  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.
