# Spatial distribution of electric field of equal probability quantum walks
based on three-level quantum system

Xiaoguang Chen  Department of communications science and engineering, School
of information science and engineering, Fudan University, Shanghai 200433,
China<EMAIL_ADDRESS>Department of communications science and
engineering, School of information science and engineering, Fudan University,
Shanghai 200433, China

###### Abstract

Based on the three-level quantum system, when it is in resonance, according to
any two lattice points closest to Hamiltonian coupling, electrons transition
from high energy level to low energy level and release photons; Or absorb
photons and transition from low energy level to high energy level, thus
obtaining the physical process of quantum walking along a straight line under
the condition of equal probability. Then, the optical radiation in the quantum
walk is mapped into a Gaussian pulse of the electric field, and the Maxwell’s
equation is solved by the three-dimensional finite-difference time-domain
method to obtain the spatial electric distributio. Finally, the physical
process of quantum walking on two parallel lines is further discussed,
involving some physical properties such as electromagnetic coupling or
coherence, quantum state exchange and so on. The electric field coupling
between two lines can be calculated by FDTD, which provides a useful tool for
the design and analysis of quantum devices.

##  1 Introduction

Quantum walk is the counterpart of classical random walk in quantum mechanics.
Quantum walk first appeared in Feynman’s article on quantum mechanical
computer [1]. Today, it can be regarded as the earliest continuous quantum
walk [2] model. Later, the quantum random walk proposed by Aharonov et al.
[3]. can be regarded as the earliest discrete quantum walk model. Quantum walk
can also be divided into atomic[4, 5], ionic [6], and photonic[7]. When single
photons and entangled photons operate in the integrated circuit of the chip,
their behavior can be coupled into harmonic oscillator or waveguide
structures[8, 9], which may be placed on a chip. This will greatly encourage
people to study the transmission of non-classical light and the interference
in the transmission process[10]. . In particular, for example, some structures
can be regarded as the nearest Hamiltonian coupling, which is similar to the
tightly constrained Hamiltonian, which is very famous in the field of
condensed matter physics. Moreover, the integrated structure also can be
allowed to do the operation of quantum logic. In addition, quantum
electromagnetism, or the quantum effect in electromagnetism, or the
quantization of electromagnetic field, has been given new significance by W.
C. Chew et al.[11, 12]. At the same time, it is driven by the single photon
source and measurement, the effectiveness of Bell theory, and the rapid
development of nanofabrication technology. We know that in computational
theory, three state units are the most effective for classical computers[13].
In the quantum world, multi-level systems are very common. We conjecture that
the future quantum processor is a quantum system based on multiple energy
levels. Therefore, based on the three-level quantum system, this paper will
analyze the specific physical process of equal probability quantum walking. As
a preliminary knowledge, we simple review the quantum walks [2, 14, 15, 16],
the three-dimensional finite-difference time-domain method[17], and then
construct the Hamiltonian operator of the open quantum system[13], mapping the
trajectory of the quantum walk into the displacement of the electric field
pulse. Through the three-dimensional finite-difference time-domain method, the
Maxwell equation is solved, and then the spatial electromagnetic distribution
is obtained.

##  2 Quantum walks

Quantum walks are typically introduced by analogy with classical random walks.
The discrete quantum walks are usually called coined quantum walks [2, 25], if
considering walk on an infinite line as an example, one can define an
amplitude of shifting to the left adjacent site or to the right adjacent site.
In this case the wave function of a walking particle, initially localized at
site “0” is

$|\psi(t+\Delta t)\rangle=A|-1\rangle+B|+1\rangle,|A|^{2}+|B|^{2}=1$ (1)

Where time interval $\Delta t$ counts steps. One way to achieve this starting
with state $|0\rangle$ is to use the state of a qubit (two-quantum system),
typically referred to as “quantum coin”, to supply amplitudes for the two
different directions

$|\psi(t+\Delta t)\rangle=A|-1,0\rangle+B|+1,1\rangle$ (2)

Here the second state index denotes the basis states of the coin. In the
paper, we mainly consider the discrete quantum walk and replaces the coin
operator with equal probability, it will be explained in detail later.

