# Tunable quantum interference effects in Floquet two- and three-level systems

Yingying Han  Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China  Shenzhen Key
Laboratory of Ultraintense Laser and Advanced Material Technology, Center for
Advanced Material Diagnostic Technology, and College of Engineering Physics,
Shenzhen Technology University, Shenzhen 518118, China  Minchen Qiao  Key
Laboratory of Artificial Micro- and Nano-structures of Ministry of Education,
and School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072,
China  School of integrated circuits, Tsinghua University, Beijing 100084,
China  Xiao-Qing Luo  Hunan Province Key Laboratory for Ultra-Fast Micro/Nano
Technology and Advanced Laser Manufacture, School of Electrical Engineering,
University of South China, Hengyang, 421001, China;  Tie-Fu Li  School of
Integrated Circuits and Frontier Science Center for Quantum Information,
Tsinghua University, Beijing 100084, China  Beijing Academy of Quantum
Information Sciences, Beijing 100193, China  Wenxian Zhang  Key Laboratory of
Artificial Micro- and Nano-structures of Ministry of Education, and School of
Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China  Xiu-Hao
Deng<EMAIL_ADDRESS>Shenzhen Institute for Quantum Science and
Engineering, Southern University of Science and Technology, Shenzhen 518055,
China  Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen, 518055, China  J.Q.
You  Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou 310027, China  Dapeng Yu  Shenzhen Institute for Quantum Science and
Engineering, Southern University of Science and Technology, Shenzhen 518055,
China  Guangdong Provincial Key Laboratory of Quantum Science and Engineering,
Southern University of Science and Technology, Shenzhen, 518055, China

###### Abstract

Quantum interference effects in the un-modulated quantum systems with light-
matter interaction have been widely studied, such as electromagnetically
induced transparency (EIT) and Autler-Townes splitting (ATS). However, the
similar quantum interference effects in the Floquet systems (i.e.,
periodically modulated systems), which might cover rich new physics, were
rarely studied. In this article, we investigate the quantum interference
effects in the Floquet two- and three-level systems analytically and
numerically. We show a coherent destruction tunneling effect in a lotus-like
multi-peak spectrum with a Floquet two-level system, where the intensity of
the probe field is periodically modulated with a square-wave sequence. We
demonstrate that the multi-peak split into multiple transparency windows with
tunable quantum interference if the Floquet system is asynchronously
controlled via a third level. Based on phenomenological analysis with Akaike
information criterion, we show that the symmetric central transparency window
has a similar mechanism to the traditional ATS or EIT depending on the choice
of parameters, additional with an extra degree of freedom to control the
quantum interference provided by the modulation period. The other transparent
windows are shown to be asymmetric, different from the traditional ATS/EIT
windows. These non-trivial quantum interference effects open up a new scope to
explore the applications of the Floquet systems.

##  I Introduction

Floquet systems, which could be characterized by periodically modulated
Hamiltonian, display rich dynamics and novel phenomena that are absent in
their un-modulated counterparts, such as quantum dynamical decoupling [1, 2,
3], time crystal [4, 5], Mach-Zehnder interferometer [6, 7], time-domain
Fresnel lenses [8], time-domain grating [9] and Floquet topological phase
[10]. The studies of periodically modulated systems are also known as Floquet
engineering [11, 12, 13, 14, 15]. Moreover, for the Floquet systems, the
dynamic steady-states are periodic steady-states, which emerge in a balance of
the energy injection by the periodic driving and the relaxation processes [16,
17, 18]. To explore the stability properties of the periodic steady-states,
the time-average values of observable physical quantities are usually observed
experimentally [19, 9], and stroboscopic evolution of a periodically driven
quantum system in steps of the modulation period is usually adopted in
theoretical studies [20].

Quantum interference effect (QIE) is the key to the quantum nature of a
system. In an atom-field interacting quantum system, QIE enables rich
interesting physics and applicable phenomena, such as electromagnetically
induced transparency (EIT) [21, 22] and Autler-Townes splitting (ATS) [23].
Both of them are observed with a transparency window induced by the un-
modulated coherent drive fields, though they originate from controversial
mechanisms that have been studied over decades in various systems and
scenarios [24, 25, 26]. In the above un-modulated systems, the properties of
QIE can be adjusted by manipulating the properties of the system, which is
relatively difficult. Few studies have been done to alter the properties of
QIE by using tunable auxiliary energy levels or phase-modulated fields [27,
28, 29]. Recently, with the rising interest in Floquet systems, rich novel
physics are discovered, where the Floquet parameters (such as modulation
period, modulation scheme, etc.) may be useful tools for tuning the properties
of QIEs. However, to our knowledge, the QIEs in the Floquet controlled systems
have not been adequately studied [30], especially the effect of the modulation
period on the properties of QIEs.

In this work, we first study the Floquet two-level system, where the intensity
of the probe field has the form of square-wave periodic sequence, which is a
basic model of direct frequency comb spectroscopy [31] and is different from
the sine/cosine pulse trains in the Mach-Zehnder interferometer [6, 32]. Note
that here the modulation period is shorter than the system’s coherence time.
We explore the steady excitation probability over a wide driving strength
range and give the analytically results with some special scenarios. A lotus-
like multi-peak spectrum is observed and the coherent destruction of tunneling
effect is found in the strong driving regime. When the Floquet two-level
system assisted with a third energy level and a periodically modulated control
field, each peak splits into two, resulting in multiple transparency windows.
Here the modulation pulse of the control field is the same as that of the
probe field, but they are asynchronous. An intriguing finding is that the
central transparency window (CTW) in the Floquet three-level system has a
similar profile to the traditional EIT or ATS in the un-modulated systems. We
use the Akaike information criterion (AIC) method [33] to discern the CTW from
EIT and ATS by evaluating their relative AIC weights for different modulation
periods and find that the CTW could be EIT-like or ATS-like in different
parameter regimes. Moreover, the quantum interference of the CTW can be
modified by the modulation period, which as an additional degree of freedom
increases the tunable space of CTW. Therefore, the CTW may provide a superior
platform for the explosion of quantum technology than the traditional EIT/ATS.

