# Efficient frequency conversion based on resonant four-wave mixing

Chin-Yao Cheng1 Zi-Yu Liu1 Pi-Sheng Hu1 Tsai-Ni Wang1 Chung-Yu Chien1 Jia-Kang
Lin1 Jz-Yuan Juo1 Jiun-Shiuan Shiu1 Ite A. Yu2,5 Ying-Cheng Chen3,5 Yong-Fan
Chen1,4,5<EMAIL_ADDRESS>1Department of Physics, National Cheng Kung
University, Tainan 70101, Taiwan
2Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
3Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617,
Taiwan
4Center for Quantum Frontiers of Research & Technology, Tainan 70101, Taiwan
5Center for Quantum Technology, Hsinchu 30013, Taiwan

###### Abstract

Efficient frequency conversion of photons has important applications in
optical quantum technology because the frequency range suitable for photon
manipulation and communication usually varies widely. Recently, an efficient
frequency conversion system using a double-$\Lambda$ four-wave mixing (FWM)
process based on electromagnetically induced transparency (EIT) has attracted
considerable attention because of its potential to achieve a nearly 100%
conversion efficiency (CE). To obtain such a high CE, the spontaneous emission
loss in this resonant-type FWM system must be suppressed considerably. A
simple solution is to arrange the applied laser fields in a backward
configuration. However, the phase mismatch due to this configuration can cause
a significant decrease in CE. Here, we demonstrate that the phase mismatch can
be effectively compensated by introducing the phase shift obtained by two-
photon detuning. Under optimal conditions, we observe a wavelength conversion
from 780 to 795 nm with a maximum CE of 91.2$\%$ $\pm$ 0.6$\%$ by using this
backward FWM system at an optical depth of 130 in cold 87Rb atoms. The current
work represents an important step toward achieving low-loss, high-fidelity
EIT-based quantum frequency conversion.

###### pacs:

42.50.Gy, 42.65.Ky, 03.67.-a, 32.80.Qk

##  I Introduction

Quantum frequency conversion of photons is important in the development of
long-distance quantum communication and effective optical quantum computing
QFC by Kumar ; DLCZ ; QC by Zeilinger . In nonlinear optics, the wave mixing
effect away from resonance can not only prevent the quantum noise caused by
vacuum fluctuations but also convert the photon frequency to a larger range
and bandwidth Boyd NLO . Therefore, in the past few decades, researchers have
often used the wave mixing effect far from resonance to achieve efficient
frequency conversion. Most of the experiments to achieve high-efficiency
frequency conversion are performed in solid materials along with sum-frequency
generation SFG1 ; SFG2 ; SFG4 ; SFG6  or Bragg-scattering four-wave mixing
(FWM) BS1 ; BS2 ; BS3 ; BS4 . In nonlinear optical systems, the highest
internal conversion efficiency (CE) has reached thus far has been of >90% by
using sum frequency mixing in a nonlinear crystal and Bragg-scattering FWM in
an optical fiber SFG1 ; BS4 . In nonlinear crystals, a strong pump light is
usually required to achieve a high-efficiency QFC. However, under strong pump
light conditions, additional noise photons are often generated due to
spontaneous Raman or parametric conversion effects, which can cause
difficulties in the practical application of QFC Parametric Noise . Although
the required pump power can be reduced by using waveguides, cavities or
fibers, this results in the coupling loss of incident photons, thereby
reducing the overall CE of QFC.

Another promising approach for implementing low-loss frequency converters is
the use of a resonant FWM system based on electromagnetically induced
transparency (EIT) Harris FWM2 ; Kang BFWM ; Yu FWM ; Liu BFWM ; Juo FWM .
Because EIT can greatly enhance nonlinear interactions between photons and
considerably suppress vacuum field noise under ideal conditions, many EIT-
based quantum applications have been proposed and demonstrated at the single-
photon level; these applications include quantum memory Hsiao QM ; Laurat QM ;
Zhu QM , photonic transistors PTransister1 ; PTransister2 ; PTransister3 ,
optical phase gates XPM2 ; XPM3 ; XPM4 , and frequency beam splitter Yu FBS .
An efficient FWM based on double-$\Lambda$ EIT system has attracted much
attention recently because it can achieve nearly 100% CE and can protect the
quantum state of frequency-converted photons effectively Chen QFC . To obtain
such a high CE, effective suppression of the spontaneous emission loss in this
resonant-type FWM system is essential. A simple solution involves backward
configuration of the applied laser fields Kang BFWM ; Liu BFWM . However, the
phase mismatch due to the backward configuration can cause significant
decreases in CE. Here, we demonstrate that the phase mismatch can be
effectively compensated by introducing a phase shift obtained through two-
photon detuning. Under optimal conditions, we observe a wavelength conversion
from 780 to 795 nm with a maximum CE of 91.2$\%$ $\pm$ 0.6$\%$ by using this
backward FWM system in cold rubidium (87Rb) atoms. To our knowledge, this is
the highest CE that has been achieved in an EIT-based FWM system thus far.

