Entanglement production in the decay of a metastable state

Abstract

When a metastable state decays into radiation, there must be entanglement between the radiation and the decaying system, as well as between radiation collected at late and early times. We study the interplay between these two types of entanglement in simple Gaussian models. We define, via a windowed Fourier transform, multimode quantum states associated with radiation fragments produced at different times and compute the corresponding entanglement entropy increments. On the basis of these results, we argue that such entropy increments are useful entanglement measures, especially in cases, such as Hawking radiation, where one wishes to separate the radiation into ``old'' and ``new.''

Figures (3)

• Figure 1: Radiation entropy increments as functions of time for the case when the resonator ($A$) starts in the squeezed vacuum corresponding to (\ref{['init']}) with $r=5$, while radiation ($B$) starts in the trivial vacuum. The solid line is the entanglement entropy of the resonator, $S_A(t)$. Eq. (\ref{['cons']}) is verified here for two values of $t_0$ ($t_0 = 1.5$ and $7.5$), with $S_{B_2A}(t_0,t)$ represented in both cases by empty circles. $S_{B_1 B_2}(0,t_0,t)$ (empty squares) is found to be independent of $t_0$, and is shown here for $t_0 =3$. As a function of $t$, it follows $S_A(t)$, in accordance with (\ref{['ii']}).
• Figure 2: Same as in Fig. \ref{['fig:lim']}, but for a case when the initial state of $A$ is mixed (a squeezed thermal state with population $N_{th}$). $S_{B_1 B_2}(0,t_0,t)$ (empty squares) no longer follows $S_A(t)$ (solid line) but still approaches $S_A(0)$ at large $t$, in accordance with (\ref{['ff']}).
• Figure 3: Conservation of uncertainty, Eq. (\ref{['cons']}), for the case when the resonator ($A$) starts in a thermal state, and radiation ($B$) starts in vacuum, and both are amplified by parametric resonance. As before, empty circles represent $S_{B_2 A}(t_0,t)$ for two values of $t_0$ (here, $t_0 = 2$ and $5.5$). The plots verify the relation (\ref{['cons']}) for a case when neither the initial nor the final state of the resonator is pure.