VI-7 dictates eergy coservatio, the mode (, must gai a quatum upo the atomic trasitio, i.e. the iitial ad fial states of the field-atom system are, as depicted i Fig. (VI-4: φ = u ; φ = + u i, f, (VI-7 The trasitio rate is drive by the iteractio Hamiltoia, which ca be writte as i t i t Hit H e ω ω = + ( H e (VI-73 H fi i Fermi s golde rule stads for the matrix elemet of Ĥ or ( H for ad upward or dowward trasitio, respectively (as it is apparet from Eq. V-0 i the derivatio of the gold e rule. The iteractio Hamiltoia describig the iteractio betwee the (, mode of the field ad the atoms H = ee (, r t r it, (VI

16 Substitutig (VI-69 ito (VI-74 ad comparig the latter with (VI-73 yields i the electric dipole approximatio [r 0 i (VI-69] hω ( H = ie a r, e, ε0v (VI-75 which yields the trasitio matrix elemet relevat for dowward trasitio H =φ H φ = + u H u = fi f ( i,, (,, hω = i + e ε V 0,, μ (VI-76 where μ= u er u is the electric dipole matrix elemet, see Eq. (V-, ad we utilized, which follows from Eq. (VI-5. With (VI-76 the trasitio rate iduced by oe mode, + a,, =, + of the radiatio field is give by πω W = ( + e μ δ( E E + hω = dowward/mode,, ε0v = W + W iduced/mode spot/mode (VI-77 where πω W e μ E E h (VI-78, iduced/mode =, δ( + ω ε0 V πω W = e μ δ( E E + h ω (VI-79 spot/mode, ε0v are the trasitio rate iduced by the field ad the spotaeous trasitio rate, respectively. It ca be readily show by the same approach (exercise that for a upward trasitio the spotaeous trasitio rate is zero, that is there is o spotaeous upward trasitio. The iduced trasitio rate obtaied i (VI-78 ca be show to be equivalet to (V-6 derived withi the semiclassical, F theory by maig the replacemets ε0 ε0εr = ε0 ; ad ( e, μ μ with the latter V c 3 implied by esemble averagig for radomly orieted atoms (exercise

17 The spotaeous emissio rate ca be calculated by summig (VI-79 over all modes of the field 4 3 μω W 0 spo = π 3 3 c ε0 h (VI-80 Summig this for all possible dowward trasitios from a arbitrary iitial eigestate u of the atomic Hamiltoia Ĥ 0 yields the radiative lifetime of this particular state τ r = Wspo, m (VI-8 m= 0 which has bee itroduced pheomeologically i the rate-equatio modellig of light-matter iteractios. 4 The derivatio ca be foud e.g. i A. Yariv, Quatum Electroics, 3d Editio, Wiley & Sos, 989, p