Figure 1:  Energy level diagram and experimental settings. (a) Energy levels
of the 87Rb $D_{1}$ and $D_{2}$ lines used in the backward FWM experiment.
Here, two-photon detuning, $\delta$, has the same magnitude and opposite
direction as does coupling detuning. Coupling detuning is defined as
$\omega_{c}-\omega_{23}$, where $\omega_{c}$ and $\omega_{23}$ are the
frequencies of the coupling field and the $|2\rangle\leftrightarrow|3\rangle$
transition, respectively. (b) Angle set between the light fields, plotted on
the upper right. (c) Schematic of the apparatus setup. ECDL, external cavity
diode laser; DL, diode laser; EOM, electro-optic modulator; AOM, acousto-optic
modulator; SMF, single-mode fiber; POL, polarizer; PBS, polarization beam
splitter; QWP, quarter-wave plate; L, lens; PH, pinhole; PMT, photo-multiplier
tube.

##  II Experiment Details

Here, we investigate a resonant-type frequency conversion based on backward
FWM in a Doppler-cooled 87Rb atomic system. The arrangement of the light
fields and relative energy levels are shown in Fig. 1(a). The probe field
($\Omega_{p}$ indicates its Rabi frequency) with an intensity of approximately
1 $\mu$W/cm2 drives the transition of the state
$|1\rangle\leftrightarrow|3\rangle$ and forms a $\Lambda$-type EIT system
alongside the coupling field ($\Omega_{c}$), which drives the transition of
the state $|2\rangle\leftrightarrow|3\rangle$ with an intensity of
approximately 10 mW/cm2. Under the EIT condition, the driving field
($\Omega_{d}$) drives the transition of the state
$|2\rangle\leftrightarrow|4\rangle$ and induces the FWM process to generate
the signal field ($\Omega_{s}$) during the transition of the state
$|1\rangle\leftrightarrow|4\rangle$. The wavelength consequently converts from
780 nm ($\Omega_{p}$) to 795 nm ($\Omega_{s}$). All the light fields are
right-circularly polarized ($\sigma+$), and the spatial configuration of these
light fields is shown in Fig. 1(b). The angle between the probe field and the
coupling field is set at 0.5∘ to suppress unwanted light from leaking into the
photon detector. The driving field is counter-propagating to the probe field
as a backward FWM condition. Because of the phase match condition, the
generated signal field is counter-propagating to the coupling field. Notably,
this backward configuration causes a non-negligible phase mismatch effect and
significantly reduces CE. However, as long as the two-photon resonance of the
built-in EIT mechanism (formed by $\Omega_{p}$ and $\Omega_{c}$) in the
backward FWM system is slightly destroyed to cause a phase shift in the probe
field, this phase mismatch effect can be effectively compensated.

Figure 1(c) illustrates the experimental setup. The probe and coupling fields
are produced from the diode laser 1 (DL1) and DL2, respectively. The coupling
field is directly injection locked by the external cavity diode laser 1
(ECDL1). One beam from the ECDL1 is sent through a 6.8-GHz electro-optic
modulator (EOM). The probe field is injection locked by an intermediate laser
seeded with the high-frequency sideband of the EOM output. The above
arrangement can eliminate the influence of the EOM output carrier on DL1. Both
the probe and coupling fields are sent to their respective single-mode fibers
(SMFs) to obtain spatial mode matching. The driving field is generated from
ECDL2, another ECDL. All these light fields are turned on and off by their
respective acousto-optic modulators (AOMs). We use AOM1 to control the probe
pulse width. The coupling and driving fields are turned on and off via AOM2
and AOM3, respectively. AOM4 is used to change the frequency of the coupling
field and thereby adjust the two-photon detuning in the FWM experiment.

