Lab. 1. Entanglement and Bell inequalities

Transcription

1 Lab.. Entanglement and Bell inequalitie In quantum mechanic, two particle are called entangled if their tate cannot be factored into ingle-particle tate: Ψ Ψ Ψ Meaurement performed on the firt particle give reliable information about the tate of the econd particle, no matter how far apart they may be. Entangled A B A B Thi i the tandard Copenhagen interpretation of quantum meaurement which ugget nonlocality of the meauring proce. In difference with claical correlation, quantum correlation do not depend on the bai. Changing the bai doe not change the tate of quantum entanglement.

2 The idea of entanglement wa introduced into phyic by Eintein-Podolky-Roen [Phy. Rev., 47, 777 (935)]

3 EPR gedanken experiment (Bohm interpretation) According to EPR who conider ytem of two eparate particle: A meaurement performed on one quantum particle ha a nonlocal effect on the phyical reality of another ditant particle - pooky action at a ditance (entanglement), in which EPR did not believe; Quantum mechanic i incomplete [ome extra ( hidden ) variable are needed] uggeted by EPR. Bohr replied to the EPR paradox that in a uch quantum tate one could not peak about the individual propertie of each of the particle, even if they were ditant from one another. Ψ = [ + + ] 3

4 In the mid-ixtie it wa realized that the nonlocality of nature i a tetable hypothei Bell Inequalitie [(Phyic,, 95 (964)]: John Bell howed that the locality hypothei with hidden variable lead to a conflict with quantum mechanic. He propoed a mathematical theorem containing certain inequalitie. An experimental violation of hi inequalitie would ugget the tate obeying the quantum mechanic with nonlocality. iφ iθ I = I e + I e = I + I + co(φ θ ) I I Subequent experimental realization of EPRB gedanken experiment confirmed the quantum prediction (Bell inequalitie were violated):. S. Freedman and J. Clauer, Phy. Rev. Lett. 8, 938 (97) 4. A. Apect et al., Phy. Rev. Lett. 47, 460 (98); 49, 9 and 804 (98),

5 Creation of Polarization Entangled Photon: Spontaneou Parametric Down Converion Type I BBO crytal V (77.6 nm) (363.8 nm) (77.6 nm) Ψ SPDC = V V i + e iφ H H H i With phae Φ equal 0 or π (which can be compenated uing a quartz plate) and with a normalization ΨEPR = (V V i ± H H i) 5 In thi tate H-polarized SPDC light cone overlap V-polarized light cone, o SPDC light in thi tate i not polarized

6 Rotation angle β ΨBell = Polarizer Detector B (V Ψ Bell A Detector A Source of photon pair V V α H B H i) i If we meaure the polarization of ignal and idler photon in the H, V bai, there are two poible outcome: both vertical and both horizontal. Polarizer Rotation angle α V i+ H We could intead meaure the polarization with polarizer rotated by an angle α, which i arbitrary. We ue the rotated polarization bai V Vα = co α V + in α H α and H α = in α V + co α H In thi bai the tate i H α ΨBell = ( Vα Vα i+ Hα Hα i ) So if we meaure in thi rotated bai, we obtain the ame reult, becaue the tate i rotation invariant. By placing polarizer rotated to arbitrary angle α and β in the idler and ignal path, we meaure the polarization of the downconverted photon. Meaured coincidence count N (α, β ) ~ PVV (α, β ) Coincidence count (per ec) ο Serie α = 45 α = 35 Serie ο where PVV (α, β ) = V α i Vβ ΨBell and VV ubcript on P indicate the meaurement outcome Vα Vβ, both photon vertical in the bae of their repective polarizer Polarizer B angle β (in degree) It i eay to how that for the EPR tate PVV (α, β ) = co ( β α ) 6

7 More generally, there are four poible outcome: co ( β α ) PVH (α, β ) = in ( β α ) PVV (α, β ) = co ( β α ) PHV (α, β ) = in ( β α ) PHH (α, β ) = We can find thi probabilitie in the experiment by meauring coincidence count N(α,β ) N (α, β ) N (α, β ) = N (α, β ) + N (α, β ) + N (α, β ) + N (α, β ) N total N (α, β ) PHH (α, β ) = N (α, β ) + N (α, β ) + N (α, β ) + N (α, β ) PVV (α, β ) = N (α, β ) N (α, β ) + N (α, β ) + N (α, β ) + N (α, β ) N (α, β ) PHV (α, β ) = N (α, β ) + N (α, β ) + N (α, β ) + N (α, β ) PVH (α, β ) = where Ntotal i the total number of pair detected and α and β are the polarizer etting α + 90o, β Let u introduce E (α, β ) P VV + PHH PVH PHV It i eay to how that E (α, β ) = co (α β ) For completely correlated event E (α = β ) = and for complete anticorrelation E (α β = π / ) = 7

