# Layered Complex Networks as Fluctuation Amplifiers

Melvyn Tyloo  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545 USA.

###### Abstract

In complex networked systems theory, an important question is how to evaluate
the system robustness to external perturbations. With this task in mind, I
investigate the propagation of noise in multi-layer networked systems. I find
that, for a two layer network, noise originally injected in one layer can be
strongly amplified in the other layer, depending on how well-connected are the
complex networks in each layer and on how much the eigenmodes of their
Laplacian matrices overlap. These results allow to predict potentially harmful
conditions for the system and its sub-networks, where the level of
fluctuations is important, and how to avoid them. The analytical results are
illustrated numerically on various synthetic networks.

Introduction.– Complex-networked systems are widely present in physical and
man-made applications ranging from neurons in the brain to large-scale power
transmission networks Barabási (2016). These systems are made of individual
elements with their own internal dynamics, that are interacting together. The
interplay between the internal degrees of freedom and the coupling typically
gives birth to organized collective dynamics such as consensus or
synchronization Strogatz (2004); Pikovsky  _et al._ (2003). The coupling
between the elements is usually modelled by complex networks. Recently a lot
of effort has been put in the refinement of this approach in order to account
for specific structures that might exist in the coupling Kurant and Thiran
(2006); De Domenico  _et al._ (2013); Bianconi (2015, 2018). Indeed, many
complex systems are interdependent or hierarchically influence each other Xiao
_et al._ (2021), as e.g. in the brain where various regions with different
functional networks interact with each other Pedersen  _et al._ (2018), or in
power networks where layers with different voltage levels are connected
Machowski  _et al._ (2008), or in social networks where people are part of
interacting communities Murase  _et al._ (2014). For such applications, the
extension to multi-layer networks as depicted in Fig. 1(a) is particularly
relevant. The latter are composed of many different layers of networks
interacting with one another. The nodes in the different layers might
represent the same individual element taking part in different coupled
dynamics, or distinct elements belonging to separate systems that somehow
interact together. Also, these systems are unavoidably subject to noise coming
from the interacting elements or from the environment van Kampen (1976). Such
noisy conditions might have different effects on the coupled system. Indeed,
depending on its amplitude and time-dependence, it might induce fluctuations
around a stationary state or even escapes from an initial basin of attraction
DeVille (2012); Hindes and Schwartz (2016); Tyloo  _et al._ (2019a). Both
outcomes are potentially harmful to the system as fluctuations might damage
some network components on the long run, and basin escapes can lead to large
system failures. Therefore, to prevent such events and in order to devise more
resilient interdependent complex systems Ronellenfitsch  _et al._ (2018);
Nagpal  _et al._ (2022), it is of particular interest to understand how noise
in one layer impact the dynamics of other layers.

In this letter, I consider two-layer systems where only one layer is subjected
to noise. Interestingly, I find that, depending on the smallest eigenvalues of
the sub-network Laplacian matrices and the overlap of their eigenmodes,
fluctuations can be significantly amplified or reduced in the second layer. I
show the latter analytically by comparing the second moment of the degrees of
freedom in each layer. I also illustrate the theory numerically on different
networks and extend the principle to networks with more than two layers.

Figure 1: (a) Illustration of a two-layer system. Each grey area corresponds
to a layer with its own coupling network. The two layers are interacting
through some coupling function represented by dotted lines. (b)-(d)
Trajectories $x_{i}$ and $y_{i}$ respectively in layer 1 and 2 for the same
network in both layers: (a) cycle networks of size $n=50$, (b) cycle networks
of size $n=100$, (d) Erdős-Rényi networks of size $n=50$ with edge probability
$p=0.15$. For readability, trajectories are only shown for a limited time
interval. Amplification of the fluctuations for cycle networks with (b)
$\lambda_{2}\cong 0.0158$ and (c) $\lambda_{2}\cong 0.004$ . (d) Reduction of
the fluctuations for Erdős-Rényi networks $\lambda_{2}\cong 2.87$ . For each
panel, the mean amplification factor is given by $\frac{\sum_{i}\langle
y_{i}^{2}\rangle}{\sum_{i}\langle x_{i}^{2}\rangle}=1209$ (b), $20181$ (c),
$0.02$ (d).

Two layers networked system.– I consider two layers of diffusively coupled
agents, $\\{x_{k}\\}$, $\\{y_{k}\\}$. Each layer has $n$ nodes and its own
undirected complex coupling network. The two layers are coupled together in a
directed way that is detailed later. The dynamics of the $2n$ agents is
governed by the following set of differential equations,

$\displaystyle\dot{x}_{i}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\mathbb{L}_{ij}^{(1)}\,x_{j}+\eta_{i}\quad
i=1,...n\,,$ (1) $\displaystyle\dot{y}_{i}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\mathbb{L}_{ij}^{(2)}\,y_{j}+f_{i}(\\{x_{k}\\},\\{y_{k}\\})\quad
i=1,...n\,,$ (2)

where $\\{x_{k}\\},\\{y_{k}\\}$ are the nodal degrees of freedom respectively
in layer 1 and 2, $\mathbb{L}_{ij}^{(l)}$ is the Laplacian matrix of the
undirected coupling network in the $l$-th layer, and $f_{i}$ is a coupling
function between the two layers. As set in Eq. (2), the inter-layer
interaction is directed, i.e. Eq. (1) does not depend on $\\{y_{k}\\}$ , while
Eq.(2) does depend on $\\{x_{k}\\}$. To investigate the propagation of noise
from the first to the second layer, only the first one is subjected to ambient
noise $\eta_{i}$ that is taken as white, i.e.
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\delta_{ij}\,\eta_{0}^{2}\,\delta(t-t^{\prime})$
rem (a). Without loss of generality, I take the initial conditions to be
$x_{i}(0)=0$ , $y_{i}(0)=0$ for $i=1,...n$ . Indeed, in the following I
consider the long time limit for the second moment which is independent of the
initial conditions. Importantly, Eqs. (1), (2) can represent the linear
approximation of a nonlinear system around a stable fixed point.

In the following, I denote $\lambda_{\alpha}^{(k)}$ , $u_{\alpha,i}^{(k)}$ the
eigenvalues and corresponding eigenvectors respectively, of the Laplacian
matrix $\mathbb{L}^{(k)}$ . As I consider connected undirected networks in
each layer, I have
$0=\lambda_{1}^{(k)}<\lambda_{2}^{(k)}\leq...\leq\lambda_{n}^{(k)}$ for
$k=1,2$ with the first eigenvector being constant i.e.
$\bm{u}_{1}^{(k)}=(1,...,1)/\sqrt{n}$ for $k=1,2$ . The second smallest
eigenvalue, namely $\lambda_{2}^{(k)}$ is usually called algebraic
connectivity of the network, with its corresponding eigenvector
${\bm{u}}_{2}^{(k)}$ called Fiedler mode.

