Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................ 5 3.2 HBT..................................... 6 3.3 Mandel.................................. 7 3.4................................................ 9 4 Hanbury-Brown Twiss 1 4.1................................. 1 4.2.................................... 11 4.3.......................................... 11 4.4...................................... 13 4.5........................................ 13 4.6............ 14 4.7 HBT................................................. 15 4.8 g 2....................................... 15 4.9 g 2..................................... 16 4.1 g 2........................................ 16 5 16 5.1 Helmholtz-Kirchhoff.................................... 16 A 17 1
1 1 Hanbury-Brown Twiss Mark Fox Quantum Optics An Introduction : : kawahara@eps.s.u-tokyo.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) 1 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) 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(rm2 Rm1)/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(r 2 R 1 )/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(r2 R1)/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 (13) 3
x X 2 X 1, y Y 2 Y 1 P 1, P 2 ( ) mutual coherence ( ) P 1 P 2 2.2 mutual coherence mutual coherence 12 ρ b γ(p 1, P 2, ) 2J 1(ν) e iψ (14) ν ν kρb (15) γ(p 1, P 2, ) 2J 1(ν) ν mutual coherence (16) b.61 λ ρ (17) 6.3 mas ( ).5 µ m.5 1 6.61 1m.63π/(36 18) (18) 1 m 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) 4
(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 ) (25) τ visibility γ(p 1, P 2, ) P2 Q P1 2: 3 Hanbury-Brown Twiss ( ) visibility mutual coherence ( ) mutual coherence phase P2 P1 3: 3.1 5 Hanbury-Brown () 5
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 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 (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) 2 Γ(r 1, r 2, τ) Γ (r 1, r 2, τ) 6
beam spliter 2 PhotoMultiplier 1 PhotoMultiplier correlator 4: HBT 3.3 Mandel Mandel HBT Mandel t-t + t I(t) V (t)v (t) α P (t) αi(t) t (35) t t + T n p(n, t, T ) T T/ t 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 j1 [1 αi(t (36) rj)δt] r1 r2 1 3 lim { 2} t n! 1 rn rr1 (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) 7
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) 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 1 n 2 p 2 (n 2, t, T ) α 1 α 2 W 1 W 2 (54) 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 8
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) 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 n P ω (n) n Bose-Einstein P ω (n) 1 n + 1 ( ) x n n ( 1 1 x 1 exp (ħω/kt ) 1 ) (64) (65) (66) (67) (68) ( ) n n (69) n + 1 ( n) 2 (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 9
4 Hanbury-Brown Twiss HBT 4.1 H ψ, ϕ H (ψ, ϕ) ψ ϕ (72) {e i } ψ ϕ n ξ i e i (73) i1 n η i e i (74) i1 ψ, ϕ {ξ 1, ξ 2,,,, ξ n } {η 1, η 2,,,, η n } (72) {ξ1, ξ2,,,, ξn} T.{η 1, η 2,,,, η n } (75) ψ, ϕ ϕ ϕ H ( ϕ H) H (dual space) H ϕ C ψ ( ψ H ) H C (72) ( ψ, ϕ ) ψ ϕ C (76) H ψ H ψ ϕ ϕ ( ϕ ) ϕ (77) ( ϕ ) ϕ (78) 1
