Lifetime renormalization of weakly anharmonic superconducting qubits: I. Role of number non-conserving terms

An outstanding challenge in superconducting quantum computing is the determination of an accurate effective model for a particular experiment. In practice, the dynamics of a superconducting qubit in a complex electromagnetic environment can be described by an effective multimode Kerr Hamiltonian at sufficiently weak excitation. This Hamiltonian can be embedded in a master equation with losses determined by the details of the electromagnetic environment. Recent experiments indicate, however, that when a superconducting circuit is driven with microwave signals the observed relaxation rates appear to be substantially different from expectations based on the electromagnetic environment of the qubit alone. This issue has been most notorious in the optimization of superconducting qubit readout schemes. We claim here that an effective master equation with drive-power dependent parameters is the most resource-efficient approach to model such quantum dynamics. In this sequence of papers we derive effective master equations whose parameters depend on the excitation level of the circuit and the electromagnetic environment of the qubit. We show that the number non-conserving terms in the qubit nonlinearity generally lead to a renormalization of dissipative parameters of the effective master equation, while the number-conserving terms give rise to a renormalization of the system frequencies. Here, in Part I, we consider the dynamics of a transmon qubit that is prepared in an initial state of a certain excitation level, but is not driven otherwise. For two different electromagnetic environments, an infinite waveguide and an open resonator, we show that the renormalized parameters display a strong dependence on the details of the electromagnetic environment of the qubit. The perturbation technique based on unitary transformations developed here is generalized to the continuously driven case in Part II.