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arXiv:2202.06454

Reply to Comment on "Phase-space consideration on barrier transmission in a time-dependent variational approach with superposed wave packets"

Akira Mei Ono

Published Feb 14, 2022• 0 citations
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Abstract

In the paper by Hasegawa, Hagino and Tanimura (HHT) [Phys. Lett. B 808 (2020) 135693, arXiv:2006.06944], they concluded that quantum tunneling was simulated by a time-dependent generator coordinate method (TDGCM). In contrast, difficulties of TDGCM in describing quantum tunneling were pointed out by Ono in the paper [Phys. Lett. B 826 (2022) 136931, arXiv:2201.02966]. Recently, HHT submitted a comment [arXiv:2202.00513v1], by which they appear to give some counterarguments to Ono's paper. Here I examine their arguments, reviewing the main points of Ono's paper. The confusion in HHT's arguments may be mostly resolved by recognizing how the wave packets interfere with each other when they are coherently superposed.

DOI
10.48550/arxiv.2202.06454

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Comment on "Phase-space consideration on barrier transmission in a time-dependent variational approach with superposed wave packets (arXiv:2201.02966)"

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We reply to the criticisms of our publication (N. Hasegawa, K. Hagino, and Y. Tanimura, Phys. Lett. B808, 135693 (2020)) made by A. Ono in his recent article, arXiv:2201.02966.

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Phase-space consideration on barrier transmission in a time-dependent variational approach with superposed wave packets

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A known limitation of time-dependent mean-field approaches is a lack of quantum tunneling for collective motions such as in sub-barrier fusion reactions. As a first step toward a solution, a time-dependent model is considered using a superposition of Gaussian wave packets, to describe the relative motion between two colliding nuclei, which may be simplified to a problem for one particle in one dimension. In this article, how the model describes the potential-barrier transmission is investigated by paying attention to the time evolution of the phase space distribution, which in particular reveals that the behavior of the free propagation of the incoming state is not trivial, depending on the number of superposed wave packets. Passage over the barrier can occur due to the high-momentum components in the incoming state corresponding to energies above the barrier height, which is, however, of classical nature and needs to be distinguished from the true quantum tunneling. Although a transmitted wave packet in some case may end up with an energy lower than the barrier, a difficulty is noticed in guaranteeing the energy conservation when the energies of different exit channels, e.g. of transmission and reflection, are individually measured. To overcome these issues for a description of quantum tunneling is still a challenging problem. This article mainly treats the same system with the same model as in the paper Hasegawa, Hagino and Tanimura (2020) [1]. However, we reach different conclusions.

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In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

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Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

Mostafazadeh, Ali

I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

arXiv:0711.1078•Nov 7, 2007•4 citations
In the paper by Hasegawa, Hagino and Tanimura (HHT) [Phys. Lett. B 808 (2020) 135693, arXiv:2006.06944], they concluded that quantum tunneling was simulated by a time-dependent generator coordinate method (TDGCM). In contrast, difficulties of TDGCM in describing quantum tunneling were pointed out by Ono in the paper [Phys. Lett. B 826 (2022) 136931, arXiv:2201.02966]. Recently, HHT submitted a comment [arXiv:2202.00513v1], by which they appear to give some counterarguments to Ono's paper. Here I examine their arguments, reviewing the main points of Ono's paper. The confusion in HHT's arguments may be mostly resolved by recognizing how the wave packets interfere with each other when they are coherently superposed.
A known limitation of time-dependent mean-field approaches is a lack of quantum tunneling for collective motions such as in sub-barrier fusion reactions. As a first step toward a solution, a time-dependent model is considered using a superposition of Gaussian wave packets, to describe the relative motion between two colliding nuclei, which may be simplified to a problem for one particle in one dimension. In this article, how the model describes the potential-barrier transmission is investigated by paying attention to the time evolution of the phase space distribution, which in particular reveals that the behavior of the free propagation of the incoming state is not trivial, depending on the number of superposed wave packets. Passage over the barrier can occur due to the high-momentum components in the incoming state corresponding to energies above the barrier height, which is, however, of classical nature and needs to be distinguished from the true quantum tunneling. Although a transmitted wave packet in some case may end up with an energy lower than the barrier, a difficulty is noticed in guaranteeing the energy conservation when the energies of different exit channels, e.g. of transmission and reflection, are individually measured. To overcome these issues for a description of quantum tunneling is still a challenging problem. This article mainly treats the same system with the same model as in the paper Hasegawa, Hagino and Tanimura (2020) [1]. However, we reach different conclusions.
In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.
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.
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.