} = 0 [ III-2b ] [ ] a exp [ i k r. [ ] a t. { } n, a r. ( ) = i ˆ. ( ) = i. ( ) exp i r. a exp i k r. a, etc...

Transcription

1 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 19 III. REPRESENTATIONS OF PHOTON STATES 1. Fock or Number State:. 11 A we have ee, the Fock or umber tate { } r k σ r k σ [ III-1 ] are complete et eigetate r of a importat group of commutig obervable -- viz. H rad,n ad M. Reprie of Characteritic ad Propertie of Fock State: a. The expectatio value of the umber operator ad the fractioal ucertaity aociated with a igle Fock tate: N = [ III-a ] = [ "ucertaity" ] = { N N } = 0 [ III-b ] b. Expectatio value of the field aociated with a igle mode: For oe mode Equatio [ II-4a ] ad [ II-4b ] reduce to r E r, t = i ˆ [ ] a exp [ i k r r i ω t] e E a exp i k r r i ω t [ III-3a ] r H r, t = i ε 0 [ ] a t E k µ ˆ e ˆ 0 exp i r [ k r i ω t] a t exp i r [ k r i ω t ] [ III-3b ] 11 I what follow, for implicity we drop the meaig that a, etc... { }, a r r k k r k ubcript o the operator ad tate vector with the obviou R. Victor Joe, May,

2 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 0 where E= hω ε 0 V r E = 0 r H = 0 [ III-4a ] E = H = { E r E r E r } = hω ε 0 V + 1 { H r H r H r }= E H = c hω ( V + 1 ) = ε 0 E ( + 1 ) µ 0 1 hω ( µ 0 V + 1 ) 1 = = E ( + 1 ) 1 ε 0 E( + 1 ) 1 µ 0 [ III-4b ] c. Phae of field aociated with igle mode: To obtai omethig aalogou to the claical theory we would like to eparate the creatio ad detructio operator (ad, thu, the electric ad magetic field operator) ito a product of amplitude ad phae operator. Followig Sukid ad Glogower, 1 we defie a phae operator, Φ uch that a ( N + 1)1 exp( i Φ) a exp( i Φ) N +1 1 [ III-5 ] Defied i thi way, the baic propertie of the phae operator may be evaluated from kow propertie of the creatio, detructio ad umber operator. Ivertig, we obtai 1 Sukid, L. ad Glogower, J., Phyic, 1, 49 (1964) R. Victor Joe, May, 000

3 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 1 exp( i Φ) ( N +1) 1 a exp( i Φ) a ( N +1) 1 ad ice a a =N + 1, it follow that [ III-6 ] exp( i Φ) exp( i Φ) = 1 [ III-7 ] but oly i thi order! Operatig o umber tate with the phae operator, we obtai from Equatio [ I-6 ] exp( i Φ) = ( N +1) 1 a = ( N +1) 1 ( ) 1 exp( i Φ) = a ( N +1) 1 = a ( +1) 1 1 = 1 = +1 [ III-8 ] Coequetly, the oly ovaihig matrix elemet of the phae operator are 1 exp i Φ = exp( i Φ) = 1 [ III-9 ] The phae operator defied by Equatio [ III-36 ] do have the felicitou or claically aalogou property of revealig magitude idepedet iformatio, but ufortuately they are ohermitia operator -- i.e. 1 exp( i Φ) exp( i Φ) 1 -- ad, hece, caot repreet obervable. However, they may be paired ito operator that are obervable -- viz. R. Victor Joe, May, 000 1

4 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE coφ = 1 exp i Φ { + exp( i Φ) } iφ = 1 { exp( i Φ) exp( i Φ) } i [ III-10 ] which have the followig ovaihig matrix elemet: 1 coφ = coφ 1 = 1 [ III-11 ] 1 i Φ = i Φ 1 = 1 i Thee early commutig operator 13 may be adopted a the quatum mechaical operator which repreet (a we will demotrate ao) the obervable phae propertie of the electromagetic field. For the Fock tate: co Φ = iφ = 0 [ III-1a ] co Φ = i Φ= { co Φ coφ } = 1 coφ i Φ = 1 [ III-1b ] [ III-1c ] c. The coordiate or Schrödiger repreetatio of tate: Recall from Equatio [ I-10a ] ad [I-31] that 13 Alo, it may be eaily etablihed that the matrix elemet of their commutator are give by [ co Φ,iΦ] = i δ δ 0 R. Victor Joe, May, 000