Noise propagation.– First, one has to choose an inter-layer coupling function.
The simplest choice is
$f_{i}(\\{x_{k}\\},\\{y_{k}\\})=x_{i}-n^{-1}\sum_{j}x_{j}$ , i.e. the first
layer tunes the natural velocities of the second one, orthogonal to the zero
mode $\bm{u}_{1}^{(2)}$ rem (b). Using this coupling function together with a
modal decomposition over the eigenvectors of $\mathbb{L}^{(1)}$ and
$\mathbb{L}^{(2)}$ , one obtains as solution of Eqs. (1),(2) (see Tyloo  _et
al._ (2018, 2019b) for details),

$\displaystyle x_{i}(t)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}e^{-\lambda_{\alpha}^{(1)}t}\int_{0}^{t}e^{\lambda_{\alpha}^{(1)}t^{\prime}}\,\sum_{j}\eta_{j}\,u_{\alpha,j}^{(1)}\,{\mathrm{d}t^{\prime}}\,u_{\alpha,i}^{(1)}\,,$
(3) $\displaystyle y_{i}(t)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}e^{-\lambda_{\alpha}^{(2)}t}\int_{0}^{t}e^{\lambda_{\alpha}^{(2)}t^{\prime}}\,\sum_{j}x_{j}\,u_{\alpha,j}^{(2)}\,{\mathrm{d}t^{\prime}}\,u_{\alpha,i}^{(2)}\,.$
(4)

Therefore, the response of the first layer depends on the scalar product
between ${\bm{\eta}}$ and $\bm{u}_{j}^{(1)}$ , while in the second layer it is
between ${\bm{x}}$ and $\bm{u}_{j}^{(2)}$ . In order to investigate how noise
is transmitted from one layer to the other, one may calculate the second
moment of the degrees of freedom, namely $\\{x_{i}\\}$ and $\\{y_{i}\\}$.
Using Eq. (3) one has in the long time limit for the first layer Tyloo  _et
al._ (2019b),

$\displaystyle\langle x_{i}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\eta_{0}^{2}}{2}\sum_{\alpha}\frac{{u_{\alpha,i}^{(1)}}^{2}}{\lambda_{\alpha}^{(1)}}\,,$
(5)

which can be related to the inverse of the resistance centrality of the $i-$th
node Klein and Randić (1993); Tyloo  _et al._ (2019b). This means the highest
resistance centrality, the least fluctuations. In the model considered here,
the $x_{i}$’s modify the natural velocities in the second layer. In order to
calculate the second moment in the second layer, one needs the two-point time
shifted correlators of the $x_{i}$’s. Due to the coupling in the first layer,
the latter is then non-trivial and reads,

$\displaystyle\langle
x_{i}(t)x_{j}(t^{\prime})\rangle=\frac{\eta_{0}^{2}}{2}\sum_{\alpha}\frac{{u_{\alpha,i}^{(1)}}u_{\alpha,j}^{(1)}}{\lambda_{\alpha}^{(1)}}e^{-\lambda_{\alpha}^{(1)}|t-t^{\prime}|}.$
(6)

Therefore, for the second layer, the input is a time- and space-correlated
noise. For equal times, i.e. $t=t^{\prime}$ , one recovers the pseudo-inverse
of $\mathbb{L}^{(1)}$ Ren  _et al._ (2010). Then, plugging Eq. (6) into Eq.
(4) yields for the second moment in the second layer,

$\displaystyle\begin{split}\langle
y_{i}^{2}\rangle=\frac{\eta_{0}^{2}}{2}\sum_{\alpha,\beta,\gamma}\sum_{k,l}\frac{{u_{\gamma,k}^{(1)}}u_{\gamma,l}^{(1)}{u_{\alpha,k}^{(2)}}u_{\beta,l}^{(2)}[2\lambda_{\gamma}^{(1)}+\lambda_{\alpha}^{(2)}+\lambda_{\beta}^{(2)}]}{\lambda_{\gamma}^{(1)}(\lambda_{\alpha}^{(2)}+\lambda_{\beta}^{(2)})(\lambda_{\gamma}^{(1)}+\lambda_{\alpha}^{(2)})(\lambda_{\gamma}^{(1)}+\lambda_{\beta}^{(2)})}{u_{\alpha,i}^{(2)}}u_{\beta,i}^{(2)}\,.\end{split}$
(7)

The fluctuation of $y_{i}$ is thus a non-trivial function of the overlap
between $\bm{u}_{\gamma}^{(1)}$ and $\bm{u}_{\beta}^{(2)}$ and their
corresponding eigenvalues. Equation 7 is obtained for two different complex
networks in the two layers. It is easier to understand what is going on by
considering the specific case of similar layers, i.e. with the same complex
network. In this particular setting, one has
$\bm{u}_{\alpha}^{(1)}=\bm{u}_{\alpha}^{(2)}=\bm{u}_{\alpha}$,
$\lambda_{\alpha}^{(1)}=\lambda_{\alpha}^{(2)}=\lambda_{\alpha}$
$\forall\alpha$ . Then Eq. (7) simplifies to,

$\displaystyle\begin{split}\langle
y_{i}^{2}\rangle=\frac{\eta_{0}^{2}}{4}\sum_{\alpha}\frac{u_{\alpha,i}^{2}}{\lambda_{\alpha}^{3}}\,.\end{split}$
(8)

Now to see how the noisy first layer affects the second one, it is important
to compare Eq. (8) to Eq. (5). Indeed, while in Eq. (5), one has the first
power of $\lambda_{\alpha}$ at the denominator, in Eq. (8) it is the third
power of $\lambda_{\alpha}$ . This seemingly tiny difference might have
serious implications. In particular, the slowest modes of the coupling
network, i.e. the $\lambda_{\alpha}$’s that are the closest to zero, primarily
determine whether the noise is amplified or reduced in the second layer.
Therefore, one expects poorly connected networks with low algebraic
connectivity to be good candidates to amplify the fluctuations, while well-
connected networks might efficiently reduce the fluctuations.

Numerical Results.– First, I illustrate the case with two times the same
coupling network in both layers, i.e. $\mathbb{L}^{(1)}=\mathbb{L}^{(2)}$ .
Figure 1(b)-(d) show the trajectories of $x_{i}$’s and $y_{i}$’s respectively
in layer 1 and 2. Following Eq. (8), to obtain the amplification of the
fluctuations, I choose cycle networks which have an algebraic connectivity
that decreases with the size, namely $\lambda_{2}=2-2\cos(2\pi/n)$ . This is
shown in panel Fig. 1(b) where one clearly sees that the fluctuations in the
second layer (bottom panel) are strongly amplified, compared to the input
noise of the first layer (top panel). In the thermodynamic limit, the
amplification factor for such systems scales with the size as $\langle
y_{i}^{2}\rangle/\langle x_{i}^{2}\rangle\propto n^{4}$ . The latter is
observed by comparing panels (b) and (c) where the network size has been
doubled and further confirmed by the amplification factors given in the
caption of Fig. 1. To obtain the other counter effect, namely the reduction of
the fluctuations, one needs a network with larger algebraic connectivity.
Therefore, I simulate two layers of the same Erdős-Rényi network with edge
probability $p=0.15$ which has considerably larger $\lambda_{2}$ [see panel
(d)]. The second moment in the second layer is significantly reduced compared
to the first one.

