Nonexponential decay of an unstable quantum system: Small-$Q$-value $s$-wave decay

We study the decay process of an unstable quantum system, especially the deviation from the exponential decay law. We show that the exponential period no longer exists in the case of the $s$-wave decay with small $Q$ value, where the $Q$ value is the difference between the energy of the initially prepared state and the minimum energy of the continuous eigenstates in the system. We also derive the quantitative condition that this kind of decay process takes place and discuss what kind of system is suitable to observe the decay.

Figure 1

The shape of the potential.

Figure 2

Examples of decay. Here we set $G=20$, $G_0=0$ [or $Q=8.97365{\left(2m{a}^2\right)}^{-1}$], $u={10}^{-4}$ on the left and $G=20$, $G_0=8.957335$ [or $Q=6.55445\times {10}^{-4}{\left(2m{a}^2\right)}^{-1}$], $u={10}^{-4}$ on the right. Each quantity in the figure is averaged over a short time interval. The example on the right corresponds to the SQS decay.

Figure 3

The analytic structure of the complex $z$ plane and the contours of the integrals. The contours and the pole shown by the broken line are on the second Riemann sheet.

Figure 4

The typical shape of the spectral function $\rho \left(E\right)$. The SQS decay is realized when the $Q$ value is smaller than the width of the peak.

Figure 5

The value of Eq. (26) is depicted as a contour map in the $(G,Q)$ plane.

Figure 6

The modified potential.