Phases and Entanglement in Neutron Interference Experiments

Transcription

1 Phases and Entanglement in Neutron Interference Experiments S. Filipp, J. Klepp, H. Lemmel, S. Sponar Y. Hasegawa, H. Rauch Atominstitut der Österreichischen Universitäten QMFPA, December 26 p. 1/34

2 Outline Neutron Interferometry Basics Interferometry Polarimetry UCNMR Phases Confinement induced phase Geometric Phase (spin, spatial, mixed, stability) Dephasing Spin-Path Entanglement State Tomography Bell Inequality Test - Quantum Contextuality QMFPA, December 26 p. 2/34

3 Neutron Interferometry Self interference of massive particle (phase space density 1 14 ) H. Rauch, W. Treimer and U. Bonse, PLA (1974). QMFPA, December 26 p. 3/34

4 Neutron Interferometry Time-Independent Schrödinger Equation: h 2 2m [ + 2m h 2 ( E V( r)) ] ψ( r) = QMFPA, December 26 p. 4/34

5 Neutron Interferometry Time-Independent Schrödinger Equation: h 2 2m [ + 2m h 2 ( E V( r)) ] ψ( r) = Wavefunction: ψ( r) = 1 (2π) 3/2 a( k)e i k r d 3 k ψ spin QMFPA, December 26 p. 4/34

6 Neutron Interferometry Time-Independent Schrödinger Equation: h 2 2m Wavefunction: ψ( r) = 1 (2π) 3/2 Typical Potentials V( r): a( k)e i k r d 3 k ψ spin [ + 2m h 2 ( E V( r)) ] ψ( r) = Scalar Fermi Potential (Nuclear Interaction): V N ( r) = 2π h2 m phase shift ψ e inbcλd ψ b cn Vector Potential (Magnetic interaction): V M ( r) = µ n B Spinor rotation ψ spin U ψ spin, (U = e i ω L t 2 σ n SU(2)) QMFPA, December 26 p. 4/34

7 Neutron Interferometry Time-Independent Schrödinger Equation: h 2 2m Wavefunction: ψ( r) = 1 (2π) 3/2 Typical Potentials V( r): a( k)e i k r d 3 k ψ spin [ + 2m h 2 ( E V( r)) ] ψ( r) = Scalar Fermi Potential (Nuclear Interaction): V N ( r) = 2π h2 m phase shift ψ e inbcλd ψ b cn Vector Potential (Magnetic interaction): V M ( r) = µ n B Spinor rotation ψ spin U ψ spin, (U = e i ω L t 2 σ n SU(2)) I I = 1 2 ψ τ + e iη ψ I = 1 + ν cos(η + Φ) (ν =.6, Φ = 6 ) I = 1 + cosη = 1+ ψ ψ τ cos(η argψ }{{} ψ τ) }{{} ν Φ 1..5 Φ H. Rauch and S. A. Werner, Neutron Interferometry Clarendon Press, Oxford (2). ν η [ ] QMFPA, December 26 p. 4/34

8 Polarimetry Spin superposition, instead of different paths ( +x = 1 2 ( +z + z )) π 2 -flip = beamsplitter Phase is encoded in the final polarization Intensity: I = +z Û π/2 out Û 2nπ η L Û (ξ,δ,ζ) Û η L Ûπ/2 in +z 2 Measured phase: φ arg ψ ψ = arg +z Û +z QMFPA, December 26 p. 5/34

9 UCNMR polarized Ultra Cold Neutrons (UCN s) low velocity 5m/s (1 7 ev ) Fermi potential e. g ev for stainless steel reflection at material wall neutron bottle wrap coils around for magnetic field polarimetric measurement 1) vacuum chamber 2) Helmholtz coils 3) storage vessel 4) shutter z x y 5) neutron guide QMFPA, December 26 p. 6/34

10 Confinement Induced Phase Shift H. Rauch and H. Lemmel. Measurement of a confinement induced neutron phase. Nature 417, (22) transverse confinement discrete transverse energy levels increase of transverse momentum decrease of longitudinal momentum phase shift neutron wavelength λ = m neutron energy E =.23eV level energies E n =.42, 1.7, 3.7,...peV wall potential V = Nb c 2π h 2 /m 54neV wall material silicon channel width a = 2µm channel length L = 2cm number of channels 25 J. M. Lévy-Leblond. Phys. Lett. A 125, (1987); D. M. Greenberger, Physica B 151, (1988) QMFPA, December 26 p. 7/34

