Dawn and Twilight Time in Quantum Tunneling

TL;DR

This paper develops a real-time, flux-based framework for metastable quantum decay by decomposing the real-time kernel into pole and branch contributions, enabling the extraction of two computable time scales: dawn time, when a dominant resonance begins governing decay, and twilight time, when the universal power-law tail takes over. The twilight time admits a closed-form expression in terms of the Lambert W function, confirming a transparent parametric dependence without fitting. For square, modified square, and Pöschl–Teller barriers, the authors derive thick-barrier formulas and establish the relation $ΓT = T_{\rm trans}$ in the thick-barrier limit, connecting decay rate, oscillation period, and transmission probability. The spectral picture naturally extends to periodic systems and quantum-field-theoretic vacuum decay, providing a framework that bridges real-time dynamics with Euclidean bounce methods.

Abstract

Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert $W$ function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation $ΓT = T_{\text{trans}}$ between the decay rate $Γ$, oscillation period $T$, and transmission probability $T_{\text{trans}}$, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.

Figures (10)

• Figure 1: Two examples of a potential $V(x)$ with a metastable region L, a destination region R, and a barrier B. For case (a), in which R is finite, the wave function initially localised in L will not completely decay, as eventually a strong back-reaction will kick in, such that the probability will oscillate between regions L and R. For case (b), in which R extends to $+\infty$, no significant back-reaction will occurs, and an initial state localized in L will continue decaying into R until it decays completely.
• Figure 2: Contour $\mathcal{C}$ and the poles of the kernel in the complex energy plane. The arc at infinity does not contribute because the integrand decays exponentially in this limit.
• Figure 3: Twilight time plot for the square barrier and modified square barrier.
• Figure 4: Twilight time plot for Pöschl-Teller potential in Appendix \ref{['Pöschl-Teller potential']} with varying dimensionless parameter $\omega_{\mathrm{C}}=b\sqrt{2mU_0}/\hbar$.
• Figure 5: Qualitative time evolution of the probability $P_L$ of finding the wave function in the left region $L$ for the initial wave function $\Psi_{0}=\sqrt{2/a}\sin(2\uppi\mu x/a)$. The relevant times $t_{\rm WKB}$, $t'_\circlebottomhalfblack$, $t_\circlebottomhalfblack$ and $t_\circletophalfblack$ are explained in the text.