5 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 3 q = 1! = 1! m hω ω π h H ω q h d m dq q 0 m ω h q exp m ω h q [ III-13 ] Therefore, the probability P ( q) of eigevalue q for a give Fock tate i give by P ( q) = q = 1! ω π h H m ω h q exp m ω d. Approximate localizatio of a photo: 14 h q Of coure a plae wave i ditributed or de-localized i both time ad pace. [ III-14 ] Defiig the wave fuctio for a photo i a tak fraught with dager,15 but the impler tak of defiig a wave fuctio approximately localized at a give itat i relatively traight forward -- viz. r ψ k r k ( r 0 ) 0 r k 0 r =C exp k k r exp[ i k r r 0 ] 0,0,0,K r =1,K,0,0,0 [ III-14 ] r k k. Photo State of Well-defied Phae: Coider the tate defied by ϕ lim + 1 [ III-15 ] 1 exp[ i ϕ] = 0 14 See Sectio i Leoard Madel ad Emil Wolf, Optical Coherece ad Quatum Optic, Cambridge Pre (1995), ISBN See Sectio i Marla O. Scully ad M. Suhail Zubairy, Quatum Optic, Cambridge Pre (1997), ISBN R. Victor Joe, May, 000 3

6 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 4 Clearly, ϕ ϕ =1 give the orthoormal propertie of the umber tate. Eetial quetio: I thi tate a eigetate of the phae operator? To awer the quetio we eed to coider the followig potetial eigevalue equatio: coφ ϕ = 1 lim +1 1 exp[ i ϕ] exp[ i Φ] + exp[ i ϕ] exp[ i Φ] =0 =0 [ III-16a ] Uig Equatio [ III-10 ] ad [ III-10 ], we obtai coφ ϕ = 1 lim ( +1 1 ) exp i ϕ =1 = 1 lim ( +1 1 ) exp iϕ = coϕ ϕ + 1 lim ( +1 1 ) [ ] 1 + exp[ i ϕ] +1 1 = 0 exp[ i ν ϕ ] ν + exp[ i ϕ] exp[ i ν ϕ] ν ν = 0 +1 ν = 1 { exp[ i ϕ] +1 exp[ i ( + 1) ϕ] exp[ i ϕ] 0 } [ III-16b ] o that the tate ϕ fail to be a trict eigeket of coφ by term that dimiih fater 1 a. Similarly, we ca ee that diagoal matrix elemet of coφ tha + 1 ad iφ are give by { } 1 ϕ co Φ ϕ = coϕ 1 lim + 1 { } 1 ϕ i Φ ϕ = iϕ 1 lim + 1 coϕ [ III-17a ] iϕ [ III-17b ] R. Victor Joe, May, 000 4

7 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 5 Reprie of Characteritic ad Propertie of Phae State: a. The expectatio value of the umber operator ad the fractioal ucertaity aociated with a tate of well-defied phae: ϕ N ϕ = lim = lim +1 = 0 1 ( +1) fractioal ϕ N ϕ ϕ N ϕ ucertaity = ϕ N ϕ = = { } lim 1 +1 =0 1 lim 6 + lim lim +1 lim =0 = lim 1 = 1 3 =0 [ III-18a ] [ III-18b ] b. Expectatio value of the field aociated with a igle mode: From Equatio [ III-3a ] ϕ r E ϕ = lim h ω e ε 0 V ˆ i k r r ω t + ϕ diverge a for large! 1 ( + 1) = 0 [ III-19 ] R. Victor Joe, May, 000 5

