Neutron Interferometry

Transcription

1 Neutron Interferometry F. E. Wietfeldt Fundamental Neutron Physics Summer School 2015 June 19, 2015

2 Michelson Interferometer Mach-Zender Interferometer

3 The Perfect Crystal Neutron Interferometer

4 Perfect Crystal LLL Neutron Interferometer Bragg condition: nλ = 2d sinθ d = lattice spacing

5 Perfect Crystal LLL Neutron Interferometer

6 Perfect Crystal LLL Neutron Interferometer

7 Perfect Crystal LLL Neutron Interferometer

8 Nuclear Phase Shift

9 Nuclear Phase Shift index of refraction: n = 1 Nbλ 2 2π relative phase shift: Δχ = k 0 l nk 0 l = Nbλ D cosθ

10 Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1?

11 Interferogram

12 Interferogram O beam: I O = A[ 1 + f cos(χ 2 χ 1 )] H beam: I H = B A f cos(χ 2 χ 1 ) contrast f = C max C min C max + C min (O-beam)

13 Precision Phase Shift Measurement Δχ = Nbλ D cosθ Example: aluminum sample, λ = 2.70 & A, 111 reflection: D = 100 µm Δχ = 2π

14 Non-Dispersive Geometry path length l = D sinθ Δχ = 2NbdD independent of λ

15 Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1?

16 Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1? Answer: Special relativity light E = pc v = dω dk = de dp = E p = ω k v 1 k neutron E = p 2 c 2 + m 2 c 4 v = dω dk = de dp = pc2 E pc2 mc 2 = h m k v k n k k 0

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18 net phase shift: Θ ε 0,γ 0 ( ) = 248π (7) radians b coh = (21) fm

19 Skew-Symmetric Neutron Interferometer

20 NIST perfect crystal silicon interferometers

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22 S18 Neutron Interferometer at the Institut Laue-Langevin

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24 Kinematic Bragg Diffraction

25 Kinematic Bragg Diffraction

26 Kinematic Bragg Diffraction

27 Kinematic Bragg Diffraction

28 Dynamical Diffraction Theory r H = Bragg vector r H = 2πn d r K 0 = internal forward scattered wave r K H = internal Bragg scattered wave Bragg condition: r K H r K 0 = r H Solve Schrödinger Eqn. inside crystal: 2 2 ( + k 0 )Ψ( r ) = v( r )Ψ( r ) with v( r ) = 4π b i δ r r i i ( ) = v Hn n e i r H n r r

29 Dynamical Diffraction Theory Dispersion Equation: K 2 2 ( K 0 ) K ( K H ) = v H approximate: ( K K 0 )( K K H ) = v 2 H 4k 0 2 quadratic equation 2 solutions for K 0

30 Dynamical Diffraction Theory

31 Dynamical Diffraction Theory internal wave function: Ψ( r r ) = ψ 0 α e i r K 0 α rr + ψ 0 β e i r K 0 β rr + ψ H α e i r K H α rr + ψ H β e i r K H β r r

32 Dynamical Diffraction Theory ψ α 0 = ψ β 0 = ψ α H = 1 2 ψ β H = y 1+ y 2 A 0 y 1+ y 2 A y 2 A y 2 A 0 y = k 0 sin2θ B 2ν H δθ misset parameter internal wave function: Ψ( r r ) = ψ 0 α e i r K 0 α rr + ψ 0 β e i r K 0 β rr + ψ H α e i r K H α rr + ψ H β e i r K H β r r

33 Dynamical Diffraction Theory Transmitted wave: Ψ trans ( r r ) = ψ tr 0 e ir k 0 rr + ψ tr H e ir k H r r iy ψ tr 0 = cosφ 1+ y sinφ φ0 ) A 2 ei(φ1 0 iy ψ tr H = 1+ y sinφ +φ0 ) A 2 e i(φ1 0 with φ 0 = ν 0 D cosθ B, φ 1 = ν H D cosθ B 1 D Φ = ν H 1+ y 2 cosθ B Transmitted intensities: 2 2 I 0 = ψ tr 0 = A 0 cos 2 Φ + y2 1+ y 2 sin2 Φ I H = ψ tr H = A 0 1+ y 2 sin2 Φ

