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

Akira Ono<EMAIL_ADDRESS>Department of Physics, Tohoku
University, Sendai 980-8578, Japan

(January 9, 2022)

###### Abstract

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
Phys. Lett. B 808 (2020) 135693, arXiv:2006.06944v1 by N. Hasegawa, K. Hagino
and Y. Tanimura. However, the conclusion of the present work disagrees with
their quick conclusion that quantum tunneling was simulated by the model.
Comments are made on this.

###### keywords:

quantum tunneling, fusion, time-dependent mean-field theory, antisymmetrized
molecular dynamics, wave packet

††journal: Physics Letters B

##  1 Introduction

It is a long-standing problem to describe many-particle quantum tunneling in
time-dependent mean-field approaches [1, 2, 3, 4], such as for sub-barrier
fusion of colliding nuclei [5]. To predict emergence of multiple reaction
channels, e.g. of fusion and scattering, is a general desire in time-dependent
approaches for many-body systems. In some cases, stochastic extension of a
model can allow channels to emerge as a consequence of time evolution, such as
in transport models for heavy-ion collisions [6, 7, 8, 9, 10, 11, 12, 13].
Tunneling further requires true quantum description for the translational
motion of a nucleus, which is often not straightforward because the center-of-
mass wave function is enforced to be localized in space, like a wave packet,
when the nucleus is described in a mean-field model.

In this situation, a possible idea to go beyond is to use a coherent
superposition of Slater determinants

$|\Psi\rangle=\sum_{a}f_{a}|\Phi(z_{a})\rangle,$ (1)

where each Slater determinant $|\Phi(z_{a})\rangle$ is specified by a set of
parameters $z_{a}$. It was suggested by Ref. [1] to determine the time
dependence of these parameter sets $\\{z_{a}(t)\\}$ and the coefficients
$\\{f_{a}(t)\\}$ by the time-dependent variational principle, to do a
simulation starting with a given initial condition. This formalism was called
a time-dependent generator coordinate method (TDGCM). Among the models which
may be called by the same name [14, 15, 16, 17, 18, 19, 20, 21], the model
used in Ref. [1] is unique in that both $\\{z_{a}(t)\\}$ and $\\{f_{a}(t)\\}$
are simultaneously determined.

For the Slater determinants $|\Phi(z_{a})\rangle$ in TDGCM, we use in this
article the wave functions of antisymmetrized molecular dynamics (AMD) [22,
23], i.e., Slater determinants of Gaussian wave packets of a fixed width. One
of the benefits of using an AMD wave function for $|\Phi(z_{a})\rangle$ is
that the center-of-mass motions of composite particles are separated from
their internal degrees of freedom. Therefore, e.g. for a collision of
${}^{4}\mathrm{He}+{}^{4}\mathrm{He}$, the many-body state can be rewritten as

$\Psi=\mathcal{A}\bigl{[}\psi(x)\Phi_{\alpha}(\xi_{1})\Phi_{\alpha}(\xi_{2})\bigr{]}\Phi_{\text{c.m.}}(X_{\text{c.m.}}),$
(2)

when the internal excitations of each nucleus are ignored. Here the center-of-
mass wave function $\Phi(X_{\text{c.m.}})$ and the internal wave functions of
individual ${}^{4}\mathrm{He}$ nuclei, $\Phi_{\alpha}(\xi_{1})$ and
$\Phi_{\alpha}(\xi_{2})$, are all written with Gaussian functions in AMD.
Then, in this simplified example, only the time evolution of the wave function
for the relative coordinate $x$ between the two nuclei

$\psi(x,t)=\sum_{a}f_{a}(t)\,e^{-\nu_{r}\bigl{(}x-\frac{z_{a}(t)}{\sqrt{\nu_{r}}}\bigr{)}^{2}}$
(3)

needs to be considered in the many-body state of Eq. (2). The Gaussian centers
$z_{a}(t)$ take complex values. It should be noted that $z_{a}(t)$ do not
always have direct physical meaning when non-orthogonal wave packets are
coherently superposed. The width parameter for the relative motion is
$\nu_{r}=2\nu$ in this example, where $\nu$ is that of single-particle wave
packets.