Second, I consider the case of different networks in each layer. Here, going
back to Eq. (7), fluctuations in the second layer crucially depend on the
overlap between the Laplacian eigenvectors of the two layers. In particular,
if the eigenvectors do not overlap enough, only little fluctuations will
propagate in the second layer. To demonstrate that, I consider the overlap
between the Fiedler modes of each layer and the amplification factor for all
combinations of five coupling networks of size $n=50$ , namely, a single
cycle, two Watts-Strogatz (WS) networks with different rewiring probabilities
Watts and Strogatz (1998), a Barabási-Albert (BA) network with $m=4$ Barabási
(2016), and an Erdős-Rényi (ER) network Newman (2018). This is shown in Tab.
1. One observes a strong amplification for combinations with the cycle and WS
I. This is due to the small algebraic connectivity in each layers and the
overlap that is still significant (0.405) due to the low rewiring probability
for the WS I. One also notice a strong reduction of the fluctuations for
combinations of BA and ER networks which satisfy both
$\lambda_{2}^{(1)},\lambda_{2}^{(1)}>1$ and
$({\bm{u}}_{2}^{(1)}\cdot{\bm{u}}_{2}^{(2)})^{2}\ll 1$ . Interestingly, for
the networks investigated here, having BA or ER in the first layer only allows
reduction of the fluctuations in the second layer, while having a cycle, WS I
or WS II in the first layer can lead to both amplification or reduction,
depending on the type of network in the second layer.

$y_{i}$ $x_{i}$ |  Cycle $\lambda_{2}^{(1)}=0.0158$ |  WS I $\lambda_{2}^{(1)}=0.0156$ |  WS II $\lambda_{2}^{(1)}=0.273$ |  BA $\lambda_{2}^{(1)}=2.302$ |  ER $\lambda_{2}^{(1)}=3.02$
---|---|---|---|---|---
Cycle $\lambda_{2}^{(2)}=0.0158$ |  $1$ $819$ |  $0.405$ $744$ |  $0.286$ $23.856$ |  $0.131$ $0.905$ |  $2.8\times 10^{-4}$ $0.766$
WS I $\lambda_{2}^{(2)}=0.0156$ |  $0.405$ $725$ |  $1$ $673.51$ |  $2.5\times 10^{-5}$ $19.81$ |  $0.0197$ $0.881$ |  $0.0113$ $0.738$
WS II $\lambda_{2}^{(2)}=0.273$ |  $0.286$ $3.1$ |  $2.5\times 10^{-5}$ $2.673$ |  $1$ $1.346$ |  $0.126$ $0.0733$ |  $0.0123$ $0.0535$
BA $\lambda_{2}^{(2)}=2.302$ |  $0.131$ $0.048$ |  $0.0197$ $0.0504$ |  $0.126$ $0.0329$ |  $1$ $0.0372$ |  $8.5\times 10^{-4}$ $0.0169$
ER $\lambda_{2}^{(2)}=3.02$ |  $2.8\times 10^{-4}$ $0.0251$ |  $0.011$ $0.0253$ |  $0.0123$ $0.0173$ |  $8.5\times 10^{-4}$ $0.0123$ |  $1$ $0.0182$
Table 1: Comparison of the overlap (first row in each cell) between the
Fiedler modes in each layer, i.e.
$({\bm{u}}_{2}^{(1)}\cdot{\bm{u}}_{2}^{(2)})^{2}$ and the amplification factor
(second row), i.e. $\frac{\sum_{i}\langle y_{i}^{2}\rangle}{\sum_{i}\langle
x_{i}^{2}\rangle}$, for a single cycle; two Watts-Strogatz network with first-
and second-nearest neighbors coupling and rewiring probability
$p_{\mathrm{rewiring}}=0.05$ (WS I), $0.35$ (WS II); a Barabási-Albert network
with $m=4$ (BA); an Erdős-Rényi with edge probability $p_{\mathrm{edge}}=0.15$
(ER). All networks have size $n=50$ . Red cells: The overlap between
${\bm{u}}_{2}^{(1)}$ and ${\bm{u}}_{2}^{(2)}$ is small for the pair WS I/WS
II. However the sum of the overlaps between
$\\{{\bm{u}}_{2}^{(1)},{\bm{u}}_{3}^{(1)}\\}$ and
$\\{{\bm{u}}_{2}^{(2)},{\bm{u}}_{3}^{(2)}\\}$ is about $0.5$ thus explaining
the amplification observed.

More than two layers.– So far I considered two-layer systems. However, it is
interesting to go one step further and have more than two layers. For example,
one may add a third layer that is influenced by the second one exactly as the
latter is influenced by the first one. The dynamics of that third layer would
then read,

$\displaystyle\dot{z}_{i}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{n}\mathbb{L}_{ij}^{(3)}\,z_{j}+f_{i}(\\{y_{k}\\},\\{z_{k}\\})\quad
i=1,...n\,,$ (9)

where $f_{i}$ is the inter-layer coupling function defined above. Even though
it is possible to calculate analytically the second moment of the $z_{i}$’s,
here I only show numerical results. I illustrate two interesting effects in
the case of three layers. First, the amplification of the fluctuations
obtained in Fig. 1(b),(c) for the same network with low algebraic connectivity
in each layer becomes even stronger when a third layer of the same network is
added. Indeed, Fig. 2(a) shows the case of three layers of the same cycle
network with nearest neighbors. Fluctuations in the third layers are strongly
amplified compared to the first one. Significant amplification factors might
then be achieved simply by adding layers of the same network with low
algebraic connectivity. Second, amplification of the fluctuations in the third
layer is still possible even if they were reduced in the second one. The
latter is shown in Fig. 2(b) where the first and the third layer are cycle
networks with nearest neighbors and the second layer is an Erdős-Rényi network
with a higher algebraic connectivity. Due to the only marginal overlap between
$\bm{u}_{2}^{(1)}$ and $\bm{u}_{2}^{(2)}$ (see Tab.1), fluctuations are
reduced in the second layer. However, in the third layer, fluctuations are
still amplified compared to the first one. This phenomenon can be understood
as follows. As the second layer reacts faster than the first one, i.e.
$\lambda_{\alpha}^{(2)}\gg\lambda_{\alpha}^{(1)}$ , $\alpha\geq 2$ , it is
able to follow closely the slowest modes of the first layer and therefore, to
transmit them to the third layer. Then, as the third layer has the same
coupling network with low algebraic connectivity as the first layer,
fluctuations may still be amplified.

Figure 2: (a)-(b) Three-layer system trajectories $x_{i}$, $y_{i}$ and $z_{i}$
respectively in layer 1, 2 and 3. Amplification of the fluctuations obtained
for (a): the same cycle network of size $n=50$ in each of the three layers,
i.e. $\mathbb{L}^{(1)}=\mathbb{L}^{(2)}=\mathbb{L}^{(3)}$ ; (b): the same
cycle network of size $n=50$ in layers 1 and 3, i.e.
$\mathbb{L}^{(1)}=\mathbb{L}^{(3)}$ , and the same Erdős-Rényi network as in
Tab. 1 for layer 2.

