2. Model and the Hamiltonian
As shown in
Figure 1, we consider a large Λ-type three-level atom coupled with a single one-dimensional waveguide at two points.
and
are the two kinds of transformation of the atoms, and the waveguide is coupled through two coupling points located at
and
, where
,
, and
are ground state, intermediate state, and excited state, respectively. The atom–waveguide coupling coefficients are
and
, respectively. The locally coupled phases
and
and the coupling intensities
and
can produce some interesting interference effects on the scattering properties, which will be discussed below. We can use superconducting quantum devices, and the locally coupled phase can be introduced through Josephson rings of external flux threads [
35].
Under rotating wave approximation (RWA), the real-space Hamiltonian of the model can be written as (
)
where
,
represents the free Hamiltonian of the waveguide mode, and
is the group velocity of the photons in the waveguide. In addition,
(
) are the boson annihilation (production) operators of the right and left photons in the waveguide, respectively.
is the frequency of the waveguide [
1,
36].
is the Hamiltonian of the atom, where
is the frequency between the ground state
and intermediate state
transition,
is the frequency between the ground state
and excited state
transition.
is the rate of external atomic dissipation due to non-waveguide modes in the environment.
describes the interaction between atoms and waveguides, where the Dirac functions
and
indicate that the atom–waveguide coupling occurs at
and
, respectively. In addition, there is an accumulated photon phase between two coupling points
and
, where
and
are renormalized wave vectors that satisfy the linearized dispersion relationship, and
.
Due to the conservation of the total excitation number in RWA, the eigenstates of the system can be represented as
where
[
] is in the position of
waveguide to create the probability amplitude of the right-moving (left-moving) photons, and the atoms are in the state
eventually. In addition,
is the probability amplitude of excited atoms. By solving the Schrödinger equation
, we can obtain the following probability amplitude equation:
We first assume that the wave vector
of a single photon from the left of the waveguide, and the atom is initialized in the ground state
. The wave function can be written as
where
, and
is the Heaviside step function. In addition,
and
(
and
) are the probability amplitudes of a right-moving photon and a left-moving photon in the
region, respectively. Finally, the atoms are in the state
(
). For cases where the frequency is fixed (i.e., the final state of the atom is
), we define
and
as the transmission and reflection amplitudes of the input photons, respectively. For frequency conversion conditions (i.e., the final state of the atom is
), we define
and
as the conversion amplitudes of the output photons of the wave vectors
and
, respectively.
We substitute Equation (4) into Equation (3) and obtain the following equations:
Next, we simplify the equations above and obtain
where
is the frequency detuning between the incident photon and the frequency of the atomic transition (
). Then, by solving Equation (6), the transmission amplitude can be obtained as (see
Appendix A for more details)
where
and
are the accumulated phase of photons transmitted between the two coupled points by the wave vectors
and
, respectively. Here,
(
) is the radiative decay rate from the excited state
to a lower-energy state
(
) contributed from each atom–waveguide coupling point [
34]. When
, the transmission amplitude in Equation (6) can be simplified as
It restores the energy of a small Λ-type atom [
37]. Because there are two coupling points, the radiation decay rate here is quadrupled. In addition, the transmission amplitude becomes
In the case of
(i.e.,
), this is exactly the same as a giant two-level atom [
38]. Similarly, we can obtain the other scattering amplitudes
where
is the phase difference between the two atomic waveguide coupling channels. Similarly, when
and
, the amplitude in Equations (10a,b) can be reduced to the amplitude of the small Λ atom and that of the large two-level atom, respectively. Obviously, photon number is conserved (i.e.,
), and the model has internal symmetry (i.e.,
). Now, we define
and
, where
and
are the transmission rate and reflection rate, respectively, and the conversion efficiency is
.
3. Result Analysis
We first investigate the dependence of the scattering probabilities on
and
, both of which can be adjusted experimentally.
Figure 2 is a three-dimensional plot of the scattering probability with detuning
and phase difference
. In
Figure 2, we can see that the position and minimum (maximum) of
(
and
) with
vary periodically with a period of
. We find that the transmission rate
varies periodically with
, and the period is just
. Similarly, the reflection rate
and the conversion efficiency
also have these properties.
In
Figure 2a, we find that the minimum value of
occurs at
(i.e.,
) when
and
or
. In addition, for a fixed value
, as the value of
increases, the value of the transmission rate
increases.