Ψ Â Ψ ( Ψ e iθ Ψ ) : Ψ Â Ψ Ψ e iθ Âe iθ Ψ Ψ Â Ψ (79) { Ψ e iθ Ψ π θ π} (8) Fiber( ) Fiber 1 1 ˆρ Ψ Â Ψ i Ψ e i e i Â Ψ i e i Â Ψ Ψ e i Tr(ˆρ) Tr(ˆρÂ) (81) { e i } ˆρ Ψ Ψ Ψ Ψ (82) Tr( ˆBĈ) e i  ˆBĈ e i e i  e j e j ˆBĈ e i i i j e j ˆBĈ e i e i  e j e j ˆBĈ e j Tr( ˆBĈÂ) (83) j i j Trace { Ψ 1, Ψ 2,... Ψ k,... Ψ N } p k p k Ψ k Â Ψ k p k Ψ k e i e i Â Ψ k ( ) e i  p k Ψ k Ψ k e i k k i i k Tr(ˆρ) Tr(ˆρÂ) (84) ˆρ p k Ψ k Ψ k k (85) 4.2 4.3 α â α α α (86) 11
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 α n n α (87) n n α (86) n 1 n 1 â α (â n 1 ) α ( n n ) α n n α (88) α n 1 α (89) n α α n n 1 α αn n! α (9) α α n α n n! n (91) 12
α 1 α α 2 α 2 n m n m α 2 ( α 2 ) n n! n α 2 exp ( α 2 ) α exp ) ( α 2 2 α n (α ) m n! m! m n α n (α ) m n! m! δ m,n (92) (93) α exp ) ( α 2 α n n (94) 2 n n! 4.4 4.5 ˆρ n P n n n P n e βħωn n e βħωn (1 e βħω )e βħωn β 1 kt (95) Thermal light n Tr(ˆρ ˆn) m ˆρ ˆn m mp n m n 2 mp n δ n,m np n m n m n n n ne βħωn (1 e βħω ) e βħω 1 e βħω (96) 13
n xn (1 x) 1 n nxn x(1 x) 2 e βħω n 1 + n (97) Thermal light ˆρ 1 1 + n m ( ) m 1 m m (98) 1 + n 4.6 ˆρ dα 2 P (α) α α (99) P P (α) 1 2 π n e α / n (1) Gaussian 1 ˆρ dα 2 e α 2 / n α α π n 1 2π dr dθe r2 / n e r2 rm+n+1 e i(m n)θ m n π n m n m! n! 2π dre (1+ n )r 1 2 r 2m+1 m m π n m! m 1 n ( ) m n dse s s m 1 m m n 1 + n 1 + n m! 1 1 + n m m ( n 1 + n ) m m m (11) (94) α re iθ 2π e i(m n)θ 2πδ m,n s r 2 (1 + n )/ n dse s s m m! 14
4.7 HBT HBT 4 g 2 (τ) g 2 (τ) n 1(t)n 2 (t + τ) n 1 (t) n 2 (t + τ) (12) 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) (13) normal ordering (Mandel & Wolf 95) â 1 â / 2 (14) â 2 â / 2 (15) â 1â1 ψ â â ψ /2 ψ ˆn ψ /2 (16) â 2â2 ψ â â ψ /2 ψ ˆn ψ /2 (17) â 1â 2â2â 1 ψ â â ââ ψ /4 (18) ψ â (â â 1)â ψ /4 (19) ψ ˆn (ˆn 1) ψ /4 (11) (111) g 2 (τ) ˆn(ˆn 1) ˆn 2 (112) ( ) 4.8 g 2 ψ photon number state n g 2 (τ) n(n 1) n 2 < 1 (113) ( n) 2 n (ˆn n) 2 n (114) n ˆn 2 n n 2 (115) 15
4.9 g 2 coherent state α â α α α α â â ââ α α 4 (116) α â â α α 2 (117) g 2 (τ) 1 (118) ( n) 2 α (ˆn n) 2 α (119) α ˆn 2 α n 2 (12) α â ââ â α n 2 (121) α â â + â â ââ α n 2 (122) (n + n 2 ) n 2 n (123) 4.1 g 2 5 5.1 Helmholtz-Kirchhoff Helmholtz-Kirchhoff P U(P ) V (x, t) U(x)e iωt (124) ( 2 + k 2 )U (125) U, U Green dσ (U U n U U ) dv (U 2 U U 2 U) (126) n σ V σ V σ n 125 P U (x) eiks s (127) 16
5: s P 5 σ S + S S P S ( ds U (eiks /s) (e iks /s) U ) n n [ ds (e iks /s)(ik 1/s)U (e iks /s) U ] S n 4πU(P ) [radius of S ] (128) Helmholtz-Kirchhoff U(P ) 1 ds 4π S [ U n ( e iks s ) ( e iks s ) ] U n (129) A V (r) (t) a(ν) cos (ϕ(ν) 2πνt) V (r) (t) 3 V (t) V (r) (t) + iv (i) (t) V (i) (t) 3 dν a(ν) cos (ϕ(ν) 2πνt) (13) dν a(ν)e i(ϕ(ν) 2πνt) (131) dν a(ν) sin (ϕ(ν) 2πνt) (132) 17
ν ν δν/ν 1 ν ν ν V (t) A(t)e i(φ(t) 2πνt) (A(t) e iφ(t) ) e 2πiνt (133) A Φ (131) (133) A(t) e iφ(t) { V (t) (A(t) e iφ(t) ) e 2πiνt µ [ dµ a(µ) e iϕ(µ)] e 2πµt (134) µ ν ν (135) µ dµ [a(µ) e iϕ(µ)] } e 2πµt e 2πiνt (136) a(µ) µ ν ν {} µ ν ν ν e 2πiνt ν µ (133) A(t) e iφ(t) A(t) Φ(t) ν A(t) Φ(t) 18