Detecting a single microwave photon requires converting its arrival into a reliable electrical signal. In the detectors developed by the Quantronics group at CEA Saclay, the photon leaves a record in an internal transmon qubit. This internship will explore how Josephson nonlinearities can amplify that record and enable fast, faithful readout without a travelling-wave parametric amplifier. The central idea is an asymmetric measurement: preserve the transmon’s ground state to suppress false clicks, while using its excited state to trigger a strongly distinguishable nonlinear resonator response. A dedicated Josephson element, separate from the transmon junction, will provide control over the readout nonlinearity. Combining circuit modelling, phase-space analysis, quantum simulations and cryogenic measurements, the project will investigate how bifurcation and the onset of chaos affect signal discrimination and measurement backaction. The goal is to improve photon detection while preserving transmon coherence and allowing rapid detector reset. The internship is intended to lead to a CIFRE PhD with Qudigy, a spin-off of the Quantronics group that plans to begin commercialising single microwave photon detectors in 2027.