8 Calculation of Bell Inequality Clauer, Horne, Shimony and Holt (CHSH), Phy. Rev. Lett., 3, 880 (969) Bell inequality define the degree of polarization correlation under meaurement at different polarizer angle. The proof involve two meaure of correlation: E (α, β ) = PVV (α, β ) + PHH (α, β ) PVH (α, β ) PH V (α, β ), and S = E(a, b) E(a, b' ) + E(a', b) + E(a', b' ), where N (α, β ) + N (α, β ) N (α, β ) N (α, β ) N (α, β ) + N (α, β ) + N (α, β ) + N (α, β ) The above calculation of S require a total of 6 coincidence meaurement (N), at polarization angle α and β : E (α, β ) = α β a = -45o a = 0o a = 45o a = 90o b = -.5o b =.5o b = 67.5o b =.5o Violation of Bell inequality if S > 8

9 It i very important to know that Bell inequality i violated only for ome angle α and β. For other angle both quantum theory and claical phyic (HVT) give the ame value of S. S = E ( a, b ) E ( a, b' ) + E ( a ', b) + E ( a ', b ' ) = co (a b) co (a b' ) + co (a' b) + co (a' b' ) For the particular choice of polarizing angle a + + = π /4, a = 0, b=-π /8, b = π /8, S max = =.88 Comparion of HVT (dahed line) and quantum theory (olid line) we obtain the reult Dependence of S on polarizer angle β = b' S (b ) S (b ) b (in degree) b (in degree) If in our experiment we meaure N (α,β ) with fringe viibility E (α, β ) = V co (α β ) and S = V max V = N max N min, different from, N max + N min, o if we have V Bell inequality will not be violated at any angle. = 0.7, 9 Quantum correlation (entanglement) i a very rare event and it i difficult to oberve!!!

10 Proof of CHSH Bell inequality E (α, β ) E (α, β ' ) + E (α ', β ) + E (α ', β ' ) Thi proof i baed on a trivial mathematical relation a + b + c +... a + b + c +... In a claical cae, the ditribution of the hidden variable λ i decribed by a function ρ (λ ), where ρ (λ ) 0 ρ (λ )dλ = The aumption of locality and realim are embodied in the aumption that for the idler photon the outcome of meaurement i determined completely by λ and the meaurement angle α, and for the ignal photon by λ and β. Thee outcome are pecified by the function B(λ, β ) = ± A(λ, α ) = ± for detection a Vβ and Hβ. for detection a Vα and Hα, and Probabilitie of particular outcome are given by the integral PVV (α, β ) = P VH (α, β ) = + A(λ, α ) + B (λ, β ) ρ ( λ ) dλ + A(λ, α ) B (λ, β ) ρ ( λ ) dλ PHH (α, β ) = PHV (α, β ) = A(λ, α ) B(λ, β ) ρ (λ )dλ A(λ, α ) + B(λ, β ) ρ (λ )dλ It i eay to how that E (α, β ) P VV + PHH PVH PHV = A(α, λ ) B( β, λ ) ρ (λ )dλ 0

11 a + b + c +... a + b + c +... [ A(α, λ ) B(β, λ ) A(α, λ ) B(β ', λ )] ρ (λ )dλ A(α, λ ) B ( β, λ ) A(α, λ ) B ( β ', λ ) ρ (λ )dλ = A(α, λ )[ B ( β, λ ) B ( β ', λ )] ρ (λ ) dλ = B ( β, λ ) B ( β ', λ ) ρ ( λ ) dλ E (α, β ) E (α, β ' ) = We ued relation A(α, λ ) = E (α ', β ) + E (α ', β ' ) B( β, λ ) + B( β ', λ ) ρ (λ )dλ In the ame manner E (α, β ) E (α, β ' ) + E (α ', β ) + E (α ', β ' ) [ B ( β, λ ) B( β ', λ ) + B( β, λ ) + B( β ', λ ) ]ρ (λ )dλ Becaue B = ± B( β, λ ) B( β ', λ ) + B( β, λ ) + B( β ', λ ) = Bell inequality in the form of Clauer, Horne, Shimony and Holt (CHSH), Phy. Rev. Lett., 3, 880 (969) E (α, β ) E (α, β ' ) + E (α ', β ) + E (α ', β ' )

12 Recent advance in quantum communication Quantum communication in Space QUEST = QUantum Entanglement in Space ExperimenT (ESA) For the propoed International Space Station experiment, the entangled photon would be beamed from orbit to two ditant ground tation, allowing them to communicate uing the quantum key. A. Zeilinger. Oct. 0, 008. Photonic Entanglement and Quantum Information Plenary Talk at OSA FiO/LS XXIV 008, Rocheter, NY.