34 Transmitted Intensities For the (111) reflection in Si at λ=2.70 Å: y = arcsec

35 Some Consequences of Dynamical Diffraction Pendellösung interference 1 Φ = ν H 1+ y 2 D cosθ B Anomalous transmission Angle amplification

36 Angle Amplification For small δ (~10-3 arcsec): Ω δ 106

37 Practical Neutron Interferometer

38 4π Rotational Symmetry of Spinors Rotation operator: Spin-1/2 particle: Rˆn (α) = e i h α ˆn r S r S = 1 2 h r σ so Rˆn (α) = e i α 2 ˆn r σ Rotations about z-axis: R z (α) = e iα /2 0 iα /2 0 e Symmetry: R z (2π ) χ = χ R z (4π ) χ = χ

39 Larmor precession phase: Δφ = ±2πµ n m n λbl / h 2

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41 Quantum Phase Shift Due To Gravity (COW Experiments) Δφ = 2πλgA h 2 m in m grav A = Hl = area of parallelogram m in = neutron inertial mass m grav = neutron gravitational mass test of weak equivalence principle at the quantum limit

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43 Δφ grav = 2πλgA 0 h 2 m in m grav sinα = qsinα A 0 = area of parallelogram at α = 0 a measured: q = 54.3 theory: q = 59.6

44 Systematic Effects in the COW Experiments q COW = 1 ( q grav (1+ ε) + q ) 2 2 bend + q 2 Sagnac dynamical diffraction correction bending of interferometer Earth s rotation Sagnac effect: Δφ Sagnac = 2m in h r Ω r A due to Earth's rotating frame bending effect: repeat experiment with x rays, different wavelengths

45 Littrell, et al. (1997) results: experiment q COW theory [rad] q COW meas. [rad] discrepancy (%) SS, (5) 50.18(5) data from Werner, et al. (1988) SS, (10) (10) LLL, (10) (15) LLL, (10) (30) Layer and Greene (1991): x rays do not fill the Borrmann fan as completely as neutrons Possible improvement (A. Zeilinger, S. Werner, FEW, et al.): Suspend interferometer inside chamber filled with ZnBr 2 +D 2 O (floating COW)

46 Precision Neutron Interferometric Measurements of Few-Body Neutron Scattering Lengths R. Haun, C. Shahi, F.E. Wietfeldt Tulane University M. Arif, W.C. Chen, T. Gentile, M. Huber, D.L. Jacobson, D. Pushin, S.A. Werner, L. Yang NIST T. C. Black University of North Carolina, Wilmington H. Kaiser, K. Schoen University of Missouri-Columbia W. M. Snow Indiana University

47 Semi-phenomological nucleon-nucleon potential model AV18 Great success with NN scattering lengths, but unable to predict 3 He, T binding energies Data from Wiringa et al., Phys. Rev. C 51, 38 (1995)

48 NN Potential Models π π π π π

49 Motivation Precision few-body neutron scattering lengths provide an additional challenge for nuclear potential models. Few body nuclear effective field theories (EFT) require precision experimental measurements to constrain short-range mean field potentials.

50 Precision neutron interferometric measurement of the n-d coherent scattering length at NIST (2003) b c = ± fm Schoen, et al., Phys. Rev. C 67, (2003)

51 Precision neutron interferometric measurement of the n- 3 He coherent scattering length at NIST (2004) b c = ± fm Huffman, et al., Phys. Rev. C 70, (2004)

52 n- 3 He Scattering Lengths 3.8 ILL (2004) 3.6 ILL (2002) NIST (2004) AV18 a 1 (fm) 3.4 ILL (1979) AV18+UIX R-matrix a 0 (fm)

53 A measurement of the n- 3 He spin-incoherent scattering length at NIST ( )

54 Spin-dependent neutron scattering total scattering length: b = b c + 2b i I(I + 1) r I r σ n coherent: b c = I + 1 2I + 1 b + + I 2I + 1 b incoherent: b i = I(I + 1) 2I + 1 ( b + b )

55 Polarized 3 He gas target: Spin Exchange Optical Pumping Spin is transferred from optically polarized alkali atoms to 3 He nuclei via the hyperfine interaction in collisions. The cell is polarized offline and then transferred to the neutron interferometer.

