# A Method for Measuring the Weak Value of Spin for Metastable Atoms

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## Abstract

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## 1. Introduction

## 2. Details of the Experimental Apparatus to Determine Weak Values of Spin

#### 2.1. Overview

#### 2.2. Stern–Gerlach Simulation Using the Impulse Approximation

#### 2.3. Initial Conditions

#### 2.4. Theory of the Weak Stage Process

#### 2.5. Extracting the Weak Value of Spin

#### 2.6. Free Evolution of the Gaussian Wave Packet at the Detector

#### 2.7. The Limit and Its Validity

## 3. Method for the Weak Measurement of Spin for Atomic Systems: Experimental Realisation

#### 3.1. Schematic Lay-Out of the Apparatus

#### 3.2. Experimental Data Confirming the Correct Functioning of the Last (Post-Selection) Stage

- It has a lifetime of approximately 8000 s [33], being unable to decay via electric dipole transitions and the Pauli exclusion principle, i.e., its decay is doubly forbidden. This lifetime is clearly large enough for the atoms to pass through all the stages of the apparatus before decaying. Furthermore, this allows scope for increasing the flight distance with no depreciable effects.
- Metastable helium atoms have an internal energy of 19.6 eV, the highest of any metastable noble gas species. Upon collision with any surface, it will easily ionise, and the emitted electron is observed with higher efficiency at the microchannel plate (MCP) detector.

#### 3.3. The Functioning of the Hexapole Stage

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- 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.
**1990**, 41, 11–19. [Google Scholar] [CrossRef] - Wiseman, H. Grounding Bohmian mechanics in weak values and Bayesianism. Phys. Lett. A
**2003**, 311, 285–291. [Google Scholar] [CrossRef] - Leavens, C.R. Weak Measurements from the point of view of Bohmian Mechanics. Found. Phys.
**2005**, 35, 469–491. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J. The Undivided Universe: An Ontological Interpretation of Quantum Mechanics; Routledge: London, UK, 1993. [Google Scholar]
- Flack, R.; Hiley, B.J. Feynman Paths and Weak Values. Entropy
**2018**, 20, 367. [Google Scholar] [CrossRef] - Schwinger, J. The Theory of Quantum Fields III. Phys. Rev.
**1953**, 91, 728–740. [Google Scholar] [CrossRef] - Feynman, R.P. Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys.
**1948**, 20, 367–387. [Google Scholar] [CrossRef][Green Version] - 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] - Hiley, B.J.; Aziz Mufti, A.H. The Ontological Interpretation of Quantum Field Theory Applied in a Cosmological Context, Fundamental Problems in Quantum Physics; Ferrero, M., van der Merwe, A., Eds.; Kluwer: Dordrecht, The Netherlands, 1995; pp. 141–156. [Google Scholar]
- Flack, R.; Hiley, B.J. Weak Values of Momentum of the Electromagnetic Field: Average Momentum Flow Lines, Not Photon Trajectories. arXiv
**2016**, arXiv:1611.06510. [Google Scholar] - Mahler, D.H.; Rozema, L.A.; Fisher, K.; Vermeyden, L.; Resch, K.J.; Braverman, B.; Wiseman, H.M.; Steinberg, A.M. Measuring Bohm trajectories of entangled photons. In Lasers and Electro-Optics (CLEO); IEEE: Piscataway, NJ, USA, 2014; pp. 1–2. [Google Scholar]
- Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, II. Phys. Rev.
**1952**, 85, 180–193. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B.J.; Kaloyerou, P.N. An Ontological Basis for the Quantum Theory: II-A Causal Interpretation of Quantum Fields. Phys. Rep.
**1987**, 144, 349–375. [Google Scholar] [CrossRef] - Holland, P.R. The de Broglie-Bohm theory of motion and quantum field theory. Phys. Rep.
**1993**, 224, 95–150. [Google Scholar] [CrossRef] - Kaloyerou, P.N. The Causal Interpretation of the Electromagnetic field. Phys. Rep.
**1994**, 244, 287–358. [Google Scholar] [CrossRef] - Morley, J.; Edmunds, P.D.; Barker, P.F. Measuring the weak value of the momentum in a double slit interferometer. J. Phys. Conf. Ser.
**2016**, 701, 012030. [Google Scholar] [CrossRef] - Sponar, S.; Denkmayr, T.; Geppert, H.; Lemmel, H.; Matzkin, A.; Tollaksen, J.; Hasegawa, Y. Weak values obtained in matter-wave interferometry. Phys. Rev. A
**2014**, 92, 062121. [Google Scholar] [CrossRef] - Bohm, D.; Schiller, R.; Tiomno, J. A Causal Interpretation of the Pauli Equation (A). Nuovo Cim. Supp.
**1955**, 1, 48–66. [Google Scholar] [CrossRef] - Bohm, D.; Schiller, R. A Causal Interpretation of the Pauli Equation (B). Nuovo Cim. Supp.
**1955**, 1, 67–91. [Google Scholar] [CrossRef] - Hiley, B.J.; van Reeth, P. Quantum Trajectories: Real or Surreal? Entropy
**2018**, 20, 353. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A. What happens in a spin measurement? Phys. Lett. A
**1986**, 119, 259–267. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A. A Causal Account of Non-local Einstein-Podolsky-Rosen Spin Correlations. J. Phys. A Math. Gen.
**1987**, 20, 4717–4732. [Google Scholar] [CrossRef] - Dewdney, C.; Holland, P.R.; Kyprianidis, A.; Vigier, J.-P. Spin and non-locality in quantum mechanics. Nature
**1988**, 336, 536–544. [Google Scholar] [CrossRef] - Monachello, V.; Flack, R. The weak value of spin for atomic systems. J. Phys. Conf. Ser.
**2016**, 701, 012028. [Google Scholar] [CrossRef][Green Version] - Bohm, D. Quantum Theory; Prentice Hall: New York, NY, USA, 1951. [Google Scholar]
- 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. A
**1989**, 40, 2112–2117. [Google Scholar] [CrossRef] - Ballentine, L.E. Quantum Mechanics: A Modern Development; World Scientific Publishing: New York, NY, USA, 1998. [Google Scholar]
- Pan, A.K.; Matzkin, A. Weak values in nonideal spin measurements: An exact treatment beyond the asymptotic regime. Phys. Rev. A
**2012**, 85, 022122. [Google Scholar] [CrossRef] - Halfmann, T.; Koensgen, J.; Bergmann, K. A source for a high-intensity pulsed beam of metastable helium atoms. Meas. Sci. Technol.
**2000**, 11, 1510–1514. [Google Scholar] [CrossRef] - Bleaney, B.I.; Bleaney, B. Electricity and Magnetism; Oxford University Press: London, UK, 1965. [Google Scholar]
- Baldwin, K. Metastable helium: Atom optics with nano-grenades. Contemp. Phys.
**2005**, 46, 105–120. [Google Scholar] [CrossRef] - Hodgman, S.S.; Dall, R.G.; Byron, L.J.; Baldwin, K.G.H.; Buckman, S.J.; Truscott, A.G. Metastable helium: A new determination of the longest atomic excited-state lifetime. Phys. Rev. Lett.
**2009**, 103, 053002. [Google Scholar] [CrossRef] [PubMed] - Halbach, K. Design of permanent multipole magnets with oriented rare earth cobalt material. Nuclear Instrum. Meth.
**1980**, 169, 1–10. [Google Scholar] [CrossRef]