The 1/e2 diameter of the probe beam waist is approximately 300 $\mu$m, whereas
the diameters of the coupling beam and driving beam are 3 and 4 mm,
respectively. The probe and signal pulses are detected by a photo-multiplier
tube module 1 (PMT1) and PMT2, respectively. In the backward FWM experiment,
the approximate overall detection efficiencies for the probe and signal fields
are 65% and 75%, respectively.

Regarding the timing sequence arrangement, we denote the time of firing the
probe pulse as $t=0$ ms. The magnetic field of the magneto-optical trap (MOT)
is first switched off at $t=-1.6$ ms. Next, the repumping laser of the MOT is
turned off at $t=-200~{}\mu$s and the coupling and driving fields are turned
on immediately. Before sending the probe pulse, the MOT’s trapping laser is
turned off at $t=-100~{}\mu$s. This timing arrangement of the optical fields
ensures that the entire atomic population is prepared at the ground state
$|1\rangle$; moreover, the repetition rate for each measurement is maintained
at 100 Hz throughout the experiment. In addition, to ensure that the probe
pulse is long enough to reach steady-state conditions, the spectrum width of
the probe pulse is required to be smaller than the width of the EIT
transparent window; therefore, the duration of the probe pulse is set to 30
$\mu$s in this experiment.

Figure 2:  Observation of the slow light effect based on EIT. The black
(higher) and blue (lower) lines show the incident probe pulse and the probe
pulse propagating through the medium, respectively. The solid lines represent
the experimental data, whereas the dashed lines represent the theoretical
curves with the following parameters: (a) $\alpha=45$,
$\Omega_{c}=0.60\Gamma$, and $\gamma_{21}=2\times 10^{-4}\Gamma$; (b)
$\alpha=130$, $\Omega_{c}=1.20\Gamma$, and $\gamma_{21}=7\times
10^{-4}\Gamma$.

##  III Theoretical Model

To theoretically analyze the propagation behavior of the probe and signal
pulse in the backward FWM medium, we use Maxwell–Schrödinger equations (MSEs):

$\displaystyle\frac{\partial\Omega_{p}}{\partial
z}+\frac{1}{c}\frac{\partial\Omega_{p}}{\partial
t}=i\frac{\alpha_{p}\gamma_{31}}{2L}\rho_{31},$ (1)
$\displaystyle-\frac{\partial\Omega_{s}}{\partial
z}+\frac{1}{c}\frac{\partial\Omega_{s}}{\partial t}-i\Delta
k\Omega_{s}=i\frac{\alpha_{s}\gamma_{41}}{2L}\rho_{41},$ (2)

where $\rho_{ij}$ is the slowly varying amplitude of the coherence between
$|i\rangle$ and $|j\rangle$; $\gamma_{31}$ and $\gamma_{41}$ represent the
total coherence decay rates from the excited states $|3\rangle$ and
$|4\rangle$, respectively; $\Delta k=k_{p}-k_{c}+k_{d}-k_{s}$ is the phase-
mismatch term (in which the plus and minus signs indicate the processes of
light field absorption and emission by the medium, respectively);
$\alpha_{p(s)}=n\sigma_{13(14)}L$ represents the optical depth (OD) of the
probe (signal) transition (where $n$ and $L$ are the density and length of the
medium, respectively). In this experiment, $\sigma_{13}\approx\sigma_{14}$,
because the Clebsch–Gordan coefficients of the probe and signal transitions
are symmetric when considering the degenerate Zeeman sublevels; moreover,
their wavelengths (780 and 795 nm, respectively) and the spontaneous decay
rates ($2\pi\times 6.06$ and $2\pi\times 5.75$ MHz, respectively) are also
close to each other. Thus, for simplicity, we assume
$\alpha_{p}=\alpha_{s}=\alpha$ and $\gamma_{31}=\gamma_{41}$. Notably,
$\gamma_{31}$ and $\gamma_{41}$ in this experiment are mainly attributed to
the spontaneous decay rate, and both are represented by $\Gamma\approx
2\pi\times 6$ MHz.