56 Polarized 3 He Cells Target cells: Boron-free GE-180 glass 4 mm flat windows 40 mm long, 25 mm dia. 1.5 atm 3 He (with 4% N 2 ) Two cells: Pistachio (115 hours) Cashew (35 hours)

57 Measuring the Scattering Length b + b = 2 φ φ ( ) N 3 λz P 3

58 Measuring the Scattering Length b + b = 2 φ φ ( ) N 3 λz P 3

59 Measuring the Scattering Length b + b = 2 ( φ φ ) N 3 λz P 3 measured simultaneously from the asymmetry in counter C4

60 Measuring N 3 λz P 3

61 Measuring N 3 λz P 3 C4 asymmetry = N N N + N = 1 (1+s)P tanh x 2 n 1+ 1 (1 s)p tanh x 2 n x = σ σ 0 1 N λz P 3 3 4λ th

62 Measuring N 3 λz P 3 P 3 = 3 He polarization P n = neutron polarization (flipper off) s = P (flipper on) n P n (flipper off) C4 asymmetry = N N N + N = 1 (1+s)P tanh x 2 n 1+ 1 (1 s)p tanh x 2 n x = σ σ 0 1 N λz P 3 3 4λ th

63 Measuring N 3 λz P 3 P 3 = 3 He polarization P n = neutron polarization (flipper off) s = P (flipper on) n P n (flipper off) 3 He (n, p) cross section: C4 asymmetry = N N N + N = x = σ σ 0 1 N λz P 3 3 4λ th 1 (1+s)P tanh x 2 n 1+ 1 (1 s)p tanh x 2 n σ th = 1 4 σ σ 1 σ 1 σ = 5333(7) barns (Hofmann and Hale, 2003) the dominant systematic error in this experiment

64 Neutron Polarimetry Use optically thick (Nσz ~ 3) 3 He cell, with known polarization, in place of neutron interferometer. Measure neutron count rate with both flip states and both directions of P 3 P n = ±.0008 s =.9951 ±.0003

65 Neutron Polarization Correction sin Δφ Δφ meas. = arctan η + cos Δφ arctan η sin Δφ 1+ η cos Δφ η = 1 P n 1+ P n e 2χ η = 1 sp n 1+ sp n e+2χ

66 The Data

67 Error budget: The Result Δb = b + b - = ± 0.031(stat) ± 0.039(sys) fm# # Δa = a 1 a 0 = ± 0.023(stat) ± 0.029(sys) fm# # 2008 Parameter 2013 σ (fm) σ (fm) φ 0 /ξ Fit (Statistical) Triplet absorption cross section σ Total absorption cross section σ un Total systematic from cross sections Phase instabilities Neutron polarization P n Spin-flipper efficiency s Total non-cross-section systematic Total statistical Total systematic Huber, et al. PRL 102, (2009) PRL 103, (2009) PRC 90, (2014)

68 n- 3 He Scattering Lengths 3.8 ILL (2004) 3.6 ILL (2002) NIST (2004) AV18 a 1 (fm) 3.4 ILL (1979) AV18+UIX R-matrix a 0 (fm)

69 n- 3 He Scattering Lengths 3.8 ILL (2004) 3.6 ILL (2002) NIST (2004) AV18 a 1 (fm) 3.4 ILL (1979) AV18+UIX R-matrix 3.2 NIST (2014) a 0 (fm)

70 Light gas scattering lengths to measure precisely using neutron interferometry: Completed: n-h n-d n- 3 He n- 3 He (spin incoherent) - new result (2014) In Progress: n- 4 He n-t

71 Quantum Cheshire Cat Well! I ve often seen a cat without a grin, thought Alice, but a grin without a cat! It s the most curious thing I ever saw in all my life! Lewis Carroll, Alice s Adventures in Wonderland

