# Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Neutron Interferometer Experiments in History

## 3. Spin Weak Measurements

#### 3.1. Theoretical Framework

#### 3.2. Neutron Optical Approach

#### 3.3. Experimental Results

## 4. The Quantum Cheshire-Cat

#### 4.1. Theory

#### 4.2. Experiment

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AAV | Aharonov, Albert and Vaidman |

AB | Aharonov-Bohm |

AC | Aharonov-Casher |

DC | direct current |

DOF | degrees of freedom |

GHZ | Greenberger-Horne-Zeilinger |

ILL | Institute Laue-Langevin |

IFM | interferometer |

LLL | triple Laue |

RF | radio frequency |

VCN | very cold neutrons |

## References

- Tonomura, A. Applications of electron holography. Rev. Mod. Phys.
**1987**, 59, 639–669. [Google Scholar] [CrossRef] - Feynman, R.; Leighton, R.; Sands, M. The Feynman Lectures on Physics, 2nd ed.; Addison-Wesley: Boston, MA, USA, 1963; Volume 1. [Google Scholar]
- Arndt, M.; Ekers, A.; von Klitzing, W.; Ulbricht, H. Focus on modern frontiers of matter wave optics and interferometry. New J. Phys.
**2012**, 14, 125006. [Google Scholar] [CrossRef] - Cronin, A.D.; Schmiedmayer, J.; Pritchard, D.E. Optics and interferometry with atoms and molecules. Rev. Mod. Phys.
**2009**, 81, 1051–1129. [Google Scholar] [CrossRef] - Popescu, S. Dynamical quantum non-locality. Nat. Phys.
**2010**, 6, 151–153. [Google Scholar] [CrossRef] - Bertlmann, R.A.; Zeilinger, A. Quantum [Un]speakables, from Bell to Quantum Information; Springer Verlag: Heidelberg, Germany, 2002. [Google Scholar]
- Maier-Leibnitz, H.; Springer, T. Ein Interferometer für langsame Neutronen. Z. Phys.
**1962**, 167, 386. [Google Scholar] [CrossRef] - Mezei, F. Neutron spin echo: A new concept in polarized thermal neutron techniques. Z. Phys.
**1972**, 25, 146. [Google Scholar] [CrossRef] - Rauch, H.; Treimer, W.; Bonse, U. Test of a single crystal neutron interferometer. Phys. Lett. A
**1974**, 47, 369–371. [Google Scholar] [CrossRef] - Rauch, H.; Werner, S.A. Neutron Interferometry; Clarendon Press: Oxford, UK, 2000. [Google Scholar]
- Rauch, H.; Zeilinger, A.; Badurek, G.; Wilfing, A.; Bauspiess, W.; Bonse, U. Verification of Coherent Spinor Rotation of Fermions. Phys. Lett. A
**1975**, 54, 425–427. [Google Scholar] [CrossRef] - Werner, S.A.; Colella, R.; Overhauser, A.W.; Eagen, C.F. Observation of the Phase Shift of a Neutron Due to Precession in a Magnetic Field. Phys. Rev. Lett.
**1975**, 35, 1053–1055. [Google Scholar] [CrossRef] - Klein, A.G.; Opat, G.I. Observation of 2π Rotations by Fresnel Diffraction of Neutrons. Phys. Rev. Lett.
**1976**, 37, 238–240. [Google Scholar] [CrossRef] - Grigoriev, S.; Kraan, W.; Rekveldt, M.Th. Observation of 4-pi periodicity of the spinor using neutron resonance interferometry. Europhys. Lett.
**2004**, 66, 164–170. [Google Scholar] - Colella, R.; Overhauser, A.W.; Werner, S.A. Observation of Gravitationally Induced Quantum Interference. Phys. Rev. Lett.
**1975**, 34, 1472–1474. [Google Scholar] [CrossRef] - Sakurai, J.J. Modern Quantum Mechanics; Addison-Wesley: New York, NY, USA, 1994. [Google Scholar]
- Littrell, K.C.; Allman, B.E.; Werner, S.A. Two-wavelength-difference measurement of gravitationally induced quantum interference phases. Phys. Rev. A
