Stabilizing Driven Steady States of Superconducting Qubits

Floquet states generated by time-periodic Hamiltonians are an attractive candidate for superconducting qubit implementations -- largely due to their dynamically-induced insensitivity to various noise channels while maintaining tunability. However, engineering stable and controllable Floquet states given a static circuit Hamiltonian and drive coupling remains a considerable challenge; experimental implementations often resort to costly parameter sweeps in order to find stable operating points (called dynamical sweet spots). Here, we employ recent progress in geometric Floquet theory and Floquet linear response theory to characterize the properties of a Floquet qubit in terms of time-dependent expectation values of observables described in the original static basis. In particular, we derive analytic conditions for achieving minimal relaxation and dephasing rates, restricting the tunable parameters entering the Floquet engineering problem. Moreover, we show that the existence of dynamical sweet spots may be tied to properties of both the time-averaged dynamics and sub-period micromotion of the qubit state, further informing the form of the stabilizing drive.