This work is outlined as follows. We explore the QIEs in the Floquet two-level
system in Sec. II and Floquet three-level system in Sec. III. Conclusions and
discussions are given in the last section of the article.

##  II The Floquet two-level system

Figure 1: (a) Schematic of a two-level system. A probe-field couples levels
$|0\rangle$ and $|1\rangle$ with a periodically modulated Rabi frequency
$\Omega_{p}(t)=\Omega_{p}(t+\tau)$ and a detuning $\Delta$. $\gamma_{10}$ is
the population damping rate from level $|1\rangle$ to $|0\rangle$.
$\gamma_{1}^{\phi}$ is the dephasing rate of state $|1\rangle$. (b) The steady
excitation probability $\rho_{11}$ as a function of detuning and the probe
field power. The graph is obtained by numerically calculating the Lindblad
master equation in Eq. (5) for $\gamma_{10}=1$, $\gamma_{1}^{\phi}=0.4$, and
$\tau=0.05$. Note that in this work we normalized the parameters in terms of
$\gamma_{10}$, such as $\gamma_{1}^{\phi}/\gamma_{10}=0.4$,
$\tau~{}\gamma_{10}=0.05$, and they are simplified to $\gamma_{1}^{\phi}=0.4$
and $\tau=0.05$. The white dashed lines show the excitation minima
[i.e.,$\sqrt{\Omega_{p}^{2}+\Delta^{2}}=2n\omega,~{}(n=1,2\cdots)$] with
modulation frequency $\omega=1/\tau$. The vertical red dashed line at
$\Omega_{p}=\omega=-13\mathrm{dBm}$ divides the weak and strong coupling
ranges. The vertical yellow dashed line denotes
$\Omega_{p}=1\approx-38.6\mathrm{dBm}$.

In this section, we first explore the QIEs in a Floquet two-level system
denoted by $|0\rangle$ and $|1\rangle$ with energy $\omega_{0,1}$, as shown in
Fig. 1(a). A probe field with frequency $\omega_{p}$ and periodically
modulated Rabi frequency $\Omega_{p}(t)=\Omega_{p}(t+\tau)$ couples levels
$|0\rangle$ and $|1\rangle$. Here $\tau$ is the modulation period and the
detuning between the transition frequency (i.e.,
$\omega_{10}=\omega_{1}-\omega_{0}$) and the frequency of the probe field is
$\Delta=\omega_{10}-\omega_{p}$. To simplify the calculation, we consider the
square-wave periodic sequence as the modulation scheme in this work, i.e.,

$\displaystyle\Omega_{p}(t)=\left\\{\begin{array}[]{ll}\Omega_{p},&~{}~{}~{}~{}~{}t\in[n\tau,(n+\frac{1}{2})\tau]\\\
0,&~{}~{}~{}~{}~{}t\in[(n+\frac{1}{2})\tau,(n+1)\tau]\end{array}\right.$ (3)

with $n=0,1,2\cdots$. The Hamiltonian of such system is

$\displaystyle H=\Delta(-|0\rangle\langle 0|+|1\rangle\langle
1|)/2-[\Omega_{p}(t)|0\rangle\langle 1|+h.c.]/2,$ (4)

with $\hbar=1$. Note that for our time-dependent Hamiltonian, here we apply
rotating-wave approximation (RWA) by assuming $\omega=1/\tau\ll 2\omega_{10}$,
$\Omega_{p}\ll\omega_{10}$, which are also the parameter ranges of many
experimental studies. Our earlier experimental work has shown the validity of
the RWA with a periodically driven superconducting qutrit [9]. Moreover, in
Appendix A, we also theoretically verified the validity of RWA in detail. With
the assumption of Markovian noise background, the Floquet systems’ density
matrix evolves as the Lindblad master equation [34, 35],

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i[H(t),\rho]+\frac{\gamma_{10}}{2}(2\sigma_{01}\rho\sigma_{10}-\sigma_{11}\rho-\rho\sigma_{11})$
(5) $\displaystyle+$
$\displaystyle\gamma_{1}^{\phi}(2\sigma_{11}\rho\sigma_{11}-\sigma_{11}\rho-\rho\sigma_{11}),$

where $\sigma_{ij}=|i\rangle\langle j|$ ($i,j=0,1$) is the projection
operators and $H(t)$ is shown in Eq. (4). To observe the steady-state
characteristics of the periodically driven systems, we only observe the data
at the end of each modulation period (i.e., the time evolution step is
$\tau$), and ignore the micro-dynamics within one modulation period. The
dynamics start from the ground state $|0\rangle$ and then evolve to the steady
states (see also Fig. 5), which are observed to study the steady-state
characteristics of the periodically driven systems.

Figure. 1(b) shows the steady excitation probability $\rho_{11}$ as a function
of detuning and the probe field power. Note that here the unit of the probe
field is $\mathrm{dBm}$, where
$1\mathrm{dBm}=10\mathrm{log(C}~{}\Omega_{p}^{2})$ with $\mathrm{C}=1.38\times
10^{-4}$ determined by experimental data [9, 36]. We find multi-peak
phenomenon in the weak driving range ($\Omega_{p}<-13\mathrm{dBm}$) and the
sidebands are well separated, which is caused by the Fourier components of the
square-wave modulated $\Omega_{p}(t)$ in Eq. (3), i.e.,

$\Omega_{p}(t)=\frac{\Omega_{p}}{2}-\sum_{n=1}^{\infty}(-1)^{n}\Omega_{pn}\cos(\omega_{n}t),$
(6)

where

$\displaystyle\Omega_{pn}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{p}}{(2n-1)\pi},$ (7) $\displaystyle\omega_{n}$
$\displaystyle=$ $\displaystyle(2n-1)\omega,(n=1,2,3\cdots).$ (8)

These expressions clearly demonstrate that the square-wave modulated field is
equivalent to employing many frequency-tunable fields and the distance between
them can be adjusted by the modulation frequency $\omega$ with frequency
separation $\omega_{n+1}-\omega_{n}=2\omega$.

