Coherence of a Josephson phase qubit under partial-collapse measurement

We discuss quantum evolution of a decaying state in relation to a recent experiment of Katz et al. Based on exact analytical and numerical solutions of a simple model, we identify a regime where the qubit retains coherence over a finite time interval independently of the rates of the three competing decoherence processes. In this regime, the quantum decay process can be continuously monitored via a “weak” measurement without affecting the qubit coherence.

Figure 1

(a) Schematic of the phase qubit: superconducting loop interrupted by a Josephson junction. (b) Potential profile and level structure of the phase qubit. Upper level ∣1⟩ decays with the rate $\Gamma$. (c) Idealized level structure used here. Left-well levels with energies $E_0$ and $E_1$ form a qubit. The state ∣1⟩ can decay into the right well where the average level spacing is $\Delta$ and the bandwidth is $\Lambda$. The tunneling amplitude to the state $\mid n⟩$ is $f_n$.

Figure 2

(Color online) Simulation results for the evolution projected onto the left well beginning with the state with $⟨{\sigma }_x⟩=1$; parameters as shown. Open symbols represent the purity $P\left(t\right)$ which starts at $P\left(0\right)=1$, closed symbols represent the longitudinal qubit polarization $-⟨{\sigma }_z⟩$; the two curves merge near $t\Delta =4$. Blue squares: equidistant spectrum in the right well, $E_n=\delta E+n\Delta$; numerical results are almost on top of the analytical result with $N\to \infty$ (dashes). The broad minima at $t⩾2\pi ∕\Delta$ and $t⩾4\pi ∕\Delta$ correspond to arrival of incoherent echoes from the right well. Green circles: data with added spectrum nonlinearity, $\beta =0.02$ (see text). Red triangles: random level spacing, $E_{n+1}-E_n=(0.5+{\xi }_n)\Delta$, where ${\xi }_n$ are uniformly distributed random numbers, $0⩽{\xi }_n<1$.