72 Weak Measurement  = ψ f  ψ i w ψ f ψ i Aharanov, Albert, and Vaidman (1988) Use quantum interference to measure the expectation value of a weakly coupled operator without measuring the state vector. Real information is obtained, while leaving the QM state undisturbed. Requires post-selection = interpretational questions

73 ARTICLE NATURE COMMUNICATIONS DOI: /ncomms5492 ^ j i. The computation specific subspace. Then of course it would only be possible to neutrons spin component on path j is h^ sz! w ^ I i ¼1 and h^ ^ II i ¼0. On make any inferences by appealing to counterfactual reasoning25. of the weak values yields h^ sz! sz! w w average, a weak interaction coupling with a probe on path II does not affect the state of that probe, as if there was effectively no spin Experimental setup. The experiment was performed at the S18 component travelling along the path. interferometer beam line at the research reactor of the Institut Note that in principle, it is possible to perform the weak Laue-Langevin32. The setup is shown in Fig. 2. measurements jointlyarticle along each path. We, however, carried A monochromatic neutron beam with a wavelength of them out separately (that is, in different runs). This is justified l ¼ 1.92 Å passes magnetic birefringent prisms, which polarize Received 11 Mar 2014 Accepted 24 Jun 2014 Published 29 Jul 2014 OPEN DOI: /ncomms5492 because a weak measurement made along a given path involves a the neutron beam. To avoid depolarization, a magnetic guide field local coupling, which in the weak limiting case only minimally pointing in the þ z direction is applied around the whole setup. A disturbs the subsequent evolution of the quantum state along that spin turner rotates the neutron spin by p/2 into the xy plane. The path. It does not affect the evolution of the quantum state along neutron s spin wavefunction is then given by Sx; þ i. Subsethe other path. This is in clear contrast with standard projective quently, the neutrons enter a triple-laue interferometer28, ,2 3 measurements, whichtobias would fundamentally disturb1, Stephan the system interferometer, spin rotator in each beam path allows Denkmayr, Hermann Geppert Sponar1, Inside Hartmutthe Lemmel, Alexandre amatzkin, 4 1 evolution by projecting quantum along both paths to a the generation of the preselected state jci i. & Yujistate Hasegawa Jeff the Tollaksen Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment z quantum theory has been revealing extraordinary and counterfrom its very beginning, intuitive phenomena, such as wave-particle duality, Schro dinger cats and quantum nony phenomenon found within the framework of quantum locality. Another paradoxical GF mechanics is the quantumxcheshire Cat : if a quantum systemps is subject to a certain pre- and ABS are spatially separated. It has been postselection, it can behave as if a particle and its property H-Det. suggested to employ weak measurements in order to explore the Cheshire Cat s nature. Here we report an experiment in which SRswe send neutrons through a perfect silicon crystal interferometer and perform weak measurements to probe the location of the particle and its magnetic moment. The experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other. P ST1 ST2 A O-Det. Figure 2 Illustration of the experimental setup. The neutron beam is polarized by passing through magnetic birefringent prisms (P). To prevent depolarization, a magnetic guide field (GF) is applied around the whole setup. A spin turner (ST1) rotates the neutron spin by p/2. Preselection of the system s wavefunction jci i is completed by two spin rotators (SRs) inside the neutron interferometer. These SRs are also used to perform the weak! "! "! "! " ^ I and! ^ II are determined. The phase shifter (PS) ^ I and s ^ II. The absorbers (ABS) are inserted in the beam paths when! ^z! ^z! measurement of s w w w w

