Enhanced observability of quantum postexponential decay using distant detectors

We study the elusive transition from exponential to postexponential (algebraic) decay of the probability density of a quantum particle emitted by an exponentially decaying source in one dimension. The main finding is that the probability density at the transition time, and thus its observability, increases with the distance of the detector from the source up to a critical distance beyond which exponential decay is no longer observed. Solvable models provide explicit expressions for the dependence of the transition on resonance and observational parameters, facilitating the choice of optimal conditions.

Figure 1

(Color online) Probability density versus $t$ for a fixed value of $k_0=1-0.5i$. Three detector positions $x$ have been considered: $x=1.5$ (red dotted line), $x=0.4$ (blue solid line), and $x=0.1$ (green dashed line). The dimensionless variables and the model are explained in Sec. 2.

Figure 2

$k$ plane with contour deformations along the SDPs (the straight lines) at times $t_2>t_1$ and saddle points $k_{s1}$, $k_{s2}$. At $t_1$, the $k_0$ pole has not yet been crossed, while at $t_2$ the pole has been crossed and encircled by the keyhole contour. The dashed line is the SDP for $t\sim \infty$, showing that the pole at $-k_0$ is never crossed.

Figure 3

(Color online) Overview of the log probability density in the $t,x$ plane for fixed $k_0=1-0.15i$.

Figure 4

(Color online) Probability density versus time, for three $x$ values, at fixed $k_0=1-0.3i$. The crossing point between the pole (straight line) and saddle terms defines the transition time $t_p$. Dotted (red) line: $x=0.1$, crossing at $t_p=12$; solid (blue) line: $x=2.5$, crossing at $t_p=7$; and dashed (green) line: $x=12$, crossing at $t_p=9$.

Figure 5

(Color online) Probability density versus $t_p\left(x\right)$ at the transition point for fixed $k_0=1-0.3i$. The drawing covers $x$ running from 0.0001 to 13. The dots correspond to the $x$ values considered in Fig. 4.

Figure 6

(Color online) Largest $x$ value where an exponential to postexponential transition is found, satisfying the conditions $R=1$ and $\text{Im}{u}_0^{(+)}>0$, versus $k_{0I}$ (normalized waves).

Figure 7

(Color online) Density of normalized waves at the largest $x$ value where the transition is defined, as function of $k_{0I}$. Solid (green) line: density of the approximate solution; dotted (black) line: value from the exact wave function.