To solve MSEs, we use the optical Bloch equations (OBEs) to obtain the
equations of motion of $\rho_{31}$ and $\rho_{41}$. Considering the case where
both the probe and signal fields are weak enough to be considered the
perturbation fields (i.e., $\rho_{11}\approx 1$), the density matrix elements
with slowly varying amplitudes of the OBEs are given by

$\displaystyle\frac{d}{dt}\rho_{21}=\frac{i}{2}\Omega^{\ast}_{c}\rho_{31}+\frac{i}{2}\Omega^{\ast}_{d}\rho_{41}+(i\delta-\frac{\gamma_{21}}{2})\rho_{21},$
(3)
$\displaystyle\frac{d}{dt}\rho_{31}=\frac{i}{2}\Omega_{p}+\frac{i}{2}\Omega_{c}\rho_{21}+(i\Delta_{p}-\frac{\gamma_{31}}{2})\rho_{31},$
(4)
$\displaystyle\frac{d}{dt}\rho_{41}=\frac{i}{2}\Omega_{s}+\frac{i}{2}\Omega_{d}\rho_{21}+(i\Delta-\frac{\gamma_{41}}{2})\rho_{41},$
(5)

where $\gamma_{21}$ is the dephasing rate between ground states $|1\rangle$
and $|2\rangle$, $\Delta_{p}=\omega_{p}-\omega_{31}$ represents one-photon
detuning, $\delta=(\omega_{p}-\omega_{c})-\omega_{21}$ is two-photon detuning,
and $\Delta=(\omega_{p}-\omega_{c}+\omega_{d})-\omega_{41}$ is three-photon
detuning. We determine the value of each parameter in Eqs (1)–(5) from
additional experiments as follows: $\Omega_{c}$ is measured on the basis of
the separation of the two absorption peaks in the EIT spectrum. OD is
determined using the delay time of the slow light pulse. $\Omega_{d}$ is
obtained by simulating the generated signal pulse in the backward FWM system.

According to the theoretical predictions, when the Rabi frequencies of the
coupling and driving fields are the same, the backward FWM system can achieve
the highest CE Liu BFWM . Therefore, for simplicity, we assume the conditions
of $\Omega_{c}=\Omega_{d}=\Omega$, $\Delta_{p}=\Delta=0$, and $\gamma_{21}=0$.
By the boundary conditions of $\Omega_{p}(z=0)=\Omega_{p0}$ and
$\Omega_{s}(z=L)=0$, we solve Eqs. (1)–(5) and obtain the steady-state
solutions for the probe and signal fields:

$\displaystyle\Omega_{p}(z=L)=\frac{2q}{2q\cos(\beta/2)+\kappa\sin(\beta/2)}\Omega_{p0}e^{-i\Delta
kL/2},$ (6)
$\displaystyle\Omega_{s}(z=0)=\frac{\alpha}{\kappa+2q\cot(\beta/2)}\Omega_{p0},$
(7)

where

$\displaystyle\kappa=\left(\alpha-\frac{2\Delta
kL\delta\Gamma}{|\Omega|^{2}}\right)-i2\xi,$ (8)
$\displaystyle\beta=\sqrt{\frac{(\Delta kL+i\alpha)(\Delta
kL\delta\Gamma+i|\Omega|^{2}\xi)}{i|\Omega|^{2}+\delta\Gamma}},$ (9)
$\displaystyle q=\sqrt{\left(\xi-\frac{i\Delta
kL\delta\Gamma}{|\Omega|^{2}}+i\alpha\right)\left(\xi-\frac{i\Delta
kL\delta\Gamma}{|\Omega|^{2}}\right)},$ (10) $\displaystyle\xi=\Delta
kL+\frac{\delta\alpha\Gamma}{|\Omega|^{2}}.$ (11)

Notably, the parameter $\xi$ can be used to estimate the amount of two-photon
detuning needed to compensate for the phase-mismatch effect. In addition, from
Eq. (7), the steady-state CE of the backward FWM process can be determined:

$\displaystyle\texttt{CE}=\left|\frac{\Omega_{s}(z=0)}{\Omega_{p0}}\right|^{2}=\left|\frac{\alpha}{\kappa+2q\cot(\beta/2)}\right|^{2}.$
(12)

If both $\delta$ and $\Delta kL$ are equal to 0, then Eq. (12) here can be
simplified to Eq. (32) in the reference Chen QFC , which describes a backward
FWM system under phase matching and two-photon resonance.