In
Figure 2b, the maximum point of
occurs at
and
or
, where
. Different from the transmission rate
, the reflection rate
gradually decreases with an increase in
at a given value of
. In
Figure 2c, we find that the conversion efficiency
, which is regardless of the change in the value of
when
Additionally, when
, this case also occurs at
or
. Comparing
Figure 2a–c, we find that the total reflection is achieved at
when
and
or
. Similarly, when the phase
increases to
and
or
, as shown in
Figure 2d–f, the total reflection point deviates, occurring at
. In fact, we find that the reflection rate, transmission rate, and conversion efficiency all change with a period of
at different phases.
To more intuitively observe the transmission rate
, reflection rate
, and conversion efficiency
, we plot the variation trends of these values with detuning
and we choose the phase difference
(see
Figure 3).
Figure 3a shows the transmission rate with detuning
at phase
. Obviously, the transmission rate curve is convex downward. The minimum value of the transmission rate occurs at
and
, which is 0.
Figure 3b,c show the values of the reflection rate and conversion efficiency curves with detuning
for phase
, respectively. In contrast, in
Figure 3a–c, we find
,
, and
when
and
. These results show that total reflection occurs. In this case, the transition of
is completely suppressed due to the destructive interference of the two corresponding decay channels, i.e.,
. Thus, the model can be reduced to the two-level atom coupled with the one-dimensional waveguide, where the resonant incident photon is fully reflected. For
, the total reflection condition is
,
, as shown in
Figure 3d–f.
In fact, as long as
, we can observe the total reflection. More interestingly, when
, no matter how the value of
changes, the system achieves frequency-independent perfect transmission (FIPT), i.e.,
and
. In addition, based on the pink dotted curves in
Figure 3c,f, we can see that no matter how the value of
changes,
. This phenomenon actually comes from the conservation of the number of photons in the system.
We further change the phase difference [
] and investigate the change in the transmission rate
, reflection rate
, and conversion efficiency
with the change in
(see
Figure 4).
In
Figure 4a, we find that when
and
, the minimum value of the transmission rate
is 0, the maximum value of the reflection rate
is 1, and the maximum value of the conversion efficiency
is 0. The results indicate that the system achieves total reflection. When the phase
increases to
[see
Figure 4b], the total reflection occurs at
. With
increasing to
[see
Figure 4c], the minimum value of the transmission rate
is 1, the maximum value of the reflection rate
is 0, and the maximum value of the conversion efficiency
is 0. They do not change with
. And we find that when
, the transmission rate of the system is always 1 and the reflection rate is always 0. This shows that the incident photon is completely transmitted and the system achieves perfect transmission, which is independent of frequency (FIPT). In
Figure 4d, when
and
, the minimum value of the transmission rate
is 0, the maximum value of the reflection rate
is 1, and the maximum value of the conversion efficiency
is 0. Total reflection occurs at
.
Previous studies have shown that in the case of small atoms, the scattering probability is completely determined by the decay ratio
[
37,
39]. However, in the case of giant atoms, the scattering probability is not only determined by the decay ratio
but is also dependent on phase. Therefore, we will continue to plot the scattering probability with respect to the decay ratio
and the phase difference
(see
Figure 5).
Figure 5 depicts the scattering probability as a function of detuning
and decay rate
. In
Figure 5a, we find that when
and
, the minimum point of the transmission rate occurs at
(i.e.,
). In addition, when
is a fixed value, the transmission rate
increases with an increase in
. In
Figure 5b, the maximum of the reflection rate occurs at
and
. At this time, the system has
. Different from the transmission rate
, the reflection rate
decreases with an increase in
when
is fixed. In
Figure 5c, we find that the best frequency transition (
) occurs at
and
. The phenomenon is the same as that of the small atoms. In addition, with an increase in
, the conversion efficiency
decreases gradually.
When the phase difference
increases to
, the minimum of the transmission rate occurs at
and
[see
Figure 5d], and the value is 0. In addition, the transmission rate
increases with an increase in
. In
Figure 5e, we find that the maximum of the reflection rate (i.e.,
) also occurs at
and
. As in the case of
, the minimum value of the transmission rate and maximum value of the reflection rate of the system occur at
and
. The difference is that the optimal frequency conversion of the system has deviated. The best transition of the frequency occurs at
and
[see
Figure 5f].
When
[see
Figure 5g–i], we find that the transmission rate
, reflection rate
, and conversion efficiency
are independent of
. This phenomenon indicates that the frequency conversion is completely suppressed.
