# Reply to Comment on “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

(February 14, 2022)

###### 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.

††journal: arXiv

##  1 Background

In Ref. [1], Hasegawa, Hagino and Tanimura (HHT) employed a time-dependent
generator coordinate method (TDGCM) aiming at a description of quantum
tunneling. Although the final goal is to establish a method for many-particle
problems, the illustrative example studied in their article [1] is equivalent
to a problem of one particle in one dimension. Their conclusion was that
quantum tunneling could be simulated by TDGCM.

On the other hand, in Ref. [2], Ono investigated the same problem under the
same TDGCM model, by paying attention to the time evolution of the phase space
distribution, which is easy for such an illustrative problem of one particle
in one dimension. In particular, it was noticed that the free propagation of
the incoming state is already non-trivial in the TDGCM solution, which affects
how the potential barrier is passed over. One of the main conclusions was that
the barrier passage can be of classical nature due to the high-momentum
components in the prepared initial state. Such classical barrier passage
cannot be regarded as quantum tunneling, contrary to the conclusion of Ref.
[1]. Based on the understanding in the phase space, the reason why HHT reached
the different conclusion was explained in Ref. [2] in the form of comments or
criticisms on Ref. [1].

Now a comment to Ref. [2] has been submitted to arXiv by HHT [3], in which
they apparently give counterarguments to Ono’s comments in Ref. [2]. My
comments on their individual arguments are given below in this article.
However, I first point out here that their arguments [3] do not take into
account the main point of Ref. [2], which is the consideration based on the
phase space distribution. Without that consideration, it is hardly possible to
understand the behavior of the solution of TDGCM. Of course, I will try to
follow HHT’s arguments as far as possible, but it is sometimes necessary to
review the phase-space consideration in Ref. [2].

In the first paragraph of Ref. [3], Ono’s comments in Ref. [2] are mentioned
by HHT as “The claim is based on the fact that the initial wave function
employed in Ref. [1] has a broad momentum distribution.” This statement is not
rigorously correct because Ono’s claim is based on the observed time evolution
of the phase space distribution, i.e., the solution of the TDGCM equation.
What is directly affected by the initial broad momentum distribution is the
solution of TDGCM, without any room for personal opinions to enter.

##  2 Does the momentum width play any role?

In the second paragraph of Ref. [3], HHT agree that the illustrative problem
treated in Refs. [1, 2] is equivalent to one particle in one dimension, in
which the TDGCM model expresses the wave function (for the relative motion of
the two colliding nuclei) by a superposition of Gaussian wave packets,

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

Here the wave packet center $z_{a}(t)$ of each component $a$ takes a complex
value, and its real and imaginary parts have information on the spatial and
momentum centers, $x_{a}(t)$ and $p_{a}(t)$, respectively. The width parameter
$\nu_{r}$ is a fixed constant, e.g., $\nu_{r}=1\ \text{fm}^{-2}$, which
determines the spatial and momentum widths, $\Delta x=1/(2\sqrt{\nu_{r}})$ and
$\Delta p=\hbar\sqrt{\nu_{r}}$; a Gaussian wave packet has the minimum
uncertainty $\Delta x\Delta p=\frac{1}{2}\hbar$. Figure 4 shows the momentum
distributions of the two Gaussian wave packets used in the initial state of
the calculation of Ref. [1]. This is the same figure as Fig. 1 of Ref. [2],
which is put in this article for the convenience. (The same for the other
three figures.)

It was properly mentioned in Ref. [2] that the width parameter $\nu_{r}$ is a
fixed constant, and therefore the Gaussian wave packets do not change their
shape, though they can, of course, move by the changes of $x_{a}(t)$ and
$p_{a}(t)$. The second paragraph of Ref. [3] seems to be mainly focusing on
such time evolution of the individual Gaussian wave packets. However, such a
consideration is insufficient to understand the wave function $\psi(x,t)$ of
Eq. (1). The effect of interference between Gaussian wave packets needs to be
considered, in addition to the “classical” motions of the Gaussian wave
packets. For example, through the interference, the momentum width can affect
the motions of the Gaussian wave packets, as we can find examples in Ref. [2]
and also as reviewed below.

We should keep in mind that the Gaussian wave packets are used as basis
functions to express the wave function $\psi(x,t)$ which represents the
physical state of the system at the specified time $t$. In Ref. [2], the term
‘wave function’ is used for the physical state $\psi(x,t)$, but not for the
Gaussian wave packet in each term in the right-hand side of Eq. (1). Generally
speaking, one cannot necessarily find physical meaning in individual basis
functions. In particular, in TDGCM, the Gaussian wave packets are not
orthogonal with each other, which makes it further difficult to find physical
meaning in individual superposed components. This was already warned, e.g., in
Ref. [2].