11 Measured phase shift incoming beam monochromator phase shifter channel stack perfect silicon interferometer 5 cm H beam O beam Calculated Phase Shift: φ = 2.5 Measured Phase Shift: φ = 2.8(4) QMFPA, December 26 p. 8/34

12 From falling cats... A falling cat always lands on its feet. R. Montgomery, Commun. Math. Phys. 128, 565 (199). QMFPA, December 26 p. 9/34

13 ...to Quantum Geometric Phases Adiabatic, Cyclic Evolution of eigenstates Berry phase [Proc. R. S. Lond. A 392, 45 (1984)] H(R(t)) n(r(t)) = E n (t) n(r(t)) B()=B( ) n() = n(τ) ψ(τ) = e iφ d e iφ g(c) n(r(τ)) φ d (τ) = 1 h τ dte n (t) φ g (C) = dr n(r) R n(r) Ω B(t) n(t) φ g = Ω 2 QMFPA, December 26 p. 1/34

14 Geometric Phase in a Neutron Interferometer Spin-Flipper in one path rotates spin about axis n in x-y-plane Direction of axis depends on the phase j i θ=π n j i QMFPA, December 26 p. 11/34

15 Geometric Phase in a Neutron Interferometer Spin-Flipper in one path rotates spin about axis n in x-y-plane Direction of axis depends on the phase In addition: Photon exchange hω = 2µB E kin j i j i j i hω j i RF-Flip j i j i 2hω QMFPA, December 26 p. 11/34

16 Geometric Phase in a Neutron Interferometer Spin-Flipper in one path rotates spin about axis n in x-y-plane Direction of axis depends on the phase In addition: Photon exchange hω = 2µB 1 st spin flipper: sinωt; 2 nd spin flipper: sin( ωt 2 + φ) Different frequencies: Compensation of different energies QMFPA, December 26 p. 11/34

17 Orange-slice geometric phase Alltogether: Orange-slice shaped path geometric phase 3 " " Geometric Phase (deg) C Ω ' RF Phase (deg) 25 3 # # S. Sponar, J. Klepp, R. Loidl, S. Filipp and Y. Hasegawa (in preparation). QMFPA, December 26 p. 12/34

18 Spatial geometric phase ψ r BS1 BS2 PS1 (η) p ψ t BS3 BS4 A(T) p PS2 ( χ) BS5 BS6 D O 2-loop interferometer, unpolarized neutrons Geometric phase due to spatial degrees of freedom (2-level system) Phase Shifts e iχ 1, e iχ 2; Aborber (Transmission T ) QMFPA, December 26 p. 13/34

19 Spatial geometric phase ψ r BS1 BS2 PS1 (η) p ψ t BS3 BS4 A(T) p PS2 ( χ) BS5 BS6 D O 2-loop interferometer, unpolarized neutrons Geometric phase due to spatial degrees of freedom (2-level system) Phase Shifts e iχ 1, e iχ 2; Aborber (Transmission T ) I = ψ r ψ r + ψ t f ψ t f +2 ψ r ψ t f cos(η arg ψ r ψ t f ) Parallel transport condition ( ψ(ξ) = e icd 1ξ p + Te icd 2ξ p ): ψ(ξ) ψ(ξ+δ) R e icd 1δ + Te icd 2δ = N ϕ(s) e iá s ϕ(s+ s) eiá d 1 = T d 2 Φ d = [ )( Φ g = arg ψ r ψ t f = χ )] 1+χ 2 2 arctan tan( χ2 χ 1 1 T T 2 P C QMFPA, December 26 p. 13/34

20 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p Superposition of Path 1 ( p ) & Path 2 ( p due to beamsplitter, absorber (T ) and phase shifter ( χ = χ 2 χ 1 ): ψ t f p +e i χ T p S. Filipp, Y. Hasegawa, R. Loidl, and H. Rauch. PRA (R) (25) (eprint: quant-ph/41238) QMFPA, December 26 p. 14/34

21 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p ψ t = p (1) QMFPA, December 26 p. 14/34

