Quantum Time-evolution in Qubit Readout Process with a Josephson   Bifurcation Amplifier
  We analyzed the Josephson bifurcation amplifier (JBA) readout process of a superconducting qubit quantum mechanically. This was achieved by employing numerical analyses of the dynamics of the density operator of a driven nonlinear oscillator and a qubit coupled system during the measurement process. In purely quantum cases, the wavefunction of the JBA is trapped in a quasienergy-state, and bifurcation is impossible. Introducing decoherence enables us to reproduce the bifurcation with a finite hysteresis. Moreover, we discuss in detail the dynamics involved 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 of 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. 
Hayato Nakano, Shiro Saito, Kouichi Semba, and Hideaki Takayanagi