Medição da Fase Topológica Fracionária em qudits fotônicos codificados espacialmente

Abstract

In this work we performed two experiments to observe the fractional topological phases in two qudits photonic systems. The qudits are encoded in the transverse position degree of freedom of photon pairs generated by the spontaneous parametric down conversion after being transmitted through multiple slits. The photons in this phenomenon are quantum correlated in momentum and energy. Initially we review the theory about the origin of the phases acquired by the evolving quantum states of one and two qubits, where we identify the dynamic, the geometric and the topological phases. Then we examine the generalization of topological phases for two qudits, and discuss its relation with the concept of connectedness of a space. Next we show a way to measure the topological phase, considering that it must appear as a relative phase between two interfering quantum states. To realize the measurement we build an optical interferometer with a spatial light modulator inside it to implement the necessary unitary operations. The polarization is used as an auxiliary degree of freedom for the operations to act conditionally. The results are close to the theoretical predictions, showing the dependence of the phase with the Hilbert space dimension of the state. In a second setup we measure the topological phase for qudits without using a longitudinal interferometer. A new source of polarization entangled photon pairs is used in this new setup. The entanglement in the auxiliary degree of freedom, polarization, is used as a resource to eliminate the interferometer, propitiating a significant improvement in the visibility of the measured interference curves, and a large increase in the signal to noise ratio. In this new setup, we are also capable of realize the unitary operations in both parts of the two qubit system.