The aim of this paper is to investigate how the TDGCM model predicts barrier
transmission in a simplified problem for one degree of freedom, still keeping
the source of the issue in many-body problems that the state usually has to be
described by localized wave packets. To allow intuitive understanding, in Sec.
4, we pay attention to the time evolution of the phase-space distribution
function which directly represents the information on the wave function
$\psi(x,t)$ of Eq. (3). In particular, already in the incoming state, we will
realize important roles played by the momentum distribution. Considerations
are made in Sec. 5 based on the phase space, to clarify how the barrier
transmission occurs in TDGCM, and what problems have to be overcome to extract
the information on quantum tunneling, if any, from TDGCM simulations.

This study mainly considers the same setup of the reaction system with the
same TDGCM model as the study of Ref. [1]. In fact, numerical results
published there are partly used here in Secs. 4 and 5. However, the
consequence of the present study disagrees with the quick conclusion made by
Ref. [1] that TDGCM could simulate quantum tunneling. Comments are made on
this in Sec. 6.

##  2 Problem setup

Our consideration here is limited to the cases which are reduced to problems
of one particle in one dimension, including the case of
${}^{4}\mathrm{He}+{}^{4}\mathrm{He}$ in one dimension. We also assume that
the antisymmetrization $\mathcal{A}$ in Eq. (2) is not essential. We will use
the terminology for one-particle problems below for simplicity. The
translation to the case of the relative motion of two nuclei is
straightforward.

For the potential barrier $V(x)$, we should allow various possibilities of the
barrier height and width for the purpose to test the model. However, also for
the simplicity of terminology, we assume that $V(x)$ is sufficiently smooth
compared to the spatial width of the wave packets, so we do not distinguish
$V(x)$ and its expectation value for a wave packet centered at $x$.

The model uses compact wave packets for the expansion in Eq. (3). Gaussian
wave packets form an overcomplete non-orthogonal basis, so $\psi(x,t)$ of Eq.
(3) can express the exact solution of the Schrödinger equation if a
sufficiently large number of wave packets can be superposed. On the other
hand, the study in Ref. [1] was done when two wave packets are used for
expansion as

$\psi(x,t)=f_{1}(t)\,e^{-\nu_{r}\bigl{(}x-\frac{z_{1}(t)}{\sqrt{\nu_{r}}}\bigr{)}^{2}}+f_{2}(t)\,e^{-\nu_{r}\bigl{(}x-\frac{z_{2}(t)}{\sqrt{\nu_{r}}}\bigr{)}^{2}}.$
(4)

A result with 10 wave packets was also reported briefly in Ref. [1]. It is not
clear whether one should ideally superpose as many wave packets as possible
until the result converges, or one should choose a suitable number of wave
packets for the best prediction of quantum tunneling. We will consider both
cases of philosophy of the model.

The position and momentum centers of each Gaussian wave packet, denoted here
by $x_{a}(t)$ and $p_{a}(t)$, are related to the parameter $z_{a}(t)$ by

$z_{a}(t)=\sqrt{\nu_{r}}\,x_{a}(t)+i\frac{p_{a}(t)}{2\hbar\sqrt{\nu_{r}}}.$
(5)

We choose the same width parameter $\nu_{r}=1\ \text{fm}^{-2}$ as Ref. [1]. As
an example of the initial condition, we may consider the same condition as in
Ref. [1] by taking

$x_{1}-x_{2}=0.1\ \text{fm}\quad\mbox{and}\quad p_{1}-p_{2}=1.21\
\text{MeV}/c$ (6)

for the two wave packets at $t=0$. The coefficients are taken to be
$f_{1}=f_{2}$ at $t=0$.

Figure 1:  The momentum distributions for the two Gaussian wave packets which
were used in the initial state in the calculation of Ref. [1]. The gray area
indicates the region that is relevant to quantum tunneling ($E<0.13$ MeV and
$p>0$).

