Action principle for continuous quantum measurement

We present a stochastic path integral formalism for continuous quantum measurement that enables the analysis of rare events using action methods. By doubling the quantum state space to a canonical phase space, we can write the joint probability density function of measurement outcomes and quantum state trajectories as a phase space path integral. Extremizing this action produces the most likely paths with boundary conditions defined by preselected and postselected states as solutions to a set of ordinary differential equations. As an application, we analyze continuous qubit measurement in detail and examine the structure of a quantum jump in the Zeno measurement regime.

Figure 1

The numerical medians (black dots) of $x,y,z,r$ and the most likely path from Eq. (9) (colored lines) $\overline{x},\overline{y},\overline{z},\overline{r}$ are shown for two scenarios: (a) $\Delta =0,\epsilon =0.5{\tau }^{-1},{\mathit{q}}_I=(1,0,0),{\mathit{q}}_F\approx (0.7,0.2,0.7);$ (b) $\Delta =-0.5{\tau }^{-1},\epsilon =0,{\mathit{q}}_I=(1,0,0),{\mathit{q}}_F\approx (0.9,0,0.5)$. The postselected ensemble size is ${10}^4$ with $\lambda =0.02$, $\delta t=0.01\tau$, and $T=0.6\tau$ (see text). The small dotted lines show the fortieth and sixtieth percentile paths.

Figure 2

The doubled state space phase portrait for different constant stochastic energies: $E<-{\Delta }^2\tau /2$ (red dotted), $E=-{\Delta }^2\tau /2$ (black solid), $E=0$ (blue dashed) and $E>0$ (gray dotted), for $\Delta =0.2{\tau }^{-1}$. The dynamics is invariant under $\theta \to \theta +\pi$.

Figure 3

The exponential of the extremized action Eq. (C2) as a function of $z_F$ for the QND case when $z_I=0.2$ plotting at three different time $T=0.01\tau$ (dotted) $T=0.5\tau$ (dashed) and $T=2\tau$ (solid).