The wave function $\psi(x,t)$ of Eq. (1) for the physical state does not have
the same spatial and momentum widths as the individual Gaussian wave packets,
not only due to the distribution of the Gaussian centers but also as a
consequence of constructive and destructive interference between Gaussian wave
packets. As mentioned in Ref. [2], Gaussian wave packets can form an
overcomplete basis. Therefore, when many Gaussian wave packets are suitably
superposed, one can, in principle, reproduce the exact solution of the
Schrödinger equation which is displayed in Fig. 4 for the case of free
propagation, in the form of the phase space distribution. Such a time
evolution of $\psi(x,t)$ is consistent with the idea of HHT [3] on “a quantum
mechanical wave packet, whose spatial width broadens as a function of time
according to its momentum distribution.” Note that, in this ideal case, the
momentum distribution in the initial state is fully reflected in the time
evolution of $\psi(x,t)$ in TDGCM. This is therefore a counter-example to
HHT’s assertion [3] “even if the wave function (1) may have a broad momentum
distribution, that will not be reflected fully in the time-dependent
dynamics.”

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
$p>0$). This figure is Fig. 1 of Ref. [2].

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]. This figure is Fig. 2 of Ref. [2].

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. This figure is
Fig. 3 of Ref. [2].

Figure 4:  Husimi function for the TDGCM state of Eq. (LABEL:eq:gcmwf) 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. 4. The open circles
represent the locations of the wave packet centers,
$\pm(\tilde{x},\tilde{p})$. This figure is Fig. 4 of Ref. [2].

##  3 Why are trajectories entangled?

Because of the interference between Gaussian wave packets, the time evolution
of the wave packet centers, $x_{a}(t)$ and $p_{a}(t)$, is highly non-trivial,
as demonstrated for the free propagation in Ref. [2], in the case of two wave
packets superposed with equal weights. Figure 4 shows the energy contour lines
in the boosted frame in which
$(x_{1}(t),p_{1}(t))=-(x_{2}(t),p_{2}(t))\equiv(\tilde{x}(t),\tilde{p}(t))$. A
solution of the TDGCM equation follows one of the contour lines, e.g.,
cyclically on the red dashed line in the case of the calculation of Ref. [1].
The momentum centers $p_{a}(t)$ are changing quite a lot, of the order of 100
MeV/$c$, already in this free propagation without any effect of the potential,
which is, of course, counter-intuitive since one would naturally expect that
the momentum and the energy should be conserved for the free propagation. I do
not know whether HHT have accepted this fact of the drastic change of
$p_{a}(t)$ during the free propagation.

Such non-trivial motions of the wave packet centers may be considered as due
to some entanglement between different trajectories, e.g.,
$(x_{1}(t),p_{1}(t))$ and $(x_{2}(t),p_{2}(t))$, as mentioned by HHT in the
third paragraph of Ref. [3]. They think, “the idea of the time-dependent
generator coordinate method (1) is nothing more than that of the entangled
trajectory molecular dynamics (ETMD) developed in quantum chemistry [4, 5].”
In my opinion, this analogy to ETMD needs to be taken with cautions because
there are some differences between TDGCM and ETMD. In ETMD, the test particles
are used as elements to represent the phase space distribution $f(x,p,t)$.
Each test particle is localized in a small region in the phase space, and
$f(x,p,t)$ is expressed as a simple summation of the positive contributions
from test particles. On the other hand, in TDGCM, each wave packet occupies a
relatively broad region in the phase space determined by the minimum
uncertainty $\Delta x\Delta p=\frac{1}{2}\hbar$, and the contribution to
$f(x,p,t)$ is a complicated summation of complex values due to interference
between wave packets. For example, see Eq. (12) of Ref. [2] for TDGCM, which
is in stark contrast to Eq. (5) of Ref. [4] for ETMD. Furthermore, the
entanglements are not from the same origin, e.g., as we can see that free
propagation is described in ETMD naturally with no entanglement occurring,
while the wave packets are non-trivially entangled in TDGCM already for free
propagation.

Nevertheless, HHT argues [3], “In ETMD, a quantum tunneling is simulated with
many classical trajectories by making them entangled during the time
evolution. This corresponds to taking a linear superposition of many Slater
determinants in the time-dependent generator coordinate method. Such
entanglement is a key to recover the quantum mechanical behavior of time-
dependent dynamics.” However, if we consider the cautions mentioned above, the
partial success of ETMD for quantum tunneling does not guarantee an automatic
success of TDGCM for the same problem. As already commented in Ref. [2], the
results of TDGCM have to be carefully analyzed, e.g., similarly to what has
been done for ETMD or as suggested by Ref. [2]. Nothing has been reported on
such tests by HHT [3].

