# Neutron Interferometer Experiments Studying Fundamental Features of Quantum Mechanics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Cheshire Cats

#### 2.1. Initial Quantum Cheshire Cat

#### 2.2. Delayed-Choice Quantum Cheshire Cat

#### 2.3. Three-Path Quantum Cheshire Cat

#### 2.4. Exchange of Grins in Photonic System

## 3. Path Presence

## 4. Direct Test of Commutation Relation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Abs | absorber |

BARC | Bhabha Atomic Research Centre |

BBO | barium borate |

DC | direct-current |

ILL | Institut Laue-Langevin |

LHS | left-hand side |

MURR | University of Missouri Research Reactor Center |

PS | phase shifter |

RF | radio-frequency |

RHS | right-hand side |

SR | spin-rotator |

USA | United States of America |

## References

- Cohen-Tannoudji, C.; Diu, B.; Laloë, F. Quantum Mechanics, 1st ed.; Wiley: New York, NY, USA, 1977; Trans. of: Mécanique quantique. Paris: Hermann, 1973. [Google Scholar]
- Schiff, L. Quantum Mechanics; Courier Corporation: Tokyo, Japan, 1968; Available online: https://books.google.at/books/about/Quantum_Mechanics.html?id=3aMTzgEACAAJ&redir_esc=y (accessed on 13 May 2023).
- Sakurai, J.J. Modern Quantum Mechanics (Revised Edition), 1st ed.; Addison Wesley: Boston, MA, USA, 1993. [Google Scholar]
- Wheeler, J.A.; Zurek, W.H. Quantum Theory and Measurement; Princeton University Press: Princeton, NJ, USA, 1983. [Google Scholar]
- Holland, P.R. The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics; Cambridge University Press: Cambridge, MA, USA, 1993. [Google Scholar] [CrossRef]
- Feynman, R.P.; Leighton, R.B.; Sands, M.L. The Feynman Lectures on Physics; Addison-Wesley: Boston, MA, USA, 1965. [Google Scholar]
- Merli, P.G.; Missiroli, G.F.; Pozzi, G. On the statistical aspect of electron interference phenomena. Am. J. Phys.
**1976**, 44, 306–307. [Google Scholar] [CrossRef] [Green Version] - Tonomura, A. Applications of electron holography. Rev. Mod. Phys.
**1987**, 59, 639–669. [Google Scholar] [CrossRef] - Sonnentag, P.; Hasselbach, F. Measurement of Decoherence of Electron Waves and Visualization of the Quantum-Classical Transition. Phys. Rev. Lett.
**2007**, 98, 200402. [Google Scholar] [CrossRef] [Green Version] - Pan, J.W.; Chen, Z.B.; Lu, C.Y.; Weinfurter, H.; Zeilinger, A.; Żukowski, M. Multiphoton entanglement and interferometry. Rev. Mod. Phys.
**2012**, 84, 777–838. [Google Scholar] [CrossRef] - Leibfried, D.; Blatt, R.; Monroe, C.; Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys.
**2003**, 75, 281–324. [Google Scholar] [CrossRef] [Green Version] - Wineland, D.J. Nobel Lecture: Superposition, entanglement, and raising Schrödinger’s cat. Rev. Mod. Phys.
**2013**, 85, 1103–1114. [Google Scholar] [CrossRef] [Green Version] - Cornell, E.A.; Wieman, C.E. Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. Rev. Mod. Phys.
**2002**, 74, 875–893. [Google Scholar] [CrossRef] [Green Version] - Ketterle, W. Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser. Rev. Mod. Phys.
**2002**, 74, 1131–1151. [Google Scholar] [CrossRef] [Green Version] - 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] - 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] - Sala, S.; Ariga, A.; Ereditato, A.; Ferragut, R.; Giammarchi, M.; Leone, M.; Pistillo, C.; Scampoli, P. First demonstration of antimatter wave interferometry. Sci. Adv.
**2019**, 5, eaav7610. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement; Oxford University Press: Oxford, UK, 2015. [Google Scholar] [CrossRef]
- 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] - Colella, R.; Overhauser, A.W.; Werner, S.A. Observation of Gravitationally Induced Quantum Interference. Phys. Rev. Lett.