8 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 6 c. Phae of field aociated with igle mode: ϕ coφ ϕ = coϕ ϕ iφ ϕ = iϕ [ III-0a ] co Φ = i Φ= { ϕ co Φ ϕ ϕ coφ ϕ } = 0 [ III-0b ] d. Probability of photo umber: Fially, we may eaily deduce the probability of fidig photo (i.e. the photo tatitic) i a particular tate of well defied phae -- viz. P = ϕ lim [ III-50 ] We ee that there i a equal, but mall probability of ay umber: thi agree with the ituitio that the magitude of the field i completely udetermied if the phae i preciely kow! 3. Coheret Photo State: 16 It would, ideed, be ueful to have eigetate of the detructio operator (electric or magetic field) -- viz. a r k α r k = α r k α r k [ III-51 ] Reprie of Characteritic ad Propertie of Coheret State: a. The Fock tate repreetatio of the coheret tate: 16 The coheret tate i a Harvard ivetio! See R. J. Glauber, Phy. Rev. 131, 766 (1963). R. Victor Joe, May, 000 6

9 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 7 Sice. a = ad a a = N +1, the a = ad we are able to write a repreetative of the ought tate i the umber tate bai -- viz. a α = α = α α [ III-5a ] or α = α 1 α = α! 0 α [ III-5b ] Uig the expaio of the idetity operator, the eigeket become α = α = 0 α α!. [ III-53 ] To ormalize the eigeket write α α = α 0 0 α α α! = α 0 0 α exp α [ ] = 1 [ III-54 ] o that α 0 = 0 α = exp 1 α. Fially, we ee that α = exp 1 α α! [ III-55 ] i a ormalize repreetatio of the eigeket of the detructio operator. R. Victor Joe, May, 000 7

10 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 8 b. The expectatio value of the umber operator ad the fractioal ucertaity aociated with a coheret tate: α N α = α [ III-56a ] { } fractioal α N α α N α ucertaity = α N α = 1 exp α α α! α 4 = 1 exp α α = α 1 α [ ( 1)+] α 4! [ III-56b ] Thu, we ee that the fractioal ucertaity dimiihe with mea photo umber! c. Expectatio value of the electric field aociated with a igle mode: From Equatio [ III-3a ] α r E α = h ω ε 0 V ˆ e α i k r r ω t+ϑ [ III-57a ] where α= α exp( i ϑ). E = { α E r E r α α E r α } = h ω ε 0 V 17 [ III-57b ] 17 Similarly H= 1 c µ 0 h ω ε 0 V for the coheret tate, o that E H = c h ω V. R. Victor Joe, May, 000 8

11 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 9 d. Probability of photo umber: From the repreetatio of the coheret tate give i Equatio [ III-55 ] we may eaily deduce the probability of fidig photo (the photo tatitic) i a particular coheret tate i give by a Poio ditributio characterized by the mea value = α. -- viz. P = α = exp[ α ] α! [ III-58 ] S AMPLE POISSON DISTRIBUTIONS - COHERENT STATE PHOTON STATISTICS R. Victor Joe, May, 000 9

12 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 30 e. Phae of field aociated with igle mode: α co Φ α = 1 exp 1 α = 1 exp 1 α = α coϑ exp 1 α α! α +1 α + α α +1 {( + 1)!! } ( N +1 ) 1 a +a ( N + 1 ) 1 [ ] α! α! ( +1) [ III-59a] Ufortuately, it i ot poible to evaluate thi ummatio aalytically. However, Carruther 18 ha give a aymptotic expaio which i valid for a large mea umber of photo -- viz. α co Φ α = coϑ α +K α >> 1 [ III-59b] f. Coheret tate a a bai: A we will ee preetly, the coheret tate are very ueful i decribig the quatized electromagetic field, but, ala, there i a complicatio -- the coheret tate are ot truly orthogoal! From Equatio [ III-6 ] we ee that β α = exp 1 α 1 β β α = exp 1 α 1 β +αβ! [ III-60 ] o that 18 Carruther, P. ad Nieto, M. M., Phy. Rev. Lett. 14, 387 (1965) R. Victor Joe, May,

