  By harnessing multiple degrees of freedom (DoFs) within a single photon,
controlled quantum unitaries, such as the two-qubit controlled-NOT (CNOT) gate,
play a pivotal role in advancing quantum communication protocols like dense
coding and entanglement distillation. In this work, we devise and realize a
CNOT operation between polarization and frequency DoFs by exploiting
directionally dependent electro-optic phase modulation within a fiber Sagnac
loop. Alongside computational basis measurements, we validate the effectiveness
of this operation through the synthesis of all four Bell states in a single
photon, all with fidelities greater than 98%. This demonstration opens new
avenues for manipulating hyperentanglement across these two crucial DoFs,
marking a foundational step toward leveraging polarization-frequency resources
in fiber networks for future quantum applications.
