Arrival times, complex potentials, and decoherent histories

We carry out a decoherent-histories analysis of the arrival-time problem, taking advantage of a recently demonstrated connection between time-ordered strings of projection operators and evolution in the presence of a complex potential of step-function form. We concentrate on the limit of a weak potential, in which the resulting arrival-time distribution function is closely related to the quantum-mechanical current. We first consider the analogous classical arrival-time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. We use the path-decomposition expansion to give a derivation of the standard arrival-time distribution defined using a complex potential. This derivation is then used in the decoherent-histories analysis to obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time). We show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent-histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival-time problem. We also find some interesting connections between backflow and decoherence.

Figure 1

The quantum arrival-time problem. We prepare an initial state localized entirely in $x>0$ and consisting entirely of negative momenta. What is the probability that the particle crosses the origin during the time interval $\left[t_1,t_2\right]$?

Figure 2

An initial state consisting of a superposition of wave packets may have significant crossings in at least two different time intervals. If the initial packets are orthogonal and the time intervals are sufficiently large (greater than the Zeno time), the packets will remain orthogonal after passing through the time intervals and the corresponding histories will be decoherent.

Figure 3

The function $f\left(u\right)$. It oscillates around zero for $u⪢0$ and oscillates around $\pi$ for $u⪡0$. As a function of $q$, $f\mathbf{\left(}u\left(p_0,q\right)\mathbf{\right)}$ differs from 0 or $\pi$ only in a region of size $1/\left|p_0\right|$ around $q=0$.

Figure 4

The path-decomposition expansion. Any path from $x_0>0$ at $t=0$ to a final point $x_1<0$ at $t=T$ has a first crossing of $x=0$ at $t_1$ and a last crossing at $t_2$. The propagator from $\left(x_0,0\right)$ to $\left(x_1,T\right)$ may be decomposed into three parts: (A) restricted propagation entirely in $x>0$, (B) free propagation starting and ending on $x=0$, and (C) restricted propagation entirely in $x<0$. The corresponding path-decomposition expansion formula is given in Eq. (B6).