In the strong driving range ($\Omega_{p}>-13\mathrm{dBm}$), the peaks get
broader and overlap with each other, eventually forming a lotus pattern, which
is too complex to get the analytic solution. However, we can obtain the steady
solutions of $\rho_{11}$ with $\Delta=0$ by calculating the optical Bloch
equations (see Appendix B),

$\displaystyle\rho_{11}=\frac{1}{2}-\frac{\gamma_{10}}{2}\sum_{n=-\infty}^{\infty}\frac{\gamma_{1}\Omega_{n}^{2}}{\gamma_{1}^{2}+(\Omega_{p}/2-n\omega)^{2}},$
(9)

where $\gamma_{1}=3\gamma_{10}/4+\gamma_{1}^{\phi}/2$ and n are integers.
$\Omega_{n}$ is a series of Bessel functions with variable $\Omega_{p}/\omega$
[see Eq. (35)]. When $\tau\rightarrow 0$, $\Omega_{0}\rightarrow 1$ and
$\Omega_{n\neq 0}\rightarrow 0$, then Eq. (9) can be reduced to
$\rho_{11}\approx\frac{1}{2}-\frac{\gamma_{10}}{2}\frac{\gamma_{1}}{\gamma_{1}^{2}+(\Omega_{p}/2)^{2}}$,
which is close to the conventional result of the un-modulated system
$\rho_{11}=\frac{1}{2}-\frac{\gamma_{10}}{2}\frac{\gamma_{1}^{\prime}}{\gamma_{1}^{\prime}\gamma_{10}+\Omega_{p}^{2}}$
with $\gamma_{1}^{\prime}=\gamma_{10}/2+\gamma_{1}^{\phi}$, except the
effective Rabi frequency of the probe field to be $\Omega_{p}/2$. These
indicate that when $\tau\rightarrow 0$, the central peak of the multi-peak
phenomenon is similar to the single resonant peak in the un-modulated system.
However, when $\tau$ away from $0$, $\Omega_{n}$ significantly modifies the
signal and induces a non-trivial phenomenon, coherent destruction of tunneling
[37, 38, 39], where the steady excitation probability $\rho_{11}$ is partially
suppressed and arising from the superposition of degenerate Floquet states.
When $\Delta=0$, from Eq. (9) one finds that the conditions for $\rho_{11}$ to
take the local minimum values are $\Omega_{p}=2n\omega,~{}(n=1,2\cdots)$.
Expanding to the more general cases, the positions of the coherent destruction
of tunneling are determined by the relationship between the effective Rabi
frequency and the modulation frequency, i.e.,
$\sqrt{\Omega_{p}^{2}+\Delta^{2}}=2n\omega,~{}(n=1,2\cdots)$, as the white
dashed lines shown in Fig. 1(b).

##  III The Floquet three-level system

Figure 2: (a) Schematic of a three-level system. Based on the above two-level
system, a control-field resonantly couples levels $|1\rangle$ and $|2\rangle$
with a periodically modulated Rabi frequency
$\Omega_{c}(t)=\Omega_{c}(t+\tau)$. $\gamma_{21}$ is the population damping
rate from level $|2\rangle$ to $|1\rangle$. $\gamma_{2}^{\phi}$ is the
dephasing rate of state $|2\rangle$. (b) A contour map of $\rho_{11}$ as a
function of $\Delta$ and $\tau$. The graph is obtained by numerically
calculating the Lindblad master equation Eq. (13) for $\gamma_{10}=1$,
$\gamma_{21}=1.4$, $\gamma_{1}^{\phi}=0.4$, $\gamma_{1}^{\phi}=0.2$
$\Omega_{c}=10.8$ and $\Omega_{p}=1$ [i.e., the vertical yellow dashed line in
Fig. 1 (b)]. The white dashed lines show the positions of the resonance peaks,
i.e., $(\Delta\pm\Omega_{c}/4)\tau=2n\pi,~{}~{}(n=0,1,2\cdots)$. (c), (d) The
steady excitation probability $\rho_{11}$ as a function of the probe field
detuning with $\tau=0.027$ [i.e., the vertical red line in (b)] and
$\tau=0.695$ [i.e., the vertical black line in (b)]. Here we normalized the
parameters in terms of $\gamma_{10}$

.

In this section, we further explore the QIEs in a Floquet three-level system,
as shown in Fig. 2(a). Here we assume the modulation scheme of $\Omega_{c}(t)$
is asynchronous to that of $\Omega_{p}(t)$, i.e.,

$\displaystyle\Omega_{c}(t)=\left\\{\begin{array}[]{ll}0,&~{}~{}~{}~{}~{}t\in[n\tau,(n+\frac{1}{2})\tau]\\\
\Omega_{c},&~{}~{}~{}~{}~{}t\in[(n+\frac{1}{2})\tau,(n+1)\tau]\end{array}\right.$
(12)

with $n=0,1,2\cdots$. Similar to the above two-level system, the Lindblad
master equation can be written as

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i[H(t),\rho]+\sum_{j=1,2}\frac{\gamma_{j,j-1}}{2}(2\sigma_{j-1,j}\rho\sigma_{j,j-1}-\sigma_{jj}\rho-\rho\sigma_{jj})$
(13) $\displaystyle+$
$\displaystyle\sum_{j=1,2}\gamma_{j}^{\phi}(2\sigma_{jj}\rho\sigma_{jj}-\sigma_{jj}\rho-\rho\sigma_{jj}),$

with

$\displaystyle H(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}(-|0\rangle\langle 0|+|1\rangle\langle
1|+|2\rangle\langle 2|)$ (14) $\displaystyle-$
$\displaystyle\left[\frac{\Omega_{p}(t)}{2}|0\rangle\langle
1|+\frac{\Omega_{c}(t)}{2}|1\rangle\langle 2|+h.c.\right].$

Similar to Eq. (4), here we apply RWA by assuming $\omega=1/\tau\ll
2\omega_{10},2\omega_{21}$, $\Omega_{p}\ll\omega_{10}$ and
$\Omega_{c}\ll\omega_{21}$.

