# 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

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**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