Proposed neutron-interferometry test of Berry's phase for a circulating planar spin

The energy eigenstates of a spin-$\frac{1}{2}$ particle in a magnetic field confined to a plane, define a planar spin. If the particle moves adiabatically around a loop in this plane, it picks up a topological Berry phase that can only be an integer multiple of $\pi$. We propose a neutron-interferometry test of the Berry phase for a circulating planar spin induced by a magnetic field caused by a very long current-carrying straight wire perpendicular to the plane. This Berry phase causes destructive interference in the direction of the incoming beam of thermal neutrons moving through a triple-Laue interferometer.

Figure 1

Left panel (a) shows a narrow beam of neutrons moving at velocity $\mathbf{v}$ with an impact parameter $b$ on a very long straight wire carrying an electrical current $I_w$ perpendicular to the plane. Right panel (b) shows the local spin associated with the spin eigenstate $|-;\mathbf{r}\rangle$ (the local spin points in the opposite direction for $|+;\mathbf{r}\rangle$). The local spin is confined to the $xy$ plane with winding number $+1$. For sufficiently strong electrical current and sufficiently slow neutrons, the spin of the neutron follows adiabatically the circular magnetic field $\mathbf{B}$. This makes it possible to measure the Berry phase manifesting the nontrivial winding number associated with the circulating local spin.

Figure 2

Triple-Laue interferometer setup to test the Berry phase associated with the local spin around the interferometer loop. A beam of thermal neutrons is coherently superposed at the first crystal plate $P$. The second crystal plate $M$ acting as mirrors recombines the beam pair at the third crystal plate $Q$. The output intensity is measured at the two detectors $D_1$ and $D_2$. In order to cancel unwanted dynamical phase effects, the current-carrying wire is located on the symmetry line connecting the two beam crossing points at $P$ and $Q$. The Berry phase associated with the planar spin gives rise to a destructive interference in the direction of the incoming beam so that all neutrons are detected at $D_2$.