Neutron Interferometry

F. E. Wietfeldt

Fundamental Neutron Physics Summer School 2015 June 19, 2015 Michelson Interferometer

Mach-Zender Interferometer The Perfect Crystal Neutron Interferometer Perfect Crystal LLL Neutron Interferometer

Bragg condition: nλ = 2d sinθ

d = lattice spacing Perfect Crystal LLL Neutron Interferometer Perfect Crystal LLL Neutron Interferometer Perfect Crystal LLL Neutron Interferometer Nuclear Phase Shift Nuclear Phase Shift

Nbλ 2 index of refraction: n = 1− 2π

relative phase shift: D Δχ = k0l − nk0l = Nbλ cosθ Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1? Interferogram Interferogram

O beam: IO = A[1+ f cos(χ2 − χ1 )]

H beam: I H = B − A f cos(χ2 − χ1 )

C − C contrast f = max min (O-beam) Cmax + Cmin Precision Phase Shift Measurement

D Δχ = Nbλ cosθ

Example: aluminum sample,

λ = 2.70A& , 111 reflection: D = 100 µm ⇒ Δχ = 2π

Non-Dispersive Geometry

D path length l = sinθ

Δχ = 2NbdD

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

E = pc E = p2c2 + m2c4 dω dE E ω v = = = = dω dE pc2 pc2 dk dp p k v = = = ≈ = h k dk dp E mc2 m 1 v ∝ v ∝ k k k n ≡ k0 net phase shift: Θ(ε0 ,γ 0 ) = 248π + 0.455(7) radians

bcoh = 4.15041(21) fm Skew-Symmetric Neutron Interferometer NIST perfect crystal silicon interferometers

S18 Neutron Interferometer at the Institut Laue-Langevin

Kinematic Bragg Diffraction Kinematic Bragg Diffraction Kinematic Bragg Diffraction Kinematic Bragg Diffraction Dynamical Diffraction Theory

r H = Bragg vector r 2πn H = d r K0 = internal forward scattered wave r K = internal Bragg scattered wave H

Bragg condition: r r r K − K = H H 0

Solve Schrödinger Eqn. inside crystal: 2 2 r r r (∇ + k0 )Ψ(r ) = v(r )Ψ(r )

r r with v(r ) = 4π b δ r − r = v eiHn ⋅r ∑ i ( i ) ∑ Hn i n Dynamical Diffraction Theory

2 2 2 2 2 Dispersion Equation: (K − K0 )(K − K H ) = vH

2 vH quadratic equation approximate: K − K K − K = ( 0 )( H ) 2 2 solutions for K 4k0 0 Dynamical Diffraction Theory Dynamical Diffraction Theory

r α r r β r r α r r β r internal wave function: Ψ(r ) = ψ α eiK0 ⋅r + ψ β eiK0 ⋅r + ψ α eiKH ⋅r + ψ β eiKH ⋅r 0 0 H H Dynamical Diffraction Theory

1 ⎡ y ⎤ ψ α = ⎢1− ⎥ A 0 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ y ⎤ ψ β = ⎢1+ ⎥ A 0 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ 1 ⎤ ψ α = − ⎢ ⎥ A H 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ 1 ⎤ ψ β = + ⎢ ⎥ A H 2 2 0 ⎣⎢ 1+ y ⎦⎥

k sin2θ y = 0 B δθ 2ν H misset parameter

r α r r β r r α r r β r internal wave function: Ψ(r ) = ψ α eiK0 ⋅r + ψ β eiK0 ⋅r + ψ α eiKH ⋅r + ψ β eiKH ⋅r 0 0 H H Dynamical Diffraction Theory

r r r r r ik0 ⋅r ikH ⋅r Transmitted wave: Ψ (r ) = ψ e + ψ e trans tr 0 tr H

⎡ iy ⎤ ν D ν D i(φ1 −φ0 ) 0 H ψ = ⎢cosΦ − sinΦ⎥e A φ0 = , φ1 = tr 0 2 0 cosθ cosθ ⎣⎢ 1+ y ⎦⎥ B B ⎡ ⎤ with ⎛ 1 ⎞ D −iy −i(φ +φ ) 1 0 Φ = ν H ψ tr H = ⎢ sinΦ⎥e A0 ⎜ 2 ⎟ 2 1+ y cosθB ⎣⎢ 1+ y ⎦⎥ ⎝ ⎠