In Ref. [1], the initial momentum was chosen as $\langle
p\rangle=\frac{1}{2}(p_{1}+p_{2})=19.98$ MeV/$c$, which is much smaller than
the momentum width, $\Delta p=\hbar\sqrt{\nu_{r}}=197$ MeV/$c$, associated
with the Gaussian wave packet. In Fig. 1, the momentum distributions for the
two individual components in Eq. (4) at $t=0$ are shown by the blue dotted
line (for the first term) and by the red solid line (for the second term). The
momentum centers $p_{1}$ and $p_{2}$ are practically identical. Since the two
wave packets are superposed in phase in Eq. (4), the distribution for the
total wave function $\psi(x,t=0)$ also has the same shape. An important
question to be addressed later is whether this broad momentum width plays any
role for the time evolution of $\psi(x,t)$. For the choice of the potential
$V(x)$ in Ref. [1], the gray area indicates the region of $0<p<\sqrt{2\mu
V_{\text{B}}}$, where $V_{\text{B}}=0.13$ MeV is the barrier height. The mass
is $\mu=1876$ MeV/$c^{2}$. This narrow region is the part relevant to quantum
tunneling, where the potential barrier cannot be passed over in classical
dynamics.

##  3 Dual view on energy and momentum

To avoid confusions, let us clearly distinguish two possible viewpoints on the
interpretation of the state described by the wave function $\psi(x,t)$, or
equivalently by the parameters such as $x_{a}(t)$, $p_{a}(t)$ and $f_{a}(t)$.
Of course, in the usual sense of quantum mechanics, we can extract any
physical information from $\psi(x,t)$ using the standard recipes. Let us call
this way “View A” in this article. In the other viewpoint, which we call “View
B” here, one ignores the momentum width of individual wave packets, so that a
wave packet is regarded as having a definite momentum $p_{a}$. More generally,
the state represented by $\psi(x,t)$ might be assumed to have a definite
momentum calculated as $\langle\psi|p|\psi\rangle/\langle\psi|\psi\rangle$. A
clear difference between the two views can be identified by considering the
value of kinetic energy. In View A, the expectation value is $\langle
p^{2}/2\mu\rangle=p_{a}^{2}/2\mu+T_{0}$, where the constant term

$T_{0}=\hbar^{2}\nu_{r}/2\mu$ (7)

originates from the momentum width of the wave packet. On the other hand, in
View B, the wave packet has a definite kinetic energy $E_{a}=p_{a}^{2}/2\mu$.
The difference between the two views is not negligible; $T_{0}=10.38$ MeV in
our example.

When nuclear reactions are simulated, View B has been taken widely in time-
dependent approaches, such as TDHF and transport models including AMD, for the
center-of-mass motions of nuclei. This view is convenient because the reaction
with a fixed beam energy can be simulated by choosing the initial value
$p_{a}=\sqrt{2\mu E_{\text{beam}}}$ and also because an emitted nucleus can be
regarded as having a definite energy $E_{a}=p_{a}^{2}/2\mu$. Although View B
violates rules of quantum mechanics (e.g. Heisenberg’s uncertainty principle),
it works in many cases without causing serious inconsistencies, mainly because
$T_{0}$ is just a constant term in the Hamiltonian, so it affects neither the
equation of motion nor the energy conservation. However, to the best of the
author’s knowledge, the models that take View B have not been extended
successfully to treat coherent superposition of wave packets, though
stochastic branching of wave packets or many-body states is typically
considered in transport models [6, 7, 8, 9, 10, 11, 12, 13]. It is therefore
an important question whether View B can be taken in the TDGCM model, which
will be addressed later in detail.

We are relatively safe in View A, though a drawback is that the initial state
cannot be prepared as a momentum eigenstate.

The lack of quantum tunneling in the models with a single wave packet (i.e. a
single Slater determinant) can be understood in either view. In View B, the
classical Hamiltonian without the $T_{0}$ term can simply explain that the
barrier cannot be passed over when $p_{a}<\sqrt{2\mu V_{\text{B}}}$ ($a=1$).
View A should be based on the initial momentum distribution $f(p)$ associated
with the wave packet. When $p_{a}<\sqrt{2\mu V_{\text{B}}}$, a tail of $f(p)$
may be still extending to the high-momentum region ($p>\sqrt{2\mu
V_{\text{B}}}$) where barrier passage would be possible. However, the
probability in $f(p)$ is larger in the part of $p<\sqrt{2\mu V_{\text{B}}}$
where barrier passage is less probable even if quantum tunneling is allowed.
Since the model does not allow the wave packet to split into two parts, the
dynamics can only choose the major branch so the wave packet will be totally
reflected. On the other hand, when $p_{a}>\sqrt{2\mu V_{\text{B}}}$, the wave
packet will be totally transmitted for the same but inverted reason. In this
case, some tail component of $f(p)$ may still exist in the low-momentum
region, and therefore one could in principle argue that transmission always
occurred even when there was some probability of $p^{2}/2\mu<V_{\text{B}}$.
However, this is customarily not called quantum tunneling.