Continuous time quantum walks do not rely on auxiliary quantum coins to
propagate. It is evolution due to dynamic change of a unitary. By the
processing method of graph theory and introducing a scalar incoherent
parameter, and discussing a mixed quantum continuum and related classical
dynamics, and solves the quantum random standard equations[14, 15].

##  3 Three dimensional finite difference time domain FDTD

In 1966, Yee established a set of finite difference equations for the time-
dependent Maxwell’s curl equations system [18]. These equations can be
expressed in discrete form in space and time, using the second-order accurate
central difference formula. The discrete positions of electric and magnetic
field components in time and space are sampled. FDTD technology divides the
three-dimensional problem geometry into cells to form a grid, which is
composed of $N_{x}\times N_{y}\times N_{z}$ elements. The algorithm samples
and calculates the field at discrete time points. The material parameters
(dielectric constant and permeability) are distributed on the FDTD grid and
are related to the field component; thus, their numbers are the same as their
respective field components[17].

When the wavelength of the electromagnetic field is larger than the atomic
size or the spacing of lattice points, the macro electromagnetic theory is
still valid, just as expressed by classical electromagnetism[19, 20]. Under
the same conditions, the finite-difference time-domain method is also
applicable to the quantum world [21]. As mentioned in reference [22], quantum
information brings vitality to computational electromagnetism.

For lossless, non-dissipative and non-uniform media, we can obtain the
discrete expression of quantum FDTD:

$\displaystyle\hat{E}_{x}^{n+1}(i,j,k)=\hat{E}_{x}^{n}(i,j,k)$ (3)
$\displaystyle+\frac{\Delta t}{\varepsilon_{x}(i,j,k)\Delta
y}(\hat{H}_{z}^{n+\frac{1}{2}}(i,j,k)-\hat{H}_{z}^{n+\frac{1}{2}}(i,j-1,k))$
$\displaystyle-\frac{\Delta t}{\varepsilon_{x}(i,j,k)\Delta
z}(\hat{H}_{y}^{n+\frac{1}{2}}(i,j,k)-\hat{H}_{y}^{n+\frac{1}{2}}(i,j,k-1))$
$\displaystyle\hat{H}_{x}^{n+\frac{1}{2}}(i,j,k)=\hat{H}_{x}^{n-\frac{1}{2}}(i,j,k)$
(4) $\displaystyle+\frac{\Delta t}{\mu_{x}(i,j,k)\Delta
z}(\hat{E}_{y}^{n}(i,j,k+1)-\hat{E}_{y}^{n}(i,j,k))$
$\displaystyle-\frac{\Delta t}{\mu_{x}(i,j,k)\Delta
y}(\hat{E}_{z}^{n}(i,j+1,k)-\hat{E}_{z}^{n}(i,j,k))$

Here, the classical electromagnetic field is promoted to the quantum world.
Here, only the component representation of the x-axis direction of the quantum
electromagnetic field in the rectangular coordinate system is given, and the
components in other directions can be obtained similarly. $\Delta t$
represents the time step, $\Delta x,\Delta y,\Delta z$ which is the spatial
step in the x, y, z three directions respectively. $\varepsilon$ is dielectric
constant and $\mu$ is permeability of the material and n in the equation
represents the number of iterations.The "$\wedge$"in Eq.(3) and Eq.(4)
represents the variables of the quantum world.

##  4 Hamiltonian operator of a three-level quantum system

For a three-level system, 1, 2 and 3 are used to represent the energy of each
energy level, namely $\mathbb{E}_{1}$, $\mathbb{E}_{2}$ and $\mathbb{E}_{3}$.
The resonant monochromatic field generated by two wave envelopes, j = 1, 2,
has frequencies of and respectively. Under the rotating wave approximation
(RWA)[13, 24, 25, 26], its Hamiltonian can be written as:

$\mathbb{H}=\begin{bmatrix}{\mathbb{E}_{1}}&{u_{1}(t)e^{i\omega_{1}t}}&{0}\\\
{u_{1}(t)e^{-i\omega_{1}t}}&{\mathbb{E}_{2}}&{u_{2}(t)e^{i\omega_{2}t}}\\\
{0}&{u_{2}(t)e^{-i\omega_{2}t}}&{\mathbb{E}_{3}}\end{bmatrix}$ (5)