2 2 2 ⎡ 2 y 2 ⎤ I0 = ψ tr 0 = A0 ⎢cos Φ + 2 sin Φ⎥ ⎣ 1+ y ⎦ Transmitted intensities: 2 2 ⎡ 1 2 ⎤ I H = ψ tr H = A0 ⎢ 2 sin Φ⎥ ⎣1+ y ⎦ Transmitted Intensities

For the (111) reflection in Si at λ=2.70 Å:

y = 1 → 0.9 arcsec Some Consequences of Dynamical Diffraction

• Pendellösung interference

⎛ 1 ⎞ D Φ = ν ⎜ H 2 ⎟ ⎝ 1+ y ⎠ cosθB

• Anomalous transmission

• Angle amplification Angle Amplification

Ω For small δ (~10-3 arcsec): ≈ 106 δ Practical Neutron Interferometer 4π Rotational Symmetry of Spinors

i r − αnˆ⋅S Rotation operator: R (α) = e h nˆ

α ˆ r r 1 r −i n⋅σ Spin-1/2 particle: S = σ so R (α) = e 2 2 h nˆ ⎛ e−iα /2 0 ⎞ Rotations about z-axis: Rz (α) = ⎝⎜ 0 eiα /2 ⎠⎟

Rz (2π )χ = −χ Symmetry:

Rz (4π )χ = χ Larmor precession phase:

Δφ = ±2πµ m λB / 2 n n l h

Quantum Phase Shift Due To Gravity (COW Experiments)

2πλgA Δφ = m m m = neutron inertial mass h2 in grav in

m = neutron gravitational mass A = H = area of parallelogram grav l

test of weak equivalence principle at the quantum limit

2πλgA Δφ = 0 m m sinα = qsinα grav h2 in grav

A0 = area of parallelogram at α = 0 a

measured: q = 54.3 theory: q = 59.6 Systematic Effects in the COW Experiments

1 2 q = ⎡ q (1+ ε) + q + q2 ⎤ 2 COW ⎣( grav bend ) Sagnac ⎦

’ dynamical bending of Earth s rotation diffraction interferometer correction

2min r r Sagnac effect: ΔφSagnac = Ω ⋅ A due to Earth's rotating frame h

bending effect: repeat experiment with x rays, different wavelengths Littrell, et al. (1997) results:

q theory q meas. discrepancy experiment COW COW [rad] [rad] (%)

SS, 440 50. 97(5) 50.18(5) -1. 6

data from Werner, et al. (1988) SS, 220 100. 57(10) 99. 02(10) -1. 5

LLL, 440 113. 60(10) 112. 62(15) -0. 9

LLL, 220 223. 80(10) 221. 85(30) -0. 9

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

ZnBr2+D2O (floating COW) 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

Semi-phenomological nucleon-nucleon potential model AV18

Great success with NN scattering lengths, but unable to predict 3He, T binding energies

Data from Wiringa et al., Phys. Rev. C 51, 38 (1995) NN Potential Models

π π

π π π 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. Precision neutron interferometric measurement of the n-D coherent scattering length at NIST (2003)

bc = 6.6727 ± 0.0045 fm

Schoen, et al., Phys. Rev. C 67, 044005 (2003) Precision neutron interferometric measurement of the n-3He coherent scattering length at NIST (2004)

bc = 5.8572 ± 0.0072 fm

Huffman, et al., Phys. Rev. C 70, 014004 (2004) n-3He Scattering Lengths

3.8

ILL (2004) 3.6 ILL (2002) NIST (2004) AV18

3.4 (fm) 1 a ILL (1979) AV18+UIX R-matrix 3.2

3.0 7.0 7.2 7.4 7.6 7.8 a0 (fm) A measurement of the n-3He spin-incoherent scattering length at NIST (2008-2014) Spin-dependent neutron scattering

2bi r r total scattering length: b = bc + I ⋅σ n I(I + 1)

I + 1 I coherent: b = b + b c 2I + 1 + 2I + 1 −

I(I + 1) incoherent: b = b − b i 2I 1 ( + − ) + Polarized 3He gas target: Spin Exchange Optical Pumping

Spin is transferred from optically polarized alkali atoms to 3He nuclei via the hyperfine interaction in collisions.