To study the steady-state properties of the Floquet three-level system, only
the data at the end of each modulation period (i.e., the time evolution step
is $\tau$) is observed. The dynamics start from the ground state $|0\rangle$,
we show the contour map of $\rho_{11}$ as a function of the probe field
detuning $\Delta$ and the modulation period $\tau$ in Fig. 2(b). One finds
that each peak in Fig.1(b) splits into two when an added control field couples
$|1\rangle$ to $|2\rangle$, resulting in multiple transparency windows, which
is similar to the multi-chromatic ATS with multi-tone control fields studies
in the un-modulated systems [40, 41, 42]. The positions of the peaks, as the
white dashed lines shown in Fig. 2(b), could be demonstrated to be exactly the
maximal constructive interference of transitions
$|+\rangle\leftrightarrow|0\rangle$ and $|-\rangle\leftrightarrow|0\rangle$,
which could be achieved by calculating relative phases $\phi_{0\pm}$ in one
modulation period. Here $|\pm\rangle=(|2\rangle\pm|1\rangle)/\sqrt{2}$ is the
dressed levels induced by the strong control field. During the controlled
half-period, the system is in the dressed basis $|\pm\rangle$ with eigenenergy
$(\Delta\pm\Omega_{c})/2$. The relative phases accumulated in the first half
period are $\phi_{0\pm}^{\prime}=(\Delta\pm\Omega_{c}/2)\tau/2$. During the
non-controlled half period, the system is in bare basis. Driven by a probe
field with detuning $\Delta$, the relative phases $\phi_{0\pm}^{\prime\prime}$
accumulated are
$\phi_{0+}^{\prime\prime}=\phi_{0-}^{\prime\prime}\approx\Delta\tau/2,$ where
$\Omega_{p}\ll\Delta$ is assumed. Then the total relative phases in one period
are
$\phi_{0\pm}=\phi_{0\pm}^{\prime}+\phi_{0\pm}^{\prime\prime}=(\Delta\pm\Omega_{c}/4)\tau$.
The maximal interference requires $\phi_{0\pm}=2n\pi$, $~{}(n=0,1,2\cdots)$.
Therefore, the constraints for absorption peaks are several curves
$\Delta=2n\pi\omega\pm\Omega_{c}/4$, as the white dashed lines shown in Fig.
2(b). The multiple transparency windows may have promising applications in
quantum memory [43], ground-state cooling [44] and so on.

Figure. 2(c) and (d) show the multiple transparency windows more clearly. One
finds the asymmetry of the transparency windows except for the central one and
the multiple transparency windows move closer to the center (i.e., $\Delta=0$)
as $\tau$ increases. Moreover, the central transparency window (CTW) is
similar to the traditional EIT and ATS, but the modulation period as a new
adjustable dimension enriches the interference properties of the CTW.

For the traditional EIT and ATS in the un-modulated systems, it is widely
acknowledged that: ATS exhibits a wide non-interference transparency window
between large Stark splittings induced by a strong field [45, 46], while EIT
exhibits a narrow transparency window induced by destructive quantum
interference (QI) between the Stark splittings [47]. In the weak probe field
approximation, according to Ref. [48] the absorption of the probe field is
determined by the first-order coherence of the probe transition and is given
as

$\text{Im}(\rho_{10})_{\text{QI}}=\frac{\Omega_{p}}{4B}[(\Delta-\frac{\Omega_{c}}{2})^{2}\Gamma+(\Delta+\frac{\Omega_{c}}{2})^{2}\Gamma+2A]$
(15)

where
$A=-[(\frac{\Omega_{c}}{2})^{2}-\Delta^{2}]\Lambda+(\Gamma-\Lambda)^{2}(\Gamma+\Lambda)$
and
$B=[(\Delta+\frac{\Omega_{c}}{2})^{2}+\Gamma^{2}][(\Delta-\frac{\Omega_{c}}{2})^{2}+\Gamma^{2}]-2[(\frac{\Omega_{c}}{2})^{2}-\Delta^{2}+\Gamma^{2}]\Lambda^{2}+\Lambda^{4}$
with
$\Gamma=(\gamma_{10}+\gamma_{21})/4+(\gamma_{1}^{\phi}+\gamma_{2}^{\phi})/2$
and
$\Lambda=(\gamma_{10}-\gamma_{21})/4+(\gamma_{1}^{\phi}-\gamma_{2}^{\phi})/2$.
Here $\Lambda$ induces quantum interference between the transitions
$|+\rangle\leftrightarrow|0\rangle$ and $|-\rangle\leftrightarrow|0\rangle$
(more detail see Appendix C). Then according to the well-known ATS (i.e., non-
interference) [49], $\Lambda$ should be zero and Eq. (15) reduces to

$\text{Im}(\rho_{10})_{\text{ATS}}=\frac{\Gamma\Omega_{p}/4}{(\Delta-\frac{\Omega_{c}}{2})^{2}+\Gamma^{2}}+\frac{\Gamma\Omega_{p}/4}{(\Delta+\frac{\Omega_{c}}{2})^{2}+\Gamma^{2}}.$
(16)

