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arXiv:2604.01906

Collective quantum tunneling with time-dependent generator coordinate method

Wenmin Deng, Guangping Chen, Ganlong Ding, Sibo Wang, Jing Peng, Haozhao Liang

nucl-thquant-phSubmitted Apr 2, 2026• Announced Apr 3, 2026• 0 citations
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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.

Comments
12 pages, 5 figures

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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.
It has been known that the time-dependent Hartree-Fock (TDHF) method, or the time-dependent density functional theory (TDDFT), fails to describe many-body quantum tunneling. We overcome this problem by superposing a few time-dependent Slater determinants with the time-dependent generator coordinate method (TDGCM). We apply this method to scattering of two α particles in one dimension, and demonstrate that the TDGCM method yields a finite tunneling probability even at energies below the Coulomb barrier, at which the tunneling probability is exactly zero in the TDHF. This is the first case in which a many-particle tunneling is simulated with a microscopic real-time approach.
Quantum tunneling in many-body systems is the subject of many experimental and theoretical studies in fields ranging from cold atoms to nuclear physics. However, theoretical description of quantum tunneling with strongly interacting particles, such as nucleons in atomic nuclei, remains a major challenge in quantum physics. An initial-value approach to tunneling accounting for the degrees of freedom of each interacting particle is highly desirable. Inspired by existing methods to describe instantons with periodic solutions in imaginary time, we investigate the possibility to use an initial value approach to describe tunneling at the mean-field level. Real-time and imaginary-time Hartree dynamics are compared to the exact solution in the case of two particles in a two-well potential. Whereas real-time evolutions exhibit a spurious self-trapping effect preventing tunneling in strongly interacting systems, the imaginary-time-dependent mean-field method predicts tunneling rates in excellent agreement with the exact solution. Being an initial-value method, it could be more suitable than approaches requiring periodic solutions to describe realistic systems such as heavy-ion fusion.
In this thesis, the tunneling dynamics of a few boson system in a double-well is investigated from an ab-initio prospective using the numerically exact Multi-Configuration Time-Dependent Hartree method. We first study a system consisting of single species of bosons with a spatially modulated interaction. The main emphasis is on the role of inhomogeneity and its effect on the tunneling. The dynamics changes from Rabi oscillations in the non-interacting case to a highly suppressed tunneling for intermediate interaction strengths followed by a reappearance of tunneling near the fermionization limit. With extreme interaction inhomogeneity in the regime of strong correlations we observe tunneling between the higher bands. A richer behavior is found for systems with higher particle number. For systems with more than two bosons, the inhomogeneity of the interaction can be tuned to generate tunneling resonances. These observations are explained on the basis of the few-body spectrum and stationary eigenstates. A tilted double-well and its interplay with the interaction asymmetry is discussed next. We demonstrate that the effects of the interaction can be compensated by the tilt leading to tunneling resonances. We then explore tunneling dynamics of binary bosonic mixtures. The focus is on the role of the inter- and intra-species interactions and their interplay. The dynamics is studied for three initial configurations: complete and partial population imbalance and a phase separated state. Increasing the inter-species interaction leads to a strong increase of the tunneling time period analogous to the quantum self-trapping for condensates. The intra-species repulsion can suppress or enhance the tunneling period depending on the strength of the inter-species correlations as well as the initial configuration. Completely correlated tunneling between the two species and within the same species as well as mechanisms of species separation and counterflow are revealed. These effects are explained by studying the many-body energy spectra as well as the properties of the contributing stationary states.
The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.
This paper presents a robust computational study of the Time-Dependent Schrödinger Equation (TDSE) dynamics for a spatial Gaussian wave packet confined within a one-dimensional infinite potential well. Employing a discrete Finite Difference Method (FDM) framework, we construct a tridiagonal Hamiltonian matrix to compute the unitary real-time evolution of the system. We systematically analyze two critical quantum phenomena: (1) scattering and quantum tunneling against an engineered central rectangular potential barrier, and (2) temporal dispersion and subsequent fractional quantum revival of the localized state. The numerical solver successfully resolved a distinct fractional revival and constructive self-interference peak at t = 1.416 time units. These computational benchmarks provide a rigorous pedagogical and diagnostic baseline for grid-based matrix methods in low-dimensional quantum mechanics.
Real-time computation of time-dependent quantum mechanical problems are presented for nuclear many-body problems. Quantum tunneling in nuclear fusion at low energy is described using a time-dependent wave packet. A real-time method of calculating strength functions using the time-dependent Schrödinger equation is utilized to properly treat the continuum boundary condition. To go beyond the few-body models, we resort to the density-functional theory. The nuclear mean-field models are briefly reviewed to illustrate its foundation and necessity of state dependence in effective interactions. This state dependence is successfully taken into account by the density dependence, leading to the energy density functional. Photoabsorption cross sections in 238 U are calculated with the real-time method for the time-dependent density-functional theory.