It is still instructive to see what is causing the counter-intuitive
entanglement in TDGCM for free propagation, which was already argued in Ref.
[2]. The phase-space distribution function (Husimi function) was displayed in
Fig. 4, for the TDGCM states corresponding to eight points selected on the red
dashed contour line in Fig. 4. We can see in Fig. 4 that the evolution from
the panel (b) to (d) is quite similar to the exact time evolution displayed in
Fig. 4 from $t=-5$ to 5 fm/$c$, i.e., the TDGCM with two wave packets provides
an accurate description for this short time interval. The time evolution of
the phase space distribution can be easily understood based on the free
propagation of each phase space element. Being induced by the momentum
distribution, a correlation develops between the position and the momentum in
the phase space. In Fig. 4, the Gaussian centers are indicated by open
circles. From the state in panel (c) to (d), the momentum centers change a lot
from $p_{a}=0$ to $p_{a}\sim\pm 150$ MeV/$c$. Such a change of momenta during
a short time of 5 fm/$c$ may be counter-intuitive as a free propagation, but
it provides the natural time evolution of the phase space distribution and
therefore the wave function $\psi(x,t)$. Thus, the trajectories of Gaussian
centers may be interpreted as entangled by struggling to reproduce the free
propagation, not of the Gaussian centers themselves but of the phase space
distribution and $\psi(x,t)$, after complicated interference between them.
This is also an example to show that the momentum distribution in $\psi(x,t)$
plays an important role in the time evolution in TDGCM.

It is noteworthy that the phase space distribution is not a simple summation
of two Gaussian functions in the state of panel (d), for example. While the
total distribution is spreading in a tilted direction in the phase space, it
is shrinking in the other direction, compared to the original width at the
state of (c). Such a shrinking can be achieved only through interference. The
same is true, e.g., for the minimum energy configuration in Fig. 4 shown by
the cross symbols, at which the momentum width is considerably narrower than
the original width $\Delta p=\hbar\sqrt{\nu_{r}}$ of the Gaussian wave packet
[2].

Thus, the effort of superposing wave packets is spent already for the free
propagation of the incoming state when TDGCM is employed for a collision
problem as in Ref. [1]. This is in contrast to ETMD in which entanglements
occur to describe purely quantum effects. Furthermore, in TDGCM, the
description of free propagation is good only during a limited time interval,
and the time evolution is pathological at most of the times as pointed out by
Ref. [2], when only a few wave packets are superposed.

##  4 Is TDGCM simulating a fixed beam energy?

For the TDGCM solution calculated by HHT in Ref. [1], it seems they assume
that the simulation corresponded to an almost fixed beam energy because
$E_{1}(t_{0})=p_{1}^{2}(t_{0})/2\mu$ and $E_{2}(t_{0})=p_{2}^{2}(t_{0})/2\mu$
were chosen to be almost identical at the initial time $t_{0}=0$. Here $\mu$
is the (reduced) mass and that energy was denoted by $E(t=0)\approx
E_{1}(0)\approx E_{2}(0)$ in Ref. [2]. This picture based only on the momentum
centers $p_{a}(t)$ is called View B [2]. Roughly speaking, View B ignores the
momentum width inherent in the individual wave packets in estimating the
energy. Since the energy $E(t=0)=0.11$ MeV was chosen below the barrier height
$V_{\text{B}}=0.13$ MeV, any transmission over the barrier was regarded as
quantum tunneling in Ref. [1].

This interpretation of the simulation was criticized by Ono in Ref. [2], as
quoted here: “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 however not correct. 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 $E(t=0)$ regarded 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.”

The fourth paragraph of Ref. [3] by HHT might be a counterargument to this
criticism. However, none of the points in Ono’s criticism is rebutted by HHT.
If they were to persist in View B to regard $E(t=0)$ as the beam energy and to
justify their result [1] as an indication of quantum tunneling, they needed to
have given counterarguments to individual points in Ono’s criticism quoted
above. This was not done in Ref. [3].

