Collective quantum tunneling with time-dependent generator coordinate method

Abstract

Inspired by the work of McGlynn and Simenel [Phys. Rev. C {\bf 102}, 064614 (2020)], this study investigates the quantum tunneling of two interacting distinguishable particles in two potential wells. We first benchmark the system by reproducing key established results: the exact quantum solution and the spurious self-trapping effect that arises in the real-time mean-field dynamics for strong interactions. To exactly capture the tunneling dynamics, we apply the time-dependent generator coordinate method (TDGCM) to the model. Numerical simulations demonstrate that the TDGCM, by utilizing the real-time mean-field states as generator states, successfully overcomes the self-trapping effect, yielding tunneling dynamics in excellent agreement with the exact solution. Furthermore, we explore the expectation values of the generator coordinates from the correlated TDGCM many-body wave function. While different methods for calculating expectation values show consistent results in some cases, significant discrepancies are observed in others, providing critical insights into the emergence of collective and single-particle behaviors in interacting systems. This work also verifies the TDGCM as a robust framework for describing collective quantum tunneling and opens avenues for its application to more complex and realistic systems.

Figures (5)

• Figure 1: TDGCM (blue solid lines), exact solutions (red dashed lines), and real-time mean-field (olive dash-dotted lines) predictions of $N_L(t)$ for (a) no interaction with $\mu=0$, (b) a very weak interaction with $\mu=0.1$, (c) a weak interaction with $\mu=1$, and (d) a strong interaction with $\mu=4$, where the number of generator states is $3$.
• Figure 2: TDGCM predictions of $N_L(t)$ for the case of $\mu=1$. The results are calculated with only two generator states, and the value of $\theta_2$ is chosen as $\theta_2 = - \pi/3$, $0$, and $\pi/3$, respectively.
• Figure 3: TDGCM (blue solid lines), exact solutions (red dashed lines), and real-time mean-field (olive dash-dotted lines) predictions of $N_L(t)$ for the case of $\mu=1$, where the numbers of both generator coordinates $\theta$ and $\phi$ are chosen as $2$.
• Figure 4: Evaluations of $\langle\theta(t)\rangle$ calculated by the reduced density matrix (blue solid lines), real-time mean-field (olive dash-dotted lines), probability-based weighted average (cyan dotted lines), overlap-based weighted average (yellow dark dash-dot-dotted lines), and eigenvalue-based weighted average (black short-dashed lines) for (a) no interaction with $\mu=0$, (b) a very weak interaction with $\mu=0.1$, (c) a weak interaction with $\mu=1$, and (d) a strong interaction with $\mu=4$.
• Figure 5: Evaluations of $\langle\phi(t)\rangle$ calculated by the inversion method with mean-field approximation (red dashed lines), reduced density matrix method (blue solid lines), and real-time mean-field (olive dash-dotted lines) for (a) no interaction with $\mu=0$, (b) a very weak interaction with $\mu=0.1$, (c) a weak interaction with $\mu=1$, and (d) a strong interaction with $\mu=4$.