Figure 3:  Observation of the backward FWM process in the MOT. With the change
of the driving Rabi frequency (a) and two-photon detuning (b), the steady-
state transmittance of the probe and signal pulses propagating in the FWM
medium are measured simultaneously. The blue squares and red circles represent
the probe and signal transmissions, respectively. The solid lines show the
theoretical curves calculated by Eqs. (1)–(5) with the parameters in (a)
$\alpha=45$, $\Omega_{c}=0.60\Gamma$, $\gamma_{21}=2\times 10^{-4}\Gamma$,
$\Delta kL=0.447\pi$, and $\delta/2\pi$ = –54 kHz; (b) $\alpha=45$,
$\Omega_{c}=\Omega_{d}=0.60\Gamma$, $\gamma_{21}=2\times 10^{-4}\Gamma$, and
$\Delta kL=0.447\pi$. The experimental data show that the maximum CE of the
backward FWM reaches 81.4$\%$ $\pm$ 0.3$\%$.  Figure 4:  Observation of the
backward FWM process in the dark SPOT. With the change of the driving Rabi
frequency (a) and two-photon detuning (b), the steady-state transmittance of
the probe and signal pulses propagating in the FWM medium are measured
simultaneously. The blue squares and red circles represent the probe and
signal transmissions, respectively. The solid lines show the theoretical
curves calculated by Eqs. (1)–(5) with the parameters in (a) $\alpha=130$,
$\Omega_{c}=1.20\Gamma$, $\gamma_{21}=7\times 10^{-4}\Gamma$, $\Delta
kL=0.134\pi$, and $\delta/2\pi$ = –27 kHz; (b) $\alpha=130$,
$\Omega_{c}=\Omega_{d}=1.20\Gamma$, $\gamma_{21}=7\times 10^{-4}\Gamma$, and
$\Delta kL=0.134\pi$. The experimental data show that the maximum CE of the
backward FWM reaches 91.2$\%$ $\pm$ 0.6$\%$.

##  IV Results and Discussion

Before performing the backward FWM experiment, we measure the slow light
effect based on EIT (Fig. 2). The OD in Fig. 2(a) measured in a typical MOT is
approximately 45. To further increase the OD, we measure the slow light pulse
in a dark spontaneous-force optical trap (SPOT). Details on the similar
experimental setup of the dark SPOT can be found in our previous study Lo XPM
. The OD in the dark SPOT is measured to be approximately 130 [Fig. 2(b)].
Moreover, the measured ground-state dephasing rate $\gamma_{21}$ is
dissimilar, which is 2$\times 10^{-4}\Gamma$ and 7$\times 10^{-4}\Gamma$ in
Figs. 2(a) and 2(b), respectively. This is mainly because the coupling field
interacts with the excited state of not only $|5P_{3/2},F=2\rangle$ but also
the excited state of $|5P_{3/2},F=3\rangle$, which leads to the AC Stark
energy shift of the ground state $|2\rangle$. The energy shift of the ground
state destroys the two-photon resonance condition of EIT and thus causes the
probe field to become absorbed by the EIT medium; this is similar to a
nonresonant photon switching effect Harris PS . Although this ground state
energy shift can be effectively compensated by introducing detuning of the
coupling field, it may still damage the EIT slightly and cause the ground
state dephasing rate to increase effectively. To compensate for the energy
shift of the ground state caused by the photon switching effect, the detuning
of the coupling field in Figs. 2(a) and 2(b) is set to –27 and –106 kHz,
respectively. According to theoretical estimates, the ground state dephasing
rate due to the photon switching effect increases by approximately 1$\times
10^{-4}\Gamma$ and 5$\times 10^{-4}\Gamma$ in Figs. 2(a) and 2(b),
respectively. This is consistent with our experimental observations.

Figure 5:  Observation of the backward FWM in the pulsed regime. The black
(higher), red (middle), and blue (lower) lines show the incident probe, the
generated signal, and the transmitted probe pulses, respectively. The solid
lines represent the experimental data, whereas the dashed lines are the
theoretical curves with the following parameters: $\alpha=130$,
$\Omega_{c}=\Omega_{d}=1.20\Gamma$, $\gamma_{21}=7\times 10^{-4}\Gamma$,
$\Delta kL=0.134\pi$, and (a) $\delta/2\pi$ = –27 kHZ; (b) $\delta/2\pi$ = 93
kHz; (c) $\delta/2\pi$ = –147 kHz. The results in (a) shows that the phase
shift due to two-photon detuning in the backward FWM system effectively
compensates for the phase mismatch and achieves a maximum CE of approximately
91%. The data in (b) and (c) show that the phase mismatch is not well
compensated, resulting in a significant decrease in CE.

