Dynamical system analysis of quantum tunneling in an asymmetric double-well potential

TL;DR

This work develops an Ehrenfest-based dynamical-systems approach to study time-dependent quantum tunneling in an asymmetric double-well, reducing the dynamics of a Gaussian wave packet to a four-dimensional system for the mean $\langle x \rangle$, the variance $V$, and their derivatives, with skewness $S$ encoding the potential's asymmetry. The conserved energy $E$ serves as a control parameter, and the model includes a Gaussian closure with $K=3V^2$, yielding a closed set of equations that predicts a tunneling threshold by analyzing the stability of a classically unstable fixed point. Stability analysis reveals energy thresholds (e.g., $E\ge 8.53$ for real $V^*$ and $E=10.60$ for hill-point stability) beyond which $\langle x \rangle$ switches between wells, signaling detectable tunneling; these predictions are supported by full time-dependent Schrödinger equation simulations. The results show qualitative agreement between the reduced dynamical-system model and the TDSE, offering an interpretable framework for quantum transport through tunneling and suggesting extensions to multi-well potentials and non-Gaussian states.

Abstract

We study quantum tunneling in an asymmetric double-well potential using a dynamical systems-based approach rooted in the Ehrenfest formalism. In this framework, the time evolution of a Gaussian wave packet is governed by a hierarchy of coupled equations linking lower- and higher-order position moments. An approximate closure, required to render the system tractable, yields a reduced dynamical system for the mean and variance, with skewness entering explicitly due to the potential's asymmetry. Stability analysis of this system identifies energy thresholds for detectable tunneling across the barrier and reveals regimes where tunneling, though theoretically allowed, remains practically undetectable. Comparison with full numerical solutions of the time-dependent Schrödinger equation shows that, beyond reproducing key tunneling features, the dynamical systems approach provides an interpretable description of quantum transport through tunneling in an effective asymmetric two-level system.

Figures (6)

• Figure 1: Symmetric and asymmetric double-well potentials. Representative forms of the quartic potential $\phi(x)$ (Eq. \ref{['eq:potential']}) are shown with fixed $a=10$ and $b=4$, and varying $c$. Curves A--E illustrate the transition from a single well to symmetric and asymmetric double-well configurations, respectively, as $c$ decreases. The asymmetric double-well potential with $c=0.35$ (curve E, blue, dashed) is studied in this work.
• Figure 2: Stability analysis. (a) The solutions for $V_-^\ast$ of the quadratic equation (Eq. \ref{['eq:V_ast_quadratic']}) are real only when the discriminant is non-negative. This condition is satisfied for energies $E \geq 8.53$, which is the minimum energy needed for the existence of the fixed point $V_-^\ast$. (b) The real parts of the two eigenvalues of the stability matrix for the fixed point $(\beta_-, 0, V_-^\ast, 0)$, associated with the potential hill, are plotted against $V_-^\ast$. Both eigenvalues are negative beyond $V^\ast_- = 4.96$, corresponding to $E = 10.60$. This signifies the transition of the potential hill from a classically unstable fixed point to a stable one, indicating the onset of tunneling.
• Figure 3: No tunneling (dynamical systems analysis). The time evolution of the mean position $\langle x \rangle$ of the wave packet is shown for initializations in (a) the left well ($\langle x\rangle (0) = 0.5$, $E= 9.0$) and (b) the right well ($\langle x\rangle (0) = 5.50$, $E = \Delta + 9.0$, where $\Delta = 4.68$ is the energy difference between the two minima). In both cases, $\langle x \rangle$ oscillates but remains confined within its well. The red dashed line at $\langle x\rangle = 3.69$ indicates the maxima at the potential hill, and the gray dashed line at $\langle x\rangle = 0$ and $7.73$ mark the potential minima for the left and right wells, respectively. The variance in the right well (panel (d)) is larger than in the left well (panel (c)). This difference can be attributed to the higher energy ($E = \Delta + 9.0$) of the wave packet in the right well compared to the left well ($E = 9.0$).
• Figure 4: Tunneling (dynamical systems analysis). The time evolution of the mean position $\langle x \rangle$ of the wave packet is shown for initializations in (a) the left well ($\langle x\rangle (0) = 0.5$, $E= 14.95$) and (b) the right well ($\langle x\rangle (0) = 5.50$, $E = \Delta + 14.95$, where $\Delta = 4.68$ is the energy difference between the two minima). In panels (a) and (b), $\langle x \rangle$ switches between the two wells---indicating detectable tunneling. The red dashed line at $\langle x\rangle = 3.69$ indicates the maxima at the potential hill, and the gray dashed line at $\langle x\rangle = 0$ and $7.73$ mark the potential minima for the left and right wells, respectively. The variance in the right well (panel (d)) is larger than in the left well (panel (c)), owing to the higher total energy. Interestingly, the variance in the right well oscillates rapidly, along with frequent barrier crossings, compared to the left well where the particle makes longer excursions between the wells.
• Figure 5: No tunneling (Full Schrödinger simulation). Time evolution of mean position $\langle x\rangle$ of the wave packet initialized at (a) $\langle x\rangle (0) = 0.5$ with $E= 9.0$ in the left well and (b) at $\langle x\rangle (0) = 5.50$ with $E = \Delta + 9.0$ in the the right well, where $\Delta = 4.68$ is the depth of the right well relative to the left. The wave packet oscillates but does not cross the barrier. The red dashed line ($\langle x\rangle = 3.69$) indicates the potential hill top. The gray dashed lines at $\langle x\rangle = 0$ and 7.73 indicate the location of potential minima in the left and right wells, respectively. The curves in panels (c) and (d) show time evolution of the variance in the left and right wells, respectively. While the mean positions are confined to their respective wells, the magnitudes and temporal fluctuations of the variance are similar.