More general coupling function.– The results obtained so far are for a simple
choice of coupling function given above Eq. (3). Here I briefly show that
similar effects are possible for a more general coupling function that reads,

$\displaystyle
f_{i}(\\{x_{k}\\},\\{y_{k}\\})=-(\mu{y}_{i}-\nu\overline{x}_{i})\,,$ (10)

where $\overline{x}_{i}=x_{i}-n^{-1}\sum_{j}x_{j}$ and $\mu,\nu\in\mathbb{R}$
. Following the same steps as for the previous coupling function one
eventually obtains for the second moment in the second layer,

$\displaystyle\begin{split}\langle
y_{i}^{2}\rangle=\frac{\nu^{2}\eta_{0}^{2}}{2}\sum_{\alpha,\beta,\gamma}\sum_{k,l}\frac{{u_{\gamma,k}^{(1)}}u_{\gamma,l}^{(1)}{u_{\alpha,k}^{(2)}}u_{\beta,l}^{(2)}[2\lambda_{\gamma}^{(1)}+\lambda_{\alpha}^{(2)}+\lambda_{\beta}^{(2)}+2\mu]}{\lambda_{\gamma}^{(1)}(\lambda_{\alpha}^{(2)}+\lambda_{\beta}^{(2)}+2\mu)(\lambda_{\gamma}^{(1)}+\lambda_{\alpha}^{(2)}+\mu)(\lambda_{\gamma}^{(1)}+\lambda_{\beta}^{(2)}+\mu)}{u_{\alpha,i}^{(2)}}u_{\beta,i}^{(2)}\,.\end{split}$
(11)

Then, assuming that the two layers have the same coupling network eventually
leads to,

$\displaystyle\begin{split}\langle
y_{i}^{2}\rangle=\frac{\nu^{2}\eta_{0}^{2}}{2}\sum_{\alpha}\frac{u_{\alpha,i}^{2}}{\lambda_{\alpha}(\lambda_{\alpha}+\mu)(2\lambda_{\alpha}+\mu)}\,.\end{split}$
(12)

First, for $\mu=0$, $\nu=1$ one correctly recovers Eq. (8). Second, if
$\nu=1$, $\mu>0$, the amplification effect obtained for the simpler coupling
function is still possible as long as $|2\lambda_{2}+\mu|<1$ . Third, quite
interestingly, for $\nu=-1$, $\mu<0$ and $|\lambda_{2}+\mu|<1$, i.e. repulsive
coupling, significant amplification might occur. Apart from these three cases,
one can ensure that no amplification happens by simply choosing
$\mu>1-\lambda_{2}$ . The more complicated case of two different networks is
left to further investigations.

Colored noise.– The amplification phenomena described in previous sections was
derived analytically for $\eta_{i}$’s being white-noises, i.e. uncorrelated in
time, and spatially independent. However, it is not limited to white-noise and
might be even more important for colored noise, i.e. correlated in time. For
example, one may take noisy sequences $\eta_{i}$’s such that
$\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\delta_{ij}\,\eta_{0}^{2}\,e^{-|t-t^{\prime}|/\tau_{0}}$
, where $\tau_{0}$ is the typical correlation time. The latter are standardly
obtained from an Orstein-Uhlenbeck process. Even though the calculations are
doable, the expressions are then complicated and not very insightful. I
therefore only discuss the limit of long correlation time, i.e.
$\lambda_{2}^{(1)}\tau_{0}$, $\lambda_{2}^{(2)}\tau_{0}\geq 1$ . In such a
situation the second moment in the first and second layers are given by,

$\displaystyle\langle x_{i}^{2}\rangle$ $\displaystyle=$
$\displaystyle{\eta_{0}^{2}}\sum_{\alpha}\frac{{u_{\alpha,i}^{(1)}}^{2}}{{\lambda_{\alpha}^{(1)}}^{2}}\,,$
(13) $\displaystyle\langle y_{i}^{2}\rangle$ $\displaystyle=$
$\displaystyle{\eta_{0}^{2}}\sum_{\alpha,\beta,\gamma}\sum_{k,l}\frac{{u_{\gamma,k}^{(1)}}u_{\gamma,l}^{(1)}{u_{\alpha,k}^{(2)}}u_{\beta,l}^{(2)}}{{\lambda_{\gamma}^{(1)}}^{2}\lambda_{\alpha}^{(2)}\lambda_{\beta}^{(2)}}{u_{\alpha,i}^{(2)}}u_{\beta,i}^{(2)}\,.$
(14)

Unlike for the white-noise case, here one clearly remarks what happens, even
for two different networks in the two layers. Given that the overlap between
$\bm{u}_{\gamma}^{(1)}$ and the slowest modes $\bm{u}_{\beta}^{(2)}$ is finite
and not too small, fluctuation can be strongly amplified by having a second
layer with $|\lambda_{\alpha}^{(2)}|\ll 1$ . Comparing Eqs. (13), (14) to Eqs.
(5), (7), one concludes that colored noise with a long correlation time leads
to stronger amplification or reduction of the fluctuations than white-noise.

Conclusion.– Using a modal decomposition of a multi-layer system, I showed
that noise originally injected in one layer might be significantly amplified
when transmitted to other connected layers. In particular, the amplification
strongly depends on the overlap between the eigenmodes of the network
Laplacian matrices in each layer and their corresponding eigenvalues. On the
one hand, when the overlap is finite and the algebraic connectivity low,
fluctuations are amplified in the second layer. On the other hand, if the
overlap is marginal or if the algebraic connectivity is high, then
fluctuations are likely to be reduced in the second layer. Moreover, important
amplification factors can be achieved in multi-layer systems by simply adding
layers with the same poorly connected network. In addition to that,
amplification of the fluctuations might still happen even when they are
reduced at intermediate layers. As shown above, fluctuation amplification
might be even more significant in case of colored noise with correlation times
longer than the intrinsic system’s time-scales. These results highlight how
inter-dependent systems might be vulnerable to noisy signals. Ways to correct
such vulnerabilities would be either to make sure that the algebraic
connectivity in each layer is large enough or that the eigenmodes of connected
layers do not overlap. The latter should be considered in future research. The
amplification effect described here might also be useful to develop inference
algorithms Xue and Bogdan (2019).

Interestingly, the results presented in this letter give indications about how
noise input should be correlated in space and time in order to possibly induce
larger or smaller fluctuations in single layer diffusively coupled systems.
Extension of this work should consider the latter fact into more details, as
well as different inter-layer coupling functions. Also, investigating moments
higher than the second Tyloo  _et al._ (2022) or including inertia for the
dynamical agents to potentially uncover resonances in case of time-dependent
inter-layer coupling Baumann  _et al._ (2020) represent two research avenues
of interest.

This work has been supported by the Laboratory Directed Research and
Development program of Los Alamos National Laboratory under project number
20220797PRD2.

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# The enhancement of pair production in oscillated overlapped fields

Adiljan Sawut  State Key Laboratory for Tunnel Engineering, China University
of Mining and Technology, Beijing 100083, China  Ying-Jun Li
<EMAIL_ADDRESS>State Key Laboratory for Tunnel Engineering, China
University of Mining and Technology, Beijing 100083, China  Miao Jiang  School
of Science, China University of Mining and Technology, Beijing 100083, China
Bai-Song Xie<EMAIL_ADDRESS>Key Laboratory of Beam Technology of the
Ministry of Education, and College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China  Institute of Radiation Technology,
Beijing Academy of Science and Technology, Beijing 100875, China

###### Abstract

The influence of potential well width on electron-positron pair production has
been examined through theoretical and numerical approaches by employing the
computational quantum field theory. Quantum interference effects in pair
production is investigated in the two overlapped potential wells with varied
widths and frequencies. Several dominant processes, involving the absorption
of an integer number of photons, significantly impact on pair production.
Notably, specific multiphoton absorption processes exhibit distinct changes as
the potential well width expands, with the absorption of four photons process
displaying noteworthy effects. Additionally, the influence of the smaller
frequency to the yield of the pair production can not be ignored and the most
optimized frequencies in our overlapped fields has been studied and exhibited.

