Off-Diagonal Geometric Phase for Mixed States

We extend the off-diagonal geometric phase [Phys. Rev. Lett. 85, 3067 (2000)] to mixed quantal states. The nodal structure of this phase in the qubit (two-level) case is compared with that of the diagonal mixed state geometric phase [Phys. Rev. Lett. 85, 2845 (2000)]. Extension to higher dimensional Hilbert spaces is delineated. A physical scenario for the off-diagonal mixed state geometric phase in polarization-entangled two-photon interferometry is proposed.

Figure 1

Nodal surfaces of the off-diagonal mixed state geometric phase for a qubit. For Bures fidelity ${\mathcal{F}}_B>0$ (mixed states), there are nodes also for paths with pure state visibility $\eta \ne 1$ at various solid angles $\Omega$.

Figure 2

Franson setup for polarization-entangled photon pairs. Polarization flips are applied to the shorter arms. In the longer arms, the photons are exposed to the unitarities $U^{\parallel }$ and $V$, respectively. The shift in the coincidence interference oscillations obtained by varying the $U(1)$ phase $\chi$ is determined by the pair of unitarities in the longer arms.