##  4  Incoming state

Figure 2:  Wigner and Husimi functions for a free particle motion, at
different times from $t=-10$ fm/$c$ to 25 fm/$c$. The Wigner function is shown
by the contour lines drawn for $f=0.2,0.4,0.6,\ldots$, while the Husimi
function is represented by the color scale. The initial condition for the
Wigner function was chosen as
$f_{\text{W}}(x,p,t=0)=2\exp(-2\nu_{r}x^{2}-p^{2}/2\hbar^{2}\nu_{r})$ which is
very similar to the Wigner function of the initial state $\psi(x,t=0)$ of the
TDGCM calculation in Ref. [1].

A freely propagating state is used at the initial stage of the simulation.
Free propagation is trivial in many models, but it may not be so in a newly
invented model, so we should carefully consider it in TDGCM. In fact, we will
realize that the incoming state in the TDGCM simulation has special features.

###  4.1 Exact solution and superposition of many wave packets

The well known exact solution for a free motion is displayed in Fig. 2, in
which the Wigner function $f_{\text{W}}(x,p,t)$ [24] is shown from $t=-10$ to
$25$ fm/$c$, by solid contour lines. The figure was drawn in the boosted frame
in which $\langle p\rangle=0$ and $\langle x\rangle=0$. Importantly, the
correlation between $x$ and $p$ develops strongly as the time progresses,
which indicates that different momentum components are resolved by the time
evolution, resulting in different spatial positions. As a reference for
comparison, we will use later the Husimi function [25] defined by

$f_{\text{H}}(x,p,t)=\iint\frac{dx^{\prime}dp^{\prime}}{2\pi\hbar}e^{-\frac{(x-x^{\prime})^{2}}{2\Delta
x^{2}}-\frac{(p-p^{\prime})^{2}}{2\Delta
p^{2}}}f_{\text{W}}(x^{\prime},p^{\prime},t)$ (8)

with $\Delta x=1/(2\sqrt{\nu_{r}})$ and $\Delta p=\hbar\sqrt{\nu_{r}}$. It is
shown in Fig. 2 by the color scale.

The equation for the time evolution of $f_{\text{W}}(x,p,t)$, derived from the
Schrödinger equation, can be viewed as an extension of the Liouville equation
with quantum terms in general (see e.g. Refs. [26, 27, 28]). However, in the
particular cases of the potential of the form $V(x)=ax^{2}+bx+c$ including the
case of free propagation, the equation is exactly identical to the classical
Liouville equation. Therefore, the evolution of the distribution function can
be easily understood based on the classical trajectories in the phase space.

If a sufficiently large number of wave packets are superposed as in Eq. (3) in
a suitable way, the free propagation can be described precisely by TDGCM in
principle. For example, wave packet centers, all of which are prepared almost
at the same point in the phase space, will move to reproduce the exact
solution e.g. by aligning along the principal axis of the deformed
distribution, with suitable changes of weight factors.

Figure 3:  Contour plot for the energy $E$ of a free particle when the wave
function is approximated by a superposition of two Gaussian wave packets,
centered at $(\tilde{x},\tilde{p})$ and $(-\tilde{x},-\tilde{p})$, with equal
coefficients that are in phase. Contour lines are drawn for
$E=\hbar^{2}\nu_{r}/2\mu+k\times 0.5\ \text{MeV}$ for integers $k$. The thick
contour line is for $E=\hbar^{2}\nu_{r}/2\mu=10.38$ MeV. The energy takes the
minimum $E=4.60$ MeV at $(\tilde{x},\tilde{p})=(\pm 0.80\ \text{fm},0)$, shown
by the crosses. The open circle indicates the initial condition chosen in Ref.
[1], $(\tilde{x},\tilde{p})=(0.05\ \text{fm},\ 0.61\ \text{MeV}/c)$. The
dashed contour line is for $E=10.33$ MeV corresponding to this initial
condition. The width parameter of the Gaussian wave packets and the mass are
chosen to be $\nu_{r}=1\ \text{fm}^{-2}$ and $\mu=1876$ MeV. Figure 4:  Husimi
function for the TDGCM state of Eq. (4) in a boosted frame. Each panel
represents the Husimi function by the color scale, for the state specified by
the parameters $(\tilde{x},\tilde{p})$, whose values are chosen on the dashed
contour line in Fig. 3. The open circles represent the locations of the wave
packet centers, $\pm(\tilde{x},\tilde{p})$.

