# Pointer models for quantum measurements

We examine matter-waves passing through a double-slit. Without monitoring the path of the particles, the usual interference patterns are visible on a screen behind the double-slit. However, if we try to measure the position of the particles, this interference pattern will typically be destroyed. This topic has already been discussed in Ref.  using a different set up and an inequality  quantifies that there is a smooth transition between the visibility of the fringes and the information about the path. We also find such a transition in a conceptually simple model using pointer-based measurements.

Our pointer is an additional quantum system, which is positioned behind the double-slit. Using a suitable interaction, the pointer will be kicked by the particle and hence store information on its path. This interaction will be characterized by a coupling strength $$\kappa$$ which allows us to realize a gradual measurement of the particle’s path.

Figure 1 shows a typical probability distribution for the particle to be at position $$x$$ for a time $$t$$ behind the double slit. The interaction strength is given by $$\kappa =0.15$$. Besides this relatively strong coupling to the pointer we can still observe an interference pattern, but we have also acquired clear information about the path of the particle , as can be seen in Fig. 2. Here we quantify the information about the path by regarding the distinguishability $$D$$ between the particle passing the left or the right slit. It clearly behaves complementary to the visibility $$V$$ of the interference pattern. For $$\kappa =0.15$$ we obtain a decent level of which-way information without losing all of the visibility .

Fig. 1: Position distribution of the particle as a function of time $$t$$.

Fig. 2: Fringe visibility $$V$$ (orange line) and the which-slit distinguishability $$D$$ (blue line) as a function of the coupling strength $$\kappa$$. The green line shows the coupling used for Fig. 1.

### Contributors

M. Daiber, I. Miller, M. Freyberger

### Collaborations

W.H. Zurek (Los Alamos, USA)

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### References

 W. K. Wootters, W. H. Zurek, Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle, Phys. Rev. D 19, 473 (1979)
 B.-G. Englert, Fringe-Visibility and Which-Way Information: An Inequality, Phys. Rev. Lett. 77, 2154 (1996)
 M. Daiber, Materiewelleninterferenz und Komplementarität, Zulassungsarbeit Staatsexamen Physik, Institut für Quantenphysik, Universität Ulm, 2016
 L. F. Streiter, Komplementaritätsrelation zwischen Interferenz- und Welcher-Weg-Information, Bachelorarbeit Physik, Institut für Quantenphysik, Universität Ulm, 2016