**1975**, 34, 1472–1474. [Google Scholar] [CrossRef] - Summhammer, J.; Badurek, G.; Rauch, H.; Kischko, U.; Zeilinger, A. Direct observation of fermion spin superposition by neutron interferometry. Phys. Rev. A
**1983**, 27, 2523–2532. [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] - 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] [Green Version] - Klepp, J.; Sponar, S.; Hasegawa, Y. Fundamental phenomena of quantum mechanics explored with neutron interferometers. Prog. Theor. Exp. Phys.
**2014**, 2014, 082A01. [Google Scholar] [CrossRef] [Green Version] - Sponar, S.; Sedmik, R.I.P.; Pitschmann, M.; Abele, H.; Hasegawa, Y. Tests of fundamental quantum mechanics and dark interactions with low-energy neutrons. Nat. Rev. Phys.
**2021**, 3, 309–327. [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] [Green Version] - Brun, T.A. A simple model of quantum trajectories. Am. J. Phys.
**2002**, 70, 719–737. [Google Scholar] [CrossRef] [Green Version] - Aharonov, Y.; Popescu, S.; Tollaksen, J. A time-symmetric formulation of quantum mechanics. Phys. Today
**2010**, 63, 27–32. [Google Scholar] [CrossRef] [Green Version] - 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] [Green Version] - Hosoya, A.; Shikano, Y. Strange weak values. J. Phys. A
**2010**, 43, 385307. [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] [Green Version] - Denkmayr, T.; Geppert, H.; Sponar, S.; Lemmel, H.; Matzkin, A.; Tollaksen, J.; Hasegawa, Y. Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment. Nat. Commun.
**2014**, 5, 4492. [Google Scholar] [CrossRef] [Green Version] - Wagner, R.; Kersten, W.; Lemmel, H.; Sponar, S.; Hasegawa, Y. Quantum causality emerging in a delayed-choice quantum Cheshire Cat experiment with neutrons. Sci. Rep.
**2023**, 13, 3865. [Google Scholar] [CrossRef] [PubMed] - Danner, A.; Geerits, N.; Lemmel, H.; Wagner, R.; Sponar, S.; Hasegawa, Y. Three-Path Quantum Cheshire Cat Observed in Neutron Interferometry. arXiv
**2023**, arXiv:2303.18092. [Google Scholar] - Lemmel, H.; Geerits, N.; Danner, A.; Hofmann, H.F.; Sponar, S. Quantifying the presence of a neutron in the paths of an interferometer. Phys. Rev. Res.
**2022**, 4, 023075. [Google Scholar] [CrossRef] - Wagner, R.; Kersten, W.; Danner, A.; Lemmel, H.; Pan, A.K.; Sponar, S. Direct experimental test of commutation relation via imaginary weak value. Phys. Rev. Res.
**2021**, 3, 023243. [Google Scholar] [CrossRef] - Carroll, L. Alice’s Adventures in Wonderland; MacMillan & Co.: London, UK, 1866; pp. 89–94. [Google Scholar]
- Aharonov, Y.; Popescu, S.; Rohrlich, D.; Skrzypczyk, P. Quantum Cheshire Cats. New J. Phys.
**2013**, 15, 113015. [Google Scholar] [CrossRef] - Oreshkov, O.; Costa, F.; Brukner, Č. Quantum Correlations with No Causal Order. Nat. Commun.
**2012**, 3, 1092. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brukner, Č. Quantum Causality. Nat. Phys.
**2014**, 10, 259–263. [Google Scholar] [CrossRef] - Wheeler, J.A. The “Past” and the “Delayed-Choice” Double-Slit Experiment. In Mathematical Foundations of Quantum Theory; Marlow, A.R., Ed.; Academic Press: Cambridge, MA, USA, 1978; pp. 9–48. [Google Scholar]
- Ma, X.S.; Kofler, J.; Zeilinger, A. Delayed-choice gedanken experiments and their realizations. Rev. Mod. Phys.
**2016**, 88, 015005. [Google Scholar] [CrossRef] [Green Version] - Pan, A.K. Disembodiment of arbitrary number of properties in quantum Cheshire cat experiment. Eur. Phys. J. D
**2020**, 74, 151. [Google Scholar] [CrossRef] - Stuckey, W.; Silberstein, M.; McDevitt, T. Concerning Quadratic Interaction in the Quantum Cheshire Cat Experiment. Int. J. Quantum Found.
**2016**, 2, 17. [Google Scholar] - Liu, Z.H.; Pan, W.W.; Xu, X.Y.; Yang, M.; Zhou, J.; Luo, Z.Y.; Sun, K.; Chen, J.L.; Xu, J.S.; Li, C.F.; et al. Experimental exchange of grins between quantum Cheshire cats. Nat. Commun.
**2020**, 11, 3006. [Google Scholar] [CrossRef] - Franson, J.D. Bell inequality for position and time. Phys. Rev. Lett.
**1989**, 62, 2205–2208. [Google Scholar] [CrossRef] - Das, D.; Pati, A.K. Can two quantum Cheshire cats exchange grins? New J. Phys.
**2020**, 22, 063032. [Google Scholar] [CrossRef] - Danan, A.; Farfurnik, D.; Bar-Ad, S.; Vaidman, L. Asking Photons Where They Have Been. Phys. Rev. Lett.