Here, $\omega_{1}:=\frac{\mathbb{E}_{2}-\mathbb{E}_{1}}{\hbar}$,
$\omega_{2}:=\frac{\mathbb{E}_{3}-\mathbb{E}_{2}}{\hbar}$, the control
quantity $u_{j}(t),j=1,2$. The above formula can be further written as:

$\mathbb{H}=D+\begin{bmatrix}{0}&{\Omega_{1}(t)}&{0}\\\
{\Omega_{1}^{*}(t)}&{0}&{\Omega_{2}(t)}\\\
{0}&{\Omega_{2}^{*}(t)}&{0}\end{bmatrix}$ (6)

Here,
$\Omega_{i}(t)$=$u_{i}(t)e^{i\omega_{i}t}$, $i=1,2.$ $D:$
=$diag(\mathbb{E}_{1},\mathbb{E}_{2},\mathbb{E}_{3})$.
That is, the energy state of the system only appears on the diagonal. This
item D is called "drifting" and can be eliminated by U transformation [23].
For a three-level system, throw away the fundamental terms, there are

$\mathbb{H}=\begin{bmatrix}{0}&{\Omega_{1}(t)}&{0}\\\
{\Omega_{1}^{*}(t)}&{0}&{\Omega_{2}(t)}\\\
{0}&{\Omega_{2}^{*}(t)}&{0}\end{bmatrix}$ (7)

For complex three-level system, the system is in resonant state if and only if
the following conditions are satisfied [24]:

$\begin{cases}\Omega_{1}(t)=cos(t/\sqrt{3})e^{\frac{i}{\hbar}[(\mathbb{E}_{2}-\mathbb{E}_{1})t+\varphi_{1}]}\\\
\Omega_{2}(t)=sin(t/\sqrt{3})e^{\frac{i}{\hbar}[(\mathbb{E}_{3}-\mathbb{E}_{2})t+\varphi_{2}]}\end{cases}$
(8)

Here $\varphi_{1}$ and $\varphi_{2}$ are two arbitrary phases.

##  5 Quantum walking on one line

Suppose that in a system of three-level atoms, the electron has a transition
(such as the L structure in Fig. 1), and the release of photons and is the
time required for the electron to transition from the third energy level of
the atom to the middle energy level, and the time required for the electron to
transition from the middle (or second) energy level of the atom to the first
(or lowest) energy level. According to the nearest neighbor Hamiltonian
coupling principle[10], the following quantum walking can be obtained: It is
assumed that there are seven atomic lattice points on a straight line. As
shown in Figure 2, the atomic lattice points are located at the coordinate
position points on the one-dimensional x-axis, i.e.
$x_{1}=-3,x_{2}=-2,x_{3}=-1,x_{4}=0,x_{5}=1,x_{6}=2,x_{7}=3$.
Step 1: Assuming that at the atomic lattice point of $x_{4}=0$, the electron
transitions from the higher energy level $(\mathbb{E}_{3})$ to the middle
energy level $(\mathbb{E}_{2})$ , and releases the photon $\hbar\omega_{1}$,
consuming time is $T_{1}$. In the resonant state, the electron transitions
from the middle energy level $(\mathbb{E}_{2})$ to the lower energy level
$(\mathbb{E}_{1})$, consuming time is $T_{2}$. At this moment, the atomic wave
function can be written as:

$|\psi_{x_{4}}\rangle=|\mathbb{E}_{1}\rangle$ (9)

At the same time, the electrons located in atomic lattice $x_{3}=-1$ and
$x_{5}=1$, absorb photons $\hbar\omega_{1}$ and $\hbar\omega_{2}$ with a
probability of 1/2 respectively, completing the transition from the low energy
level to the high energy level. Its atomic wave function can be written as:

$|\psi_{x_{3}}\rangle=|\psi_{x_{5}}\rangle=c_{2}|\mathbb{E}_{2}\rangle+c_{3}|\mathbb{E}_{3}\rangle$
(10)

