= Neutral-kaon two-pion decay time dependence
{title2=$I_{\pi\pi}(t)\propto|\rho e^{-i\lambda_St}+\eta_{\pi\pi}e^{-i\lambda_Lt}|^2$}
For a coherently prepared state proportional to $\rho|K_S\rangle+|K_L\rangle$, evolve each component with $\lambda_j=m_j-i\Gamma_j/2$. The <proper time> is $t$, and $\hbar=1$. Since both components reach the same final state, their <decay amplitudes> add before squaring. Factoring out $|A_S|^2$ and a common beam normalization gives the two-<pion> intensity
$$
I_{\pi\pi}(t)=|\rho|^2e^{-\Gamma_St}
+|\eta_{\pi\pi}|^2e^{-\Gamma_Lt}
+2e^{-(\Gamma_S+\Gamma_L)t/2}
\operatorname{Re}\!\left(\rho^*\eta_{\pi\pi}e^{-i\Delta m t}\right),
\qquad \Delta m=m_L-m_S.
$$
The last term is <quantum interference>. It vanishes upon averaging over an independently random preparation phase, but cannot be discarded from a coherent preparation. Summing the two positive contributions alone would lose the phase-sensitive information.
\Image[/neutral-kaon-survival-and-two-pion-interference.png]
{title=Chosen <kaon> lifetimes and coherent two-<pion> intensity}
{description=Original dimensionless illustration with $\tau_S=1$, $\tau_L=8$, $\rho=0.35$, $\eta_{\pi\pi}=0.06e^{0.8i}$ and $\Delta m=1.6$. Upper curves are separate single-state survival factors; lower curves are intensities in an arbitrary common scale. This is not a historical fit or a normalized set of <decay channel> probabilities.}
At late <proper time>, the short-lived contribution and the <quantum interference> term die away, leaving the long-lived tail $|\eta_{\pi\pi}|^2e^{-\Gamma_Lt}$. Such a tail is easier to identify after the initially abundant short-lived component has disappeared. The useful experimental balance is between eliminating that component and retaining enough long-lived decays to observe a rare <decay channel>.
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