74 P ST1 O-Det. ST2 Figure 2 Illustration of the experimental setup. The neutron beam is polarized by passing through magnetic birefringent prisms (P). To prevent depolarization, a magnetic guide field (GF) is applied around the whole setup. A spin turner (ST1) rotates the neutron spin by p/2. Preselection of the system s wavefunction jci i is completed by two spin rotators (SRs) inside the neutron interferometer. SRs are also used to perform the weak 20%These absorber II,+! "! "! "! " ^ ^ ^ ^ ^z!ii w. The absorbers (ABS) are inserted in the beam paths when!in ^z!i w and s and!iiii w are determined. The phase shifter (PS) measurement of s I w path makes it possible to tune the relative phase w between the beams in path I and path II. The two outgoing beams of the interferometer are monitored by the H and O detector in reflected and forward directions, respectively. Only the neutrons reaching the O detector are affected by postselection using a spin turner (ST2) and a spin analyzer (A). + a + b 20% absorber SA Io in path I Path II + c I, SA SA Io Io Path I Intensity (c.p.s.) O-detector O-detector O-detector ABSORBER IN PATH I NO ABSORBER ABSORBER IN PATH II!!!!!! 0 0!! Phase shift " (rad) Phase shift " (rad) Phase shift " (rad)! "! " ^ I and P ^ II using an absorber with transmissivity T ¼ 0.79(1). The intensity is plotted as a function of the Figure 3 Measurement of P w w relative phase w. The solid lines represent least-square fits to the data and the error bars represent one s.d. (a) An absorber in path I; no significant loss in intensity is recorded. (b) A reference measurement without any absorber. (c) An absorber in path II: the intensity decreases. These results suggest that for the successfully postselected ensemble, the neutrons go through path II.! 2 0

75 b c SA IH IH PATH II IO SA Bz IO cos10 I,+ + sin10 I, II,+ H-detector H-detector IO SA 20 spin rotator in path II PATH I II,+ Intensity (c.p.s.) Bz IH a H-detector spin rotator inpath pathii 20 IN I,+" cos10 I, + "2sin10 2 " Phase shift! (rad) NO ROTATION I, " 2 0 " 2 " Phase shift! (rad) O-detector " 2 20 IN PATH II I, " 2 " 0 Phase shift! (rad) O-detector O-detector Intensity (c.p.s.) NO ROTATION 20 IN PATH I 6 " 2 0 " 2 " " 2 0 " 2 20 IN PATH II " " 2 0 " 2 " Phase shift! (rad) Phase shift! (rad) Phase shift! (rad) $ % $ % ^ I and r ^ II applying small additional magnetic fields. The intensity of the O beam (with the spin analysis) and the ^z P ^ Figure 4 Measurement of r P z w w H beam (without the spin analysis) is plotted as a function of the relative phase w. The solid lines represent least-square fits to the data and the error bars represent one s.d. (a) A magnetic field in path I; interference fringes appear both at the postselected O detector and the H detector. (b) A reference measurement without any additional magnetic fields. Since the spin states inside the interferometer are orthogonal, interference fringes appear neither in the O, nor the H detector. (c) A magnetic field in path II; interference fringes with minimal contrast can be seen at the spin postselected O detector, whereas a clear sinusoidal intensity modulation is visible at the H detector without spin analysis. The measurements suggest that for the successfully

76 Quantum Cheshire Cat So considering only post-selected events: r k O,+ Each neutron took path II but its spin (magnetic moment) took path I

77 Quantum Cheshire Cat So considering only post-selected events: r k O,+ Each neutron took path II but its spin (magnetic moment) took path I A valid interpretation of reality?

78 The Neutron by Gina Berkeley When a pion an innocent proton seduces With neither excuses Abuses Nor scorn For its shameful condition Without intermission The proton produces: a neutron is born. What love have you known O neutron full grown As you bombinate into the vacuum alone? Its spin is 1/2, and its mass is quite large - about 1 AMU but it hasn t a charge; Though it finds satisfaction in strong interaction It doesn t experience Coulombic attraction But what can you borrow Of love, joy, or sorrow O neutron, when life has so short a tomorrow? Within its Twelve minutes Comes disintegration Which leaves an electron in mute desolation And also another ingenuous proton For other unscrupulous pions to dote on. At last, a neutrino; Alas, one can see no Fulfilment for such a leptonic bambino. No loving, no sinning Just spinning and spining Eight times through the globe without ever beginning... A cycle mechanic No anguish or panic For such is the pattern of life inorganic. O better The fret a Poor human endures Than the neutron s dichotic Robotic Amours.