Figure 1: Three possible configurations of atomic energy levels, when the
system is in a resonant state, $\Delta a=\Delta b=0.$ $\Omega_{a}$ and
$\Omega_{b}$ are the frequency of photons, respectively [13]. In this paper,
we use L-type configuration to analyze the problem. Figure 2: The first
quantum walk in a three-level atomic system, where 1/2 represents the
probability of transition

Step 2: As shown in Fig.3, when the atomic lattice point is at the $x_{3}=-1$
and $x_{5}=1$, the electron transitions from the higher energy level
$(\mathbb{E}_{3})$ to the middle energy level $(\mathbb{E}_{2})$ and releases
the photon $\hbar\omega_{1}$, consuming time is $T_{1}$. In the resonant
state, the electron transitions again from the middle energy level
$(\mathbb{E}_{2})$ to the lower energy level $(\mathbb{E}_{1})$ and releases
the photon $\hbar\omega_{2}$, consuming time is $T_{2}$. At the moment, the
atomic wave function can be written as:

$|\psi_{x_{3}}\rangle=|\psi_{x_{5}}\rangle=|\mathbb{E}_{1}\rangle$ (11)

When the atom lattice point is located in $x_{2}=-2$ and $x_{6}=2$, the
electron absorbs the photon $\hbar\omega_{1}$ with a quarter probability and
transitions from the intermediate level $(\mathbb{E}_{2})$ to the higher
energy level $(\mathbb{E}_{3})$, which takes time $T_{1}$. Then, the electron
absorbs photon $\hbar\omega_{2}$ with a quarter probability and transitions
from lower energy level $(\mathbb{E}_{1})$ to intermediate energy level
$(\mathbb{E}_{2})$, which takes time $T_{2}$. At this moment, the atomic wave
function can be written as:

$|\psi_{x_{2}}\rangle=|\psi_{x_{6}}\rangle=c_{2}^{{}^{\prime}}|\mathbb{E}_{2}\rangle+c_{3}^{{}^{\prime}}|\mathbb{E}_{3}\rangle$
(12)

Figure 3: The second step of quantum walking in a three-level atomic system,
in where 1/4 represents the probability of transition.

The atomic lattice point located in $x_{4}=0$, the electron absorbs photon
$\hbar\omega_{1}$ with a probability of $(1/4+1/4)$ and transitions from the
intermediate level $\mathbb{E}_{2}$ to the high-energy level $\mathbb{E}_{3}$,
which takes time $T_{1}$. Similarly, the electron absorbs photon
$\hbar\omega_{2}$ with a probability of $(1/2+1/2)$ and transition from the
lowest energy level $\mathbb{E}_{1}$ to the middle energy level
$\mathbb{E}_{2}$, which takes time $T_{2}$. At this moment, the atomic wave
function can be written as:

$|\psi_{x_{4}}\rangle=c_{2}^{{}^{\prime\prime}}|\mathbb{E}_{2}\rangle+c_{3}^{{}^{\prime\prime}}|\mathbb{E}_{3}\rangle$
(13)

Step 3: Located in the atom lattice point $x_{4}=0$, electrons release the
photons $\hbar\omega_{1}$ and $\hbar\omega_{2}$, and the total consuming time
is $T_{1}+T_{2}$. At the same time, the electrons on the atomic lattice points
$x_{2}=-2$ and $x_{6}=2$, and release photons and, and the total consuming
time is still $T_{1}+T_{2}$. At the atomic lattice points $x_{3}=-1$ and
$x_{5}=1$, the electron absorbs the photon $\hbar\omega_{1}$ with a
probability of $1/4$ and $1/8$, respectively. Similarly, the photon
$\hbar\omega_{2}$ is absorbed with $1/4$ probability and $1/8$ probability. At
last, located in $x_{1}=-3$ and $x_{7}=3$ atomic lattice points, electrons
absorb photons with a probability of $1/8$. In addition, the atomic wave
function at this step is omitted here and will not be repeated.