###  4.2 Superposition of two wave packets

Next, let us investigate the free propagation when only two wave packets are
superposed in TDGCM. We can easily know the trajectories of the wave packet
centers from the energy contours, as shown in Fig. 3. The energy expectation
value is

$\displaystyle E=\langle H\rangle$
$\displaystyle=T_{0}+\frac{1}{N}\sum_{a}\sum_{b}\frac{p_{ab}^{2}}{2\mu}f_{a}^{*}f_{b}N_{ab}$
(9) $\displaystyle N$ $\displaystyle=\sum_{a}\sum_{b}f_{a}^{*}f_{b}N_{ab}$
(10)

for a TDGCM state, with $p_{ab}=i\hbar\sqrt{\nu_{r}}(z_{a}^{*}-z_{b})$ and the
overlap matrix $N_{ab}=e^{-\frac{1}{2}(z_{a}^{*}-z_{b})^{2}}$. For a free
motion, the condition of $f_{1}=f_{2}$ is preserved and the wave packet
centers can be expressed as $(x_{1},p_{1})=(\tilde{x},\tilde{p})$ and
$(x_{2},p_{2})=(-\tilde{x},-\tilde{p})$ at any time $t$ in the boosted frame.
The thick contour line in Fig. 3 is for $E=T_{0}$. With this line as a
separatrix, there are two kinds of trajectories of $(\tilde{x},\tilde{p})$. If
$|\tilde{p}|$ is sufficiently large, $|\tilde{x}|$ will increase, resulting
eventually in non-interfering two wave packets. On the other hand, in the
region surrounded by the separatrix, cyclic trajectories exist. They move
around the energy minima shown by the crosses in Fig. 3. The energy at the
minimum configuration is $E=4.60$ MeV, which is significantly lower than
$T_{0}=10.38$ MeV, suggesting that two wave packets can form a rigid pair by
effectively constructing a single wave packet with a narrow momentum width and
a broad spatial width.

The point of $(\tilde{x},\tilde{p})$ of the initial condition Eq. (6) is
indicated in Fig. 3 by the open circle. It was chosen inside the domain of
cyclic trajectories, but it is very close to the separatrix. Under this
initial condition, the point of $(\tilde{x},\tilde{p})$ should move cyclically
on an energy contour shown by the red dashed line in Fig. 3. This is
consistent with the trajectories in Fig. 2 of Ref. [1]. Here we are only
looking at the wave packet centers, as in View B (see Sec. 3). The physical
meaning of a cyclic motion for free propagation seems difficult to understand
in this view.

Let us then take View A by looking directly at the information in the TDGCM
wave function $\psi(x,t)$ of Eq. (4). Eight points are chosen on the red
dashed contour in Fig. 3, and the Husimi function $f_{\text{H}}(x,p)$ is
calculated for each point and displayed in Fig. 4. In each panel, the location
of the superposed two wave packets, $\pm(\tilde{x},\tilde{p})$, are shown by
open circles. The Husimi function is defined by Eq. (8) with the Winger
function

$\displaystyle f_{\text{W}}(x,p)$ $\displaystyle=\int
ds\,e^{-ips/\hbar}\,\psi^{*}(x-\tfrac{1}{2}s)\,\psi(x+\tfrac{1}{2}s)$ (11)
$\displaystyle=\frac{1}{N}\sum_{a}\sum_{b}2\,e^{-2\nu_{r}(x-x_{ab})^{2}-(p-p_{ab})^{2}/2\hbar^{2}\nu_{r}}f_{a}^{*}f_{b}N_{ab},$
(12)

with $x_{ab}=(z_{a}^{*}+z_{b}^{*})/2\sqrt{\nu_{r}}$ and
$p_{ab}=i\hbar\sqrt{\nu_{r}}(z_{a}^{*}-z_{b})$. The Wigner function is,
however, not displayed in the figure; it sometimes shows violent oscillatory
behavior and it takes negative values in some regions.