**2013**, 111, 240402. [Google Scholar] [CrossRef] - Geppert-Kleinrath, H.; Denkmayr, T.; Sponar, S.; Lemmel, H.; Jenke, T.; Hasegawa, Y. Multifold paths of neutrons in the three-beam interferometer detected by a tiny energy kick. Phys. Rev. A
**2018**, 97, 052111. [Google Scholar] [CrossRef] [Green Version] - 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] [Green Version] - Englert, B.G. Fringe Visibility and Which-Way Information: An Inequality. Phys. Rev. Lett.
**1996**, 77, 2154–2157. [Google Scholar] [CrossRef] [PubMed] - Hofmann, H.F. Direct evaluation of measurement uncertainties by feedback compensation of decoherence. Phys. Rev. Res.
**2021**, 3, L012011. [Google Scholar] [CrossRef] - Hall, M.J.W. Prior information: How to circumvent the standard joint-measurement uncertainty relation. Phys. Rev. A
**2004**, 69, 052113. [Google Scholar] [CrossRef] [Green Version] - 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] [Green Version] - Denkmayr, T.; Geppert, H.; Lemmel, H.; Waegell, M.; Dressel, J.; Hasegawa, Y.; Sponar, S. Experimental Demonstration of Direct Path State Characterization by Strongly Measuring Weak Values in a Matter-Wave Interferometer. Phys. Rev. Lett.
**2017**, 118, 010402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Heisenberg, W. Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Phys.
**1925**, 33, 879–893. [Google Scholar] [CrossRef] - Kennard, E.H. Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys.
**1927**, 44, 326–352. [Google Scholar] [CrossRef] - Robertson, H.P. The Uncertainty Principle. Phys. Rev.
**1929**, 34, 163–164. [Google Scholar] [CrossRef] - Arthurs, E.; Kelly, J.L., Jr. On the Simultaneous Measurement of a Pair of Conjugate Observables. Bell Labs Tech. J.
**1965**, 44, 725–729. [Google Scholar] [CrossRef] - Busch, P. Indeterminacy relations and simultaneous measurements in quantum theory. Int. J. Theor. Phys.
**1985**, 24, 63–92. [Google Scholar] [CrossRef] - Ozawa, M. Physical content of Heisenberg’s uncertainty relation: Limitation and reformulation. Phys. Lett. A
**2003**, 318, 21–29. [Google Scholar] [CrossRef] [Green Version] - Busch, P.; Lahti, P.; Werner, R.F. Proof of Heisenberg’s Error-Disturbance Relation. Phys. Rev. Lett.
**2013**, 111, 160405. [Google Scholar] [CrossRef] [Green Version] - Busch, P.; Lahti, P.; Werner, R.F. Colloquium: Quantum root-mean-square error and measurement uncertainty relations. Rev. Mod. Phys.
**2014**, 86, 1261–1281. [Google Scholar] [CrossRef] - Buscemi, F.; Hall, M.J.W.; Ozawa, M.; Wilde, M.M. Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach. Phys. Rev. Lett.
**2014**, 112, 050401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Branciard, C. Error-tradeoff and error-disturbance relations for incompatible quantum measurements. Proc. Natl. Acad. Sci. USA
**2013**, 110, 6742–6747. [Google Scholar] [CrossRef] [Green Version] - Allman, B.E.; Kaiser, H.; Werner, S.A.; Wagh, A.G.; Rakhecha, V.C.; Summhammer, J. Observation of geometric and dynamical phases by neutron interferometry. Phys. Rev. A
**1997**, 56, 4420–4439. [Google Scholar] [CrossRef] - Kim, Y.S.; Lim, H.T.; Ra, Y.S.; Kim, Y.H. Experimental verification of the commutation relation for Pauli spin operators using single-photon quantum interference. Phys. Lett. A
**2010**, 374, 4393–4396. [Google Scholar] [CrossRef] [Green Version] - Vaidman, L. Weak-measurement elements of reality. Found. Phys.
**1996**, 26, 895–906. [Google Scholar] [CrossRef] [Green Version] - Aharonov, Y.; Cohen, E.; Landsberger, T. The Two-Time Interpretation and Macroscopic Time-Reversibility. Entropy
**2017**, 19, 111. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic of a quantum Cheshire Cat where the properties of the neutrons are identified with parts of the cat. The cat’s body represents the particle and its grin represents the spin. Manipulations, each “identifying the location of a different property”, only yield effects in a different path of the interferometer such that one may perceive “the properties to be separated” inside the interferometer.