In order to more intuitively observe the variation trends of the transmission rate
, reflection rate
, and conversion efficiency
with respect to detuning
and decay rate
, we set
to 0, 1, 2, and 3, respectively, and draw
Figure 6.
Figure 6 shows the curves of scattering rate with detuning
at different phase differences
. In
Figure 6a, the minimum value of the transmission rate occurs at
and
, which is 0.
Figure 6b,c show the variation trend of the reflection rate and conversion efficiency with detuning
at phase difference
, respectively. In
Figure 6c, we find that the optimal transition frequency of the system occurs at
and
(dashed green lines). In contrast, in
Figure 6a–c, when the
and
, we found that
,
,
. The results show that total reflection occurs.
Figure 6d–f show the curves of scattering rate with detuning
at different phase differences
. When
, the minimum system transmission rate (
) appears at
and
[see solid blue line in
Figure 6d]. In
Figure 6e, we find that the maximum value of the system reflection rate (
) also occurs at
and
. This phenomenon shows that the total reflection of the system occurs under the conditions
and
when
. However, unlike those in the case of
, the optimal frequency conversion of the system occurs at
and
[see red dashed line in
Figure 6f]. In addition, based on the pink dotted curves in
Figure 6c,f,i, we can see that no matter how the value of
changes,
.
In fact, the best frequency conversion (
) occurs at
and
when
and
[see
Figure 6a–c]. It is the same as the behaviors of small atoms. With an increase in
, the conversion efficiency
decreases gradually. When
and
[see
Figure 6g–i], the frequency conversion is completely suppressed. The incoming photons of the system are completely reflected. In this case, all scattering probabilities are independent of
. When
and
(see
Figure 6d–f), the best frequency conversion (
) appears in the
and
.
We further change the decay ratio and study the changes of transmission rate
, reflection rate
, and conversion efficiency
with
values changing [see
Figure 7]. In
Figure 7a, we find that the minimum of the transmission rate
is monotonically increasing when
and
. Its minimum value occurs at
, which is 0. The maximum value of the reflection rate
and the maximum value of the conversion efficiency
are monotone decreasing curves. Both of the maximums occur at
and the values are 1. However, when the phase difference
increases to π/2 [see
Figure 7b], the minimum of the transmission rate
has a constant value of 0 no matter how the value of
changes. The maximum value of the reflection rate
is a monotone decline curve, and its maximum value occurs at
, which is 1.
When
continues to increase to π [see
Figure 7c], the minimum of the transmission rate
is always 0. The maximum value of the reflection rate
is always 1. And the maximum value of the conversion efficiency
is always 0, i.e.,
. They do not change with a change in
, and the system shows the total reflection phenomenon, which is independent of the value of
. In
Figure 7d,
is monotonically increasing when
and
. There is a minimum value
occurring at
.
and
are monotone decreasing curves, and the curves reach their maximum value at
, which is 1.
We plot the scattering probability with detuning
and dissipation rate
for phase
and decay rate
. In
Figure 8a, we find that there is a minimum value of transmission rate, which occurs at
and
. In addition, for the fixed value of
, the transmission rate
increases with an increase in the value of
. In
Figure 8b, the maximum of the reflection rate occurs at
and
. Different from the transmission rate
, the reflection rate
decreases with an increase in
when
is fixed. In
Figure 8c, we find that the optimal frequency conversion (
) occurs at
and
. In addition, with an increase in
, the conversion efficiency
decreases gradually.
When the phase difference
increases to
, the minimum of the transmission rate occurs at
and
[see
Figure 8d]. Compared with
Figure 8a, we find that the minimum of transmission rate decreases with an increase in phase difference. However, when
is the fixed value, the transmission rate
still increases with an increase in
. Correspondingly, the maximum reflection rate increases with an increase in phase difference, and the maximum occurs at
and
[see
Figure 8e]. At this time, the system does not have an optimal frequency conversion, i.e.,
[see
Figure 8f].
When
, we find that the transmission rate
obtains a minimum value of
at
and
[see
Figure 8g]. The reflection rate
obtains a maximum value of
at
and
[see
Figure 8h]. And the conversion efficiency
is independent of
and
, and its value is always
[see
Figure 8i]. This phenomenon indicates that the photons are completely reflected in the system.