##  I Introduction

The laser is the most powerful tool to study the strong field quantum
electrodynamics (QED) in laboratory frame Burke:1997ew  and had enormous
breakthrough since the invention of the Chirped Pulsed Amplification (CPA)
technics Strickland:1985gxr ; MacLeod:2022qid . Researchers were optimistic
Mourou:2006zz ; Chen:2002nd  to the development of the laser technology that
they had estimated the intensity of the laser would have a rapid growth and
approaches Schwinger critical field soonSchwinger:1951nm . However, the
current laser intensity is five to six orders of magnitude smaller, which is
$10^{23}\sim 10^{24}\rm{W/cm^{2}}$ ELI ; XCELS . Meanwhile, many theoretical
works have been performed, and one of them is the electron-positron pair
production that is the topic studied in this paper.

The study of electron-positron pair production Sauter:1931zz ; Affleck:1981bma
; Kim:2000un ; Dunne:2005sx  in intense electromagnetic fields is a rich area
of theoretical physics, where a variety of methods have been developed to
tackle this nonlinear QED process Su:2019nwr ; Gong:2018zqs . The Wentzel-
Kramers-Brillouin (WKB) approximation is a semi-classical approach that allows
for an analytical understanding of tunneling processes, which are essential
for describing pair production in strong fields DiPiazza:2004lsj ;
Dumlu:2010ua ; Dumlu:2011rr ; Strobel:2014tha . The Dirac-Heisenberg-Wigner
(DHW) formalism provides a phase-space description of quantum fields, offering
a comprehensive framework to study the real-time dynamics of pair production
Hebenstreit:2011wk ; Li:2015cea ; Li:2017qwd ; Olugh:2018seh ; Hu:2023pmz ;
Xie:2017xoj . The quantum Vlasov equation (QVE), a kinetic equation for the
distribution function of particles in phase space, has been adapted to include
QED effects, enabling the study of plasma oscillations and instabilities that
can occur in the presence of strong fields Kluger:1991ib ; Alkofer:2001ik ;
Hebenstreit:2009km ; Nuriman:2012hn ; Li:2014psw ; Kohlfurst:2013ura ;
Gong:2019sbw . Computational quantum field theory (CQFT) Braun:1999zza ;
Krekora:2004trv  has been very helpful in providing numerical solutions to
scenarios where analytical methods fall short, allowing for the exploration of
pair production in various field configurations Jiang:2013wha ;
Su2012MagneticCO ; Tang:2013qna ; Sawut:2021cjr ; Lv:2018sqm ; Su:2019jot ;
Zhou:2019euj . These theoretical tools each offer unique insights into the
complex nature of pair production, and their development and application
continue to enhance our understanding of quantum phenomena.

In this paper, we study the pair production process in two overlapped
oscillating laser fields with different frequencies and varied width of the
potential well by using the CQFT method. As the width between two fields gets
wider the interference effects can be observed and emerge a periodic variation
in both pair production probability and production rate. These phenomena can
be explained by the first-level perturbation theory conveniently Jiang:2013wha
. Multiphoton effects can be seen from the transition energy probability
distribution $U_{pn}$ picture specifically, each process has been influenced
by the width of the potential well and some processes play main role in this
scenario.

Our paper is organized as follows. In Sec. II, we give a brief introduction to
our theoretical framework that allow us to calculate the energy and momentum
distribution, number of created pairs in 1+1 dimensional fields. In Sec. III,
we provide the theoretical results and explanations of studying the different
features of pair production by considering the variable width of the potential
well. The Sec. IV is a summary for our work.

We use the atomic units ($\hbar=m_{e}=e=1$) throughout this paper.

##  II Theoretical framework and the background field

CQFT is a method that numerically solves the Dirac equation in all time and
space Wang:2020wyp ; Wu:2019weg ; Zhou:2021hko ; Zhou:2019euj . Thus, the
calculation begins with the Dirac equation,

$i\partial\hat{\psi}\left({z},t\right)/\partial{t}=[{c\alpha_{z}{\hat{p}}_{z}+\beta
c^{2}+V(z,t)}]\hat{\psi}\left(z,t\right),$ (1)

where $\alpha_{z}$ and $\beta$ denote the Dirac matrices, and we will focus on
a single spin due to the lack of magnetic field in our one-dimensional system.
As a results, the four-component spinor wave function becomes two components,
and the Dirac matrix $\alpha_{z}$ and $\beta$ are replaced by the Pauli
matrices $\sigma_{1}$ and $\sigma_{3}$ Tang:2013qna . The field operator
$\hat{\psi}\left({x},t\right)$ can be expanded in both time-depended and time-
independed complete bases as follows:

$\hat{\psi}\left({x},t\right)=\sum_{{p}}{\hat{b}}_{p}(t)u_{p}({x})+\sum_{n}\
{\hat{d}}_{n}^{\dagger}(t)u_{n}({x})$ (2)
$\displaystyle=\sum_{{p}}{\hat{b}}_{p}u_{p}({x,t})+\sum_{n}\
{\hat{d}}_{n}^{\dagger}u_{n}({x,t}),$ (3)

where $u_{p}({x})$ and $u_{n}({x})$ are the field-free energy eigenstates,
$u_{p}({x,t})$ and $u_{p}({x,t})$ are the time evolution of the $u_{p}({z})$
and $u_{n}({z})$ or can be explained as time-dependent wave function.
${\hat{b}}_{p}$ and ${\hat{d}}_{n}^{\dagger}$ are the electron annihilation
and the positron creation operators, the time-dependent creation and
annihilation operators can be written as follows:

${\hat{b}}_{p}(t)=\sum_{{p^{\prime}}}{{\hat{b}}_{p^{\prime}}\int
dxu_{p}^{*}({x})u_{p^{\prime}}\left({x},t\right)+\sum_{n^{\prime}}\
{\hat{d}}_{n^{\prime}}^{\dagger}\int
dxu_{p}^{*}({x})u_{n^{\prime}}\left({x},t\right),\ \ \ \ \ \ \ \ \ \ }$ (4)
${\hat{d}}_{n}^{\dagger}(t)=\sum_{{p^{\prime}}}{{\hat{b}}_{p^{\prime}}\int
dxu_{n}^{*}({x})u_{p^{\prime}}\left({x},t\right)+\sum_{n^{\prime}}\
{\hat{d}}_{n^{\prime}}^{\dagger}\int
dxu_{n}^{*}({x})u_{n^{\prime}}\left({x},t\right).\ \ \ \ \ \ \ \ \ \ }$ (5)

For the spatial distribution of electrons, it is defined by the positive
energy part of the field operator as
$\hat{\psi}_{e}\left({x},t\right)\equiv\sum_{{p}}{\hat{b}}_{p}(t)u_{p}({x})$,
and the spatial particle number density of created electrons can be written as

