Quantum tunneling from excited states in the steadyon picture

TL;DR

This paper extends the steadyon real-time path integral framework to tunneling from excited, resonance-type states in a false vacuum. It derives a direct expression for the decay rate from a generic initial state and shows that, for resonance states, the dominant saddle is a periodic steadyon whose imaginary action reproduces the Euclidean instanton result, thereby recovering the WKB tunneling rate with a regulator-independent justification. Numerical simulations in a 1D potential demonstrate a multi-resonance structure where different resonances can dominate at different times, validating the resonant decomposition and its connection to the steadyon formalism. The work clarifies how initial-state boundary conditions select the relevant saddles, provides a first-principles real-time path-integral description of tunneling from arbitrary states, and opens pathways to higher-dimensional and quantum-field generalizations, including the computation of path integral prefactors.

Abstract

Recent developments in the understanding of real-time path integrals led to the development of the ``steadyon picture'' for the semi-classical calculation of quantum tunneling rates. We discuss tunneling out of a generic localized initial state in this picture and present its application for the important example of a resonance state in a one-dimensional point particle potential. We find that the steadyon picture indeed reproduces existing results obtained using the WKB method. Our analysis furthermore demonstrates how applying this picture to physical states naturally addresses open conceptual questions regarding this framework. Finally, we perform a numerical study for a specific potential. We demonstrate in particular the existence of regimes in which the tunneling rate is dominated by higher resonances, rather than the false vacuum, as well as their importance.

Figures (10)

• Figure 1: Quantum tunneling of a point particle in a potential (black), represented by its wave function (blue). Left panel: At $t=0$, the particle is fully localised within the well $\mathcal{F}$. $x_*$ denotes the point beyond which its wave function decays exponentially. Right panel: At times larger or comparable to the potential's typical time scale $t \gtrsim t_{\rm sys}$, the particle's wave function has penetrated the potential barrier separating $\mathcal{F}$ from the adjacent basin $\mathcal{R}$ into which the particle tunnels. We assume that the exponential decay of the wave function $x_*$ remains true for long enough times to allow for "steady" tunneling. If the point beyond which the exponential decay of the wave function sets off varies with time, we call $x_*$ the largest value during the considered time interval. $x_s$ denotes the point on the other side of the barrier with $V(x_s)=V(x_*)$.
• Figure 2: Eq. \ref{['eq:P-flux-turning-point']} visualised. Left panel: For different values of the auxiliary variable $\Delta t$, the saddle points of the path integrals in Eq. \ref{['eq:P-flux-master']} are given by different steadyons. The projection of these solutions onto the imaginary-time axis can be identified with instanton defined on the imaginary-time interval $\Delta \tau \simeq \epsilon \cdot \Delta t$. Right panel: Examples for such instantons for the concrete example of the double-well potential used in appendix \ref{['app:steady-review']}, assuming a vanishing initial velocity for $x_{\rm Re}$. The instanton corresponding to the shortest depicted $\Delta \tau$, $\Delta \tau_{\rm fv}$, can be identified with a section of the instanton connecting the false vacuum to its corresponding convergence point. While this solution amounts to the smallest Euclidean action, the total integrand in Eq. \ref{['eq:P-flux-master']} is also sensitive to the wave function factor. In Sec. \ref{['sec:resonance']}, we demonstrate how this integral can be evaluated for the concrete example of a resonance state.
• Figure 3: (a) Potential used in these numeric simulations is shown as $V(x)$. The potential $V_\mathrm{res}(x)$ defined in \ref{['eq:Vres']} is also shown. The first four resonant states with $m = 6\cdot10^3$ (in natural units $\hbar = 1$), displaced vertically by their respective energies shown by the dashed lines are plotted in the figure. (b) The lowest approximate resonant state was time evolved over a total time $T = 150$, and the overlap onto the false-vacuum is plotted in black. Fits to \ref{['eq:discretePF']} are shown both for only including a single state, as well as including two states in the fit.
• Figure 4: Simultaneous fit to the time evolution of various different intial states. The inset panels on each subfigure show the initial state used in each simulation plotted on top of the potential $V(x)$. The data for all times $t \geq 60$ serves as input for the simultaneous fit, for which the energies $E_i$ and decay constants $\Gamma_i$ were held constant between the different simulations. As the three different initial states had differing amounts of contamination from (unknown) higher excited resonances, each simulation has a different time beyond which the excited resonances have decayed enough for the two-state fit to be a good description.
• Figure 5: Comparison between the theory prediction provided in \ref{['eq:thadhoc']} to the numerically extracted states in the $(E,{\rm log}(\Gamma))$ plane for two different masses. $E_\mathrm{res}$ are the energies given be exact diagonalisation of the modified potential \ref{['eq:Vres']}. As only energy shifts can be extracted by fits to $P_{\mathcal{F},n}$, all energies have been shifted such that the lowest energy fitted state has the energy given by the lowest energy $E_\mathrm{res}$.