Using circuit QED, we consider the measurement of a superconducting transmon qubit via a coupled microwave resonator. For ideally dispersive coupling, ringing up the resonator producescoherent states with frequencies matched to transmon energy states. Realistic coupling is not ideally dispersive, however, so transmon-resonator energy levels hybridize into joint eigenstate ladders of the Jaynes-Cummings type. Previous work has shown that ringing up the resonator approximately respects this ladder structure to produce a coherent state in the eigenbasis (a dressed coherent state). We numerically investigate the validity of this coherent state approximation to find two primary deviations. First, resonator ring-up leaks small stray populations into eigenstate ladders corresponding to different transmon states. Second, within an eigenstate ladder the transmon nonlinearity shears the coherent state as it evolves. We then show that the next natural approximation for this sheared state in the eigenbasis is a dressed squeezed state, and derive simple evolution equations for such states using a hybrid phase-Fock-space description.
We analyze the transfer of a quantum state between two resonators connected by a superconducting transmission line. Nearly perfect state-transfer efficiency can be achieved by usingadjustable couplers and destructive interference to cancel the back-reflection into the transmission line at the receiving coupler. We show that the transfer protocol is robust to parameter variations affecting the transmission amplitudes of the couplers. We also show that the effects of Gaussian filtering, pulse-shape noise, and multiple reflections on the transfer efficiency are insignificant. However, the transfer protocol is very sensitive to frequency mismatch between the two resonators. Moreover, the tunable coupler we considered produces time-varying frequency detuning caused by the changing coupling. This detuning requires an active frequency compensation with an accuracy better than 90% to yield the transfer efficiency above 99%.
We analyze single-shot readout for superconducting qubits via controlled catch, dispersion, and release of a microwave field. A tunable coupler is used to decouple the microwave resonatorfrom the transmission line during the dispersive qubit-resonator interaction, thus circumventing damping from the Purcell effect. We show that if the qubit frequency tuning is sufficiently adiabatic, a fast high-fidelity qubit readout is possible even in the strongly nonlinear dispersive regime. Interestingly, the Jaynes-Cummings nonlinearity leads to the quadrature squeezing of the resonator field below the standard quantum limit, resulting in a significant decrease of the measurement error.