$\rho_{e}\left({x},t\right)=\langle\langle\text{vac}\mid\mid{\hat{\psi}}_{e}^{+{\dagger}}\left({x},t\right){\hat{\psi}}_{e}^{+}\left({x},t\right)\mid\mid\text{vac}\rangle\rangle$
(6)
$=\sum_{{n}}{\mid\sum_{{p}}{U_{pn}\left(t\right)u_{p}\left({x}\right)\mid^{2},\
}}$ (7)

where $U_{p,n}\left(t\right)=\langle u_{p}(x)\mid U(t)\mid u_{n}(x)\rangle$
and $U(t)=\hat{T}{\rm exp}(-i\int Hdt)$ represents the time evolution
operator. Energy density of the electrons are expressed as

$\rho({E_{p}},t)=\sum_{{n}}{\mid{U_{p,n}\left(t\right)\mid^{2},}}$ (8)

and Eq. (7) integral over the space to obtain the total number of created
electrons as follow:

$N(t)=\int
dx\rho\left({x},t\right)=\sum_{p,n}{\mid{U_{p,n}\left(t\right)\mid^{2}.}}$ (9)

Our theoretical frameworks are fully ready for the calculation to the next
step and insert the background field to study the pair production processes.

##  III Numerical results

In this work our field configuration for the potential well can be written as
follow:

$V(z)=\frac{V_{1}}{2}S_{1}(z){\rm
sin}({\omega}_{1}{t})+\frac{V_{2}}{2}S_{2}(z){\rm sin}({\omega}_{2}{t}),$ (10)

where $S_{1}(z)$ and $S_{2}(z)$ are the shape of our Sauter field and can be
expressed as $S_{1}(z)={\rm tanh}((x-D_{1}/2)/W_{1})+{\rm
tanh}((x+D_{1}/2)/W_{1})$ and $S_{2}(z)={\rm tanh}((x-D_{2}/2)/W_{2})+{\rm
tanh}((x+D_{2}/2)/W_{2})$, $V_{1}$ and $V_{2}$ are the depth of the each well,
$D_{1}$ and $D_{2}$ are the widths of the potential wells, $\omega_{1}$ and
$\omega_{2}$ are the frequencies of the fields, $W_{1}$ and $W_{2}$ are the
widths of each electric field and we will use $W$ for both of them due to
$W_{1}=W_{2}$. The shape of the potential well can be visualized as Fig. 1.

It is shown in Eq. (10) that the two colored fields will act on the whole
process to trigger pair production via multiphoton effect. We set the
correlated parameters as, $V_{1}$=$V_{2}=2c^{2}$, $\omega_{1}=0.7c^{2}$ and
$\omega_{2}=1.7c^{2}$, $D_{1}$ and $D_{2}$ can be varied from $1/c$ to $20/c$
and $W=2/c$, where $c$ is the speed of light ($c=137.036$ a.u).

Figure 1: Shape of the two overlapped potential wells where $D_{1}$ and
$D_{2}$ are for the widths of the potential well, $W_{1}$ and $W_{2}$ are the
widths of the electric field, $V_{1}$ and $V_{2}$ are the height the well.

After numerically solving the Dirac equation by using CQFT method in
overlapped potential wells, we have shown the yield of the pair production in
time $T=0.003$ (a.u) by changing the width of the potential well in five
different sets. First set is where $D_{1}$ and $D_{2}$ vary from $1/c$ to
$20/c$ at the same time, as shown in Fig. 2(a) (black line). Second set is
$D_{2}$ varies from $1/c$ to $20/c$ and $D_{1}=20/c$ (blue line) and other
three sets are the variation of $D_{1}$ from $1/c$ to $20/c$ where $D_{2}$
takes different values, $D_{2}=20/c,10/c,5/c$, separately.

As shown in Fig. 2(a), for the first set, the yield of the pair production
increases linearly from $1/c$ to $9/c$ and starts to change periodically until
the last value of $D$, the periodic variation is much more noticeable where
the width $W$ gets smaller, see the reference Jiang:2013wha . Such a change in
the yield of creation with different potential width $D$ is a coherence
effect, which is due to the interference of the created particles from both
localized fields. The yields of pair production for second set(blue line) and
third set(red line) are indistinctive after $5/c$. However, we can be able to
discover there are slight dominant position for the third set in the yield of
pair production where $D_{2}=20/c$ and $D_{1}$ varies from $1/c$ to $20/c$. It
can be explained by the dominant frequency of $\omega_{2}=1.7c^{2}$ still has
the advantage of creating pairs and the smaller frequency
$\omega_{1}=0.7c^{2}$ also plays an important role as a dynamically assisted
field. For the potential width with $D_{2}=10/c$ and $5/c$, they both increase
and reach the highest value at $D_{1}=D_{2}$, then start to decrease until
$D_{2}=20/c$. As for the production yield reaches the highest value at
$D_{1}=D_{2}$, the two oscillated field overlapped exactly one on the other
and increase the intensity of the combined field and starts to trigger the
creation of more pairs.

There is another interesting phenomenon that the yield of pair production has
the same value for $D_{1}=15/c,D_{2}=20/c$ and $D_{1}=15/c,D_{2}=10/c$, we can
easily observe the crossed value of blue line and green line from Fig. 2(a).
To explain this phenomenon we have drawn two isolines (i) and (ii) in Fig.
2(a) and discover the yields of pair production are exactly same for
$D_{1}=5/c,D_{2}=10/c$ and $D_{1}=15/c,D_{2}=10/c$ in green line,
$D_{1}=15/c,D_{2}=20/c$ in red line(same value with blue line) and
$D_{1}=10/c,D_{2}=5/c$ for purple line. Our assumption is the yield of pair
production are the same for different overlapped fields when the absolute
value of $|D_{1}-D_{2}|\geq 5/c$. So we have demostrated the isoline (ii) to
emphasize the validity of our assumption. As for isoline (ii), the yields of
pair production are in same values for $D_{1}=3/c,D_{2}=10/c$ and
$D_{1}=17/c,D_{2}=10/c$ in green line, $D_{1}=12/c,D_{2}=5/c$ in purple line
and $D_{1}=13/c,D_{2}=20/c$ in red line(same value with blue line), where
$|D_{1}-D_{2}|=7/c$.

Figure 2: Yield of the pair production with different $D_{1}$ and $D_{2}$
sets, for (a) $D_{1}=D_{2}$ varies from $1/c$ to $20/c$ (black line),
$D_{1}=20/c$ and $D_{2}$ varies from $1/c$ to $20/c$ (blue line), other three
sets $D_{2}=20/c,10/c,5/c$ separately, and $D_{1}$ varies from $1/c$ to
$20/c$. For (b) exactly the same result for black line in Fig. 2(a) for
highlighting. Other parameters used in the simulations are $t=0.003$,
$\omega_{1}=0.7c^{2}$ and $\omega_{2}=1.7c^{2}$, $V_{1}=V_{2}=2c^{2}$ and
$W=2/c$.