Figure 4: The third step of quantum walking in a three-level atomic system,
where $1/4$ and $1/8$ represent the probability of transition.

##  6  Numerical calculation

Suppose a beam of light, with a wavelength of $804nm$ and a frequency of
$3.7\times 10^{14}Hz$, passes through the center of a $900nm\times 90nm\times
190nm$ quartz crystal, as shown in Fig.5. The first step of quantum walking is
realized in the center of the crystal, the electrons transition from the high
energy level to the middle energy level, releasing photon $\hbar\omega$ in the
form of electromagnetic pulse, while the electrons in the surrounding atomic
lattice absorb photons. We can regard this process as electric field energy
storage, that is

$\frac{1}{2}\varepsilon E_{0}^{2}=\hbar\omega$ (14)

From this, we can calculate the average electric field intensity of the energy
level, further write a simple Gaussian form of electromagnetic pulse, namely:

$E(t)=E_{0}cos(\omega_{c}(t-t_{0}))e^{-\frac{(t-t_{0})^{2}}{\tau^{2}}}$ (15)

That is, the cosine modulated Gaussian wave packet is used to represent
photons. The above formula is also the mathematical expression of the
transient electric field of photons in time domain. The mathematical
expression of photons in frequency domain can be obtained through Fourier
transform:

$E(\omega)=E_{0}(\frac{\tau\sqrt{\pi}}{2}e^{-\frac{\tau^{2}(\omega-\omega_{0})^{2}}{4}}+\frac{\tau\sqrt{\pi}}{2}e^{-\frac{\tau^{2}(\omega+\omega_{0})^{2}}{4}})$
(16)

This is consistent with the mathematical expression[10] of the electric field
when the electron is in the transition between the atomic energy levels, and
also satisfies the resonance condition-Eq.(8).

Figure 5: A beam of red light passes through a quartz crystal, assuming that
it passes through seven atomic lattice points.

To select the time delay $t_{0}=4.5\tau$, and $\tau=\frac{n_{c}\Delta
S_{max}}{2c}$, where $\Delta S_{max}$ is the maximum value of FDTD grid step
$(\Delta x,\Delta y,\Delta z)$, and $n_{c}$ is the number of grids per
wavelength, and $c$ is the speed of light.

In addition, in FDTD simulation[17], the initial condition of the field is
zero, so the electric field of the excitation source must also be zero. This
can move the time Term in the Gaussian waveform by one time unit, so that the
instantaneous value of the electric field at the beginning is zero. As shown
in Fig. 6, when the number of iterations is $n=100$, a complete Gaussian
electric field pulse can be obtained.

Figure 6: The Gaussian pulse waveform.

Assuming that the distance between atomic lattice points is 40nm, the
calculation results of the upper half of the first step of quantum walking in
Fig. 2 are shown in Fig.7.

Figure 7: The first step of quantum is the spatial electric field distribution
when the electron completes the transition, in which the abscissa is in the x
direction, the ordinate is in the z direction, and the unit is nm. The
electric field is polarized along the z direction.

Fig.8 is the second step of quantum walking, corresponding to the upper half
of Fig.3, in which, the electric field value is only 0.707 of the
corresponding value in Fig.7.

Figure 8: The distribution of electric field in space when the electron
completes the transition in the second step of quantum walking. Figure 9: The
distribution of electric field in space when the electron completes the
transition in the third step of quantum walking.

Fig.9 corresponds to the upper half of Fig.4. It can be seen from the above
figures that the quantum electromagnetic field at this time is instantaneous
and ultra-short distance. Each step of quantum walking will randomly add an
excitation source.

