Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................ 5 3.2 HBT..................................... 6 3.3 Mandel.................................. 6 3.4................................................ 8 3.5.................................. 9 A 11 1 1 Hanbury-Brown Twiss Mark Fox Quantum Optics An Introduction 1 1
kawa h@tmu.ac.jp (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2.1 van Cittert - Zernike (mutual coherence function) Γ(Q 1, Q 2, τ) V (Q 1, t)v (Q 2, t + τ) (1) (complex degree of coherence) γ(q 1, Q 2, τ) V (Q 1, t)v (Q 2, t + τ) I(Q1 )I(Q 2 ) (2) Sm Rm2 Rm1 P2 P1 1: S m P 1, P 2 1 S m A m P 1 P 2 V m1 (t) = A m (t R m1 /c) e iω(t Rm1/c) R m1 (3) V m2 (t) = A m (t R m2 /c) e iω(t Rm2/c) R m2 (4) 2
P 1 P 2 Γ(P 1, P 2, ) V (P 1 )V (P 2 ) (5) = m A m(t R m1 /c)a m (t R m2 /c) e iω(r m2 R m1 )/c R m1 R m2 (6) R m1 R m2 2πc/ ω A m Γ(P 1, P 2, ) = m (7) A m(t)a m (t) e iω(r m2 R m1 )/c R m1 R m2 (8) I(S) e iω(r2 R1)/c R 1 R 2 ds (9) R m1 R 1 R m2 R 2 1 γ(p 1, P 2, ) = I(S) e iω(r 2 R 1 )/c ds (1) I(P1 )I(P 2 ) R 1 R 2 I(S) I(P j ) Γ(P j, P j, ) = ds (11) van Cittert-Zernike (α, β) P 1,P 2 (X 1, Y 1 ), (X 2, Y 2 ) R 2 j γ(p 1, P 2, ) = eiψ dαdβi(α, β)e ik(αx+βy) dαdβi(α, β) (12) ψ k[(x2 2 + Y 2 2 ) (X 2 1 + Y 2 1 )] 2R x X 2 X 1, y Y 2 Y 1 P 1, P 2 ( ) mutual coherence ( ) P 1 P 2 (13) 2.2 mutual coherence mutual coherence 12 ρ b γ(p 1, P 2, ) = 2J 1(ν) e iψ (14) ν ν kρb (15) 3
γ(p 1, P 2, ) = 2J 1(ν) ν mutual coherence (16) b =.61 λ ρ (17) 6.3 mas ( ).5 µ m 1 m.5 1 6.61 1m (18).63π/(36 18) P 1 P 2 Q V (Q, t) = k 1 V (P 1, t t 1 ) + k 2 V (P 2, t t 2 ) (19) I(Q) = V (Q, t)v (Q, t) (2) = k 1 2 V (P 1, t t 1 )V (P 1, t t 1 ) + k 2 2 V (P 2, t t 2 )V (P 2, t t 2 ) + 2Re[ k 1 k 2 V (P 1, t t 1 )V (P 2, t t 2 ) ] (21) = k 1 2 I(P 1 ) + k 2 2 I(P 2 ) + 2 k 1 k 2 Re[Γ(P 1, P 2, t 1 t 2 )] (22) = I (1) (Q) + I (2) (Q) + 2 I (1) (Q)I (2) (Q)Re[γ(P 1, P 2, t 1 t 2 )] (23) (I (j) (Q) ) γ(p 1, P 2, t 1 t 2 ) A(t) Φ(t) δ = ντ = 2π(R 2 R 1 )/λ 2I (1) (Q)(1 ± γ(p 1, P 2, t 1 t 2 ) ) (24) (I (1) = I (2) ), mutual coherence visibility γ(p 1, P 2, τ) = I max(p ) I min (P ) I max (P ) + I min (P ) τ = visibility γ(p 1, P 2, ) (25) 3 Hanbury-Brown Twiss ( ) visibility mutual coherence ( ) mutual coherence phase 4