The panels in Fig. 4 have been arranged so that they are similar to those for
the exact solution in Fig. 2, at least for the left five panels. In fact, the
evolution in panels from (b) to (d), is quite similar to the exact evolution
from $t=-5$ to $5$ fm/$c$. The states in (a) and (e) may also be acceptable
approximations at $t\approx-10$ and $10$ fm/$c$. In this time interval, the
evolution is consistent with the correct development of the $p$–$x$
correlation. However, after the state of (e), the TDGCM solution largely
deviates from the exact solution. The distribution is unphysically separated
into two parts in (f), (g) and (h). The shape of each part is almost frozen,
and the evolution of the $p$–$x$ correlation is now pathological. In panel
(f), for example, the front part ($x>0$) of the distribution still has a weak
tendency to have positive momentum $\langle p\rangle$ on average, but this
part has to unphysically lose $\langle p\rangle$, in order to move to the
forward direction, stalling at the state of (g). The TDGCM solution further
evolves to (h) and then to (a), closing a cycle of the trajectory. In Fig. 2
of Ref. [1], the period of a cycle is found to be about 150 fm/$c$, which is
much longer than the time from panel (a) to (e) in Fig. 4. Namely, the
pathological motion, from (e) through (g) to (a), is very slow.

Coming back to View B, let us look at the momentum $p_{a}$ and the energy
$E_{a}=p_{a}^{2}/2\mu$ ($a=1$ or $2$) for this free propagation. The
consequence of the model is that both $p_{a}$ and $E_{a}$ are changing in
time, which sounds quite unrealistic as a free motion. The amount of the
change is of the order of 100 MeV/$c$ in $p_{a}$, which is similar to the
momentum width for the wave function. This implies that the origin of the
change of $p_{a}$ and $E_{a}$ is the momentum width, or the evolution of the
$p$–$x$ correlation induced by it. The effect of coherent superposition is not
easy to understand in View B.

The state studied here is used as the incoming state of a TDGCM simulation. If
one takes View B, it is at least inconvenient that the energy is changing so
much before the potential plays any role. In View A, the state has a momentum
distribution $f(p)$ which has to be taken care of in some way, but a good
feature is that the time dependence of $f(p)$ during the free propagation is
small because the mean and the variance of the distribution are exactly
conserved by the model. However, the evolution of the $p$–$x$ correlation is
pathological, as in Fig. 4.

##  5  Considerations on barrier passage

The incident particle is described as in the previous section by a time-
dependent state in TDGCM, and it now arrives at the potential barrier. We have
to carefully consider how the behavior of the incoming state may affect the
barrier passage.

###  5.1 Superposition of many wave packets

First, let us consider the case in which a large number of wave packets are
superposed. In View A, the evolution in the incoming stage will be similar to
the exact free propagation in Fig. 2. Note that the initial state has some
momentum distribution $f(p)$. The potential barrier is first encountered by
the leading front part of the phase space distribution which has a relatively
high momentum as a consequence of the developed $p$–$x$ correlation. This
leading part can pass over the barrier if it has energy higher than the
barrier $V_{\text{B}}$, and the part arriving later will tend to be reflected
if the energy is lower than $V_{\text{B}}$. Similar consideration is possible
in View B for the passage and reflection of individual wave packets, which
have been aligned to represent the developed $p$–$x$ correlation during the
free propagation.

When a moderate number (e.g. 10) of wave packets are superposed, the situation
may be similar to the above consideration (see e.g. the case with 10 wave
packets briefly reported in Ref. [1], and that in Ref. [29]). However, 10 wave
packets are probably not enough to precisely express the free propagation for
a long time (for $3000$ fm/$c$ in the case of Ref. [1]). Then interference
among wave packets may have been lost, at least partly, and therefore the
barrier passage is likely determined by independent motions of wave packets
(or e.g. of rigid pairs or groups of wave packets) when they encounter the
barrier.