**Figure 2.**Illustration of the experimental setup of the initial quantum Cheshire Cat experiment. Purple arrows give the direction of local magnetic fields, gray arrows indicate the spatial motion—translation and rotation—of the absorber and phase shifter, respectively. Red and blue arrows are the initial up and down spin polarization vectors. The fraction that is initially up polarized is rotated by the field in the coil DC1 into the x direction.

**Figure 3.**Setup and results of the neutrons’ population in the initial quantum Cheshire Cat experiment. Purple arrows are the direction of local magnetic fields, gray arrows indicate the physical rotation of the phase shifter. An absorber is inserted (

**a**) in path I or (

**c**) in path II, while (

**b**) the reference measurement is done without the absorber.

**Figure 4.**Setup and results of the neutrons’ spin component in the initial quantum Cheshire Cat experiment. Purple arrows are the direction of local magnetic fields, gray arrows indicate the physical rotation of the phase shifter. A weak magnetic field is applied (

**a**) in path I or (

**c**) in path II, while (

**b**) the reference measurement is done without the magnetic field.

**Figure 5.**Experimental setup of the delayed-choice quantum Cheshire Cat experiment. Purple arrows give the direction of local magnetic fields, gray arrows indicate the spatial rotation of the phase shifter. Red and blue arrows are the initial up and down spin polarization vectors. The fraction that is initially up polarized is rotated by the field in the coil DC1 into the x direction. The rotation angle of the spin turner DC2 switches randomly between the two values $\pm \pi /2$, realizing the post-selections of $|{\mathrm{f}}_{\pm}\rangle $.

**Figure 6.**Graphical depiction of the emergence of the quantum Cheshire Cat with the delayed-choice implementation of the post-selected states (

**a**) $|{\mathrm{f}}_{+}\rangle $ and (

**b**) $|{\mathrm{f}}_{-}\rangle $. For both (

**a**,

**b**), the top schematic gives a possible impression of the below measurement results of intensities. The upper intensities are the result of weak absorber measurements and the bottom intensities the results of weak spin rotation measurements, while tuning the relative phase $\chi $ with the phase shifter.

**Figure 7.**Schematic of a three-path quantum Cheshire Cat where the properties of the neutrons are represented with parts of the cat. The cat’s body represents the particle, its grin represents the spin and its stripes the energy. Three different manipulations, each “identifying the location of a different property” of the neutron, only yield effects in a different path of the interferometer such that one may “perceive the three properties to be separated” inside the interferometer.

**Figure 8.**Setup of the experiment for the three-path quantum Cheshire Cat. Gray arrows indicate the spatial rotation of the two phase shifters. Red and blue arrows indicate the local polarization vectors before and after the preparation stage. The local polarization vectors are symbolized by red arrows for the up spin orientation and blue arrows for the down spin orientation. The setup contains a pre- and post-selection. In between, weak interactions can be applied while the reactions in the O-beam are monitored.

**Figure 9.**Interferograms with applied weak interactions. Intensities in O-beam given over the phase shifts induced in the path indicated at the bottom. The weak interaction is specified to the left. Conspicuous differences to the interferograms with only the preparation applied are highlighted with yellow background. As this is the case for a different path for each interaction, the perception of a three-path quantum Cheshire Cat may be given.

**Figure 10.**Schematic of the setup for “the exchange of grins” in a photonic system. The subfigure 2c of “Design of experiments” by Liu et al. [46], is licenced under CC BY 4.0, see http://creativecommons.org/licenses/by/4.0/ (accessed on 13 May 2023). Two entangled photons and their spins are interfered in a Franson interferometer. A and B represent the two Cheshire Cats Anna and Belle. The two heights of the paths are indicated as up (u) and down (d). The setup contains neutral density filters (ND), polarization-sensitive density filters (PD), glass plates (GP), a beam splitter (BS), quarter-wave plates (QWP), have-wave plates (HWP), polarizing beam splitters (PBS) and interference filters (IF). The intensity is recorded at four output ports.