**1997**, 56, 1767–1780. [Google Scholar] [CrossRef] - Van der Zouw, G.; Weber, M.; Felber, J.; Gähler, R.; Geltenbort, P.; Zeilinger, A. Aharonov-Bohm and gravity experiments with the very-cold-neutron interferometer. Nucl. Instrum. Meth. A
**2000**, 440, 568–574. [Google Scholar] [CrossRef] - Klepp, J.; Sponar, S.; Hasegawa, Y. Fundamental phenomena of quantum mechanics explored with neutron interferometers. Prog. Theor. Exp. Phys.
**2014**, 2014. [Google Scholar] [CrossRef] - De Haan, V.O.; Plomp, J.; van Well, A.A.; Rekveldt, M.T.; Hasegawa, Y.H.; Dalgliesh, R.M.; Steinke, N.J. Measurement of gravitation-induced quantum interference for neutrons in a spin-echo spectrometer. Phys. Rev. A
**2014**, 89, 063611. [Google Scholar] [CrossRef] - Nielsen, M.A.; Chuang, I. Quantum Computation and Quantum Information; Cambridge Unviversity Press: Cambridge, UK, 2000. [Google Scholar]
- Bell, J.S. On the Einstein-Podolsky-Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] - Kochen, S.; Specker, E.P. The problem of hidden variables in quantum mechanics. J. Math. Mech.
**1967**, 17, 59–87. [Google Scholar] [CrossRef] - Hasegawa, Y.; Loidl, R.; Badurek, G.; Baron, M.; Rauch, H. Violation of a Bell-like inequality in single-neutron interferometry. Nature
**2003**, 425, 45–48. [Google Scholar] [CrossRef] [PubMed] - Geppert, H.; Denkmayr, T.; Sponar, S.; Lemmel, H.; Hasegawa, Y. Improvement of the polarized neutron interferometer setup demonstrating violation of a Bell-like inequality. Nucl. Instrum. Methods Phys. Res. Sect. A
**2014**, 763, 417–423. [Google Scholar] [CrossRef] [PubMed] - Hasegawa, Y.; Loidl, R.; Badurek, G.; Baron, M.; Rauch, H. Quantum Contextuality in a Single-Neutron Optical Experiment. Phys. Rev. Lett.
**2006**, 97, 230401. [Google Scholar] [CrossRef] [PubMed] - Cabello, A.; Filipp, S.; Rauch, H.; Hasegawa, Y. Proposed Experiment for Testing Quantum Contextuality with Neutrons. Phys. Rev. Lett.
**2008**, 100, 130404. [Google Scholar] [CrossRef] [PubMed] - Bartosik, H.; Klepp, J.; Schmitzer, C.; Sponar, S.; Cabello, A.; Rauch, H.; Hasegawa, Y. Experimental Test of Quantum Contextuality in Neutron Interferometry. Phys. Rev. Lett.
**2009**, 103, 040403. [Google Scholar] [CrossRef] [PubMed] - Greenberger, D.M.; Horne, M.A.; Zeilinger, A. Bell’s Theorem, Quantum Theory, and Concepts of the Universe; Kafatos, M., Ed.; Kluwer Academics: Dordrecht, The Netherlands, 1989; pp. 73–76. [Google Scholar]
- Greenberger, D.M.; Shimony, A.; Horne, M.A.; Zeilinger, A. Bell’s theorem without inequalities. Am. J. Phys.
**1990**, 58, 1131–1143. [Google Scholar] [CrossRef] - Hasegawa, Y.; Loidl, R.; Badurek, G.; Durstberger-Rennhofer, K.; Sponar, S.; Rauch, H. Engineering of triply entangled states in a single-neutron system. Phys. Rev. A
**2010**, 81, 032121. [Google Scholar] [CrossRef] - Erdösi, D.; Huber, M.; Hiesmayr, B.C.; Hasegawa, Y. Proving the generation of genuine multipartite entanglement in a single-neutron interferometer experiment. New J. Phys.
**2013**, 15, 023033. [Google Scholar] [CrossRef] - Aharonov, Y.; Albert, D.Z.; Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett.
**1988**, 60, 1351–1354. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Vaidman, L. Properties of a quantum system during the time interval between two measurements. Phys. Rev. A
**1990**, 41, 11–20. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Vaidman, L. The two-state vector formalism: an updated review. Lect. Nots. Phys.
**2007**, 734, 399–447. [Google Scholar] - Aharonov, Y.; Bergmann, P.G.; Lebowitz, J.L. Time Symmetry in the Quantum Process of Measurement. Phys. Rev.
**1964**, 134, 1410–1416. [Google Scholar] [CrossRef] - Ritchie, N.W.M.; Story, J.G.; Hulet, R.G. Realization of a measurement of a “weak value”. Phys. Rev. Lett.
**1991**, 66, 1107–1110. [Google Scholar] [CrossRef] [PubMed] - Duck, I.M.; Stevenson, P.M.; Sudarshan, E.C.G. The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100. Phys. Rev. D
**1989**, 40, 2112–2117. [Google Scholar] [CrossRef] - Hosten, O.; Kwiat, P. Observation of the Spin Hall Effect of Light via Weak Measurements. Science
**2008**, 319, 787–790. [Google Scholar] [CrossRef] [PubMed] - Dixon, P.B.; Starling, D.J.; Jordan, A.N.; Howell, J.C. Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification. Phys. Rev. Lett.
**2009**, 102, 173601. [Google Scholar] [CrossRef] [PubMed] - Starling, D.J.; Dixon, P.B.; Jordan, A.N.; Howell, J.C. Precision frequency measurements with interferometric weak values. Phys. Rev. A
**2010**, 82, 063822. [Google Scholar] [CrossRef] - Starling, D.J.; Dixon, P.B.; Williams, N.S.; Jordan, A.N.; Howell, J.C. Continuous phase amplification with a Sagnac interferometer. Phys. Rev. A
**2010**, 82, 011802. [Google Scholar] [CrossRef] - Feizpour, A.; Xing, X.; Steinberg, A.M. Amplifying Single-Photon Nonlinearity Using Weak Measurements. Phys. Rev. Lett.
**2011**, 107, 133603. [Google Scholar] [CrossRef] [PubMed] - Ota, Y.; Ashhab, A.; Nori, F. Entanglement amplification via local weak measurements. J. Phys. A
**2012**, 45, 415303. [Google Scholar] [CrossRef] - Zhou, L.; Turek, Y.; Sun, C.P.; Nori, F. Weak-value amplification of light deflection by a dark atomic ensemble. Phys. Rev. A
**2013**, 88, 053815. [Google Scholar] [CrossRef] - Kofman, A.G.; Ashhab, S.; Nori, F. Nonperturbative theory of weak pre- and post-selected measurements. Phys. Rep.
**2012**, 520, 43–133. [Google Scholar] [CrossRef] - Kocsis, S.; Braverman, B.; Ravets, S.; Stevens, M.J.; Mirin, R.P.; Shalm, L.K.; Steinberg, A.M. Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer. Science
**2011**, 332, 1170–1173. [Google Scholar] [CrossRef] [PubMed] - Lundeen, J.S.; Sutherland, B.; Patel, A.; Stewart, C.; Bamber, C. Direct measurement of the quantum wavefunction. Nature
**2011**, 474, 188–191. [Google Scholar] [CrossRef] [PubMed] - Goggin, M.E.; Almeida, M.P.; Barbieri, M.; Lanyon, B.P.; O’Brien, J.L.; White, A.G.; Pryde, G.J. Violation of the Leggett-Garg inequality with weak measurements of photons. Proc. Natl. Acad. Sci. USA
**2011**, 108, 1256–1261. [Google Scholar] [CrossRef] [PubMed] - Salvail, J.Z.; Agnew, M.; Johnson, A.S.; Bolduc, E.; Leach, J.; Boyd, R.W. Full characterization of polarization states of light via direct measurement. Nat. Photonics
**2013**, 7, 316–321. [Google Scholar] [CrossRef] - Dressel, J.; Malik, M.; Miatto, F.M.; Jordan, A.N.; Boyd, R.W. Colloquium: Understanding quantum weak values: Basics and applications. Rev. Mod. Phys.
**2014**, 86, 307–316. [Google Scholar] [CrossRef] - Ozawa, M. Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A
**2003**, 67, 042105. [Google Scholar] [CrossRef] - Rozema, L.A.; Darabi, A.; Mahler, D.H.; Hayat, A.; Soudagar, Y.; Steinberg, A.M. Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements. Phys. Rev. Lett.
**2012**, 109, 100404. [Google Scholar] [CrossRef] [PubMed] - Ringbauer, M.; Biggerstaff, D.N.; Broome, M.A.; Fedrizzi, A.; Branciard, C.; White, A.G. Experimental Joint Quantum Measurements with Minimum Uncertainty. Phys. Rev. Lett.
**2014**, 112, 020401. [Google Scholar] [CrossRef] [PubMed] - Kaneda, F.; Baek, S.Y.; Ozawa, M.; Edamatsu, K. Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement. Phys. Rev. Lett.
**2014**, 112, 020402. [Google Scholar] [CrossRef] [PubMed] - Dressel, J.; Nori, F. Certainty in Heisenberg’s uncertainty principle: Revisiting definitions for estimation errors and disturbance. Phys. Rev. A
**2014**, 89, 022106. [Google Scholar] [CrossRef] - Resch, J.; Lundeen, J.; Steinberg, A. Experimental realization of the quantum box problem. Phys. Lett. A
**2004**, 324, 125–131. [Google Scholar] [CrossRef] - Lundeen, J.S.; Steinberg, A.M. Experimental Joint Weak Measurement on a Photon Pair as a Probe of Hardy’s Paradox. Phys. Rev. Lett.
**2009**, 102, 020404. [Google Scholar] [CrossRef] [PubMed] - Yokota, K.; Yamamoto, T.; Koashi, M.; Imoto, N. Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair. New J. Phys.
**2009**, 11, 033011. [Google Scholar] [CrossRef] - Aharonov, Y.; Botero, A.; Popescu, S.; Reznik, B.; Tollaksen, J. Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A
**2002**, 301, 130–138. [Google Scholar] [CrossRef] - Bliokh, K.Y.; Bekshaev, A.Y.; Kofman, A.G.; Nori, F. Photon trajectories, anomalous velocities and weak measurements: a classical interpretation. New J. Phys.
**2013**, 15, 073022. [Google Scholar] [CrossRef] - Dressel, J.; Bliokh, K.Y.; Nori, F. Classical Field Approach to Quantum Weak measurements. Phys. Rev. Lett.
**2014**, 112, 110407. [Google Scholar] [CrossRef] [PubMed] - Sponar, S.; Denkmayr, T.; Geppert, H.; Lemmel, H.; Matzkin, A.; Tollaksen, J.; Hasegawa, Y. Weak values obtained in matter-wave interferometry. Phys. Rev. A
**2015**, 92, 062121. [Google Scholar] [CrossRef] - Wu, S.; Mølmer, K. Weak measurements with a qubit meter. Phys. Lett. A
**2009**, 374, 34–39. [Google Scholar] [CrossRef] - Aharonov, Y.; Popescu, S.; Rohrlich, D.; Skrzypczyk, P. Quantum Cheshire Cats. New J. Phys.
**2013**, 15, 113015. [Google Scholar] [CrossRef] - Denkmayr, T.; Geppert, H.; Sponar, S.; Lemmel, H.; Matzkin, A.; Tollaksen, J.; Hasegawa, Y. Experimental observation of a quantum cheshire cat in matter wave interferometry. Nat. Commun.
**2014**, 5, 4492. [Google Scholar] [CrossRef] [PubMed] - Aharonov, Y.; Cohen, E. Weak Values and Quantum Nonlocality. 2015; arXiv:1504.03797. [Google Scholar]
- Danan, A.; Danan, D.; Bar-Ad, S.; Vaidman, L. Asking Photons Where They Have Been. Phys. Rev. Lett.
**2013**, 111, 240402. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Original setup for weak spin measurement of spin $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-particles, proposed by AAV. After the weak interaction, the sub beams belonging to ${\widehat{\sigma}}_{\pm z}$ are still overlapping by a large extent.

