Neutron optical experiments exploring foundamental quantum-phenomena

Transcription

1 Neutron otical exeriment exloring foundamental quantum-henomena Yuji HASEGAWA Atomintitut der Öterreichichen Univeritäten, Wien, AUSTRIA PRESTO, Jaan Science and Technology Agency, JAPAN 1. Neutron interferometer/olarimeter 2. Recent neutron otical exeriment 2-1 Quantum contextuality Violation of a Bell-like inequality Kochen-Secker-like aradox 2-2 Quantum tate tomograhy 2-3 Geometric hae Non-cyclic atial geometric hae Geometric hae for mixed tate 3. Summary Claical and quantum inteterference 2.Oct. 25, Palacký Univ., Olomouc, Czech Re. 1

2 Neutron interferometry Neutron Samle o Phae Shifter χ m = kg = 1 2 h Ψ II H-Detector µ = J/T τ = 887 R =.7 fm u d d quark tructure Ψ I I= Ψ I +e iχ o Ψ II 2 O-Detector 2

3 Neutron interferometer 3

4 Larmor interferometry neutron olarimeter Samle o O ±Y = e ±iβ ±Y O +Z = O 1 2 +Y + Y 2β = 1 2 eiβ +Y +e iβ Y = eiβ 2 +Y +e 2iβ Y 4

5 Neutron otical exeriment exloring foundamental quantum-henomena Yuji HASEGAWA Atomintitut der Öterreichichen Univeritäten, Wien, AUSTRIA PRESTO, Jaan Science and Technology Agency, JAPAN 1. Neutron interferometer/olarimeter 2. Recent neutron otical exeriment 2-1 Quantum contextuality Violation of a Bell-like inequality Kochen-Secker-like aradox 2-2 Quantum tate tomograhy 2-3 Geometric hae Non-cyclic atial geometric hae Geometric hae for mixed tate 3. Summary Claical and quantum inteterference 21. Oct. 25, Palacký Univerity, Olomouc, Czech Reublic 5

6 From two-article to two-ace entanglement +1-1 a I II +1 b -1 Ψ = 1 2 I II + I II Entanglement between Two-Particle 2-Particle Bell-State 2-Sace Bell-State Ψ = 1 2 I II + I II Ψ = 1 2 I + II I, II rereent 2-Particle, rereent 2-Sace, e.g., in & ath (Non-)Contextuality ==>> Bell-like inequality (In)Deendent Reult for commuting Obervable 6

7 Bell-like like inequality with E (α,χ) 2 S' 2 with S' = E' α 1,χ 1 E' α 1,χ 2 +E'α 2,χ 1 +E'α 2,χ 2 where E' α,χ = N' ++ α,χ + N' α,χ N' + α,χ N' + α,χ N' ++ α,χ +N' α,χ +N' + α,χ +N' + α,χ = N' ++ α,χ +N' ++ α+π,χ+π N' ++ α,χ+π N' ++ α+π,χ N' ++ α,χ +N' ++ α+π,χ+π +N' ++ α,χ+π +N' ++ α+π,χ N' ±± α,χ = Ψ P α; ±1 P χ; ±1 Ψ Prediction by quantum theroy N' ++ qm α,χ = co α+χ a well a E' α,χ = co α+χ, then, S' = 2 2 = 2.82 > 2 for α 1 = π 2, α 2 = χ 1 = π 4, χ 2 = π 4 Remark: contrat! C >71% 7

8 Contextuality in quantum mechanic Non-contextuality: Indeendent reult: v A B = v A v B for meaurement of the commuting obervable, A,B =. Non-locality i one aect of contextuality ( P I(r I), P II(r II) =, ince r I r II ) In quantum muchanic: Non local correlation are exected Contextual 8

9 Schematic view of the exeriment 9

10 Violation of a Bell-like like inequality E' α,χ = N' ++ α,χ +N' ++ α+π,χ+π N' ++ α,χ+π N' ++ α+π,χ N' ++ α,χ +N' ++ α+π,χ+π +N' ++ α,χ+π +N' ++ α+π,χ where N' ++ α,χ = Ψ P α P χ Ψ E' α 1,χ 1 =.542 ±.7 E' α 1,χ 2 =.488 ±.12 E' α 2,χ 1 =.538 ±.6 E' α 2,χ 2 =.483 ±.12 where α 1 = α 2 =.5π χ 1 =.79π χ 2 = 1.29π ===>>> S' E' α 1,χ 1 +E'α 1,χ 2 E'α 2,χ 1 +E'α 2,χ 2 = 2.51 ±.19 > 2 Cf. Max. violaion: S =2.81>2S Y. Haegawa et al., Nature Vol. 425, Set. 4, 23 1