Instead, HTT [3] assert their opinion as “In our opinion, the energy spreading
inherent in each Gaussian wave function should not fully be included in
estimating the total energy spreading, as it does not contribute to the
dynamics when a single wave packet is involved. That is, one should make a
clear distinction between the energy spreading inherent in single-particle
wave functions and that due to a distribution of initial conditions. As the
former does not contribute to the dynamics when a single wave packet is
considered, it is not trivial to estimate appropriately the energy spreading
of initial wave packet for the TDGCM when a Gaussian function with a fixed
width is employed.” Although this statement by HHT is a little vague because
of the usage of “not fully” and “not trivial,” their opinion seems to adopt
View B to regard the distribution of $E_{a}(t_{0})=p_{a}^{2}(t_{0})/2\mu$ as
that of the beam energy. As mentioned by Ono [2], when the model uses only a
single wave packet, we know many cases in which View B works even though it is
not fully consistent with quantum mechanics. HHT misunderstand that the same
is true for TDGCM, by which they draw an incorrect conclusion. As already
demonstrated in previous sections, we have to consider the effect of the
interference between the superposed wave packets, which makes it meaningless
to divide the distribution into a frozen part and the rest. Also, as quoted
above from Ref. [2], their opinion indeed includes serious inconsistencies as
a practical problem.

In some sense, it is true that the momentum width is reflected only partially
in the dynamics when only two Gaussian wave packets are superposed in the
setup of Ref. [1]. This is because of the pathological evolution observed in
Fig. 4, and therefore shows a defect of the model rather than its ability to
predict barrier transmission in a controlled way. The argument in Ref. [2] is
quoted here: “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. 4). 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].”

##  5 Can energy conservation be violated?

Quantum tunneling is a phenomenon that cannot occur in the corresponding
classical system under the condition of energy conservation. It is, therefore,
hardly possible to argue quantum tunneling if the model does not have well-
defined conserved energy. As pointed out by Ono [2], when barrier transmission
occurred, the observed energy in the transmission (and also reflection)
channel was significantly different from the beam energy $E(t=0)$ in the TDGCM
solution of Ref. [1]. Although it may depend on the source of the problem, we
should not regard any model with such energy non-conservation as qualified for
studying quantum tunneling.

Now HHT argue in the fifth paragraph of Ref. [3]. For the interpretation about
the seriousness of the energy non-conservation mentioned above, they stated,
“a care must be taken in this interpretation, due to the well known spurious
cross-channel correlation in the time-dependent Hartree-Fock theory. This
problem may not be cured completely if the number of Slater determinants
superposed in the wave function is small.” I agree with this understanding for
TDHF; this is indeed why TDHF cannot describe quantum tunneling. The problem
can be considered as the unphysical nonlinearity caused by the limitation of
the wave function. The problem in TDHF is usually argued in many-body systems,
while TDGCM with a limited number of Gaussian wave packets suffers from this
problem already in one-particle systems. HHT seem to agree now that TDGCM does
not conserve energy due to a well-known fundamental defect of this kind of
model, and therefore it is not qualified yet as a model for quantum tunneling.

It is difficult to grasp the final sentence of HHT, “As in ETMD, only the
total energy has a clear physical meaning in considering quantum tunneling,
rather than the energy of each ‘test particle’.” In the present context, the
only possible interpretation of the “total energy” is the sum of the kinetic
and potential energies, since the Hamiltonian of this one-particle system has
only these two terms. The same word is used for this meaning, e.g., in some
papers of ETMD [5, 6, 7], where each test particle has its total energy, in
contrast to HHT’s sentence. The total energy of each test particle does not
need to be conserved, which, however, does not have a direct physical meaning.
The test particles represent the phase-space distribution function that
presumably corresponds to a quantum state. Such a quantum state treated in
phase-space approaches is usually not an energy eigenstate, which we should
interpret as such. In the asymptotic region where the potential is a constant,
it should be straightforward, in principle, to extract the distribution of the
energy from test particles in each channel.

In case the “total energy” means the energy averaged over all channels, i.e.,
the energy expectation value for $\psi(x,t)$, the statement by HHT implies
that ETMD is also not qualified, but we need to check the literature. In Refs.
[6, 7], repaying the borrowed energy by a transmitted test particle is argued,
with which one may expect that the relation of the energies of different
channels could be suitably described by ETMD. Some practical tests have been
performed for ETMD in the literature, but it does not guarantee an automatic
success of TDGCM. As mentioned before and in Ref. [2], similar and other tests
are necessary for TDGCM to claim quantum tunneling.

##  6 Summary

Individual comments from HHT [3] were examined. Since HHT did not consider the
main point of Ono’s paper [2], necessary reviews were also made here.
Probably, the confusion in HHT’s arguments can be mostly resolved if they
understand how the Gaussian wave packets interfere with each other. After all,
there is nothing to change in Ono’s paper [2], including the criticisms to the
conclusion of HHT’s paper [1].

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP17K05432 and
JP21K03528.

## References

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