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.

Figure 1

Hysteretic behavior of the oscillation amplitude $a(t)$ of a classical JBA. When the driving force $f(t)$ (broken line) is changed with time $t$, the amplitude (thick line) varies as shown (schematic).

Figure 2

Time variations of some quantum expectation values when the driving force $f$ is operated as shown in Fig. 1. (a) Number of bosons excited in JBA. (b) Purity of the JBA. (c) Fluctuation in JBA amplitude.

Figure 3

Time evolution of JBA during readout. The figures show Q representations ($⟨\alpha |{\mathrm{Tr}}_q[\rho ]|\alpha ⟩$) of the JBA oscillator states, where $|\alpha ⟩$ is the coherent state of a complex amplitude $\alpha =x+ip$ (in the rotating frame). (a) Beginning of the readout. The state of the total system is (schematically) $\rho =|{\psi }_0⟩⟨{\psi }_0|$, with $|{\psi }_0⟩=1/\sqrt{2}(|g{⟩}_q+|e{⟩}_q)\otimes |G{⟩}_J$. (b) Starting the transition. Entanglement formation and projection are carried out during this period. (c) During the transition. Entanglement has already been destroyed. (d) The entire system has become a mixture of classically correlated states.

Figure 4

Time variation of the entanglement between the JBA and the qubit. ${\tau }_i(i=1,2,3)$ corresponds to those in Eq. (4), that is, ${\tau }_1$: entanglement formation, ${\tau }_2$: projection, ${\tau }_3$: end of the readout.