13 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 31 αβ ( ( α β )) = exp( α β ) β α = exp α β +αβ +α β = exp α β [ III-61 ] That i, the eigeket are approximately orthogoal oly whe α β i large! g. The diplacemet operator: There are a growig ad igificat et of applicatio where it i ueful to expre the coheret tate directly i term of the vacuum tate 0. If we ue the umber tate geeratig rule = a -- i.e. Equatio [ I-7 ] -- the coheret tate may be writte i the form α = exp 1 α! 0 α a 0! = exp α a 1 α 0 [ III-6 ] If we make u of the Baker-Haudorff theorem, 19 we may eaily how that 19 The Baker-Haudorff theorem or idetity may be tated a [ [ ]] = B,[ A,B ] whe A, A,B { } exp{ A +B } = exp{ A } exp{ B } exp 1 A,B [ ] [ ] = 0. For a proof, ee, for example, Charle P. Slichter Priciple of Magetic Reoace, Appedix A or William Louiell Radiatio ad Noie i Quatum Electroic. R. Victor Joe, May,

14 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 3 o that A α α = A α 0 = exp α a α a 0 [ III-63 ] may be iterpreted a a creatio operator which geerate a = A ( α) i a I ome treatmet A ( α) i ) 0 ad the coheret tate coheret tate from the vacuum. (It adjoit operator A α detructio operator which detroy a tate). decribed a the diplacemet operator (writte D α are called the diplaced tate of the vacuum. 1 To explore thi poit of view (ad to give ome meaig to the phae of the coheret tate eigevalue), we may expre α i a two-dimeioal, dimeiole phae pace repreetatio. To that ed, followig Equatio [ I-16 ], we write the dimeiole coordiate a θ = mω h 1 q = a exp [ i γ ] +a exp[ i γ] [ III-64a ] ad the dimeiole mometum a π = m hω 1 p = a exp i ( γ+π ) [ ]+a exp i ( γ+π ) [ ] [ III-64b ] o that [ θ, π] = i a,a = i [ III-64c ] 0 We ca (or rather you will) how that D ( α)a D ( α) = a + α ad D ( α)a D ( α) = a + α 1 See Elemet of Quatum Optic, Pierre Meytre ad Murray Sarget III, Spiger-Verlag (1991), ISBN X. R. Victor Joe, May, 000 3

15 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 33 ad ice thee variable are caoical ( θ) ( π) 1 [ III-64d ] Sice a = 1 ( θ i π ) exp[ i γ] a = 1 ( θ +i π ) exp[ i γ] [ III-65 ] the mode field (ee Equatio [II-4a]) b r E r, t = i ˆ [ ] a exp[ i k r r + i ω t] e E a exp i k r r i ω t [ III-66a ] become r E r, t = ˆ [ III-66b ] { } e E π co( k r r ω t+γ )+θ i k r r ω t+γ Sice p ha a coordiate pace repreetatio i h d dq = i( h ω ) 1 d d θ ad q ha a mometum repreetatio i h d d p = i ( h ω) 1 d d π, 3 αa α a = α r a a + i α i a +a = [ α r d dθ+α i d dπ] [ III-67a ] Of coure, i geeral ( A) ( B) 1 [ A,B ] where ( A ) = A A 3 If thi ufamiliar, ee Equatio [ I-0 ] ad [ I- ] i the lecture ote etitled The Iteractio of Radiatio ad Matter: Semiclaical Theory. R. Victor Joe, May,

16 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 34 ad A α = exp αa α a [ ] [ III-67b ] = exp α d d r θ +α i d dπ Thu, A ( α) defie or geerate a two-dimeioal Taylor expaio whe it act o a fuctio of θ ad π. I particular, if we take the phae pace repreetatio of the groud or vacuum tate θπ a the product of two Gauia (ee Equatio [ I-10a ] ad [ I-9 ]), the A ( α) θπ repreet a hift or diplacemet of thi phae pace repreetatio -- i.e. θπ α = θπa ( α) 0 = u G ( θ α r ) u G ( π α i ) [ III-68 ] I light of Equatio [ II-3b ], α ( t) = α exp( i ω t) we ca write θπ α t where α= α exp( i φ). u G ( π α co( ω t +φ) ) [ III-69 ] = u G θ α co( ω t +φ) h. The diagoal coheret-tate repreetatio of the deity operator (Glauber-Sudarha P-repreetatio): It may be eaily etablihed that 1 = 1 π β β d β= β β d Re( β) d Im( β) [ III-70 ] o that it eem quite reeaoable to look for a repreetatio of the deity matrix i the form ρ = P ( β ) β β d β [ III-71 ] For a pure coheret tate, P i clearly a two-dimeioal delta fuctio R. Victor Joe, May,