22 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p ψ t = 1 2 ( p + p ) (2) QMFPA, December 26 p. 14/34

23 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p ψ t = 1 2 ( p + T p ) (3) QMFPA, December 26 p. 14/34

24 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p ψ t f = 1 2 ( p +e i χ T p ) (4) QMFPA, December 26 p. 14/34

25 Spatial geometric phase - Paths p p D O ψ r BS2 BS4 BS6 BS1 PS1 (η) p ψ t BS3 A(T) p PS2 ( χ) BS5 q q θ χ p p ψ t f (1+ei χ T)( p + p ) ψ r e i χ T( p + p ) I = e iη ψ r + ψ t f 2 = ψ r ψ r + ψ t f ψ t f +2 ψ r ψ t f cos(η arg ψ r ψ t f ) Φ = arg ψ r ψ t f = χ 1 + χ 2 2 [ ( ) ( χ 1 T arctan tan 2 1+ T )] (5) QMFPA, December 26 p. 14/34

26 Results - 4.1mm/.5mm & 4.1mm/4.1mm Π Π 3 2 Π 2Π 5 2 Π φ measured g χ 1 /χ 2 = 4.1/.5: =.665 ±.1; φ analytic g = p p Π Π 3 2 Π 2Π 5 2 Π 3Π Χ 1 2 Π Π 3 2 Π 2Π q q θ χ Π Π 3 2 Π 2Π 5 2 Π Χ p p χ 1 /χ 2 = 4.1/4.1: φ measured g = ±.17 rad, φ analytic g = π QMFPA, December 26 p. 15/34

27 Mixed state phase - Polarimetry J. Klepp, S. Sponar, Y. Hasegawa, E. Jericha, G. Badurek. PLA (25). ψ ϱ = i (1-r i ) ψ i ψ i 2 φ g φ ρ argtr[uρ] = arg i p i ψ i U ψ i QMFPA, December 26 p. 16/34

28 Mixed state phase - Polarimetry J. Klepp, S. Sponar, Y. Hasegawa, E. Jericha, G. Badurek. PLA (25). ψ ϱ = i (1-r i ) ψ i ψ i 2 φ g φ ρ argtr[uρ] = arg i p i ψ i U ψ i z+ z+ ϱ r Uϱ z- z- Larsson and Sjöqvist, PLA (23) QMFPA, December 26 p. 16/34

29 Production of mixed states Electronic Random Noise fluctuations produces dephasing. QMFPA, December 26 p. 17/34

30 Results Mixed state phase φ ρ : I φ ρ = arccos min (1 r)/2 ( ) I min (1 r)/2+r 2 (1+r)/2 I max Φ[rad] 3π/8 π/4 π/8 Ω=3π/4 Ω=π/2 Ω=π/4 z+ z+ ϱ r Uϱ intensity [arb. units] r=.987(4) r=.765(7) r=.69(6) r=.325(4) r=.124(2)..2.4 r z- z coil position [mm] 5 6 J. Klepp, S. Sponar, Y. Hasegawa (in preparation). QMFPA, December 26 p. 18/34

31 Dephasing... Noise in one interferometer arm leads to dephasing. [M. Baron, H. Rauch and M. Suda, J. Opt. B 5, 5241 (23).] QMFPA, December 26 p. 19/34

32 ... and how to fight it By applying equal magnetic fields in both arms interference fringes can be retrieved. QMFPA, December 26 p. 2/34

33 Geometric dephasing DeChiara and Palma [PRL 91, 944 (23)]: δb(t) B(t) Adiabatic evolution with small fluctuations of the direction of the magnetic field Perturbations in the magnetic field induce deviations in phase, but φ g is stabilizing for increasing time Basic ingredients: Spin-1/2 particles (neutrons) + controllable 3D- magnetic field. QMFPA, December 26 p. 21/34

34 Numerics Variance of geometric phase vanishes for long evolution time: δb(t) B(t) Σg.4.2 Σ n Ω rad s rad s T s QMFPA, December 26 p. 22/34