The cell is polarized offline and then transferred to the neutron interferometer. Polarized 3He Cells

Target cells: •Boron-free GE-180 glass •4 mm flat windows •40 mm long, 25 mm dia. 3 •1.5 atm He (with 4% N2)

Two cells: “Pistachio” (115 hours) “Cashew” (35 hours) Measuring the Scattering Length

2(φ ↑ − φ ↓ ) b − b = − + − N λz P 3 3 Measuring the Scattering Length

2(φ ↑ − φ ↓ ) b − b = − + − N λz P 3 3

measured simultaneously from the asymmetry in counter C4 Measuring N λz P 3 3 Measuring N λz P 3 3

↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 N ↑ N ↓ 1 + 1+ 2 (1− s)Pn tanh x

⎛ σ − σ ⎞ x = 0 1 N λz P ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ Measuring N λz P 3 3

3 P3 = He polarization

Pn = neutron polarization (flipper off)

P (flipper on) s = n P (flipper off) n

↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 N ↑ N ↓ 1 + 1+ 2 (1− s)Pn tanh x

⎛ σ − σ ⎞ x = 0 1 N λz P ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ Measuring N λz P 3 3

3 P3 = He polarization

Pn = neutron polarization (flipper off)

P (flipper on) s = n P (flipper off) n

3 He (n, p) cross section:

1 3 σ = σ + σ = 5333(7) barns th 4 0 4 1 ↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 σ N ↑ N ↓ 1 1 ≈ 0 − 2 × 10−3 + 1+ 2 (1− s)Pn tanh x σ 0

(Hofmann and Hale, 2003) ⎛ σ − σ ⎞ x = 0 1 N λz P the dominant ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ systematic error in this experiment Neutron Polarimetry

Use optically thick (Nσz ~ 3) 3He cell, with known polarization, in place of neutron interferometer.

Measure neutron count rate with both flip states and both directions of P3

Pn = 0.9291 ± .0008

s = .9951 ± .0003 Neutron Polarization Correction

⎛ sin Δφ ⎞ ⎛ η↑ sin Δφ ⎞ Δφ = arctan − arctan meas. ↓ ⎜ ↑ ⎟ ⎝⎜ η + cos Δφ ⎠⎟ 1+ η cos Δφ ⎝ ⎠

⎛ 1− P ⎞ ⎛ 1− sP ⎞ η↑ = n e−2χ η↓ = n e+2χ ⎜ 1+ P ⎟ ⎜ 1+ sP ⎟ ⎝ n ⎠ ⎝ n ⎠ The Data

2008 2014 The Result

Δb’ = b+’ – b-’ = -5.411 ± 0.031(stat) ± 0.039(sys) fm NEUTRON INTERFEROMETRIC MEASUREMENT OF THE . . . PHYSICAL REVIEW C 90,064004(2014) Δa’ = a1’ – a0’ = -4.053 ± 0.023(stat) ± 0.029(sys) fm

TABLE III. The uncertainty budget for !b!.Uncertaintiesrelated to 3He absorption cross section that are identical for both data sets Error budget: Weighted Average are summed in quadrature. = (5.87 ± 0.04) fm (Uncertainty scaled by 5.3) 2008 Parameter 2013 σ (fm) σ (fm)

2 χ 0.053 !φ0/ξ Fit (Statistical) 0.038 _____ ILL 06 NI 44.8 0.028 Triplet absorption cross section σ1 0.028 NIST 04 NI 2.9 0.007 Total absorption cross section σun 0.007 0.029 Total systematic from cross sections 0.029 IBR 81 CS ILL 79 NI 0.025 Phase instabilities 0.040 ILL 77 NI 7.7 0.005 Neutron polarization P 0.004 n Huber, et al. BNL 74 Re 0.002 Spin-ﬂipper efﬁciency s 0.004 _____ PRL 102, 200401 (2009) 55.4 0.026 Total non-cross-section systematic 0.040 PRL 103, 179903 (2009) 0.053 Total statistical 0.038 PRC5.5 90, 0640045.75 (2014)6 6.25 6.5 6.75 0.039 Total systematic 0.049 b'c (fm)