As for the periodic changes in the yield, we have investigated four different
widths of potential well in Fig. 2(b), namely
$D_{1}=D_{2}=11/c,13/c,15/c,18/c$ and present the corresponding transition
energy distribution probability $U_{pn}$ in Fig. 3. The horizontal and
vertical axis represent the positive and negative energy states, respectively.
Each intersection point of the axis corresponds to a specific negative energy
state, indicating the probability of transition to a certain positive energy
state. The color at each point represents the probability of transition when
the absorbed energy is the sum of the horizontal and vertical axis,
$E_{p}+E_{n}$. As for Fig. 3(a), it is the $U_{pn}$ contour graph for
$D_{1}=D_{2}=11/c$, (b) for $D_{1}=D_{2}=13/c$, (c) for $D_{1}=D_{2}=15/c$ and
(d) for $D_{1}=D_{2}=18/c$. The difference between those four figures is
observable.

Figure 3: Contour plot of probability distribution of created pairs transition
energy in oscillating fields with $\omega_{1}=0.7c^{2},\omega_{2}=1.7c^{2}$
,$V_{1}=V_{2}=2c^{2}$ and the width $W=2/c$. From (a) to (d), corresponding
width of the potential well are $D=11/c,13/c,15/c,18/c$. Other parameters are
same with the Fig. 2.

To provide a more precise explanation, we use Fig. 3(a) to highlight the main
processes in red, which are separately corresponded to the processes
$\omega_{1}+\omega_{2}=2.4c^{2},2\omega_{1}+\omega_{2}=3.1c^{2},3\omega_{1}+\omega_{2}=3.8c^{2},4\omega_{1}+\omega_{2}=4.5c^{2}$
from left to the right. The most dominant process in here is
$\omega_{1}+\omega_{2}$ due to the total energy of two photons are larger than
the threshold energy the pair creation,
$\omega_{1}+\omega_{2}=2.4c^{2}>2c^{2}$, pairs are created continuously at a
constant rate. Obviously, the probability distribution of this process in Fig.
3(a) is smaller than the distribution in Fig. 3(b). For the second major
process which is $2\omega_{1}+\omega_{2}=3.1c^{2}$, has indistinctive changes
from Fig. 3(a) to Fig. 3(b). However, for the third process
$3\omega_{1}+\omega_{2}=3.8c^{2}$, only in Fig. 3(b) and Fig. 3(d) is more
noticeable and these exactly correspond to the higher creation yield in Fig. 2
with the width of $D=13/c$ and $D=18/c$. The fourth process
$4\omega_{1}+\omega_{2}=4.5c^{2}$ has the lowest probability distribution and
hardly noticeable in Fig. 3(a) and Fig. 3(c). Although each processes are
influenced by the different potential width, but the third process
$3\omega_{1}+\omega_{2}=3.8c^{2}$ has more distinctive variation in the yield
of pair production.

Another interesting phenomenon is that the probability distribution of
transition energy has started to disperse into many small pieces as the
potential width gets larger, see the first and second major process
$\omega_{1}+\omega_{2}=2.4c^{2}$, $2\omega_{1}+\omega_{2}=3.1c^{2}$. For
example, the second major process changes from 6 small areas in Fig. 3(a) to 9
small areas in Fig. 3(d).

Figure 4: Momentum spectrum of the created pairs at $t=0.003$ with parameters
of $V_{1}=V_{2}=2c^{2}$, $\omega_{1}=0.7c^{2}$ and $\omega_{2}=1.7c^{2}$,
$W=2/c$, for $D_{1}=D_{2}=11/c,13/c,15/c,18/c$. The given numbers are the peak
values for $D_{1}=D_{2}=18/c$.

For the further demonstration of periodically variation in the yield, we also
share the results for the momentum spectrum for different potential well
widths in Fig. 4. For this four different sets, we are aware of that
$D_{1}=D_{2}=13/c$ and $D_{1}=D_{2}=18/c$ have the higher value of pair
production yield than $D_{1}=D_{2}=11/c$ and $D_{1}=D_{2}=15/c$ from the Fig.
2(b). The momentum spectrum for different $D$ likely have no distinctive
advantages, however, we can observe very slight optimization for
$D_{1}=D_{2}=13/c$ (red line) and $D_{1}=D_{2}=18/c$ (green line) in Fig. 4
around lowest momentum range.

As we have discussed above in detail, periodically variation in the yield and
the dispersion of the probability distribution of the transition energy are
the most interesting phenomena. Those can be explained by deriving the first-
order perturbation theory, as in reference Jiang:2013wha . The first-order
transition amplitude from negative state $\mid{n}\rangle$ to the positive
state $\mid{p}\rangle$ at arbitrary time $t$ is given by:

$C^{(1)}_{pn}=\frac{1}{i}\int_{0}^{t}\langle p\mid H^{\prime}\mid n\rangle
e^{i(E_{p}-E_{n})\tau}d\tau.$ (11)

After we insert the interaction part of the Hamiltonian, we obtain the
expression:

$C^{(1)}_{pn}=\pi WV_{0}\frac{A_{pn}}{2}{\rm csch}[\pi
W\frac{(p+n)}{2}]\cos[\frac{(p+n)D}{2}]f(t),$ (12)

where $f(t)$ is a time depended function related to the field frequency
$\omega$ and due to our study object is width $D$, the $f(t)$ is a time-
depended part which is unnecessary to be written specifically. The inner
product $A_{pn}$ can be defined as:

$A_{pn}=\frac{{\rm sgn}(n)\sqrt{E_{p}+c^{2}}\sqrt{-E_{n}-c^{2}}+{\rm
sgn}(p)\sqrt{E_{p}-c^{2}}\sqrt{-E_{n}+c^{2}}}{4\pi\sqrt{-E_{p}E_{n}}}.$ (13)

As shown in Eq. (12), periodically changes for the probability distribution of
the transition energy comes from the $\cos[(p+n)D/2]$ part. By summing over
all the states of $p$ and $n$, we can obtain the first-order perturbation
estimation of the total pair production as:

$N^{(1)}=\sum_{pn}\mid{C^{(1)}_{pn}}\mid^{2},$ (14)

the potential width $D$ related term in last expression $\cos^{2}[(p+n)D/2]$
leads to the periodical variation in yield of pair production with the changes
of the potential width $D$, the interference effect is also more pronounced.
Using equation $\cos^{2}[(p+n)D/2]$ in perturbation theory we can calculate
the oscillation periods for yields with momentum $p$ in Fig. 4 for different
width of potential well. As for $D_{1}=D_{2}=15/c$, we can easily calculate
the oscillation period as $\Delta p=28.69$ and
$p=\sqrt{({E_{p}}^{2}-c^{4})/c^{2}}=90.9$, which is the highest value of the
momentum. We can list the other peak values for momentum perturbatively and
compare the results with the peak values in Fig. 4, there has an acceptable
margin of error, for example, the momentum we have calculated $p=90.9$ and the
highest peak value $p=94.25$ shown in Fig. 4, the peak value of $p=118.68$ and
the $p+\Delta p=90.9+28.69=119.59$.

As we have mentioned above, there is a slight advance in the yield for red
line in Fig. 2 where $D_{1}$ keeps varying and $D_{2}=20/c$ than the blue line
where $D_{1}=20/c$ and $D_{2}$ varies. Our assumption is the yield of pair
production is also sensitive to the small frequency as a dynamically assisted
to the electron-positron production process.

