A non-reciprocal dispersive interaction in circuit QED: Part II, measurements and analysis*

Non-reciprocity is a valuable property for building interesting and complex quantum systems.  Previous experimental studies have primarily focused on non-reciprocal excitation transfer between linear modes, which can be fully captured by a non-Hermitian effective Hamiltonian matrix. To go beyond these semi-classical descriptions, we implement a dispersive type of non-reciprocal interaction between a transmon qubit and a superconducting cavity, where the system dynamics can be captured by a cavity-qubit nonlinear jump operator in a new effective theory of non-reciprocity. This theory is a result of adiabatic elimination and its predictions must be compared to time-averaged quantities. Thus, we experimentally measured the time-averaged qubit dispersive shifts and dephasing rates from a decaying photon population, which agree excellently with model predictions, showing a notable asymmetry between positive and negative applied magnetic fields due to the non-reciprocity. We further verify the usefulness of the model by studying the dispersive shifts and dephasing rates in the steady-state regime under a continuous cavity drive, which show good agreement with theory. Furthermore, we explore potential applications of this model, such as qubit phase gate operations.