**Figure 11.**Schematic of the exchange of grins between two quantum Cheshire Cats. The figure “Schematic illustration” by Liu et al. [46] is licenced under CC BY 4.0, see http://creativecommons.org/licenses/by/4.0/ (accessed on 13 May 2023). No changes were made. Two quantum Cheshire Cats, Anna (A) and Belle (B), exchange their grin and frown. This is a possible interpretation of observations in the presented photonic experiment. Two entangled photons and their spin “appear to exchange their spins”. The two heights of paths in the interferometer are indicated as up (u) and down (d).

**Figure 13.**Feedback compensation scheme applied to a path measurement in a neutron interferometer. The black arrow on the left is the incident beam direction. The circular black arrows give the orientation of spin rotations in the quadratic spin manipulators. Red arrows indicate the local polarization vectors during the procedure. Neutrons are initially polarized in $+x$ direction, denoted by the up arrow. The spin is rotated in path I by a small angle $\alpha $. The “compensation” rotations by ${\beta}_{\pm}$ in the exit beams can fully restore the original spin orientation.

**Figure 14.**Experimental setup of the feedback compensation scheme shown in Figure 13. Purple arrows give the direction of local magnetic fields, and gray arrows indicate the spatial rotation of the phase shifter. Red and blue arrows are the initial up and down spin polarization vectors. The fraction that is initially up polarized is rotated by the field in the coil DC1 into the x direction. The spin analysis is implemented only in one of the exit beams. This exit represents the $|+\rangle $ exit if $\chi =0$ and the $|-\rangle $ exit if $\chi =\pi $.

**Figure 15.**Experimental results of the path presence in path I, given by ${\omega}_{1\pm}\equiv {lim}_{\alpha \to 0}\phantom{\rule{0.166667em}{0ex}}{\beta}_{0\pm}/\alpha $, versus interaction strength $\alpha $ for $\alpha =\pi /4,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\pi /8$ and $\alpha =\pi /16$.

**Figure 16.**Schematic sketch for the experimental test of the non-commutativity of Pauli spin matrices using different sequences of rotations (

**a**) for neutrons from 1997 [67] and (

**b**) for photons from 2010 [68]. The gray arrow indicates the spatial rotation of the phase shifter. The red arrow indicates the neutron’s initial up spin state. Purple arrows indicate the orientation of local magnetic fields.

**Figure 17.**Setup of the experiment for the commutation relation. The purple arrows give the directions of local magnetic fields and the gray arrow indicates the spatial rotation of the phase shifter. Red and blue arrows are the initial up and down spin polarization vectors. In path I, the neutron beam passes through a radio-frequency (RF) spin flipper, where the combined spin/energy degree of freedom is used as a marker.

**Figure 18.**Experimental results of left-hand side (orange) and right-hand side (green) of the commutation relation in Equation (15) are plotted as a function of the phase shift $\chi $.

**Table 1.**Path presences for a 4:1 beam splitter. (

**a**) Preparation, and (

**b**) path presences depending on the final state.

(a) | Path I | Path II | |
---|---|---|---|

initial amplitudes | ${a}_{1}=\frac{2}{\sqrt{5}}$ | ${a}_{2}=\frac{1}{\sqrt{5}}$ | |

initial probabilities | ${p}_{1}=\frac{4}{5}$ | ${p}_{2}=\frac{1}{5}$ | |

(b) | Probability | Presence in Path I | Presence in Path II |

+ exit | ${p}_{+}=\frac{9}{10}$ | ${\omega}_{1+}=\frac{2}{3}$ | ${\omega}_{2+}=\frac{1}{3}$ |

− exit | ${p}_{-}=\frac{1}{10}$ | ${\omega}_{1-}=2$ | ${\omega}_{2-}=-1$ |

average | ${\overline{\omega}}_{1}=\frac{4}{5}$ | ${\overline{\omega}}_{2}=\frac{1}{5}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Danner, A.; Lemmel, H.; Wagner, R.; Sponar, S.; Hasegawa, Y.
Neutron Interferometer Experiments Studying Fundamental Features of Quantum Mechanics. *Atoms* **2023**, *11*, 98.
https://doi.org/10.3390/atoms11060098

**AMA Style**

Danner A, Lemmel H, Wagner R, Sponar S, Hasegawa Y.
Neutron Interferometer Experiments Studying Fundamental Features of Quantum Mechanics. *Atoms*. 2023; 11(6):98.
https://doi.org/10.3390/atoms11060098

**Chicago/Turabian Style**

Danner, Armin, Hartmut Lemmel, Richard Wagner, Stephan Sponar, and Yuji Hasegawa.
2023. "Neutron Interferometer Experiments Studying Fundamental Features of Quantum Mechanics" *Atoms* 11, no. 6: 98.
https://doi.org/10.3390/atoms11060098