In order to more intuitively observe the transmission rate
, reflection rate
, and conversion efficiency
with respect to detuning
and dissipation rate
, we take
as
and draw
Figure 9.
In
Figure 9a, we find that the minimum transmission rate for phase difference
occurs at
and
, which is
. The minimum transmission rate increases with an increase in dissipation.
Figure 9b,c show the reflection rate and conversion efficiency with detuning
at phase difference
, respectively. In
Figure 9b, the maximum reflection rate occurs at
and
, which is
. Unlike the transmission rate, the maximum reflection rate decreases with an increase in dissipation. In
Figure 9c, we find that the system has an optimal frequency transition, and the optimal frequency transition occurs at
and
(solid blue line).
Figure 9d–f show the scattering rate with detuning
at phase difference
. At this time, the minimum of the transmission rate occurs at
and
, and its value is
[see the blue solid line in
Figure 9d]. In
Figure 9e, we find that the maximum of the reflection rate also occurs at
and
, which is 0.4444. These indicate that there is no total reflection in the system when
. And unlike the case of
, there is no optimal frequency conversion, i.e.,
[see
Figure 9f].
Figure 9g–i show the scattering rate with detuning
at phase difference
. At this time, the minimum of the transmission rate appears at
and
, and its value is 0 [see the blue solid line in
Figure 9g]. In
Figure 9h, we find that the maximum of the reflection rate also occurs at
and
, which is 1. These show that the total reflection occurs when
, and the conditions for the total reflection are
and
. At this time, there is no optimal frequency conversion in the system [see
Figure 9f].
In fact, the optimal frequency conversion () occurs at and when and . This phenomenon shows that when there is no atomic dissipation in the system, the system can achieve the optimal transmission rate of photons, which is 0.25. With an increase in dissipation, the number of transmitted photons decreases. When and , if there is no dissipation, the incident photon can be completely reflected and the photon cannot pass through the system.
We further change the dissipation rate and study the extreme values of transmission rate
, reflection rate
, and conversion efficiency
as
changes [see
Figure 10]. In
Figure 10a, we find that the minimum of the transmission rate
is monotonically increasing when
and
. Its minimum value occurs at
, which is 0.25. The maximum of reflection rate
is the monotone decline curve, and its maximum value occurs at
, which is 0.25. The maximum of conversion rate
is the monotone decline curve with the maximum value of
at
. At this time, the system has the optimal frequency conversion. When the phase difference
increases to
[see
Figure 10b], the minimum transmission rate
is a monotonically increasing curve. Its minimum value occurs at
, which is 0.1111. The maximum of reflection rate
is a monotonically declining curve, and its maximum value occurs at
, which is 0.4444. The maximum of conversion rate
is a monotonically declining curve, and its maximum value occurs at
, which is
. There is no optimal frequency conversion in the system.
When
continues to increase to
[see
Figure 10c], the minimum of the transmission rate
appears at
, which is 0. The maximum of the reflection rate
appears at
, and its value is 1. The maximum of the conversion rate
is always 0, i.e.,
. At this time, the system has the total reflection at
.
Figure 10d shows the variation in the extreme values of the transmission rate
, reflection rate
, and conversion efficiency
with
at
. In
Figure 10d, we can see that the extreme values of the transmission rate
, reflection rate
, and conversion efficiency
are the same as those of the case of
.
According to Equations (7) and (10), the transmission rate
and reflection rate
are independent of the local coupling phase
. Therefore, the effects of the locally coupled phase
on the transmission rate and reflection rate are not discussed here. We analyze the changes in the conversion efficiency
with respect to detuning
and locally coupled phase
[see
Figure 11(a1–c1)]. In
Figure 11(a1), there is a maximum transmission rate when
. The value of the maximum is 0. In
Figure 11(b1), the reflection rate obtains a maximum of 0.4444 at
no matter how the value of
changes. In
Figure 11(c1), the conversion efficiency
is always 0 regardless of the value of
.
Figure 11(a2–c2) shows the conversion efficiency
with respect to detuning
when
is
, respectively. Obviously, the conversion efficiency
is independent of the local coupling phase no matter how
varies [see
Figure 11(a2–c2)]. When
, conversion efficiency
obtains a maximum value of 0.5 at
[see
Figure 11(a2)]. When
, conversion efficiency
obtains a maximum value of 0.4444 at
[see
Figure 11(b2)]. When
, the conversion efficiency is always 0, i.e.,
[see
Figure 11(c2)].