Multi entanglement in a single-neutron system

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1 Multi entanglement in a single-neutron system Yuji Hasegawa Atominstitut der Österreichischen Universitäten, Vienna, AUSTRIA PRESTO, Japan Science and Technology Agency(JST), Saitama, JAPAN I. Introduction Neutron optical experiments Neutron interferometry II. Multi entanglement in single particle Spin-path entanglement Spin-path-energy entanglement Multi energy-level entanglement III. Summary 7. September

2 Neutron interferometry Neutrons Sample o Phase Shifter χ m = kg s = 1 2 h Ψ II H-Detector µ = J/T τ = 887 s R = 0.7 fm u d d quark structure Ψ I I= Ψ I +e iχ o Ψ II 2 O-Detector 2

3 Neutron interferometers 3

Advantages of the use of neutrons Single-particle (events) Massive, composite system, no fine-structure ~100mg Following Schrödinger-equation E photon ~1eV Pure

4 Advantages of the use of neutrons Single-particle (events) Massive, composite system, no fine-structure ~100mg Following Schrödinger-equation E photon ~1eV Pure single-events (of Ferminons) ~100% detector efficiency E neutron ~1GeV Week (controllable)-coupling with an environment decoherence Storable, e.g. neutron bottle ~100t 4

5 Multi-entanglement in single-particle Muti-entanglement in neutrons bi-entanglement: spin-path tri-entanglementl: spin-path-energy multi-entanglement: energy-levels 5

6 From two-particle to two-space entanglement +1-1 a I II +1 s b -1 Ψ = 1 2 I II + I II Entanglement between Two-Particles 2-Particle Bell-State 2-Space Bell-State Ψ = 1 2 I II + I II Ψ = 1 2 s I p + s II p I, II represent 2-Particles s, p represent 2-Spaces, e.g., spin & path 6

7 Various two-level system 1/2-Spinor s = H int = µ B Larmor precession Two-level atom φ atom = e g H int = ihg k e g a k e ikr g e a k e ikr Rabi oscillation Two-path interferometer Ψ = Ψ I Ψ II H PS = e+iχ 0 0 e iχ ==>> Described by SU(2) Sinusouldal intensity oscillation 7

8 Two-particle vs. two-space entanglement 2-Particle Bell-State 2-Space Bell-State Ψ = 1 2 I II + I II Ψ = 1 2 s I p + s II p I, II represent 2-Particles Measurement on each particle s, p represent 2-Spaces, e.g., spin & path Measurement on each property A I I a = +1 P a;+1 B II II b = +1 P b;+1 I + 1 P a; 1 II + 1 P b; 1 A s α = +1 P α s B p χ = +1 P χ p s + 1 P α+π p + 1 P χ+π where P ξ; ±1 = 1 2 ± eiξ ± e iξ where P φ = 1 2 ϕ +eiφ ϕ ϕ +e iφ ϕ Then, A I,B II =0 Then, A s,b p =0 (Non-)Contextuality ==>> Bell-like inequality (In)Dependent Results for commuting Observables 8 Non-Locality: r I r II for P I(r I) & P II(r II)

9 Contextuality in quantum mechanics Non-contextuality: Independent results: v Aˆ S ( α) Bˆ P ( χ) = v Aˆ S ( α) v Bˆ P ( χ ) ˆ S for measurements of the commuting observables, ( ) ˆ P( ) A α,b χ = 0. Non-locality is one aspect of contextuality ( P I(r I), P II(r II) =0, since r I r II ) In quantum muchanics: Non local correlations are expected Contextual 9

10 Entanglement between two-spaces 2-Space Bell-State Ψ = 1 2 s I p + s II p Subsystems H = H 1 H 2 Ψ = 1 2 I II + I II Y. Hasegawa et al., Nature Vol. 425, Sept. 4,

11 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 s p P χ Ψ E' α 1,χ 1 = ± E' α 1,χ 2 = ± E' α 2,χ 1 = ± E' α 2,χ 2 = ± where α 1 =0 α 2 = 0.50π χ 1 = 0.79π 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 Cf. Max. violaion: S =2.81>2S Y. Hasegawa et al., Nature Vol. 425, Sept. 4,

