Experimental Investigation of the Quantum Measurement Process Using Electrons in Superfluid Helium

Abstract

We report on mobility measurements of electron bubbles in superfluid helium-4. Electrons are introduced into the liquid from a plasma discharge in the vapor. For electrons with energy only a small amount above the minimum energy needed to enter the liquid, the wave function is partially transmitted into the liquid. We investigate the possibility that this partial transmission results in the formation of stable electron bubbles each of which contains only a fraction of the complete electron wave function. By measuring the mobility of these bubbles we can estimate their size. Our simulation of the mobility of these bubbles is consistent with the experimental data, and supports the idea that the interaction of the electron with the liquid helium does not result in a measurement that immediately determines that an electron is, or is not, in the bubble. Other possible explanations and their difficulties are also discussed in the paper.

Appendix

To determine the effect of leakage it is necessary to find how many field lines from a charge $Q$ situated at a position $\vec{r}$ will reach the collector plate. To reach the collector these lines have to pass through the Frisch grid. This problem could possibly be addressed using standard methods based on finite element techniques. However this approach is made very difficult because of the wide range of length scales that are involved. As shown in Fig. 8, the geometry of the field lines is influenced by the presence of conducting electrodes of dimensions of the order of centimeters (such as the homogenizer disks), and also by structures with features as small as 10 $\mu$ (the wires making up the grid). This wide range of size requires a very complicated mesh.

The Monte Carlo method is as follows [81, 82]. Consider a charge located at some particular position in the cell. A “walker” starts at the position of the charge. The walker then undergoes a random walk within the volume occupied by the conducting electrodes. The random walk ends when the walker reaches an electrode. This procedure is then repeated for $N_{walkers}$ walkers, and the number $N_{collector}$ of these walkers that reach the collector, rather than some other electrode, is determined. The image charge on the collector is then given by

$$Q_{image} = - Q\,N_{collector} /N_{walkers}$$

(22)

This result as described so far would not provide a very useful algorithm because at first sight it would appear that to give an accurate result the steps of the random walk would have to be smaller than the smallest linear dimensions of any of the structures. If this were chosen to be 5 $\mu$ then for a walker to migrate a distance of 1 cm would require of the order of 10⁷ steps. However, it can be shown [81, 82] that the result of Eq. 22 still holds even if the step distance is varied during the random walk. The only requirement is that at each stage of the walk the step distance must be less than the distance $\zeta$ from the current position of the walker to the nearest electrode. Thus to make the required number of steps a minimum, in each step the walker is moved to a point randomly chosen on the surface of a sphere of radius $\zeta$ centered on the previous position. In our program the random walk is terminated when the walker reaches a position which is less than 0.1 µ from an electrode. The justification for terminating the random walk slightly outside an electrode is given in Sect. 7.2 of ref. [81]. Notice that the detailed path of the walker is not of interest.

To obtain the results shown in Fig. 9 we used $2 \times 10^{6}$ walkers with initial positions distributed uniformly over the area of the slab of ions as shown in Fig. 7. This calculation was then repeated for a series of positions of the slab giving the results shown in Fig. 9.