FIG. 12. (Color online) An ideogram of the coherent scattering !φ was inflated in quadrature by 0.043 rad. For 2013 we find 0 length measurements for n-3He taken from Refs. [24,25,63–66]. that 2φmag 21 3mrad,whichisconsistentwithboththe = ± The blue band represents the weighted average σ of the three 2008 data and the field gradient measurement. experiments with smallest quoted uncertainties.± Techniques used The weighted average of both data sets gives were neutron interferometry (NI), total cross section (CS), and reflectivity (RE). !b! [ 5.411 0.031 (stat.) 0.039 (syst.)] fm. (63) = − ± ± This corresponds to a 4σ shift of !b! compared to our previous result reported in Ref. [26]. This shift is entirely attributable to the inclusion of phase shifts from magnetic-field gradients (ii) in the event of future more accurate measurements of the in our fitting. Allowing our fit of !φ0 versus ξ an additional absorption cross sections, one can immediately update the 3 degree of freedom increased the statistical uncertainty in the spin-dependent n- He scattering length. Zimmer et al. de- scattering length by a factor of 2. However, tripling the original termined σ1 from an average of the experimental results of data set yielded a final statistical uncertainty only 20% larger Refs. [56,57], with the limitation that σ1 ! 0. Whereas, as than that reported in Ref. [26]. The uncertainty budget for !b! described in Sec. VI A,weusedatheoreticallypredictedσ1 for each individual data set is given in Table III.Theweighted but with an inflated uncertainty. Our result, stated independent average is performed by weighting both the statistical and the of the triplet absorption cross section, is systematic uncertainties unrelated to neutron absorption on 3He in quadrature. The systematic uncertainty related to 3He !b! (this work) absorption was added to the total systematic uncertainty in 4 σ1 [( 10.1929 0.0760) 10− fm/b] 1 σun. Eq. (63). = − ± × − σ ! un " (64) VIII. CONCLUSIONS AND DISCUSSION This is in disagreement with the result of Zimmer et al. of We have performed a precision measurement of the differ- ence !b [ 5.411 0.031 (stat.) 0.039 (syst.)] fm be- ! !b! (Ref. [60]) tween the triplet= − and singlet± scattering± lengths of n-3He using neutron interferometry to 0.9% relative standard uncertainty. 4 σ1 [( 10.3628 0.0180) 10− fm/b] 1 σun = − ± × − σ The ultimate precision of this technique is systematically ! un " limited by the triplet absorption cross section corresponding (65) to a relative uncertainty of 0.5%. This result is in good agreement with the only previous direct measurement of by 2σ when factoring out the absorption cross sections. !b! 5.462 0.046 fm performed by Zimmer et al. at There have been a number of experiments measuring the the Institut=− Laue-Langevin± (ILL) [60]. Reference [60]used coherent scattering length of n-3He defined by Eq. (11a)using aspin-echoapparatustomeasuretherelativedifferenceinthe techniques such as measuring neutron reflectivity, relative pseudomagnetic spin precession [61,62]betweenaneutron phase shifts, and neutron transmissions. The three most precise 3 passing though a polarized He cell and an empty reference measurements of bc! were done with neutron interferometry: beam. That technique is fundamentally different than the Kaiser et al. [64], Huffman et al. [24], and Ketter et al. [25]. technique applied here. One can state the results independent However, the two most recent results differ by more than of the triplet absorption cross section and total absorption cross 7σ .Figure12 shows an ideogram of the coherent scattering section from our results and those of Ref. [60]. This is done length measurements. Each measurement is represented by a for two reasons: (i) Both groups estimated σ1 differently and Gaussian centered about their result with a normalized area

064004-13 n-3He Scattering Lengths

3.8

ILL (2004) 3.6 ILL (2002) NIST (2004) AV18

3.4 (fm) 1 a ILL (1979) AV18+UIX R-matrix 3.2

3.0 7.0 7.2 7.4 7.6 7.8 a0 (fm) n-3He Scattering Lengths

3.8

ILL (2004) 3.6 ILL (2002) NIST (2004) AV18

3.4 (fm) 1 a ILL (1979) AV18+UIX R-matrix 3.2 NIST (2014)

3.0 7.0 7.2 7.4 7.6 7.8 a0 (fm) Light gas scattering lengths to measure precisely using neutron interferometry:

Completed: • n-H • n-D • n-3He • n-3He (spin incoherent) - new result (2014)