##  7 Description of quantum walking process and electric field distribution
of two parallel beams of light

As shown in Fig.10, the distances between adjacent crystal lattice points are
equal. It is assumed that the initial excitation source is generated at the
origin 0 of the coordinate axis, that is, the electron transitions from the
third energy level $(\mathbb{E}_{3})$ to the intermediate energy level
$(\mathbb{E}_{2})$ and releases the photon $\hbar\omega_{1}$, takes time
$T_{1}$. As shown in Fig.11, that is, they located above $x_{1}$ and $x_{2}$
axes. The double arrow straight line on the left between $x_{1}=0$ and
$x_{2}=0$, it indicates that the electron between the lattice points of the
two lines transitions and absorbs the photon $\hbar\omega_{1}$ with a
probability of $1/3$. That is, in the resonant state, the electron transitions
from the intermediate energy level $(\mathbb{E}_{2})$ to the lowest energy
level $(\mathbb{E}_{1})$ again. The time taken is $T_{2}$, as shown below the
$x_{1}$ and $x_{2}$ axis in Fig.11, and the double arrow straight line on the
right between $x_{1}=0$ and $x_{2}=0$, and indicates that the electron between
the atomic lattice points of the two lines transitions and absorbs the photon
$\hbar\omega_{2}$ with 1/3 probability. In time $T_{1}$ or $T_{2}$, the
absorption and release of photons by electrons between $x_{1}=0$ and $x_{2}=0$
atomic lattices, and it can be regarded as the exchange of quantum states.

Figure 10: Two parallel beams passing through a quartz crystal. The $x_{1}$
expresses photon $\hbar\omega_{1}$ propagates along the x-axis. In the same
way, $x_{2}$ represents photon $\hbar\omega_{2}$ propagates along the x-axis.
Figure 11: The first step of quantum walking of four photons on two parallel
lines. Figure 12:  Quantum walking of four photons on two parallel lines takes
the first step. At $t=T_{1}$ , the electric field distribution in space is
shown.

The electric field distribution corresponding to Fig.11 is shown in Fig.12. At
this moment, as can be seen from Fig.12, the electric field is strongly
coupled between $x_{1}=0$ and $x_{2}=0$. Quantum walking of two parallel line
photons takes the second step, as shown in Fig.13. In time $T_{1}$, the
electrons at $x_{1}=1$ and $x_{1}=-1$, as well as $x_{2}=1$ and $x_{2}=-1$
begin to transition from high energy level to intermediate energy level,
releasing the photon $\hbar\omega_{1}$ with a probability of $1/3$, while
$x_{1}=2$ and $x_{1}=-2$ as well as $x_{2}=2$ and $x_{2}=-2$, the electrons
absorb photon $\hbar\omega_{1}$ with a probability of $1/9$ respectively. At
this time $t=T_{1}$, the electrons at $x_{1}=1$ and $x_{1}=-1$ as well as
$x_{2}=1$ and $x_{2}=-1$, absorb the photon $\hbar\omega_{1}$ released by the
electron transition at $x_{1}=0$ and $x_{2}=0$ with a probability of $1/9$,
that is, at this time, the photons are exchanged with the nearest atomic
lattice around $x_{1}=0$ and $x_{2}=0$ with a probability of $1/9$, that is,
the exchange of quantum states. At this point, the atomic wave function can be
written as:

$\psi_{x_{1}=0}=\sqrt{\frac{3}{9}}|\mathbb{E}_{3}\rangle$ (17)
$\psi_{x_{1}=1}=\psi_{x_{1}=-1}=\sqrt{\frac{2}{9}}|\mathbb{E}_{3}\rangle$ (18)
$\psi_{x_{1}=2}=\psi_{x_{1}=-2}=\sqrt{\frac{1}{9}}|\mathbb{E}_{3}\rangle$ (19)

Similarly, when time t is in the time period $T_{2}$, quantum walking path of
photon $\hbar\omega_{2}$ is the same as the above.

Figure 13: Quantum walking of four photons on two parallel lines takes the
second step.

As can be seen from Fig. 14, the electric field coupling between atomic
lattice points is very close, and three excitation sources are generated on
each line.

Figure 14: In the second step of the quantum walking of four photons on two
parallel lines, the distribution of electric field in space is from top to
bottom, and the time step is from $ndt=40dt$ to $ndt=100dt$. The evolution
process of electric field with time is shown. Figure 15:  Quantum walking of
four photons on two parallel lines takes the third step.

