Parity-Based Time-Bin Encoding Enabling SWAP Between Polarization and Time-Bin Qubits

Multi-degree-of-freedom photonic quantum processing requires routing between degree-of-freedom (DOF) qubit encodings on a single photon. A SWAP between polarization and time-bin qubits is Multi-degree-of-freedom photonic quantum processing requires routing between degree-of-freedom (DOF) qubit encodings on a single photon. A SWAP between polarization and time-bin qubits is an advantageous primitive for such architectures, however conventional early/late time-bin encoding does not support bidirectional logical time-bin flips from late to early which limits the ability to implement certain quantum operations. We introduce a parity-based time-bin encoding in which logical vert 0 rangle_T and vert 1 rangle_T correspond to even and odd multiples of a spacing Δ t, so that a physical delay of Δ t implements vert 0 rangle_T leftrightarrow vert 1 rangle_T. This encoding is the enabling ingredient that makes a polarization-controlled delay line implement CNOT_{P → T} and aligns naturally with periodic refractive index modulation for CNOT_{T → P}. Composing three such CNOT operations sequentially results in a deterministic SWAP between polarization and time-bin degrees of freedom. We analyze field-based modulation polarization-rotation error probability and timing-resolution constraints set by both EOM drive electronics and photon detection.