Classical limit of the quantum Zeno effect by environmental decoherence

We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region—this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the $\hslash \to 0$ limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [Phys. Rev. A 88, 022128 (2013)], demonstrating the suppression of quantum-mechanical reflection by decoherence.

Figure 1

Evolution of $\langle p^2\rangle$ (solid line) for a sequence of projections with $\epsilon =0.01$. At $t=0$ an environment is introduced with (a) $D=100$; (b) $D=4000$; (c) $D=20000$. Before $t=0$ the system has reached a steady state. $\langle p^2\rangle$ is broken down into (i) ${\langle p^2\rangle }^{\mathrm{red}}$, Eq. (6.12) (dashed line); (ii) $\Delta$, Eq. (6.14) (dash-dotted line); (iii) $\Sigma$, Eq. (6.16) (dotted line).

Figure 2

Time series of various moments. The solid lines correspond to an environment with diffusion parameter $D=20000$. The dashed lines correspond to classical estimates for the steady-state condition using the same value of $D$ with absorbing boundary conditions. The time spacing between projections is $\epsilon =0.01$.

Figure 3

The Wigner function for the steady state with $D=20000$.

Figure 4

The spatial probability distribution $\rho (x,x)$ in the steady-state condition immediately following a projection for different environments.

Figure 5

The half-life ${\tau }_{1/2}$ for the state to remain in the region [$-0.5,0.5$] minus the classical decay time ${\lambda }^{-1}$, with $D$ and $\epsilon$. The surface shows the numerical results for the full quantum system; the solid line indicates the theoretical condition dividing classical from quantum behavior.

Figure 6

The behavior of the half-life ${\tau }_{1/2}$ for the state to remain in the region $[-L/2,L/2]$ with $D$ (fixed $L=1.0$) and $L$ (fixed $D={10}^4$); and the variance of $p$, $\langle p^2\rangle$ with $D$ (fixed $L=1.0$) and $L$ (fixed $D={10}^4$). The data points indicate numerical results for the full quantum system; the solid lines give the theoretical prediction assuming classical behavior. The time spacing between projections is $\epsilon =0.01$.

Figure 7

Probability flux lines for unitary evolution of a Gaussian wave packet: (a) no projections; (b) with a series of projections onto the spatial range $[-0.5,0.5]$ (marked with dashed lines). The initial Gaussian wave function in each case has width $\sigma =0.1$ in units described in the text.

Figure 8

The effect of an environment on probability flux lines. Plots show evolution from an initial Gaussian wave function of width $\sigma =0.1$ with (a) no environment; (b) environment with $D=100$; (c) environment with $D=4000$. The time spacing between projections is $\epsilon =0.01$ in units described in the text.