In Progress: • n-4He • n-T ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492

he study of fundamental quantum mechanical phenomena the system behaves as if the neutrons go through one beam path, is not only enriching our scientific knowledge, but also our while their magnetic moment travels along the other. Tunderstanding of the natural laws. This understanding lead to the development of numerous technological applications: Results quantum non-locality1–4 plays an essential role in quantum Theoretical considerations. In our experiment, the neutron plays cryptology5, the understanding of wave-particle duality6 made the role of the cat and the cat’s grin is represented by the neu- semiconductor technology possible7 and the investigation of tron’s spin component along the z direction. The system is Schro¨dinger cats8 advanced the field of quantum information initially prepared so that after entering the beam splitter its processing and communication9. quantum state is given by During the examination of the quantum measurement process, 10,11 1 1 Aharonov, Albert and Vaidman introduced the weak value, ci Sx; I Sx; II ; 2 defined as j i¼ p2 j þij iþ p2 j Àij i ð Þ c A^ c where |I (|II ) standsffiffiffi for the spatial partffiffiffi of the wavefunction A^ h f j j ii 1 along pathi I (pathi II) of the interferometer and |S ;± denotes h iw¼ c c ð Þ x i h f j ii the spin state in ±x direction. In order to observe the quantum where ci and cf are the initial (‘preselected’) and final Cheshire Cat, after we preselect the ensemble, we will next per- (‘postselected’)j i statesj i of the system and Aˆ is an observable of form weak measurements of the neutrons’ population in a given ˆ ˆ the system. A w represents information of the observable A path on the one hand and of the value of the spin in a given path between pre-h andi postselection, which can be obtained by weakly on the other. Subsequently, the ensemble is postselected in the coupling the system to a measurement device, that is, a probe, final state: without significantly altering the subsequent evolution of the 1 system. Due to the weakness of the coupling between the system c Sx; I II : 3 f p and the measurement device, the information obtained by a single j i¼ 2 j Ài½j iþj i ð Þ measurement is limited. To attain useful information about a Whenever the postselectionffiffiffi succeeds, that is, when a Cheshire quantum system, the measurement has to be repeated several Cat is created, a minimally disturbing measurement will find the ˆ times. The weak value A w then reflects the average conditioned Cat in the upper beam path, while its grin will be found in the on the pre- and postselectedh i ensemble12. lower one. The first experimental determination of a weak value showed This paradoxical behaviour can be quantified using the weak that the weak measurement scheme can be used as a means of value. We can calculate the weak values of the projection 13 amplification. Subsequently, the experimental work on weak operators on the neutron path eigenstates Å^ j j j , with j I, ^ jjih ^ ¼ measurements demonstrated that they allow information about a II. From equations (1–3), we obtain ÅI w 0 and ÅII w 1, quantum system to be obtained with minimal disturbance14,15, the first of these expressions indicatingh thati ¼ a weak interactionh i ¼ that they can be used for high-precision metrology16,17 and that coupling the spatial wavefunction to a probe localized on path I they are perfectly suited for the study of quantum paradoxes18–22. has no effect on the probe on average, as if there was no neutron A surprising effect originating from pre- and postselection of a travelling on that path. Note that this is in accordance with the system is the ability to ‘separate’ the location of a system from one following theorem18: if the weak value of a dichotomic operator of its properties23–26 as suggested by the Cheshire Cat story in equals one of its eigenvalues, then the outcome of an ideal (also Alice in Wonderland: ‘Well! I’ve often seen a cat without a grin,’ known as a strong) measurement of the operator is that same thought Alice; ‘but a grin without a cat! It’s the most curious eigenvalue with probability one. So, if we were to weakly measure 27 ^ thing I ever saw in all my life!’ . The essential property of a ÅI w and obtain 0, then, per the above theorem, since the quantum Cheshire Cat in a Mach–Zehnder-type interferometer projectionh i operator is dichotomic and the weak value is an is, that the cat itself is located in one beam path, while its grin is eigenvalue, we could also perform an ideal measurement of Å^ I 25 located in the other one . An artistic depiction of this behaviour and again obtain 0. Similarly for Å^ II. is shown in Fig. 1. We can also rely on weak measurements to determine the In this work, we prepare and measure the Cheshire Cat states location of the neutrons’ spin component. This is done by by means of neutron interferometry28, which has already been measuring the weak value of the spin component along each path successfully used to study many other purely quantum j by applying a unitary interaction on the respective path. The mechanical effects29–31. The experimental resultsQuantum suggest that Cheshireappropriate observable Cat to ascertain the weak value of the

