Direct observation of geometric phases using a three-pinhole interferometer

We present a method to measure the geometric phase defined for three internal states of a photon (polarizations) using a three-pinhole interferometer. From the interferogram, we can extract the geometric phase related to the three-vertex Bargmann invariant as the area of a triangle formed by interference fringes. Unlike the conventional methods, our method does not involve the state evolution. Moreover, the phase calibration of the interferometer and the elimination of the dynamical phase are not required. The gauge invariance of the geometric phase corresponds to the fact that the area of the triangle is never changed by the local phase shift in each internal state.

Figure 1

Coordinate system of three-pinhole interferometer. The three pinholes are irradiated by monochromatic light and the three spherical waves interfere. We can extract the geometric phase for three states, $|{\psi }_1\rangle$, $|{\psi }_2\rangle$, and $|{\psi }_3\rangle$, from the three-pinhole interferogram.

Figure 2

Experimental setup for three-pinhole interference with different polarization states. In front of the upper pinhole, we placed a film-type linear polarizer, LP${}_1$, attached to a rotatable mount with graduated scales for adjusting the angle $\theta$, whereas in front of the lower left and right pinholes we placed film-type quarter-wave plates having orthogonal fast axes, $0^{°}$ (QWP${}_1$) and ${90}^{°}$ (QWP${}_2$), respectively, behind a linear polarizer LP${}_2$ with a fixed angle of ${30}^{°}$. Incident light on the pinholes is circularly polarized, and transmittance of light through each pinhole is 50%. Under this configuration, the visibility of the fringe $P_{\mathit{ij}}(x,y)$ is more than or equal to 0.5, which is sufficient to retrieve clear ridge lines.

Figure 3

Ridge triangles and geometric phases for $\theta =0^{°}$, ${50}^{°}$, ${90}^{°}$, and ${130}^{°}$. (a) The three-pinhole interferogram obtained in our experiment. The actual size of each figure is $1.5\hspace{0.5em}\mathrm{mm}\times 1.5\hspace{0.5em}\mathrm{mm}$. (b) Ridge lines extracted from the above interferograms. (c) Corresponding spherical triangle on the Poincaré sphere. (d) Geometric phase versus area of ridge triangle. The area of the ridge triangle is normalized by the maximum area. The solid line indicates the theoretical curve, and it shows a quadratic characteristic.

Figure 4

Ridge triangles for $\theta ={90}^{°}$ with (a) no shifts, (b) shift on pinhole 1, (c) shift on pinhole 2, and (d) shift on pinhole 3. The ridge triangle is parallel displaced, but it is not deformed for a local phase shift. The actual size of each figure is $1.5\hspace{0.5em}\mathrm{mm}\times 1.5\hspace{0.5em}\mathrm{mm}$.