Equation (16) is exactly the sum of two Lorentzian peaks corresponding to
transitions $|+\rangle\leftrightarrow|0\rangle$ and
$|-\rangle\leftrightarrow|0\rangle$ with Stark splitting $\Omega_{c}$. By
comparing the values of $\mathrm{Im}(\rho_{10})_{\mathrm{ATS}}$ and
$\mathrm{Im}(\rho_{10})_{\mathrm{QI}}$ in Eqs. (15) and (16) under $\Delta=0$,
we find that when $0<\Lambda<\Gamma$ ($-\Gamma<\Lambda<0$), it induces
destructive (constructive) interference and shallows (deepens) the absorption
valley. Moreover, when $\Lambda\approx\Gamma$, absorption is almost completely
suppressed (i.e., $\mathrm{Im}(\rho_{10})_{\mathrm{QI}}|_{\Delta=0}\approx 0$)
due to complete destructive interference between transitions
$|+\rangle\leftrightarrow|0\rangle$ and $|-\rangle\leftrightarrow|0\rangle$.
Similarly, one has almost complete constructive interference for
$\Lambda\approx-\Gamma$. When the control field is getting weaker or the
decoherence rates are getting greater, the absorption dip corresponding to the
transparency might disappear. Therefore, the additional condition for the
probe transparency dip to be observed is
$\partial\mathrm{Im}(\rho_{10})_{\mathrm{QI}}/\partial\Delta^{2}|_{\Delta=0}>0$,
giving $\Omega_{c}>2\sqrt{(\Gamma-\Lambda)^{3}/(3\Gamma-\Lambda)}$.

To explore the interference properties of the CTW in our Floquet three-level
system, the AIC method [33] is used to discern the CTW from QI and ATS models
in Eqs. (15) and (16) by evaluating their relative AIC weights for different
modulation periods.

###  III.1 ATS-like profile

Figure 3: Absorption $\mathrm{Im}(\rho_{10})$ as a function of detuning
$\Delta$, and various $\tau$ with best fits to
ATS($\Omega_{c},\Omega_{p}$,$\Gamma$) (black dashed lines) and
QI($\Omega_{c},\Omega_{p}$,$\Gamma$,$\Lambda$) (blue dot dashed lines) models
calculated for $\tau=0.001$ (Bright cyan squares) with a good fit to
ATS$(5.391,0.4543,0.9958)$ as well as QI$(5.396,0.4538,0.9949,-0.0071)$,
$\tau=0.05$ (red triangles) with a better fit to QI$(5.69,0.56,1.17,-0.73)$
than ATS$(4.972,0.6535,1.326)$, $\tau=0.1$ (green circles) with a better fit
to QI$(5.626,0.7424,1.447,-0.8687)$ than ATS$(4.641,0.8731,1.607)$,
$\tau=0.15$ (Magenta asterisks) with a better fit to
QI$(5.847,1.011,1.657,-0.9817)$ than ATS$(4.763,1.231,1.89)$. Here we
normalized the parameters in terms of $\gamma_{10}$

.

In this section, we use the parameters that satisfy the ATS model in Eqs.
(16), i.e., $\gamma_{10}=1$, $\gamma_{21}=1.4$, $\gamma_{1}^{\phi}=0.4$,
$\gamma_{2}^{\phi}=0.2$, $\Omega_{p}=1$ and $\Omega_{c}=10.8$, which is also
consistent with the parameters in Fig. 2. Then $\Gamma=0.9$ and $\Lambda=0$,
where a non-interference transparency window appears for the un-modulated
system. In Fig. 3, we plot the numerical results of $\mathrm{Im}(\rho_{10})$
obtained from Eq. (13) for the Floquet three-level system. Note that here we
only focus on the properties of the CTW and the numerical results is obtained
by calculating the dynamic steady states of the Floquet three-level system
with time evolution step $\tau$. As an objective way to identify the more
appropriate model for the CTW in the Floquet three-level system, we also plot
the fitting profile for each $\tau$ using
QI($\Omega_{c},\Omega_{p}$,$\Gamma$,$\Lambda$) and
ATS($\Omega_{c},\Omega_{p}$,$\Gamma$) models with fitting parameters
$\Omega_{p}$, $\Omega_{c}$, $\Gamma$ and $\Lambda$ in Eqs. (15) and (16) as a
comparison. For small Floquet period, e.g. $\tau=0.001$, ATS and QI fitting
curves merge and both fit well with the simulated absorption. The fitted value
of the interference term $\Lambda$ is close to 0, which indicates that the CTW
in rapidly modulated system has the similar properties to the transparency in
it un-modulated counterparts. This could be understood as the quantum Zeno
effect [50]. For larger Floquet periods, such as $\tau=0.05,0.1,0.15$, we find
that the CTW fits better with the QI model. The fitted interference
$\Lambda<0$, which indicates constructive interference in the CTW. This could
be understood qualitatively as that the periodic driving heat the system and
effectively change the decay rates. However, as the modulation frequency
$\omega$ increases, the heating speed becomes slower, which is known as
Floquet prethermalization [51]. To conclude, in the ATS’s parameter regime,
the CTW is ATS-like but with significant interference adjustable in a wide
range of modulation periods, which differentiates the CTW in the Floquet
three-level system from the traditional ATS.