Figure 1 | Artistic depiction of the quantum Cheshire Cat. Inside the interferometer, the Cat goes through the upper beam path, while its grin travels along the lower beam path. Figure courtesy of Leon Filter. “Well! I’ve often seen a cat without a 2 grin,” thoughtNATURE Alice, COMMUNICATIONS “but a| 5:4492grin | DOI:without 10.1038/ncomms5492 a | www.nature.com/naturecommunications & cat! It’s2014 theMacmillan most Publishers curious Limited. thing All rights I ever reserved. saw in all my life! Lewis Carroll, Alice’s Adventures in Wonderland Weak Measurement

ˆ ψ f A ψ i Aˆ = 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 NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492 ARTICLE

^ neutrons’ spin component on path j is s^zÅj w. The computation specific subspace. Then of course it would only be possible to ^ h i ^ 25 of the weak values yields s^zÅI w 1 and s^zÅII w 0. On make any inferences by appealing to counterfactual reasoning . average, a weak interaction couplingh i with¼ a probeh on pathi ¼II 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 1.92 Å passes magnetic birefringent prisms, which polarize because a weak measurementReceived 11 Mar made 2014 along| Accepted a given 24 Jun 2014 path| Published involves 29 aJul 2014the¼ neutronDOI: 10.1038/ncomms5492 beam. To avoid depolarization,OPEN a magnetic guide field local coupling, whichObservation in the weak limitingof case a only quantum minimally pointing Cheshire in the z direction Cat is applied around the whole setup. A disturbs the subsequent evolution of the quantum state along that spin turner rotatesþ the neutron spin by p/2 into the xy plane. The path. It does not affectin the a evolution matter-wave of the quantum interferometer state along neutron’s spin experiment wavefunction is then given by |Sx; . Subse- the other path. This is in clear contrast with standard projective quently, the neutrons enter a triple-Laue interferometerþi 28,31. measurements, whichTobias would Denkmayr fundamentally1, Hermann Geppert disturb1, Stephan the system Sponar1,Inside Hartmut the Lemmel interferometer,1,2, Alexandre a Matzkin spin rotator3, in each beam path allows evolution by projectingJeff the Tollaksen quantum4 & Yuji state Hasegawa along1 both paths to a the generation of the preselected state c . j ii

From its very beginning,z quantum theory has been revealing extraordinary and counter- intuitive phenomena, such as wave-particle duality, Schro¨dinger cats and quantum non- y locality. Another paradoxical phenomenonGF found within the framework of quantum mechanics is the ‘quantumx Cheshire Cat’: if a quantum systemPS is subject to a certain pre- and postselection, it can behave as if a particle and its propertyABS are spatially separated. It has beenH-Det. suggested to employ weak measurements in order to explore the Cheshire Cat’s nature. Here we report an experiment in whichSRs we send neutrons through a perfect silicon crystal interferometer and perform weak measurements to probe the location of the particle and its A 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 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 c is completed by two spin rotators (SRs) inside the neutron interferometer. These SRs are also used to perform the weak jii ^ ^ ^ ^ measurement of s^zÅI w and s^zÅII w. The absorbers (ABS) are inserted in the beam paths when ÅI w and ÅII w are determined. The phase shifter (PS) makes it possible to tune the relative phase between the beams in path I and path II. The two outgoing beams of the interferometer are monitored w by the H and O detector in reﬂected and forward directions, respectively. Only the neutrons reaching the O detector are affected by postselection usinga spin turner (ST2) and a spin analyzer (A).

abc Path II

SA Io SA Io SA Io 1 Atominstitut, Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria. 2 Institut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France. 3 Laboratoire de Physique The´orique et Mode´lisation, CNRS Unite´ 8089, Universite´ de Cergy-Pontoise, 95302 Cergy-Pontoise, France. 4 Institute for Quantum Studies and Schmid College of Science and Technology, Chapman University, 1 University Drive, Orange, California 92866, USA. Correspondence and requests for materials should be addressed to Y.H. (email: [email protected]). Path I NATURE COMMUNICATIONS | 5:4492 | DOI: 10.1038/ncomms5492 | www.nature.com/naturecommunications 1 O-detector & 2014 Macmillan PublishersO-detector Limited. All rights reserved. O-detector

12 10 8 6 4 Intensity (c.p.s.) 2 ABSORBER IN PATH I NO ABSORBER ABSORBER IN PATH II

– ! 0 ! ! – ! 0 ! ! – ! 0 ! ! 2 2 2 2 2 2 Phase shift " (rad) Phase shift " (rad) Phase shift " (rad)

Figure 3 | Measurement of P^ I and P^ II using an absorber with transmissivity T 0.79(1). The intensity is plotted as a function of the w w ¼ relative phase . 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 w 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.

