Quantum Zeno effect appears in stages

In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for a continuous partial measurement. We show that the onset of the Zeno regime is marked by a cascade of transitions in the system dynamics as the measurement strength is increased. Some of these transitions are only apparent in the collective behavior of individual quantum trajectories and are invisible to the average dynamics. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.

Figure 1

(a) The system. A Hamiltonian induces oscillations between levels $\left|0\right.〉$ and $\left|1\right.〉$ of a qubit, which is continuously measured by a detector weakly coupled to one of the levels. (b) Dynamical flow (red and blue arrows) of $\theta \left(t\right)$ from Eq. (3) under “no-click” postselected dynamics. For sufficiently weak measurements, $\lambda <1$ (left), the dynamics is oscillatory; for $\lambda >1$ (right), stable and unstable fixed points (${\theta }_{+}$ and ${\theta }_{-}$, respectively) emerge. The states in the interval $\theta \in (-\pi ,{\theta }_{+})$ are inaccessible to the system under both the no-click and the full stochastic dynamics.

Figure 2

The stochastic dynamics of the qubit state in the $y\text{−}z$ section of the Bloch sphere exhibits transitions at relative measurement strengths $\lambda =1$, $2/\sqrt{3}$, and 2. The long-time probability distribution $P_{t=\infty }\left(\theta \right)$ at different values of $\lambda$ is shown as the height above the unit circle for the analytic result (red solid line) and for the numerical simulations (green dots). The trajectories of the expectation values ${\overline{s}}_{y,z}\left(t\right)$, for the system initialized in $\left|0\right.〉$ at $t=0$, are shown with the dashed magenta lines. Each numerical simulation involved 10 000 stochastic trajectory realizations, tracing the evolution until $t=10$, with measurements yielding random outcomes happening at intervals $dt=0.01$, using ${\mathrm{\Omega }}_s=1$; $\theta$ is binned in intervals of size $5{}^{\circ }$.

Figure 3

The decay rates $\zeta \left(\lambda \right)$ (solid) and $\overline{\zeta }\left(\lambda \right)$ (dashed) characterizing, respectively, the probability to observe no clicks $P^{\left(0\right)}\left(t\right)$ and the survival probability $[1+{\overline{s}}_z\left(t\right)]/2$. Note the respective decay rate maxima at $\lambda =1$ and 2. Inset: The time dependence of $P^{\left(0\right)}\left(t\right)$ for $\lambda =0.5$ (solid) and $\lambda =1.25$ (dashed). At long times $P^{\left(0\right)}\left(t\right)$ decays exponentially with (without) superimposed oscillations for $\lambda <1$ ($\lambda >1$).