The above observations of the CTW’s underlying mechanism are supported by
quantitative criteria. To be specific, we apply the AIC method, which uses the
relative entropy to identify the most information model[45]. The information
loss of a given model with $k$ fitting parameters to the numerical data is
quantified by $I=N\mathrm{ln}(R/N)+2k$, where $N$ is the number of numerical
data for fitting and $R$ denotes the fitting residual sum of squares. The per-
point AIC contribution is $\bar{I}=I/N$. Hence the AIC per-point weights are

$\bar{w}_{QI}=\frac{\mathrm{exp}(-\frac{1}{2}\bar{I}_{QI})}{\mathrm{exp}(-\frac{1}{2}\bar{I}_{ATS})+\mathrm{exp}(-\frac{1}{2}\bar{I}_{QI})}$
(17)

for QI model and $\bar{w}_{\mathrm{ATS}}=1-\bar{w}_{QI}$ for ATS model.
Greater weight means more likelihood fitting. For $\tau=0.001,0.05,0.1,0.15$,
$\bar{w}_{QI}=0.51,0.74,0.75,0.64$ respectively. Not surprisingly, for
$\tau=0.001$, $\bar{w}_{QI}\approx\bar{w}_{ATS}$. In general regions of
$\tau$, $\bar{w}_{QI}>\bar{w}_{ATS}$, which means that the QI model is indeed
the more appropriate model for the CTW in the Floquet three-level system.

###  III.2 EIT-like profile

Figure 4: Absorption $\mathrm{Im}~{}(\rho_{10})$ as a function of detuning
$\Delta$, and various $\tau$ with best fits to
QI($\Omega_{c},\Omega_{p}$,$\Gamma$,$\Lambda$) (black solid lines) model
calculated for $\tau=0.001,0.05,0.1,0.2$ with a good fit to
QI$(0.4809,1.812,1.875,1.817)$, QI$(0.7392,1.474,2.31,2.155)$,
QI$(1.018,1.092,2.736,2.529)$ and QI(1.109,0.5408,2.62,2.265) respectively.
Here we normalized the parameters in terms of $\gamma_{10}$

.

In this section, we further consider the parameters that satisfy the QI model
in Eqs. (15), i.e., $\gamma_{10}=1$, $\gamma_{21}=0.1$, $\gamma_{1}^{\phi}=3$,
$\gamma_{2}^{\phi}=0$, $\Omega_{p}=1$ and $\Omega_{c}=3.55$. Then
$\Gamma=1.775,\Lambda=1.725$, which corresponds to almost completely
destructive interference between transitions
$|+\rangle\leftrightarrow|0\rangle$ and $|-\rangle\leftrightarrow|0\rangle$
under $\Delta=0$. Similar to Sec. III.1, Fig. 4 shows the numerical results of
$\mathrm{Im}(\rho_{10})$ and the fitting profile for each $\tau$ using the QI
model in Eqs. (15). Note that here the poor ATS fitting results are ignored.
When $\tau=0.001$, the CTW has a narrow dip at zero detuning, which is similar
to the conventional EIT in the un-modulated system. As the modulation period
increases, the dip becomes shallow, and the modulation weakens the strength of
the destructive interference. Therefore, in the EIT parameter regime, the CTW
is demonstrated to exhibit an EIT-like profile but with interference
adjustable with the modulation period.

##  IV Discussions and conclusions

We investigate the QIEs in the Floquet two- and three-level systems. In the
Floquet two-level system, the monochromatic periodic pulses (e.g. square-wave)
generate equivalent poly-chromatic drivings, which enable lotus-like multi-
peak phenomenon and the coherent destruction of tunneling effect. In the
Floquet three-level system, where the probe and control fields are
asynchronously modulated by the same square-wave pulses, multiple transparency
windows are observed and the quantum interference of the CTW can be tuned by
the modulation periods of the external fields. And the CTW becomes EIT-like or
ATS-like by adjusting the modulation periods of the external fields without
changing the properties of the systems, which will greatly improve the
application prospects of the existing systems. Moreover, the multiple
transparency windows may provide a powerful platform beyond the applications
based on the traditional single transparent window, such as multi-frequency
all-optical switching, which can switch on/off multi-chromatic fields
simultaneously.

The modulation scheme proposed here can be easily implemented experimentally
in various three-level systems, such as atoms gases [52, 53], superconducting
quantum circuits [54], quantum dots [55], nanoplasmonics [56], optomechanics
[57, 58] and so on. For a qutrit, which is a three-level artificial atom in
the superconducting circuits, it can be manipulated by microwave fields. The
modulation period $\tau$ that can be realized experimentally is about dozens
of ns [9], which is much smaller than the coherence time of the system (about
0.5ms) [59]. For such systems, hundreds of periods can be realized within the
coherence time of the system.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grant Nos. U1930201, 91836101, U1801661 and 11934010, the Key-Area
Research and Development Program of Guangdong Province (Grant No.
2018B030326001), Guangdong Provincial Key Laboratory(Grant No.
2019B121203002), the Science, Technology and Innovation Commission of Shenzhen
Municipality (No. JCYJ20170412152620376, No. KYTDPT20181011104202253), Grant
No.2016ZT06D348, and the Natural Science Foundation of Guangdong Province
(Grant No. 2017B030308003). X.-Q.L acknowledges the support by the Foundation
of Hunan Educational Committee (Grant No. 19C1585), the Natural Science
Foundation of Hunan Province of China (Grant No. 2020JJ5466), and the National
Natural Science Foundation of China (Grants No. 12104214).

##  Appendix A The validity of rotating-wave approximation for time-dependent
Hamiltonian

Figure 5:  Comparison of numerical results of $\rho_{11}$ without RWA
[obtained from Eq. (18)] and with RWA [obtained from Eq. (20)] for (a)
$\tau=0.001$, $\Delta=0$, $\Omega_{p}=1$, (b) $\tau=0.15$, $\Delta=0$,
$\Omega_{p}=1$, (c)$\tau=0.001$, $\Delta=40$, $\Omega_{p}=1$, (d)$\tau=0.05$,
$\Delta=0$, $\Omega_{p}=200$. Other parameters are $\omega_{p}=6000$,
$\omega_{10}=\Delta+\omega_{p}$, $\gamma_{1}^{\phi}=0.4$, $\gamma_{10}=1$.
Here we normalized the parameters in terms of $\gamma_{10}$

.