NATURE COMMUNICATIONS | 5:4492 | DOI: 10.1038/ncomms5492 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492 ARTICLE

^ neutrons’ spin component on path j is s^zÅj w. The computation specific subspace. Then of course it would only be possible to ^ h i ^ 25 of the weak values yields s^zÅI w 1 and s^zÅII w 0. On make any inferences by appealing to counterfactual reasoning . average, a weak interaction couplingh i with¼ a probeh on pathi ¼II 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 jointly 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 1.92 Å passes magnetic birefringent prisms, which polarize because a weak measurement made along a given path involves a the¼ neutron beam. To avoid depolarization, a magnetic guide field local coupling, which in the weak limiting case only minimally pointing in the z direction is applied around the whole setup. A disturbs the subsequent evolution of the quantum state along that spin turner rotatesþ 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; . Subse- the other path. This is in clear contrast with standard projective quently, the neutrons enter a triple-Laue interferometerþi 28,31. measurements, which would fundamentally disturb the system Inside the interferometer, a spin rotator in each beam path allows evolution by projecting the quantum state along both paths to a the generation of the preselected state c . j ii

y GF x PS ABS H-Det.

SRs A

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 c is completed by two spin rotators (SRs) inside the neutron interferometer.20% These absorber SRs are also used to perform the weak jii ^ ^ II,+ ^ ^ measurement of s^zÅI w and s^zÅII w. The absorbers (ABS) are inserted in the beam paths when ÅinI w pathand Å IIII w are determined. The phase shifter (PS) makes it possible to tune the relative phase between the beams in path I and path II. The two outgoing beams of the interferometer are monitored w by the H and O detector in reﬂected+ and forward directions, respectively. Only the neutrons+ reaching the O detector are affected by postselection+ usinga spin turner (ST2) and a spin analyzer (A).

abc Path II 20% absorber I,− SA Io SA Io SA Io in path I

Path I

O-detector O-detector O-detector

12 10 8 6 4 Intensity (c.p.s.) 2 ABSORBER IN PATH I NO ABSORBER ABSORBER IN PATH II

– ! 0 ! ! – ! 0 ! ! – ! 0 ! ! 2 2 2 2 2 2 Phase shift " (rad) Phase shift " (rad) Phase shift " (rad)

NATURE COMMUNICATIONS | 5:4492 | DOI: 10.1038/ncomms5492 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492

A phase shifter is inserted into the interferometer to tune the oscillation. As a second step, an absorber with known transmis- relative phase w between path I and path II. Hence, a general sivity of T 0.79(1) is inserted into path I and the phase shifter 1 iw=2 iw=2 ¼ ABS postselected path state is given by e À I e II . Of the scan is repeated, which yields I . Finally, the absorber is taken p2 iþ j iÞ I two outgoing beams of the interferometer, only the O beam is out of path I and put into path II. The subsequent phase shifter À ABS affected by a spin analysis. The H beamffiffi without a spin analysis is scan allows the extraction of III . A typical measurement result is used as a reference monitor for phase and count-rate stability. depicted in Fig. 3. The spin postselection of the O beam is done using a spin turner In path I, the absorber has no effect. In comparison with the and a polarizing supermirror. Both outgoing beams are measured reference intensity, no significant change can be detected in the using 3He detectors with very high efficiency (over 99%). All count rate. As opposed to this, the very same absorber decreases measurements presented here are performed in a similar manner: the intensity when it is put in path II. This already tells us that the The phase shifter is rotated, thereby scanning w and recording neutrons’ population in the interferometer is obviously higher in interferograms. The interferograms allow to extract the intensity path II than it is in path I. for w 0. This ensures that the path postselection is indeed carried ¼ 1 Determining the location of the neutrons’ spin component. out on the state p2 I II corresponding to the O beam. ðj iþjiÞ The weak measurements of the neutrons’ spin component in each ffiffi path are achieved by applying additional magnetic fields in one or Determining the neutrons’ population. To determine the the other beam path. This causes a small spin rotation, which ^ neutrons’ population in the interferometer’s paths, Åj w are allows to probe the presence of the neutrons’ magnetic moment measured by inserting absorbers into the respective pathh jiof the in the respective path. If there is a magnetic moment present in ^ interferometer. Åj w is evaluated in three steps: first, a reference the path, the field has an effect on the measured interference intensity is measuredh i by performing a phase shifter scan of fringes. If no change in the interferogram can be detected, there is the empty interferometer to determine IREF. For the reference no magnetic moment present in the path where the additional measurement, the spin states inside the interferometer are field is applied. The condition of a weak measurement is fulfilled orthogonal. Therefore, the interferogram shows no intensity by tuning the magnetic field small enough. A spin rotation of