11 Non-locality & contextuality

12 Theory (1) Bell-tate for in & ath entanglement Ψ = 1 2 Rem.: I + II X 1 X 2 Ψ = Ψ Y 1 Y 2 Ψ = Ψ X 1 Y 2 Ψ = Y 1 X 2 Ψ Product Obervable Pauli-tye obervable for in and ath X 1 = x =A = +1 P +x X 2 = x =B = +1 P +x Y 1 = y =A π 2 = +1 P +y Y 2 = y =B π 2 = +1 P +y Here, X 1,X 2 =, Y 1,Y 2 = X 1,Y 2 =, Y 1,X 2 = + 1 P x + 1 P x + 1 P y + 1 P y C I + X 1 X 2 + Y 1 Y 2 X 1 Y 2 Y 1 X 2 The following equality hold for thee arameter C=1+v X 1 X 2 + v Y 1 Y 2 v X 1 Y 2 v Y 1 X 2 where v X j = ±1 & v Y j = ±1 12

13 Theory (2) Non-Contextuality Hidden Variable (NCHV) theory lead to C NCHV =1+v X 1 v X 2 + v Y 1 v Y 2 v X 1 v Y 2 v Y 1 v X 2 = v X 1 v X 2 1 v Y 1 v Y v Y 1 v Y 2 = ± 2 Then, C NCHV = 1 + X 1 X 2 + Y 1 Y 2 X 1 Y 2 Y 1 X 2 2 C QM =4 Indeendent reult for the commuting obervable, X 1,X 2 =, Y 1,Y 2 =, X 1,Y 2 =, Y 1,X 2 = Quantum mechanic redict C QM = Ψ I + X 1 X 2 + Y 1 Y 2 X 1 Y 2 Y 1 X 2 Ψ =1+ Ψ X 1 X 2 Ψ + Ψ Y 1 Y 2 Ψ Ψ X 1 Y 1 Y 2 X 2 Ψ =1+ Ψ X 1 X 2 Ψ =4 + Ψ Y 1 Y 2 Ψ + Ψ Z 1 Z 2 Ψ 13

14 Exerimental etu Magnetic Prim B Mu-Metal Sin-Turner II ƒ (Guide Field) Phae Shifter (c) Helmholtz Coil H-Detector B I ƒ Æ Beam toer Neutron Interferometer B Larmor Accelerator Sin Rotator (a) Heuler Crytal O-Detector 14

15 Intenity ( neutron /12 ) Reult Intenity (neutron/12) P +x Exerimental Reult (a) Ex -com onent (b) Ey -com onent (c) Ez-com onent (i) x-comonent I (ii) y-comonent (iii) z-comonent +x> I I +y> I > 5 π P π x 2π 3π Phae hift, χ ( rad) π π 2π 3π C x = Ψ X 1 X 2 Ψ = Ψ P +x Ψ P +x P χ= Phae hift, χ (rad) Ψ + Ψ P x P χ=π =.61 ±.8 I x> Intenity ( neutron /12 ) Ψ Ψ P x P χ=π Ψ P v= C y = Ψ Y 1 Y 2 Ψ Ψ Intenity (neutron/12) = Ψ P +y Ψ P +y P χ= π 2 P χ= 3π 2 =.667 ±.8 I y> Ψ + Ψ P y P +y P y π π 2π 3π Phae hift, χ ( rad) π π 2π 3π Phae hift, χ (rad) Ψ Ψ P y Intenity ( neutron /12 ) P χ= 3π Ψ P χ= π 2 Intenity (neutron/12) π π π 2π P II Sinor rotation, α ( rad) 2π π π 2π C z = Ψ Z 1 Z 2 Ψ Ψ = Ψ P χ= +π 2 Ψ P χ= π Sinor rotation, α (rad) P I π P 2 II =.861 ±.1 I II> Ψ + Ψ P χ= π P I 2 Ψ Ψ P χ= +π P II Ψ +π P 2 I Ψ C meaured =1+C x +C y +C z = ±.15 > 2 15

