Noncyclic geometric phase due to spatial evolution in a neutron interferometer

We present a split-beam neutron interferometric experiment to test the noncyclic geometric phase tied to the spatial evolution of the system: The subjacent two-dimensional Hilbert space is spanned by the two possible paths in the interferometer, and the evolution of the state is controlled by phase shifters and absorbers. A related experiment was reported previously by Hasegawa et al. [Phys Rev A 53, 2486 (1996)] to verify the cyclic spatial geometric phase. The interpretation of this experiment, namely to ascribe a geometric phase to this particular state evolution, has met with severe criticism from Wagh [Phys. Rev A 59, 1715 (1999)]. The extension to a noncyclic evolution manifests the correctness of the interpretation of the previous experiment by means of an explicit calculation of the noncyclic geometric phase in terms of paths on the Bloch-sphere.

Figure 1

(Color online) Experimental setup utilizing a double-loop perfect-crystal neutron interferometer: One loop is used for the state manipulation with a phase shifter (PS2) together with a beam attenuator $\left(A\right)$ and the other one provides a reference beam with adjustable phase by use of the phase shifter (PS1).

Figure 2

(Color online) Paths on the Bloch sphere corresponding to the evolution of the state $\mid {\psi }_t^0⟩$ (a) in the cyclic and (b) noncyclic case. The enclosed solid angle as seen from the origin of the sphere is proportional to the geometric phase observed in the experiment.

Figure 3

(Color online) Observed phase shift $\Phi$ for a noncyclic evolution of the state vector parameterized by the relative phase shift $\Delta \chi$. The dotted line indicates the theoretical prediction for the geometric phase assuming hundred percent visibility, whereas the solid line takes the diminished visibility into account. For a cyclic evolution $(\Delta \chi =2\pi )$ we obtain ${\Phi }_g=-0.684\left(48\right)$ radians for the transmission ratio $T_2∕T_1=0.120\left(5\right)$.