Analytical Fock-State Generation and SWAP using a Rabi-Driven Transmon
Deterministic Fock-state generation and inter-mode SWAP are foundational primitives for bosonic quantum computing, yet most implementations rely on numerically optimized pulses, per-state calibration, strong dispersive coupling, or higher transmon levels, each adding control overhead that grows with system size. We present an analytical, calibration-light protocol operating entirely within the two-level g-e manifold of a weakly dispersively coupled transmon. A Rabi drive on the qubit, combined with a single sideband tone per mode, synthesizes an on-demand Jaynes-Cummings interaction whose entire family of pulse times follows the closed-form scaling $\tau_n=\tau_1/\sqrt{n}$. Once the single base time $\tau_1$ is set, every higher-$n$ operation is fixed analytically, with no per-state retuning, shelving, or numerical optimization. On a superconducting flute cavity with two high-Q modes, we deterministically prepare Fock states through $|n{=}5\rangle$, realize an inter-mode SWAP characterized on vacuum, single-photon, and coherent-state inputs, and generate and coherently swap the dual-rail Bell state $(|1,0\rangle+|0,1\rangle)/\sqrt{2}$, confirming that the operation preserves inter-mode coherence. Because the pulses are constant-amplitude and free of per-state optimization, the achievable fidelity is set directly by ancilla coherence and drive-ramp duration; a master-equation analysis isolates these hardware factors and shows that the analytical scaling itself imposes no obstacle to high-fidelity operation at high $n$. Requiring only one sideband line per mode and a single Rabi drive, the protocol is well suited to weakly coupled, high-$Q$ 3D architectures where calibration economy and analytical pulse design are at a premium.