35 Numerics Variance of geometric phase vanishes for long evolution time: δb(t) B(t) Σg.4.2 Σ n Ω rad s rad s T s Mean Geometric Phase φ g equal to noisefree geometric phase (1 st order). But: φg theory s r 112 s r 35 s r 11 s r 3.5 f(x) f(φ) φ 1 Φg Φg rad 5.3 x T[s] Π 2 Θ rad Π QMFPA, December 26 p. 22/34

36 Setup PF2, ILL, Grenoble: QMFPA, December 26 p. 23/34

37 Spin superposition Ramsey - Fringes: (Guide-Field: 89 mg, ω = 1829(1) rad/s) pol. neutrons/duty cycle t [ms] 1 15 Decoherence still too dominant (T 2 = 2(5) ms)! QMFPA, December 26 p. 24/34

38 Spin-Path Entanglement Photons 2-particle entanglement Ψ = 1 2 ( H 1 V 2 + V 1 H 2 ) photo: Kwiat and Reck QMFPA, December 26 p. 25/34

39 Spin-Path Entanglement Photons 2-particle entanglement Neutrons single particle spin-path entanglement, paths in interferometer analogous to 2 nd particle Ψ = 1 2 ( H 1 V 2 + V 1 H 2 ) Ψ = 1 2 ( I + II ) photo: Kwiat and Reck QMFPA, December 26 p. 25/34

40 State Tomography Matrix-representation of Bell state Ψ + = 1 2 ( I + II ) Ψ + Ψ + = Basis: { I, II, I, I } In general, a (not normalized) bipartite state is determined by 16 parameters QMFPA, December 26 p. 26/34

41 Measurement Scheme Observables: Projective Measurements P a = 1/2(1+ a a ) { χ =, π 2 P p +x,pp +y { α =,π P s +z,p s z Path: beamblock I,II P p +z,pp z Spin: α = π/2 P s +x,ps +y Example: < P p +x Ps ±z >,< Pp +y Ps ±z > < Pp ±z Ps +x >,< Pp ±z Ps ±z > 16 combinations QMFPA, December 26 p. 27/34

42 Measurement Schemes Three different states: 1. Ψ 1 = +x I + x II 2. Ψ 2 = x I + +x II 3. Ψ 3 = +x ( I + II ) QMFPA, December 26 p. 28/34

43 Results Ψ 1 = +x I + x II Ψ 2 = x I + +x II Ψ 3 = +x ( I + II ) F=.785 F=.749 F=.98 Y. Hasegawa, J. Klepp, S. Filipp, R. Loidl and H. Rauch (in preparation). QMFPA, December 26 p. 29/34

44 Bell Inequalities z z E 4 E 2 θ 5 E1 θ1 θ 2 x E 3 θ 4 x Entanglement leads to violation of Bell-type Inequality spin-path entangled state also violates a Bell Inequality CHSH- Inequality: 2 < S = E(E 1,E 3 )+E(E 1,E 4 )+E(E 2,E 3 ) E(E 2,E 4 ) < 2 E(E i,e j ) = Ψ (n i σ) (n j σ) Ψ = N i j(+,+)+n i j (, ) N i j (+, ) N i j (,+) N i j (±,±) N i j (±,±) = Ψ P s i Pp j Ψ QMFPA, December 26 p. 3/34

45 Bell Inequality - Observables Photons: Polarization of particle 1 and 2 along n 1,2 -axis Neutrons: Spin-polarization along n-axis and which-path by rotation of phase shifter Quantum Mechanics predicts non-local/contextual correlations Smax = 2 2 (Tsirelson Bound) QMFPA, December 26 p. 31/34

46 Measurements Path: Spin: QMFPA, December 26 p. 32/34

47 Violation of BI with Neutrons S = 2.51 ±.19 Hasegawa, Loidl, Badurek, Baron and Rauch. Nature 425 (23) 45. QMFPA, December 26 p. 33/34

48 Summary and Outlook Measurement of phases with different neutron interferometry techniques: Interferometer: Confinement induced phase shift, Spinor geometric phase, Spatial geometric phase, Dephasing Polarimeter: Mixed state geometric phase UCNMR: Dephasing of the geometric phase Spin-Path entanglement State tomography Bell inequality test Forthcoming experiments: Dephasing of the geometric phase Geometric phase and entanglement Quantum Contextuality - Kochen-Specker-like contradictions Irreversibility Phase of non-unitary transformation QMFPA, December 26 p. 34/34