12 Bell s s inequality & Kochen-Specker theorem 12 Bell s inequality S 2 ( classical) or S = ( quantum) ( ) ( ) ( ) ( ) where S E a, b + E a, b E a, b + E a, b Remark: statistical violation Kochen Specker theorem Contradiction between a hidden variable (HV) theory and the quantum theory: namely, the HV theory assuming (1) all observables have definite values of all time and (2) the values of those variables are intrinsic and independent of the measurement device, i.e., non-contextual hidden variables (NCHVs) v Aˆ v Bˆ = + 1 v Oˆ =± 1& v Bˆ v Cˆ =+ 1 v Cˆ v Aˆ = 1 Remark: no-go theorem, all vs nothing (AVN) results Non-contextual assumption: v AB ˆ ˆ = v Aˆ v Bˆ if A ˆ, Bˆ = 0 A B A C

13 Kochen-Specker experiment with neutrons (1)Contradiction E Xˆ Xˆ = E x y 1 2 Yˆ Yˆ = E E = x y ' Xˆ Yˆ Yˆ Xˆ % E = (2) Violation with statistical probabilities { ( )} ( ) E' = p + p p + p = E E + 1 = x y x y x y (3) Violation with a product observables C' = 1 E E E' = x y 13

Quantum state tomography of entangled 2-qubits2 1 0 0 1 2 2 0 0 0 0 ρ 0 0 0 0 1 0 0

14 Quantum state tomography of entangled 2-qubits ρ Ψ= 1 H H + V V = 2 Re Im D.F. James et al., Phys. Rev. A 64 (2001)

15 Quantum state tomography of neutron s s Bell-state Ψ = I + II 1 real part imaginary part »ØI\» I\»ØII\» II\»ØII\» I\» II\»ØI\ F = Ψ ρ Ψ = 0»ØI\» I\»ØII\ 0.79» II\»ØII\» I\» II\»ØI\ 15

16 Schematic view of the experiment Incident, with spin-turner Ψ = I + 1 II Incident, with spin-turner Ψ = I + 2 II Incident, without spin-turner Ψ = I + 0 II 16

17 Quantum state tomography --- results 0.5 Ψ 1 = I + II Ψ 2 = I + II Ψ 0 = I + II real part »ØI\» I\»ØII\» II\»ØII\» I\» II\»ØI\»ØI\» I\»ØII\» II\»ØII\» I\» II\»ØI\»ÆI\» I\»ÆII\» II\»ÆII\» II\» I\»ÆI\ »ØI\» I\»ØII\ imaginary part» II\»ØII\» I\» II\»ØI\ F=0.785»ØI\» I\»ØII\» II\»ØII\» I\» II\»ØI\ F=0.749»ÆI\» I\»ÆII\» II\»ÆII\» II\» I\»ÆI\ F=

18 Multi-entanglement in single-particle Muti-entanglement in neutrons bi-entanglement: spin-path tri-entanglementl: spin-path-energy multi-entanglement: energy-levels 18

19 Radio-Frequency (RF) Spin-Flipper Flipper: : energy transfer 19

20 Stationary interference-pattern pattern: : energy compensation 20 S. Sponar et al.

21 Experimental setup (1) 21

22 Experimental setup (2) 22

23 Experimental setup (3) 23

24 Experimental setup (4) 24

25 Experimental setup (5) 25

Multi entanglement (GHZ state) Phase shifter ( χ) Spin rotator ( α) Ψ Neutron Ψ = Ψ = ΨI Ψ( E 0) Ψ Neutron { Path Spin Energy 0 0 r Energy ( ) ( ) ( )} iχ iα iγ + e ΨII e e Ψ ( E0 + hωr ) Ψ Path

