Population-Based Discrimination of Classical and Few-Photon Microwave Fields with a Flux-Tunable Transmon Qudit

Interactions between few-photon microwave fields and superconducting artificial atoms provide a method to study photon statistics and manipulate quantum states for photonic quantum information processing. Superconducting transmons couple to microwave fields through an anharmonic ladder of energy levels, producing a response beyond the two-level description. Here, we propose and analyze a detector based on a flux-tunable transmon qudit in circuit quantum electrodynamics. Within a Kerr-type model, we calculate its population response to classical-field excitation and quantized Fock and coherent states. In the four-photon example, a selected detuning produces periodic, nearly complete transfer to the fourth transmon level for a Fock input, while a coherent input of equal mean energy gives a population of approximately 0.2 at the same interaction time. In the linear regime, we derive a target-population bound for classical inputs. From single-tone and two-tone spectroscopy, analyzed using the cosine transmon Hamiltonian, we extract circuit parameters and resolve higher transitions. Ramsey interferometry characterizes dephasing on the fundamental transition. We propose a detection protocol combining flux-selected interaction with dispersive population readout. Our results establish a population-based framework for microwave-state discrimination using a superconducting artificial atom.