Quantum Time Evolution in a Qubit Readout Process with a Josephson Bifurcation Amplifier

We analyzed the Josephson bifurcation amplifier (JBA) readout process of a superconducting qubit quantum mechanically by calculating the dynamics of the density operator of a driven nonlinear oscillator and a qubit coupled system during the measurement process. In purely quantum cases, bifurcation is impossible. Introducing decoherence enables us to reproduce the bifurcation with a finite hysteresis. When a qubit is initially in a superposition state, we have observed the qubit-probe (JBA) entangled state, and it is divided into two separable states at the moment the JBA transition begins. This corresponds to projection. To readout the measurement result, however, we must wait until the two JBA states are macroscopically well separated. The waiting time is determined by the strength of the decoherence in the JBA.