Berry Phase in Pathangled Systems

Abstract

We introduce pathangled quantum states, spatially correlated systems governed via production angles, to achieve geometric control of entanglement beyond spin/polarization constraints. By driving the system through cyclic adiabatic evolution of an external parameter in Mach-Zehnder interferometers, we demonstrate that Berry phases and production angles become additional degrees of freedom for Bell correlations. We identify an approximate critical angle $24.97^\circ$ that geometrically manifests the Bell-limit for certain measurement settings, delineating boundaries between local-hidden-variable theories and quantum mechanics. This framework simplifies state preparation while enabling geometry-driven entanglement control, thus providing distinct experimental advantages.

Figures (2)

• Figure 1: Schematic representation of pathangled systems. In the first scenario, particles adiabatically driven by $R_1(t)$ acquire a geometric phase and are detected after $P_{LR_1}$ and $BS_1$. In the second scenario, cyclic evolution under $R_2(t)$ with $P_{LR_2}$ enables geometric/topological phase observation in closed MZ trajectories. Bell limit occurs at production angle $(\alpha)$ equals $\alpha_c\approx24.97^\circ$, quantum nonlocality manifests for $\alpha_c <\alpha<\pi/2-\alpha_c$ in blue regions with certain retarder settings.
• Figure 2: Correlation function $S(\alpha,\gamma)$ under distinct scenarios. (Left) Plots of $S_{\rm I}(\alpha,\gamma)= \sqrt{2}+\sqrt{2}C(\alpha)\left|\cos{2\gamma}\right|$ vs $\gamma$ for fixed $C(\alpha)$ values. The shared control of $C(\alpha)$ and $\gamma$ over the same term bounds the minimum of $S$ at $\sqrt{2}$. (Center) Plots of $S_{\rm II}(\alpha,\gamma)=C\sqrt{2}+\sqrt{2}\left|\cos{2\gamma}\right|$ vs $\gamma$ for fixed $C(\alpha)$, with colors matching the left panel. Independent control over $C(\alpha)$ and $\gamma$ enables access to all correlation parameters in the range $(0,2\sqrt{2})$. (Right) Both scenarios attain the Bell-limit $(S=2)$ at $C\approx 0.4$ (corresponding critical angle $\alpha_c \approx 24.97^\circ$). For $\alpha<\alpha_c$, the Bell-limit $S=2$ cannot be exceeded regardless of $\gamma$. For $\gamma=0$, the blue region here is consistent with that in Fig. \ref{['Fig1']}, exhibiting identical quantum correlations. Within $\alpha_c < \alpha < \pi/2 - \alpha_c$, particles exhibit nonlocal QM behaviors. Thus, $\alpha_c$ constitutes an operational analog to certain retarder configurations for $S$. Accordingly, at $\alpha=\pi/4$, the Tsirelson bound $S=2\sqrt{2}$ is saturated.