P2 Q P1 2: P2 P1 3: 3.1 5 Hanbury-Brown ( ) I(r j, t) I(r j, t) I(r j, t) (26) I(r 1, t) I(r 2, t + τ) = (I(r 1, t) I(r 1, t) )(I(r 2, t + τ) I(r 2, t + τ) ) (27) = I(r 1, t)i(r 2, t + τ) I(r 1, t) I(r 2, t + τ) (28) = V (r 1, t)v (r 1, t)v (r 2, t + τ)v (r 2, t + τ) V (r 1, t)v (r 1, t) V (r 2, t + τ)v (r 2, t + τ) (29) V x j Lsserlis x 1x 2 x 3x 4 = x 1x 2 x 3x 4 + x 1x 4 x 2 x 3 I(r 1, t) I(r 2, t + τ) = V (r 1, t)v (r 2, t + τ) V (r 2, t + τ)v (r 1, t) (3) = Γ(r 1, r 2, τ)γ(r 1, r 2, τ) (31) = Γ(r 1, r 2, τ) 2 (32) 2 2 Γ(r 1, r 2, τ) = Γ (r 1, r 2, τ) 5
3.2 HBT Hanbury-Brown Twiss Hanbury-Brown Narrabri Stellar Intensity Interferometer 32 (Hanbury-Brown, Davis, Allen 1974) 32 mas ζp up.41 ±.3 mas (1969 ) 4 Bigot et al. 211.445 mas beam spliter 2 PhotoMultiplier 1 PhotoMultiplier correlator 4: HBT (33) g 2 ( ) g 2 1 g 2 (τ) = γ(r 1, r 2, τ) 2 + 1 (33) g 2 (τ) I(r 1, t)i(r 2, t + τ) (34) I(r 1, t) I(r 2, t + τ) 3.3 Mandel Mandel HBT Mandel t-t + t I(t) = V (t)v (t) P (t) = αi(t) t (35) α t t + T n p(n, t, T ) T T/ t 6
t r1,..., t rn ( ) 1 T/ t T/ t T/ t rn T/ t p(n, t, T ) = lim... (α t) n I(t t n! r ) i= [1 αi(t i)δt] n j=1 [1 αi(t (36) rj)δt] r1= r2= 1 3 = lim { 2} t n! 1 rn= r=r1 (37) 1 1 no(δt) 1 (38) n [ T/δt ] n t+t 2 αi(t r1 )δt α I(t )dt (39) 3 exp r 1= [ α t+t t I(t )dt ] t (4) p(n, t, T ) = 1 n! [αw (t, T )]n e αw (t,t ) (41) W (t, T ) t+t t I(t )dt (42) I W p(w ) { } 1 P (n, t, T ) = p(n, t, T ) = dw p(w ) n! [αw (t, T )]n αw (t,t ) e (43) p(n, t, T ) = dw p(w )P p (n, W ) (44) ( dw p(w )f(w ) f(w ) ) Mandek {} P p (n, W ) t t + T n = = np(n, t, T ) = n= dw p(w ) np p (n, W ) (45) n= dw p(w )αw (46) = αw (47) n 2 = np(n, t, T ) = dw p(w ) n 2 P p (n, W ) (48) = n= n= dw p(w )(αw + α 2 W 2 ) (49) = αw + α 2 W 2 (5) ( n) 2 = n 2 n 2 = αw + α 2 W 2 α 2 W 2 (51) = n + α 2 [ W ] 2 (52) 7