ab c

I H PATH II I H I H Bz

SA IO SA IO SA IO

B z cos10° I,+ + sin10° I,− 20° spin rotator PATH I in path II II,+ II,+ H-detector H-detector H-detector 160 + + + 150

140 130 20° spin rotator

Intensity (c.p.s.) 20° INin PATH path II NO ROTATION 20° IN PATH II I, 120 I,− − ––" 0 " – " ––" 0 " – " ––" 0 " – " cos10° I,− +2sin10° I,+2 2 2 2 2 Phase shift ! (rad) Phase shift ! (rad) Phase shift ! (rad)

O-detector O-detector O-detector 16 14 12 10 8 Intensity (c.p.s.) 20° IN PATH I NO ROTATION 20° IN PATH II 6 ––" 0 " – " ––" 0 " – " ––" 0 " – " 2 2 2 2 2 2 Phase shift ! (rad) Phase shift ! (rad) Phase shift ! (rad) ^ ^ Figure 4 | Measurement of r^zPI w and r^zPII w applying small additional magnetic fields. The intensity of the O beam (with the spin analysis) and the H beam (without the spin analysis) is plotted as a function of the relative phase . The solid lines represent least-square fits to the data and the error bars w 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 postselected ensemble (only the O detector) the neutrons’ spin component travels along path I.

4 NATURE COMMUNICATIONS | 5:4492 | DOI: 10.1038/ncomms5492 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492

Figure 1 | Artistic depiction of the quantum Cheshire Cat. Inside the interferometer, the Cat goes through the upper beam path, while its grin travels along the lower beam path. Figure courtesy of Leon Filter. r So considering only post-selected events: k ,+ 2 NATURE COMMUNICATIONS | 5:4492 | DOI:O 10.1038/ncomms5492 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. Each neutron took path II but its spin (magnetic moment) took path I ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5492

Figure 1 | Artistic depiction of the quantum Cheshire Cat. Inside the interferometer, the Cat goes through the upper beam path, while its grin travels along the lower beam path. Figure courtesy of Leon Filter. r So considering only post-selected events: k ,+ 2 NATURE COMMUNICATIONS | 5:4492 | DOI:O 10.1038/ncomms5492 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. Each neutron took path II but its spin (magnetic moment) took path I

A valid interpretation of reality? Te Neuton by Gina Berkeley

When a pion an innocent protn seduces Witin its Wit neiter excuses Twelve minuts Abuses Comes disintgraton Nor scorn Which leaves an electon in mut desolaton For its shamefl conditon And also anoter ingenuous protn Witout intrmission For oter unscrupulous pions t dot on. Te protn produces: a neuton is born. At last, a neutino; What love have you known Alas, one can see no O neuton fl grown Fulﬁlment for such a leptnic bambino. As you bombinat int te vacuum alone? No loving, no sinning Its spin is 1/2, and its mass is quit large Just spinning and spining -about 1 AMU Eight tmes trough te globe witout ever beginning... but it hasn’t a charge; A cycle mechanic Tough it ﬁnds satsfacton in stong intracton No anguish or panic It doesn’t experience Coulombic atracton For such is te patern of life inorganic. But what can you borrow O beter Of love, joy, or sorrow Te fet a O neuton, when life has so short a tmorrow? Poor human endures Tan te neuton’s dichotc Robotc Amours.