In this section, we verify the validity of RWA with the Floquet two-level
system described in the main text. The exact Hamiltonian is

$\displaystyle
H(t)=H_{0}-[\frac{\Omega_{p}(t)(1+e^{2i\omega_{p}t})}{2}|0\rangle\langle
1|+h.c.]$ (18)

with $H_{0}=\Delta(-|0\rangle\langle 0|+|1\rangle\langle 1|)/2$. Here
exp$(2i\omega_{p}t)$ is the counter-rotating term proportional to exp$[\pm
i(\omega_{10}+\omega_{p}-\Delta)t]$. Substituting Eq. (6) in the main text
into Eq. (18), the coupling term between levels $|0\rangle$ and $|1\rangle$
becomes

$\displaystyle\left[\frac{\Omega_{p}}{2}-\sum_{n=1}^{\infty}(-1)^{n}\Omega_{pn}\cos(\omega_{n}t)\right](1+e^{2i\omega_{p}t}).$
(19)

The counter-rotating term exp$(2i\omega_{p}t)$ causes rotating terms
proportional to exp$[\pm i(\omega_{n}\pm 2\omega_{p})t]$, which is usually a
fast rotating term ($2\omega_{p}\gg\omega_{n}$) that can be ignored. For
example in superconducting circuit (atom) experiments, the frequency of near-
resonant microwave (laser) is several GHz (THz) [60, 61], however, the
modulation frequency is about dozens of MHZ [9]. Another condition for RWA to
be applicable in Eq. (19) is $\Omega_{pn}\ll\omega_{n}+\omega_{p}$, which can
be simplified to $\Omega_{p}\ll\omega_{p}\approx\omega_{10}$, and this is the
scenario we considered. Note that here we focus on the situation for small n,
because the corresponding Rabi frequency $\Omega_{pn}$ decreases sharply as n
increases. Therefore, in our Floquet system, the RWA is validly and in
$\omega_{p}$’s rotating frame, Eq. (18) becomes

$\displaystyle
H(t)_{\text{RWA}}=H_{0}-[\frac{\Omega_{p}(t)}{2}|0\rangle\langle 1|+h.c.].$
(20)

In Fig. 5, we compare the numerical results of $\rho_{11}$ obtained from Eq.
(18) and Eq. (20) for the extensive parameters that we study in the main text,
and find that the RWA results agree well with the exact results. Similarly,
for our Floquet three-level system, the conditions for the RWA to be
applicable are $\omega\ll 2\omega_{10},2\omega_{21}$,
$\Omega_{p}\ll\omega_{10}$ and $\Omega_{c}\ll\omega_{21}$.

##  Appendix B Analytical results of the Floquet two-level system

Figure 6: $(\Omega_{n})^{2}$ [obtained from Eq. (35)] as a function of $\tau$
with $\Omega_{p}=1$. Here we normalized the parameters in terms of
$\gamma_{10}$

.

In the weak driving range ($\Omega_{p}<\omega$), the sidebands are well
separated and show clearly resonance peaks caused by the different frequency
components. The analytical results of these sidebands can be easily obtained
as

$\displaystyle\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{\frac{\gamma^{\prime}_{1}}{2\gamma_{10}}(\frac{\Omega_{p}}{2})^{2}}{(\gamma^{\prime}_{1})^{2}+\Delta^{2}+\frac{\gamma^{\prime}_{1}}{\gamma_{10}}(\frac{\Omega_{p}}{2})^{2}}$
(21)
$\displaystyle+\sum_{n=1}^{\infty}\frac{\frac{\gamma^{\prime}_{1}}{2\gamma_{10}}\Omega_{pn}^{2}}{(\gamma^{\prime}_{1})^{2}+(\Delta+\omega_{n})^{2}+\frac{\gamma^{\prime}_{1}}{\gamma_{10}}\Omega_{pn}^{2}}$

with $\gamma_{1}^{\prime}=\gamma_{10}/2+\gamma_{1}^{\phi}$. Equation (21)
consists of a series of independent Lorentizians, each one with width
$\sqrt{(\gamma^{\prime}_{1})^{2}+\gamma^{\prime}_{1}\Omega_{pn}^{2}/\gamma_{10}}$.

In the strong driving range ($\Omega_{p}>\omega$), the pattern becomes too
complex to get the analytic solution. However, we can get the analytical
expression for the resonance condition ($\triangle=0$) and expand to the
general case. When $\triangle=0$, the optical Bloch equations are

$\displaystyle\frac{\partial V(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma^{\prime}_{1}V,$ (22) $\displaystyle\frac{\partial
U(t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{10}}{2}-\frac{1}{2}(\gamma_{10}+\gamma^{\prime}_{1})U(t)+i\Omega_{p}(t)U(t)$
(23) $\displaystyle-$
$\displaystyle\frac{1}{2}(\gamma_{10}-\gamma^{\prime}_{1})W(t),$
$\displaystyle\frac{\partial W(t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{10}}{2}-\frac{1}{2}(\gamma_{10}+\gamma^{\prime}_{1})W(t)-i\Omega_{p}(t)W(t)$
(24) $\displaystyle-$
$\displaystyle\frac{1}{2}(\gamma_{10}-\gamma^{\prime}_{1})U(t),$

where

$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\rho_{10}(t)+\rho_{01}(t)\right],$ (25)
$\displaystyle U(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\rho_{11}(t)-\rho_{00}(t)+\rho_{01}(t)-\rho_{10}(t)],$
(26) $\displaystyle W(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\rho_{11}(t)-\rho_{00}(t)-\rho_{01}(t)+\rho_{10}(t)].$
(27)