Figure 5: The total pair production as a function of time in a.u., (a) for
$\omega_{1}$ decreases from $0.6c^{2}$ to $0.2c^{2}$ while the $\omega_{2}$
increases from $1.7c^{2}$ to $2.2c^{2}$, but the sum of the two frequencies
keep constant where $\omega_{1}+\omega_{2}=2.4c^{2}$. (b) for
$\omega_{1}=0.2c^{2}$ and $\omega_{2}$ varies from $1.9c^{2}$ to $2.3c^{2}$.
With the potential well width of $D_{1}=D_{2}=20/c$, and other parameters are
same with the Fig. 4.

To observe this assumption, we have investigated the yield of the pair
production with different frequencies $\omega_{1}$ and $\omega_{2}$, but the
sum of them keep constant $\omega_{1}+\omega_{2}=2.4c^{2}$ in Fig. 5(a). As a
result, the set $\omega_{1}=0.2c^{2},\omega_{2}=2.2c^{2}$ has the highest pair
production yield. To explain the sensitive of small frequency to the
production rate, we have also demonstrated the yield for $\omega_{1}=0.2c^{2}$
and $\omega_{2}$ has been varied from $1.9c^{2}$ to $2.3c^{2}$ in Fig. 5(b).

As shown in Fig. 5(a), the yield of pair production increases as the
$\omega_{1}$ starts to decrease, but the same time $\omega_{2}$ increase due
to the sum of the two frequencies keep constant in the overlapped fields. The
influence of higher frequency $\omega_{2}$ is vital, but the role of smaller
frequency $\omega_{1}$ can not be ignored. In Fig. 5(b), we set the
$\omega_{1}=0.2c^{2}$ and varies the $\omega_{2}$ from $1.9c^{2}$ to
$2.3c^{2}$. Clearly, the $\omega_{1}=0.2c^{2},\omega_{2}=2.2c^{2}$ set has the
highest pair production yield even higher than the
$\omega_{1}=0.2c^{2},\omega_{2}=2.3c^{2}$, see Fig. 5(b). Generally,
$\omega_{1}=0.2c^{2}$ and $\omega_{2}=2.2c^{2}$ are the best fit for frequency
selection to create highest yield of pair production in our overlapped fields
due to the lowest frequency $\omega_{1}$ plays an important role as a
dynamically assisted field.

Figure 6: Contour plot of probability distribution of created pairs transition
energy corresponding to the Fig. 5. From (a) to (e), the frequency
$\omega_{1}$ varies from $0.6c^{2}$ to $0.2c^{2}$ and $\omega_{2}$ varies from
$1.8c^{2}$ to $2.2c^{2}$, the sum of them keep constant
$\omega_{1}+\omega_{2}=2.4c^{2}$, but (f) is for $\omega_{1}=0.2c^{2}$ and
$\omega_{2}=2.3c^{2}$. Other parameters are same with the Fig. 2.

In order to get the result for further study, we have presented the
corresponding transition energy distribution probabilities $U_{pn}$ for those
sets shown in Fig. 5. We can be able to differ the variation of the pair
production processes through $U_{pn}$ conveniently, see Fig. 6.

Now we can compare these two figures Fig. 5(a) and Fig. 6 to get more precise
information that the yield of pair production in different frequencies can be
affected by some processes and leads to significant enhancement. There have
emerged new processes to create more electron-positron pairs as the
$\omega_{1}$ decreases from $0.6c^{2}$ to $0.2c^{2}$ and $\omega_{2}$
increases from $1.8c^{2}$ to $2.2c^{2}$ at the same time in order to keep
$\omega_{1}+\omega_{2}$ constant as $2.4c^{2}$. Obviously, this results are
conform to the yield of the pair production which is also increased exactly in
same condition. In Fig. 6(a) where $\omega_{1}=0.6c^{2},\omega_{2}=1.8c^{2}$
and (b) where $\omega_{1}=0.5c^{2},\omega_{2}=1.9c^{2}$, there are
respectively three
($\omega_{1}+\omega_{2},2\omega_{1}+\omega_{2},3\omega_{1}+\omega_{2}$) and
four
($\omega_{1}+\omega_{2},2\omega_{1}+\omega_{2},3\omega_{1}+\omega_{2},4\omega_{1}+\omega_{2}$)
main processes are responsible for the creation of pairs. However, there have
emerged many new processes after $\omega_{1}=0.4c^{2},\omega_{2}=2.0c^{2}$
shown in Fig. 6(c) and more than four-photon processes have started to be very
clear to be observed, even have become the dominant processes, see the Fig.
6(d)(e)(f).

As for the reason why the set of $\omega_{1}=0.2c^{2}$ and
$\omega_{2}=2.2c^{2}$ has the higher pair production yield than the set of
$\omega_{1}=0.2c^{2}$ and $\omega_{2}=2.3c^{2}$, we have observed that there
is slight optimization for $\omega_{1}=0.2c^{2}$ and $\omega_{2}=2.2c^{2}$ in
$\omega_{1}+\omega_{2},2\omega_{1}+\omega_{2},3\omega_{1}+\omega_{2}$
processes after comparing the $U_{pn}$ result in Fig. 6(e) and (f).

##  IV Summary

The details regarding the influence of potential well width on electron-
positron pair production have been thoroughly examined in this study.
Analytical and mathematical approaches within the CQFT framework were employed
to obtain numerical solutions to the Dirac equation across all time and space.
The investigation delves into variations in production yield during the
multiphoton pair production process, with a specific focus on dominant
processes that involve the absorption of an integer number of photons.

Notably, distinct changes in specific multiphoton absorption processes were
observed as the potential well width expanded, with particular attention given
to the $3\omega_{1}+\omega_{2}$ process, which exhibited noteworthy effects.
The yield of pair production has increased linearly as the $D_{1}=D_{2}$
increase to $9/c$ and starts to varied periodically. We have explained this
phenomenon by solving the first level perturbative expression. The highest
yield of pair production with condition $D_{1}=D_{2}$ has been explained by
the exact overlapped of the two oscillated fields. As the width of potential
well increase, the yield of pair production reaches the highest value at
$D_{1}=D_{2}$ and they have same value of yield when $|D_{1}-D_{2}|\geq 5/c$
for different $D_{1}$ and $D_{2}$ sets.

Also, there is slight optimization in the yield of pair production for the
smaller frequency $\omega_{1}$ when $D_{1}$ and $D_{2}$ change separately in
Fig. 2(a) (red line and blue line). We have investigated the pair production
yield with different set of frequencies $\omega_{1}$ and $\omega_{2}$ in order
to study the influence of the smaller frequency. In result, the higher
frequency is not only the main reason for the enhancement of the pair
production, but the lower frequency of the oscillated field is also playing an
important role in pair production process. The set $\omega_{1}=0.2c^{2}$ and
$\omega_{2}=2.2c^{2}$ has the highest pair production yield comparing to the
other sets due to the effective of small number photons absorption in
electron-positron pair production process.

##  V Acknowledgments

Some helpful discussions with A. Orkash and C.K. Li are acknowledged. This
work was supported by the National Natural Science Foundation of China(NSFC)
under Grants No. 11974419, No. 12375240, No.11935008 and the Strategic
Priority Research Program of Chinese Academy of Sciences under Grant No.
XDA25051000.

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