26 Multi entanglement (GHZ state) Phase shifter ( χ) Spin rotator ( α) Ψ Neutron Ψ = Ψ = ΨI Ψ( E 0) Ψ Neutron { Path Spin Energy 0 0 r Energy ( ) ( ) ( )} iχ iα iγ + e ΨII e e Ψ ( E0 + hωr ) Ψ Path = { ΨI, ΨII } iχ e = N bc λ D : phase shifter where Ψ Spin = {, iα } where e = ω L t : spin rotator iγ e = Ψ ω t = { Ψ: zero ( E ) field, Ψ ( Eprecession + hω r ) } Zero field precession ( γ) 26

27 Mermin s inequality for GHZ state Neutron's GHZ-state { Ψ Ψ Ψ( E ) GHZ = I 0 +Ψ Ψ ( E + hω ) II 0 r } Mermin'sinequalityfor GHZ-state M 2 according to non contextual theory NC where M E σ σ σ E σ σ σ p s e p s e x x x x y y E σ σ σ E σ σ σ p s e p s e y x y y y x Relative phases are manipulated: { Ψ Ψ Ψ( E ) Neutron = I 0 iχ iα iγ ( e II ) ( e ) ( e ( E0 hωr ) )} + Ψ Ψ + iχ e = N bc λ D : phase shifter iα where e = ω L t : spin rotator iγ e = ω t : zero field precession r In contrast quantum theory predicts M Quantum = 4 for Ψ GHZ Phase shifter ( χ) Spin rotator ( α) Zero field precession ( γ) 27

Neutron ΨI Ψ( E0) i i i ( e II ) ( e ) (

28 Mermin s inequality: result 1 Phase shifter ( χ) Spin rotator ( α) Ψ = { Neutron ΨI Ψ( E0) i i i ( e II ) ( e ) ( e ( E0 r) )} χ α γ hω + Ψ Ψ + iχ e = N b λ D: phase shifter c iα where e = ω t : spin rotator L iγ e = ω t : zero field precession r Zero field precession ( γ) α=0 α=π/2 α=π α=3π/2 R. Loidl et al. 28 γ=0 γ=π/2 γ=π γ=3π/2

29 Mermin s inequality: result 2 Mermin'sinequalityfor tri-ghz-state M 2 according to non contextual theory NC where In contrast quantum theory predicts M Quantum M E σ σ σ E σ σ σ = 4 for Ψ p s e p s e x x x x y y E σ σ σ E σ σ σ p s e p s e y x y y y x GHZ Phase shifter ( χ) Spin rotator ( α) We obtained the values: E E E E Finally, M σ σ σ = p s e x x x p s e σ σ σ = x y y p s e σ σ σ = y x y p s e σ σ σ = Measured y y x = 2.62 ± 0.08 > 2 R. Loidl et al. Zero field precession ( γ) 29

30 Multi entanglement: discussions1 Energyshifts: E E shift shift ( f = 57MHz) = 0.47µeV c ( f = 58kHz) = 0.48neV 58kHz B.Alefeld et al. Z. Phys B41 (1981)

31 Multi entanglement: discussions2 Multi degrees-of-freedoms entanglement Ψ = Ψ Ψ Ψ Neutron {, } Path I II Spin Path Ψ = Ψ Ψ { } where Ψ =, Spin { ( E ), ( E hω r ) } Ψ = Ψ Ψ +... Energy Energy Energy Multi energy-level entanglement Ψ=ΨEnergy ΨEnergy ω ω Ψ ω { E hω E hω } where Ψ Energy = Ψ( 0 ), Ψ ( 0+ ) ω n Bell s inequality Energy + hω 2 + hω 1 hω 2 E 0 + hω 3 hω ω ω2 ω ω ω2 ω ω ω2 ω3 1 + ω ω2 ω3 hω 1 M M M M M M M M M M M M 1 31

32 Investigations with neutrons: entanglement Muti-entanglement in neutrons bi-entanglement: spin-path tri-entanglement: spin-path-energy multi-entanglement: energy-levels Co-workers: S.Sponar, J.Klepp, R.Loidl, S.Filipp H. Rauch 32

33 33

Neutron optical experiments exploring foundamental quantum-phenomena

Neutron optical experiments exploring foundamental quantum-phenomena 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