16 Contradiction in violation? Ex =.61 Ey =.667 } Eexected = +.47 (63%) Emeaured = Ideally Eex = +1 erfect contradiction Emr = -1 16

17 Neutron otical exeriment exloring foundamental quantum-henomena Yuji HASEGAWA Atomintitut der Öterreichichen Univeritäten, Wien, AUSTRIA PRESTO, Jaan Science and Technology Agency, JAPAN 1. Neutron interferometer/olarimeter 2. Recent neutron otical exeriment 2-1 Quantum contextuality Violation of a Bell-like inequality Kochen-Secker-like aradox 2-2 Quantum tate tomograhy 2-3 Geometric hae Non-cyclic atial geometric hae Geometric hae for mixed tate 3. Summary Claical and quantum inteterference 21. Oct. 25, Palacký Univerity, Olomouc, Czech Reublic 17

18 Quantum tate tomograhy of entangled 2-qubit ρ Im Ψ= 1 H H + V V = 2 Re D.F. Jame et al., Phy. Rev. A 64 (21)

19 Exerimental etu { } { } +x +y ±z +x +y ±z,,,, :16-variable to be meaured! Magnetic Prim B B Mu-Metal Sin-Turner II ƒ (Guide Field) Phae Shifter (c) Helmholtz Coil H-Detector I ƒ Æ Beam toer Neutron Interferometer B Larmor Accelerator Sin Rotator (a) Heuler Crytal χ = & π / 2( + x Path I( + z ) Path II( z ) & + y ) α = & π / 2( + x & u in( + z ) down in( z ) & + y ) O-Detector 19

20 Tyical reult --- χ ocillation at ±z +z Intenity (neutron/12) ,5-1, -,5,,5 1, 1,5 Rotation of hae hifter (deg.) i + x + z + x -z i & i i + y + z + y -z are determined. z Intenity (neutron/12) ,5-1, -,5,,5 1, 1,5 Rotation of hae hifter (deg.) 2

21 Tyical reult --- α ocillation at ±z +z Intenity (neutron/12) Current for inor rotation (ma) i +z +x -z +x i & i i +z ±z -z ±z are determined. z Intenity (neutron/12) Current for inor rotation (ma) 21

22 Analyi + z + z + y + z + x + z x + z + z z + y z ( ) (1) = in9, ( ) (2) = co9, ( ) (3) = in88, ( ) (4) = in88, ( ) (5) = in19, ( ) (6) = co19 + x z x z ( ) (7) = in88, ( ) (8) = in88,, + z + x + y + x + x + x x + x + z + y + y + y ( ) (9) = co11, ( ) (1) = in6, ( ) (11) = in11, ( ) (12) = in11, ( ) (13) = in11, ( ) (14) = co 6, + x + y x + y ( ) (15) = in11, ( ) (16) = in11, Thee 16-value denity matrix, ρ 22

23 > > IQuantum tate tomograhy of neutron Bell-tate real art Ψ = + 1 I II imaginary art,4,3,2,1,4,3,2,1, > I >> I >, > I I> >> II > I > I > II> > I>,5 >I> >-,1 >> I II> >> II > > II> > I>,5 > I> II>79 F = Ψ ρ Ψ =. 23

24 Schematic view of the exeriment Incident, with in-turner Ψ = I + 1 II Incident, with in-turner Ψ = I + 2 II Incident, without in-turner Ψ = I + II 24

25 25 imaginary art real art Quantum tate tomograhy Quantum tate tomograhy comarion comarion > II> > II> > I> > I>,,1,2,3,4,5 > I> > I> > II> > II> > II> > II> > I> > I>,,1,2,3,4,5 > I> > I> > II> > II> > II> > II> > I> > I>,,1,2,3,4,5 > I> > I> > II> > II> > II> > II> > I> > I>,,1,2,3,4,5 > I> > I> > II> > II> > II> > II> > I> > I>-,1,,1,2,3,4,5 > I> > I> > II> > II> > II> > II> > I> > I>-,1,,1,2,3,4,5 > I> > I> > II> > II>1 I II Ψ = + 2 I II Ψ = + I II Ψ = + F=.79 F=.75 F=.91