**Figure 2.**Schematic illustration of a triple Laue (LLL) neutron interferometer experiment for a weak measurement of the spin operator ${\widehat{\sigma}}_{z}$. The setup consists of three stages: (i) pre-selection (

**green**) using a magnetic field prism for polarisation and a $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-spinrotator the initial state$|{\psi}_{\mathrm{i}}\rangle =|{\mathrm{S}}_{x};+\rangle $ is prepared; (ii) weak interaction (

**blue**) in the interferometer a weak spin rotation by $\pm \phantom{\rule{0.166667em}{0ex}}\alpha $ is applied in arm I and II, respectively; (iii) post-selection (

**brown**) a combination of a spin-rotator and analysing supermirrow is used to post-select on the final state $|{\psi}_{\mathrm{f}}(\theta ,\varphi )\rangle $, before count rate detection.

**Figure 3.**Experimentally determined real (

**top**panel) and modulus (

**bottom**panel) and a direct measurement of the imaginary component (

**central**panel), of the weak value of ${\widehat{\sigma}}_{z}$, together with the theoretical predictions (

**blue**line). Bloch-sphere representations are given for pre- and post-selected spin state (

**a**) $\varphi =0$; and (

**b**) $\varphi =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. For $\varphi =0$ and $\theta =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ the weak value of ${\widehat{\sigma}}_{z}$ equals the expectation value of ${\widehat{\sigma}}_{z}$ (

**red**arrow).

**Figure 4.**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.

**Figure 5.**Measurement of ${\langle {\widehat{\Pi}}_{\mathrm{I}}\rangle}_{\mathrm{w}}$ and ${\langle {\widehat{\Pi}}_{\mathrm{II}}\rangle}_{\mathrm{w}}$ using an absorber. The intensity is plotted as a function of the relative phase χ. The solid lines represent least-square fits to the data and the error bars represent one standard deviation (

**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 post-selected ensemble, the neutrons behave as going through path II.

**Figure 6.**Measurement of ${\langle {\widehat{\sigma}}_{z}{\widehat{\Pi}}_{\mathrm{I}}\rangle}_{\mathrm{w}}$ and ${\langle {\widehat{\sigma}}_{z}{\widehat{\Pi}}_{\mathrm{II}}\rangle}_{\mathrm{w}}$ applying small additional magnetic fields. The intensity is plotted as a function of the relative phase χ. 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; (

**b**) a reference measurement without any additional magnetic fields, where due to the orthogonal spin states no interference fringes are observed; (

**c**) a magnetic field in path II; interference pattens do not differ significantly from reference measurement, suggesting the neutrons’ spin component travels along path I.

Population | Magnetic Moment |
---|---|

${\langle {\widehat{\Pi}}_{\mathrm{I}}\rangle}_{\mathrm{w}}$ = 0.14(4) | $|{\langle {\widehat{\sigma}}_{z}{\widehat{\Pi}}_{\mathrm{I}}\rangle}_{\mathrm{w}}{|}^{2}$ = 1.07(25) |

${\langle {\widehat{\Pi}}_{\mathrm{I}}\rangle}_{\mathrm{w}}$ = 0.96(6) | $|{\langle {\widehat{\sigma}}_{z}{\widehat{\Pi}}_{\mathrm{II}}\rangle}_{\mathrm{w}}{|}^{2}$ = 0.02(24) |

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**MDPI and ACS Style**

Sponar, S.; Denkmayr, T.; Geppert, H.; Hasegawa, Y.
Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat. *Atoms* **2016**, *4*, 11.
https://doi.org/10.3390/atoms4010011

**AMA Style**

Sponar S, Denkmayr T, Geppert H, Hasegawa Y.
Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat. *Atoms*. 2016; 4(1):11.
https://doi.org/10.3390/atoms4010011

**Chicago/Turabian Style**

Sponar, Stephan, Tobias Denkmayr, Hermann Geppert, and Yuji Hasegawa.
2016. "Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat" *Atoms* 4, no. 1: 11.
https://doi.org/10.3390/atoms4010011