The barrier passage speculated above is of classical nature, at least partly,
because passage occurs mainly when the energy is higher than $V_{\text{B}}$.
In View A, barrier passage occurred, simply because the initial momentum
distribution $f(p)$ had included high-momentum components. In View B, the
momenta $p_{a}(t)$ are changing in time during the free propagation, and the
barrier passage may be determined by the value of $p_{a}^{2}(t)/2\mu$ when
each wave packet arrives at the barrier. However, the interpretation has to be
more complicated if wave packets still interfere around the barrier, which is
beyond the scope of View B.

We can still expect that some probability of true quantum tunneling is
contained in the total probability of barrier passage. It is in principle
possible to investigate such possibility by analyzing the energy $E$ of the
transmitted particle. In View A, due to the distribution $f(p)$, the energy
may be $E>V_{\text{B}}$ in many cases but some probability may be found in
$E<V_{\text{B}}$, which can be identified as quantum tunneling. In the other
View B, it is expected that the energies $E_{a}=p_{a}^{2}/2\mu$ of the
transmitted wave packets are usually higher than $V_{\text{B}}$, which however
does not deny a possibility that some transmitted wave packets end up with
lower energies if wave packets are still interfering around the barrier (see
an example in Sec. 5.2). To obtain a converged result, dependence on the
conditions needs careful investigation, such as the number of wave packets and
the timing of arrival at the barrier.

In the approach called entangled trajectory molecular dynamics [30, 31, 32],
where quantum effects are taken into account by coupling the phase space
trajectories, the calculated transmission probability is often compared with
the classical result obtained from the same initial condition of the phase
space distribution (in View A). This allows identification of the genuine
quantum contribution to the barrier transmission. Similar analysis should also
be performed for the TDGCM solution.

Another strict check for tunneling is to analyze the dependence on the
thickness of the potential barrier $V(x)$. If the transmission is quantum
tunneling, its probability should decrease when the width of the barrier is
increased. The transmission is not quantum tunneling if it occurs for the
infinite width [e.g. $V(x)=0$ for $x<0$ and $V(x)=V_{\text{B}}$ for $x>0$].
With a result of such a check, quantum tunneling would be clearly identified.

###  5.2  Superposition of two wave packets

Next, let us investigate the case in which only two wave packets are
superposed. We stay in View A here (see Sec. 6 for the case of View B). The
behavior of the incoming state is already nontrivial as discussed in the
previous section, and the cyclic motion in the incoming state is very slow in
the pathological phase [from (e) through (g) to (a) in Fig. 4]. Therefore, the
potential barrier will be encountered with a large chance in the pathological
phase. For example, if the barrier is encountered in the stage of (f), the
front part of the distribution will pass over the barrier when the average
momentum $\langle p\rangle$ in this part is high enough. In fact, each of the
front and rear parts is similar to a Gaussian wave packet like the initial
state in panel (c), and even a slightly larger $\langle p\rangle$ than the
initial one may be sufficient for barrier passage (as reminded by Fig. 1).
Such a barrier passage by a part with a frozen shape is not called tunneling
in general. The same can occur in usual AMD and TDHF when $\langle p\rangle$
is high enough. This consideration on the above example seems to be consistent
with the case shown in Fig. 2 of Ref. [1].

After the front part passed over the potential barrier, the two parts may
still continue to interfere weakly in a similar way to the evolution from (f)
to (h) in Fig. 4. Then the transmitted part can reduce $\langle p\rangle$
resulting in an energy lower than $V_{\text{B}}$, as it should be in quantum
tunneling. A question is whether the energy is guaranteed to result in a right
value. In the example reported in Ref. [1], this transmitted component had an
energy expectation value lower than the ${}^{4}\mathrm{He}+{}^{4}\mathrm{He}$
threshold by about 2 MeV (see Fig. 4 of Ref. [1]), which is possible in a
sense due to the attractive interaction between the nuclei. However, as the
initial quantum state did not include components lower than the threshold,
such an event with an inconsistent energy needs more investigation. This
manifests the problem that the energy distribution, for the total state
$\psi(x,t)$, is not conserved in TDGCM with the restricted form of Eq. (4),
even though the conservation of the energy expectation value is guaranteed.

##  6  Comments on Ref. [1]

Our investigations in the previous sections show that quantum tunneling is not
yet easy to argue in the TDGCM model, which disagrees with the quick
conclusion of Ref. [1] that TDGCM could simulate quantum tunneling. Origins of
this erroneous conclusion by Ref. [1] are explained below.

