Beyond Heisenberg s uncertainty principle:

Transcription

1 Theory München 21. May 2013 Beyond Heisenberg s uncertainty principle: Error-disturbance uncertainty relation studied in neutron s successive spin measurements Yuji HASEGAWA Atominsitut, TU-Wien, Vienna, AUSTRIA I. Introduction: neutron optical experiments II. Uncertainty relation for error-disturbance V. Summary

2 The neutron Particle Feels four-forces Wave m = (1) x kg s = 1 2 = (18) x J/T = 887(2) s R = 0.7 fm = 12.0 (2.5) x10-4 fm3 u - d - d - quark structure CONNECTION de Broglie B H( h m. v Schrödinger r,t) i ( r,t) t & boundary conditions c h = (20) x m m.c For thermal neutrons = 2 Å, 2000 m/s, 20meV h B = 1.8 x m m.v 1 c 10-8 m 2k p v. t 10-2 m v (5) x 106 m d 0 2 (4) m... mass, s... spin,... magnetic moment, c... Compton wavelength, B decay lifetime, R... (magnetic) confine- debroglie wavelength, c... ment radius,... electric polarizability; all other -B coherence length, p... packet measured quantities like electric charge, magnetic two level system length, d... decay length, k. monopole and electric dipole moment are com- momentum width, t... chopper patible with zero B opening time, v... group velocity, phase.

3 Neutrons s in quantum mechanics Particle and wave properties p = mv = h/λ (L. De Broglie) Schroedinger equation ( r, t) i H( r, t) t (E. Schrödinger) Uncertainity Δx Δp h/4π M.C.Escher, 1938 (W. Heisenberg)

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

5 Neutron interferometer

6 Neutron interferometer family

7 First results First results TRIGA reactor Vienna 1974 detectors interferometer X-ray system Graphite monochromator H.Rauch, W.Treimer, U.Bonse, Phys.Lett.A47(1974)369

8 Neutron interferometer experiment(1) Hamiltonian H = B = B Wave function t = exp iht h 0 = exp ibt h 0 = exp i symmetry of spinor wavefunction Where = 2 h Bdt H. Rauch et al., PL A54 (1975) 425.

9 Neutron interferometer experiment(2) Gravitationally induced quantum phase Energy of neutron gravitational potential k0 k E mgh ( ) 0 2m 2m and k ( k k Phase shift COW 0 m ) 2 gh sin k 2 2 m ga sin R. Colella et al., PRL 34 (1975) 1472; J.L. Staudenmann et al., PR A21 (1980) 1419.

10 Neutron interferometer experiment (3) General framework of spin- 1 2 system Superposition of spinor wavefunctions (i) 22 Pauli Spin Matrices: x = 01 10, y = 0 i i 0, z = (ii) Eigenvectors: s x = , s y = i, s z = 1 0, 0 1. Superposition 1 2 s +z +e i s z = e i s +x s +y s x s y J. Summhammer et al., PL A27 (1983) 2532.

11 Geometric/Topological phases Berry Phase (adiabatic & cyclic evolution) [ Berry; Proc.R.S.Lond. A 392, 45 (1984)] Ω (for 2-level systems) Non-adiabatic & non-cyclic evolution [Samuel & Bhandari, PRL 60, 2339 (1988)] S. Filipp, et al., PRA72 (2005)

12 Two-particle vs. two-space entanglement 2-Particle Bell-State = 1 2 I II + I II I, II represent 2-Particles Quantum contextuality 2-Space Bell-State = 1 2 s I p + s II p s, p represent 2-Spaces, e.g., spin & path Violation of Bell-like inequality S' E' 1, 1 +E' 1, 2 E' 2, 1 +E' 2, 2 = > 2 Hasegawa et al., Nature2003, NJP2011 Kochen-Specker-like contradiction 1 63% E ˆ ˆ ˆ ˆ xey E' X1Y2Y 1X Hasegawa et al., PRL2006/2009 Tri-partite entanglement (GHZ-state) Neutron I ( E0) i i i e II e e ( E0 r ) M Measured Hasegawa et al., PRA2010

13 W- and GHZ- states in a single neutron system W state: GHZ state: D. Erdösi et al. New J. Phys. 15 (2013)

14 Neutron polarimetry - Bell-Test, PLA (2010) - Leggett-Test, NJP (2012) - GHZ-entanglement, NJP (1012)

15 Neutron Detector

16 Uncertainty relation: historical 1 In 1927 Heisenberg postulated an uncertainty principle: γ-ray thought experiment p1q1 h with q 1 (mean error) & p 1 (discontinuous change) in modern treatment for commuting observables: ( Q) ( P) 2 :error of the first measurmen ( Q) : disturbanceon the second measurement (P)

17 Uncertainty relation: historical 2 Kennard considered the spread of a wave function ψ : standard deviations Robertson generalized the relation to arbitrary pairs of observables in any states ψ 1 ( A) ( B) A, B 2 dependent on the state but independent of the appartus 1 Is ( A) ( B) A, B generally valid? 2

18 Ozawa s s Universally Valid Uncertainty Relation 1

19 Ozawa s s Universally Valid Uncertainty Relation 2

20 Definition of error & disturbance

21 Error and disturbance for projective measurement

22 Universally valid uncertainty relation by Ozawa :error of the first measurmen ( A) : disturbanceon the second measurement ( B) :standarddeviations First term: error of the first measuremt, disturbance on the second measurement second and third terms: crosstalks between spreads of wavefunctions and error/disturbance M. Ozawa, Phys. Rev. A 67, (2003).

23 Experimental scheme Successively measurement of 2 noncommuting observables A and B Apparatus 1 measures O A, Apparatus 2 measures B

24 Theoretical predictions 1 For error and disturbance: For the standard deviations:

25 Theoretical predictions 2 -Heisenberg s relation - new uncertainty relation

26 Experimental setup

27 Polarizer for neutrons 1 Stern-Gerlach apparatus Multilayer polarizer

28 Polarizer for neutrons 2

29 Spin rotator: Larmor precession

30 Experimental setup

31 Adjustment Unitary transformation: U=e iασ/2 Determination of U(±π) - Scanning of the current in x-direction Determination of U(±π/2) - Position scanning Contrast of each DC coil adjustement: ~98% Contrast of the whole system: ~96%

32 Experimental data

33 Experimental determination where I + and I - represent the positive and negative projections Projection operator: Error of A (projective measurment):

34 Results: error-disturbance trade-off ( A) A O O A A O A ( A I) O ( A I) A A A

35 Results: new/old uncertainty relation New uncertainty principle :errorof the first measurmen ( A) : disturbanceon the second measurement ( B) :standard deviations Heisenberg product J. Erhart et al., Nature Phys. 8, (2012)

36 Results 1: incident spin-state state ( s>= >) s 0, 0

37 Results 2: polar angle of O A [O A ] θ(o A ) = /12 θ(o A ) = /3 New sum is always above border!

38 Results 3: polar angle of B [[ B] (B) = /6 (B) = /4 Asymmetry appears!

39 Results 4: azimuthal angle of B [[ B] (B) = 2/3 (B) = 5/6 Sum touches the border!

40 Publications by other groups 1

41 Publications by other groups 2 ArXiv; ArXiv;

42 Concluding remarks: error-disturbance uncertain relation Universally valid uncertainty-relation by Ozawa is experimentally tested! - Neutron s spin measurement confirmed the new error-disturbance uncertainty relation. - New sum is always above the limit! Heisenberg product is often below the limit! - Error & disturbance are determined from data. Projective measurements are exploited.

43 J. Erhart et al., Nature Phys. 8, (2012)