**Figure 1.**Schematic view of the experimental technique [25]. Helium atoms in the ${m}_{S}=+1$ metastable state enter from the left, with spin vector angle $\theta $. The atoms pass through the weak and strong S-G magnets before reaching the detector. The displacement due to the weak interaction is ${\Delta}_{w}$, which is a function of the chosen pre-selected spin state. For simplicity, the ${m}_{S}=0$ spin state is not shown.

**Figure 2.**The pulsed helium gas enters from the left. Preparation of the metastable atoms occurs in the first two chambers. In the next chamber, the hexapole magnet (HM) pre-selects the ${m}_{S}=+1$ state, which moves onto the weak stage (WS), which is comprised of the magnet, and then on to the strong stage (SS) involving the magnet. Finally, the atoms are detected using a micro-channel plate detector (MCP). This figure is reproduced from [25].

**Figure 3.**A series of plots showing how the displacement, ${\Delta}_{w}$, of the Gaussian wave packet is constrained by various limits. The red curve is the first order approximation, which is dominated by $\mathrm{tan}(\theta /2)$. The blue curve is the exact treatment of the system taking into account all higher order terms. The red and blue curves coincide when the limit $L={L}_{o}=0.37$; this is the maximum limit for which the first order approximation holds.

**Figure 4.**The S-G magnet showing the various grades/shapes of the Nd-Fe-B magnets in the setup in order to achieve a constant field gradient, $dB/dx$, of 100 T/m.

**Figure 5.**Distribution of three metastable species along the x-axis as they travel through a strong S-G magnet and are detected via an MCP detector. From top to bottom, metastable helium (He*) in the ${2}^{3}{S}_{1}$ triplet state with ${m}_{S}=\pm 1,0$, metastable neon (Ne*) and argon (Ar*) in the $3{P}_{2}$ state with ${m}_{J}=\pm 2,\pm 1,0$. The states are clearly delineated, indicating that they would be good candidates for measuring weak values of angular momentum. The central peak contribution is larger for all cases due to the double contribution from the $m=0$ state and photons.

**Figure 7.**Simulation of a He* beam travelling through the designed hexapole magnet; the dashed red lines signify the ${m}_{s}=-1$ defocused state, while the blue solid lines signify the ${m}_{s}=+1$ focused state.

**Figure 8.**Distribution of the ${m}_{S}=+1$ and ${m}_{S}=0$ spin states of the system along the x-axis. When a He* beam travels through a permanent hexapole magnet, the ${m}_{S}=-1$ spin state is defocused and lost to the magnet and the vacuum chamber walls. Note: the width of the atom beam is larger here due to the removal of the collimation region before the S-G magnet for test purposes.

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**MDPI and ACS Style**

Flack, R.; Monachello, V.; Hiley, B.; Barker, P.
A Method for Measuring the Weak Value of Spin for Metastable Atoms. *Entropy* **2018**, *20*, 566.
https://doi.org/10.3390/e20080566

**AMA Style**

Flack R, Monachello V, Hiley B, Barker P.
A Method for Measuring the Weak Value of Spin for Metastable Atoms. *Entropy*. 2018; 20(8):566.
https://doi.org/10.3390/e20080566

**Chicago/Turabian Style**

Flack, Robert, Vincenzo Monachello, Basil Hiley, and Peter Barker.
2018. "A Method for Measuring the Weak Value of Spin for Metastable Atoms" *Entropy* 20, no. 8: 566.
https://doi.org/10.3390/e20080566