17 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 35 P( β )= δ ( ) ( β α ) =δ 1 Example 1 -- Coheret tate ( Re ( α) ) δ 1 Re β ( Im( α) ) [ III-7 ] Im β I geeral, uig Equatio [ III-60 ] -- i.e. β α = exp 1 α 1 β +αβ [ III-60 ] we may fid a imple procedure for fidig the P-repreetatio by writig α ρ α = P ( β ) α β β α d β = exp α Thu, α ρ α exp α fuctio P β P β exp β P ( β ) = 1 π exp β exp β [ ] d β exp αβ β α i the two-dimeioal Fourier traform 0f the ad we may write α ρ α exp ( α ) [III-73 ] exp [ αβ +β α ] d α [ III-74 ] A a ecod example, coider a thermal radiatio field decribed by a caoical eemble ρ = where H = hω a a + 1. Thu, exp H k B T [ ] Tr exp H k B T [III-75 ] ρ = 1 exp h ω hω k B T exp [III-76 ] k B T R. Victor Joe, May,

18 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 36 ad = Tr ρ a a 1 hω = exp 1 k B T [III-77 ] o that ρ = [III-78 ] Thu, we ca write ρ = [III-79 ] ( 1+ ) +1 ad α ρ α = α α ( 1+ ) +1 exp ( α ) ( α ) = 1+! exp α = exp α 1+ 1 [III-80 ] Fially, we ee that Example -- Thermal radiatio - a chaotic tate = exp α P α = exp β π ( 1+ ) 1 π exp ( α ) 1+ 1 exp βα* +αβ * d β [III-81 ] A a third example, coider Fock or umber tate. From Equatio [ III-55 ] we ee that R. Victor Joe, May,

19 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 37 α ρ α = α α = exp ( α )! ( α ) [III-8a ] ad P ( β ) = 1! 1 π exp β = exp β ( α )! 1 β * β π exp [ αβ +β α ] d α exp[ αβ +βα ] d α [ III-8b ] o that Example 3 -- Pure Fock or umber tate P( β )= exp( β )! β * β δ β [ III-8b ] i. The Glauber-Sudarha-Klauder optical equivalece theorem: Suppoe we have ome ormally ordered fuctio a,a = c m a a m [III-83 ] f N m The expectatio value i give by a,a = Tr f N ρ f ( N ) a,a [III-84 ] R. Victor Joe, May,

20 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 38 Uig Equatio [ III-71 ] we ee that a,a = Tr P ( α ) c m α α a a m m = P ( α) c m α a a m α f N = P α m c m α * α m d α m d α d α [III-85a ] or, fially, the optical equivalece theorem a,a = f N P ( α ) f ( N ) α, α * [III-85b ] j. The Ucertaity Relatiohip for { θ, π}: Sice a,a =1 we ee from Equatio [ III-64a ] that θ = θ θ = a a exp [ i γ] + a a exp[ i γ] + a a + a a a exp [ i γ] a exp [ i γ] a a [ III-86 ] = : θ : +1 where : A : ymbollize the ormally ordered expectatio value of the operator A. From Equatio [III-85b ] : θ a,a : = P ( α ) θ α, α * d α [ III-87 ] R. Victor Joe, May,

21 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 39 : θ a,a : = [ ] d α P ( α ) α * exp ( i γ) + α exp( i γ) [ III-88 ] If we chooe γ (ad P α π >1(queezed tate)! ) uch that : θ a,a : < 0, the θ >1 ad R. Victor Joe, May,