26 Neutron otical exeriment exloring foundamental quantum-henomena Yuji HASEGAWA Atomintitut der Öterreichichen Univeritäten, Wien, AUSTRIA PRESTO, Jaan Science and Technology Agency, JAPAN 1. Neutron interferometer/olarimeter 2. Recent neutron otical exeriment 2-1 Quantum contextuality Violation of a Bell-like inequality Kochen-Secker-like aradox 2-2 Quantum tate tomograhy 2-3 Geometric hae Non-cyclic atial geometric hae Geometric hae for mixed tate 3. Summary Claical and quantum inteterference 21. Oct. 25, Palacký Univerity, Olomouc, Czech Reublic 26

27 Geometric hae Cyclic evolution: Ψ T =e iφ Ψ Rewriting with a eriodic function, ( φ T = φ ) Ψ t =e i f (t) φ t with f (T) f () = φ Pancharatnam ( 56) Berry ( 84) Simon ( 83) Aharonov & Anandan ( 87) Samuel & Bhandari ( 88) Multilying and integrating the Schrödinger equation, Ψ t H Ψ t dt =ih Ψ t d dt Ψ t dt Thu, one obtain total hae φ = 1 h = δ + γ T Ψ t H Ψ t dt T +i φ t d dt φ t dt dynamical geometric 27

28 Geometric hae for 1/2-in ytem Examle: 1/2-in of neutron in a magnetic field Ψ T = ex iµbt h co( θ 2 θ 2 ) ex iµbt h in( θ 2 ) θ: olar angle from the direction of the magnetic field B geodeic For a eriodic evolution, ( T= nπh µb ) δ =nπ coθ γ =nπ 1 coθ dynamical geometric θ γ=ω/2 28

29 Geometric hae in lit-beam exeriment Y.Haegawa et al., PRA53 (1996)

30 Non-cyclic evolution Cyclic evolution Non-cyclic evolution 3

31 Exerimental reult (1) Rotation of PS2 (deg) 31

32 Exerimental reult (2) Cyclic evolution Non-cyclic evolution S. Fili, Y. Haegawa, R. Loidl, and H. Rauch, quant-h/41238; PRA72 (25) 2162(R) 32

33 Geometric hae for mixed tate z y x Z DETECTOR I = + z Ux( π / 2) POLARIZER U z( η) U P-FLIPPER U z(2π n η) Ux( π / 2) + z L B X /2 SPIN-TURNER 2 L エ L = L+ L エ ANALYZER -B X /2 SPIN-TURNER B a SU(2)-COILS (ξ,δ,ζ) ˆ e U φ = arg = arcco U e I coξ I I min max e iδ iζ = iζ iδ inξ e + I inξ coξ min J. Kle, S. Sonar, Y. Haegawa, E. Jericha and G. Badurek, quant-h/5529; PLA 342 (25) = co 2 ξ co 2 δ + in 2 ξ in 2 ( ζ η) I I min max = co = co 2 2 ξ co ξ co 2 2 δ, δ + in 2 ξ

34 Exerimental reult ψ () ψ () ψ (τ ) ψ (τ ) 34

35 Work in rogre --- new ohiticated etu z y x Z DETECTOR POLARIZER P-FLIPPER Lエ L L = L+ Lエ B X /2 SPIN-TURNER ANALYZER -B X /2 SPIN-TURNER B a SU(2)-COILS (ξ,δ,ζ) 35

36 Work in rogre --- roduction of mixed tate (1) Random arrival time: B(trnd) P(t) (2) Random magnetic field: Brnd(t) P(t) (3) Mixture of ure and comlete mixed tate (Werner tate analogy): Ψure><Ψure +(1 ) diag(1/2,1/2) (4) Inhomogenou magnetic field: B(r) P(r) B P 1 = P (t)dt T (3) P MONOCHROMATOR RANDOM LY M AGNETIZED FILM (1) B oz SO URC E z y PO LARIZED NEUTRON BEAM PO LARIZER (4) x 36

37 Summary Quantum contextuality: violation of a Bell-like inequality Kochen-Secker-like aradox Quantum tate tomograhy: Thee tomograhical technique will be alied to, decoherence/deolarization roce, characterization of robutne of geometric hae, non-unitary evolution of neutron entangled tate, etc. eecially from the olarized incident! Geometric hae: Error tolerant and robut hae 37

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