As shown in Fig.15, the electrons on the lattice points of $x_{1}=3$ and
$x_{1}=-3$ as well as $x_{2}=3$ and $x_{2}=-3$ absorb the photon
$\hbar\omega_{1}$ with a probability of $1/27$, while the electrons on the
lattice points of $x_{1}=2$ and $x_{1}=-2$ as well as $X_{2}=2$ and
$X_{2}=-2$, transition from high energy level to intermediate energy level
with a probability of $1/9$, releasing the photon $\hbar\omega_{1}$. The
electrons on the lattice points of $x_{1}=1$ and $x_{1}=-1$ as well as
$x_{2}=1$ and $x_{2}=-1$, absorb photon $\hbar\omega_{1}$ with the probability
of $(1/9+1/27)$ and release photon $\hbar\omega_{1}$ with the probability of
$1/9$, that is, the quantum states with the probability of $1/9$ exchange with
each other. Finally, the electrons on the lattice point at $x_{1}=0$ and
$x_{2}=0$, release the photon $\hbar\omega_{1}$ from the high energy level to
the intermediate energy level with a probability of$1/9$. At the same time,
they exchange photons with the electrons on the nearest lattice point with a
probability of $1/9$, that is, quantum exchange. The electric field
distribution and its evolution with time are shown in Fig.16. The evolution
process of electric field with a time $T_{1}$ . As can be seen from Fig.16,
the coupling of electric field between grid points begins to weaken and spread
along the straight line, and five excitation sources are generated on each
line. At this time, the wave function on the atomic lattice point can be
written as

$\psi_{x_{1}=0}=\frac{1}{3}|\mathbb{E}_{3}\rangle$ (20)
$\psi_{x_{1}=1}=\psi_{x_{1}=-1}=\sqrt{\frac{1}{9}+\frac{1}{27}}|\mathbb{E}_{3}\rangle$
(21)
$\psi_{x_{1}=2}=\psi_{x_{1}=-2}=\sqrt{\frac{2}{27}}|\mathbb{E}_{3}\rangle$
(22)
$\psi_{x_{1}=3}=\psi_{x_{1}=-3}=\sqrt{\frac{1}{27}}|\mathbb{E}_{3}\rangle$
(23)

Similarly, when time $t$ is in the time period $T_{2}$, the quantum walks of
photons $\hbar\omega_{2}$ is the same as the above.
In the above discusstion, we suppose that, two quantized harmonic oscillator
modes interact with three energy levels, these modes intersect with only one
quantized harminic oscillator every two energy levels, and there is only one
direct interaction between two of the three possible energy levels. Only for a
quantum walk on a straight line, its physical process is similar to the
traditional discrete-time quantum walk[27]. However, the quantum walk based on
two parallel lines has rich physical content. As can be seen from Fig.14 and
Fig.16, the third step of quantum walking produces two pairs of excitation
sources than the second step of quantum walking, and the excitation sources
between the two lines interact with each other and produce the exchange of
quantum states. In addition, the time evolution reflects the radiation process
of light pulse, that is, from weak to strong, and finally disappear. At the
same time, it also shows the instantaneity and ultrashort distance of
electromagnetic field.

##  8 Conclusions

Based on the three-level atomic structure and the geometric control theory of
quantum computing, this paper gives the equal probability model of quantum
walking on a straight line in the resonant state of the system. Through the
quantum finite-difference time-domain method, the spatial electric field
distribution generated by the first step, the second step and the third step
of quantum walking are calculated respectively, and the physical process of
each step and the mathematical expression of atomic wave function are given.
Next, we give the physical evolution process of the quantum walk on two
parallel lines under the condition of synchronization at the same frequency.
Using the quantum electromagnetic calculation tool FDTD, the coupling of
quantum evolution at a time can be obtained. However, the electric field at
this time is instantaneous and ultra-short distance. Moreover, the degree of
electromagnetic coupling or electromagnetic coherence between the two wires
and the exchange of quantum states can be obtained. The combination of quantum
walking and quantum finite-difference time-domain method brings convenience to
quantum information processing and provides a practical tool for the
development of quantum devices.

Figure 16: In the third step of the quantum walking of four photons on two
parallel lines, the distribution of electric field in space is from top to
bottom, and the time step is from $ndt=40dt$ to $ndt=100dt$.

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