Next, we turn on the driving field to conduct the backward FWM experiment.
Figure 3 depicts the experimental observation of the backward FWM effect in
MOT. Throughout the backward FWM experiment, five measurement samples are used
to calculate the statistical error bars. Each measurement sample averages 1024
times. We measure the steady-state transmittance of the probe and signal
pulses propagating in the FWM medium versus the driving Rabi frequency and
two-photon detuning, as shown in Fig. 3(a) and 3(b), respectively. According
to the spatial configuration of the probe, coupling, and driving fields
mentioned in Fig. 1(b), the phase-mismatch parameter $\Delta k$ is calculated
to be approximately 0.893$\pi$ cm-1. Because the diameter of the cold atom
cloud in the MOT is approximately 5 mm, the value of $\Delta kL$ determined in
the experiment is 0.447$\pi$. Figure 3(b) shows that the maximum CE of the
incident probe light converted into FWM signal light reaches 81.4$\%$ $\pm$
0.3$\%$, where the two-photon detuning is set to –70 kHz.

To further increase the CE, we perform the backward FWM experiment in the dark
SPOT. By adjusting the intensity ratio of the six trapping laser beams in the
dark SPOT, we generate a cold atom cloud with an OD of up to 130 but with a
small diameter ($\approx$1.5 mm). Figure 4 shows the experimental observation
in the dark SPOT. Because the value of $\Delta kL$ here is smaller than that
at MOT, we use smaller two-photon detuning to compensate the phase-mismatch
effect. The experimental data in Fig. 4(b) show that at a two-photon detuning
setting of –27 kHz, the maximum CE can reach 91.2$\%$ $\pm$ 0.6$\%$. Moreover,
the bandwidth of this resonant backward FWM is approximatively 0.8 MHz, which
is mainly affected by the EIT effect and the intensity balance condition Liu
BFWM , so the bandwidth can be increased by increasing the intensity of the
coupling and driving light.

Figures 5(a), 5(b), and 5(c) present parts of the raw data in Fig. 4(b), where
two-photon detuning is –27, 93, and –147 kHz, respectively. On the basis of
the experimental conditions, we estimate that the phase shift of the probe
field caused by the detuned EIT effect is approximately –0.129$\pi$,
0.405$\pi$, and –0.704$\pi$ in 5(a), 5(b), and 5(c), respectively. Because the
phase mismatch $\Delta kL$ in this backward FWM experiment is approximately
0.134$\pi$, it can be well compensated by the two-photon detuning condition in
Fig. 5(a). By contrast, poor phase matching conditions cause a significant
reduction in CE, as shown in Figs. 5(b) and 5(c).

##  V Conclusion

Here, we demonstrated the use of EIT-based double-$\Lambda$ FWM mechanism for
efficient frequency conversion. The spatial backward arrangement of the
applied light fields can suppress the spontaneous emission loss of the
resonant-type FWM system considerably. In this study, we use the small two-
photon detuning of about tens of kHz to compensate for the phase-mismatch
effect in the backward FWM so as to achieve quasi-phase matching conditions,
which can significantly increase the CE of the EIT-based frequency converter.
We consequently observe a wavelength conversion from 780 to 795 nm with a
91.2$\%$ CE at an OD of 130 in cold 87Rb atoms. This low-loss resonant FWM
based on EIT provides a promising method for implementing a high-fidelity
quantum frequency converter Chen QFC  and therefore has potential applications
in optical quantum information technology.

## ACKNOWLEDGMENTS

We thank Pei-Chen Kuan, Wen-Te Liao, and Ray-Kuang Lee for their useful
contributions to discussions. This work was supported by the Ministry of
Science and Technology of Taiwan [grant numbers: 107-2112-M-006-008-MY3 and
108-2639-M-007-001-ASP]. We also acknowledge the support from Center for
Quantum Technology in Taiwan.

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