The important role played by the momentum distribution $f(p)$ was overlooked
by Ref. [1]. Since the barrier height $V_{\text{B}}$ is very low compared to
the width of $f(p)$ in their setup (see Fig. 1), the classical barrier passage
is dominant, and true quantum tunneling from the narrow gray region in Fig. 1
can only be a minor fraction. Nevertheless, the whole probability of barrier
passage was misinterpreted as ‘quantum tunneling’.

Ref. [1] took View B, abandoning interpretation of the wave function
$\psi(x,t)$. They chose the condition at $t=0$ so that all
$p_{a}^{2}(t=0)/2\mu$ have almost the same value $E(t=0)$, and the simulation
was regarded as corresponding to this beam energy. They then claimed that
barrier transmission was all due to quantum tunneling because
$E(t=0)<V_{\text{B}}$. This is a serious mistake made by Ref. [1]. What one
would claim depends on what time was regarded as the ‘initial’ time $t_{0}$,
because $p_{a}(t_{0})$ depend on $t_{0}$ without any effect of the potential
in the incoming state. It is actually impossible to simulate a fixed beam
energy in TDGCM. There is no foundation for View B with $E(t=0)$ as the beam
energy, when wave packets are coherently superposed. We can also expect that
the result of a simulation depends on the timing of arrival at the barrier,
which makes any statement questionable if it is based on only a single
simulation.

If $E(t=0)$ is still regarded as the beam energy, a strange problem is noticed
in Figs. 2 and 4 of Ref. [1], for the energies of ‘fusion’ and ‘elastic’ exit
channels in a ${}^{4}\mathrm{He}+{}^{4}\mathrm{He}$ collision. The energy
$E_{1}(t)=p_{1}^{2}(t)/2\mu$ in the ‘elastic’ channel was much higher than the
initial energy $E(t=0)$, by about 2 MeV. The same can be seen as the large
slope of a trajectory in Fig. 2 of Ref. [1] after $t=4000$ fm/$c$. Thus the
result of TDGCM is in contradiction to the fundamental fact that the energy in
the elastic channel must agree with the initial one. In the ‘fusion’ channel,
the energy changed from $E(t=0)$ by about $-2$ MeV, as shown in Fig. 4 of Ref.
[1]. Such deexcitaion would be possible if something like a $\gamma$ ray were
emitted, but such an effect was not included here. Thus the energy
conservation is unphysically violated, which makes it difficult to argue
quantum tunneling.

##  7 Summary and conclusion

Investigations were made to evaluate the TDGCM model proposed by Ref. [1] as a
method for many-particle quantum tunneling such as fusion of colliding nuclei,
in the simplest examples equivalent to one-particle problems in one dimension.
It was found that the description of the free propagation in the incoming
state is nontrivial, in particular when only a few wave packets are
superposed. This was visualized here by using the phase-space distribution
function. Including also the ideal case of superposing many wave packets, one
needs to consider the momentum width in the prepared initial wave function, to
understand the behaviors of the incoming state, which then affect the passage
of potential barrier. Depending on the condition of the simulation, the broad
momentum distribution allows classical barrier passage by high-momentum
components, which must be distinguished from true quantum tunneling.

In some case, a transmitted wave packet can end up with an energy lower than
the barrier height. However, to argue quantum tunneling, the energy must be
well under control by the model. In particular, consistency is required for
the energies of the individual exit channels (e.g. of transmission and
reflection). This criterion is not satisfied by the TDGCM model when a few
wave packets are superposed, which should be carefully considered in practical
applications. Generally, one has to analyze the energies in the incoming and
outgoing states, e.g. to identify any contribution of true quantum tunneling
in the calculated solutions of the TDGCM model. Also the dependence on the
thickness of the potential barrier may be a clue to extract true information
on tunneling.

Such careful analyses are left to be done. Contrary to the quick conclusion
made by Ref. [1], the description of many-particle quantum tunneling in time-
dependent mean-field approaches still remains a challenging problem.

## Acknowledgments

The author thanks Naoto Hasegawa for discussions which confirmed the author’s
ideas presented in this article. This work was supported by JSPS KAKENHI Grant
Numbers JP17K05432 and JP21K03528.

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