Intensity ( n) 2 > n ( ) ( n) 2 = n ( n) 2 < n ( ) HBT Mandel n 1 n 2 = = n 1 n 2 p 1 (n 1, t, T )p 2 (n 2, t, T ) (53) n 1 = n 2 = n 1 p 1 (n 1, t, T ) n 2 p 2 (n 2, t, T ) = α 1 α 2 W 1 W 2 (54) n 1= n 2= n 1 n 2 = n 1 n 2 n 1 n 2 = α 1 α 2 W 1 W 2 (55) W j W j W j (56) W 3.4 E n = (n + 1/2)ħω (57) n : P ω (n) = exp ( E n /kt ) n= exp ( E n/kt ) (58) = x n n= xn (59) = x n (1 x) (6) x exp ( ħω/kt ) (61) 8
n= xn = 1/(1 x) (x < 1) n = = np ω (n) (62) n= nx n (1 x) (63) n= = (1 x)x d dx = (1 x)x d dx x = 1 x ( ) x n n= ( 1 1 x ) (64) (65) (66) (67) n = 1 exp (ħω/kt ) 1 (68) P ω (n) n Bose-Einstein ( n) 2 = P ω (n) = 1 ( ) n n (69) n + 1 n + 1 (n n) 2 P ω (n) = n + n 2 (7) n= N m (Mandel & Wolf 95) ( n) 2 = n + n 2 /N m (71) 68 HBT 3.5 HBT HBT 3 3 Jam session 9
1: creation and annihilation operators : â n = n + 1 n + 1 â n = n n 1 â = [â, â ] = 1 number operator : ˆn = â â ˆn n = n n Hamiltonian : Ĥ = ħω (ˆn ) + 1 2 Ĥ ψ = ħω ( n + 2) 1 ψ HBT 4 g 2 (τ) g 2 (τ) = n 1(t)n 2 (t + τ) n 1 (t) n 2 (t + τ) (72) g 2 (τ) = â 1 (t)â 2 (t + τ)â 2(t + τ)â 1 (t) â 1 (t)â 1(t) â 2 (t + τ)â 2(t + τ) (73) normal ordering (Mandel & Wolf 95) â 1 = â / 2 (74) â 2 = â / 2 (75) â 1â1 = ψ â â ψ /2 = ψ ˆn ψ /2 (76) â 2â2 = ψ â â ψ /2 = ψ ˆn ψ /2 (77) â 1â 2â2â 1 = ψ â â ââ ψ /4 (78) = ψ â (â â 1)â ψ /4 (79) = ψ ˆn (ˆn 1) ψ /4 (8) (81) g 2 (τ) = ˆn(ˆn 1) ˆn 2 (82) 1
( ) photon number state: ψ photon number state n coherent state: g 2 (τ) = coherent state α â α = α α : n(n 1) n 2 < 1 (83) ( n) 2 = n (ˆn n) 2 n (84) = n ˆn 2 n n 2 = (85) α â â ââ α = α 4 (86) α â â α = α 2 (87) g 2 (τ) = 1 (88) ( n) 2 = α (ˆn n) 2 α (89) = α ˆn 2 α n 2 (9) = α â ââ â α n 2 (91) = α â â + â â ââ α n 2 (92) = (n + n 2 ) n 2 = n (93) 4 (?) A V (r) (t) a(ν) cos (ϕ(ν) 2πνt) V (r) (t) = 4 TeX dν a(ν) cos (ϕ(ν) 2πνt) (94) 11
2: g 2 (τ) > 1 ( n) 2 > n g 2 (τ) = 1 ( n) 2 = n g 2 (τ) < 1 ( n) 2 < n 5 V (t) = V (r) (t) + iv (i) (t) = V (i) (t) dν a(ν)e i(ϕ(ν) 2πνt) (95) dν a(ν) sin (ϕ(ν) 2πνt) (96) ν ν δν/ν 1 ν ν = ν V (t) = A(t)e i(φ(t) 2πνt) = (A(t) e iφ(t) ) e 2πiνt (97) A Φ (95) (97) A(t) e iφ(t) = { V (t) = (A(t) e iφ(t) ) e 2πiνt = µ [ dµ a(µ) e iϕ(µ)] e 2πµt (98) µ ν ν (99) µ dµ [a(µ) e iϕ(µ)] } e 2πµt e 2πiνt (1) a(µ) µ = ν ν = {} µ = ν ν ν e 2πiνt ν µ (97) A(t) e iφ(t) A(t) Φ(t) ν A(t) Φ(t) 5 12