In the absence of damping ($\gamma_{10}=\gamma^{\prime}_{1}=0$), the solutions
of Eqs. (23)and (24) show that the components $U(t)$ and $W(t)$ oscillate with
frequencies $\pm\Omega_{p}(t)$, and their oscillation frequencies differ by
$2\Omega_{p}(t)$. Therefore, in a frame oscillating with $\Omega_{p}(t)$ the
terms proportional to $(\gamma_{10}-\gamma^{\prime}_{1})/2$ oscillate with
$\pm 2\Omega_{p}(t)$. Thus, we can ignore the rapidly oscillating terms, and
obtain the solutions of $U(t)$ and $W(t)$ by direct integration [62],

$\displaystyle U(t)=-\frac{\gamma_{10}}{2}\int_{0}^{t}dt^{\prime
A(t-t^{\prime})}e^{\sum\limits_{n=1}^{\infty}B[\sin(\omega_{n}t^{\prime})-\sin(\omega_{n}t)]},$

where

$\displaystyle
A=-\frac{1}{2}(\gamma_{10}+\gamma^{\prime}_{1}-i\Omega_{p}),~{}~{}B=\frac{i(-1)^{n+1}\Omega_{pn}}{\omega_{n}}.$
(29)

It is seen from Eq. (26) that

$\displaystyle\frac{\rho_{11}(t)-\rho_{00}(t)}{2}=\mathrm{Re}[U(t)],$ (30)

Combined with $\rho_{11}(t)+\rho_{00}(t)=1$, the population of state
$|1\rangle$ is

$\rho_{11}(t)=\frac{1}{2}-\mathrm{Re}[U(t)].$ (31)

From Eq. (26), we also get

$\mathrm{Im}[\rho_{10}(t)]=-\mathrm{Im}[U(t)].$ (32)

Substituting Eq. (B) into Eq. (31) and Eq. (32), and straightforward
calculating the steady solution of $\rho_{11}$ and $\mathrm{Im}(\rho_{10})$

$\displaystyle\rho_{11}(\infty)$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{\gamma_{10}\gamma_{1}}{2}\sum_{n=-\infty}^{\infty}\frac{\Omega_{n}^{2}}{\gamma_{1}^{2}+(\Omega_{p}/2-n\omega)^{2}},$
$\displaystyle\mathrm{Im}[\rho_{10}(\infty)]$ $\displaystyle=$
$\displaystyle\frac{\gamma_{10}}{2}\sum_{n=-\infty}^{\infty}\frac{\Omega_{n}^{2}(\Omega_{p}/2-n\omega)}{\gamma_{1}^{2}+(\Omega_{p}/2-n\omega)^{2}},$
(34)

where

$\displaystyle\Omega_{n}$ $\displaystyle=$
$\displaystyle\sum_{\xi=-\infty}^{\infty}\sum_{l=-\infty}^{\infty}\cdots\sum_{g=-\infty}^{\infty}J_{\xi}\bigg{(}\frac{-2\Omega_{p}}{\omega\pi}\bigg{)}J_{l}\bigg{(}\frac{2\Omega_{p}}{9\omega\pi}\bigg{)}\cdots$
(35) $\displaystyle
J_{g}\bigg{(}\frac{(-1)^{q}2\Omega_{p}}{(2q-1)^{2}\omega\pi}\bigg{)},$

with $\xi=n-3l-\cdots-(2q-1)g$ and
$\gamma_{1}=(\gamma^{\prime}_{1}+\gamma_{10})/2$. From Eqs. (B)-(35), we find
that the steady solutions of $\rho_{11}$ and $\mathrm{Im}(\rho_{10})$ are
related with $\Omega_{n}^{2}$, which is a series of Bessel functions with
variable $\Omega_{p}/\omega$. In fig. 6, we plot $(\Omega_{n})^{2}$ as a
function of $\tau$ with $\Omega_{p}=1$ and find that only $\Omega_{0}^{2}\neq
0$, when $\tau\rightarrow 0$.

##  Appendix C Derivation of QI and ATS models in Eqs. (15) and (16)

In this section, we derive the QI and ATS models in Eqs. (15) and (16) with an
un-modulated Hamiltonian

$H=\frac{\Delta}{2}(|1\rangle\langle 1|+|2\rangle\langle 2|-|0\rangle\langle
0|)-(\frac{\Omega_{p}}{2}|0\rangle\langle
1|+\frac{\Omega_{c}}{2}|1\rangle\langle 2|+h.c.)$ (36)

The dynamics affected by the control field could be better studied in dressed
state representation by replacing the $H(t)$ in Eq. (13) with Eq. (36) [48].
The elements in the density matrix associated with the probed transition
dynamics are

$\displaystyle\dot{\rho}_{-0}$ $\displaystyle=$
$\displaystyle-\left[i(\Delta+\frac{\Omega_{c}}{2})+\Gamma\right]\rho_{-0}-\Lambda\rho_{+0}$
(37) $\displaystyle+$
$\displaystyle\frac{i\Omega_{p}}{2\sqrt{2}}(\rho_{00}-\rho_{--}-\rho_{-+}),$
$\displaystyle\dot{\rho}_{+0}$ $\displaystyle=$
$\displaystyle-\left[i(\Delta-\frac{\Omega_{c}}{2})+\Gamma\right]\rho_{+0}-\Lambda\rho_{-0}$
(38) $\displaystyle+$
$\displaystyle\frac{i\Omega_{p}}{2\sqrt{2}}(\rho_{00}-\rho_{++}-\rho_{+-}).$

From Eqs. (37) and (38), one finds that $\Lambda$ cross-couples the two
dressed states’ dynamics and induces quantum interference between $\rho_{+0}$
and $\rho_{-0}$. Then we give the first-order steady solutions of $\rho_{+0}$
and $\rho_{-0}$ with the weak probe field approximation, where the steady-
state zero-order populations and coherence are
$\rho^{0}_{00}=1,~{}\rho^{0}_{++}=\rho^{0}_{--}=\rho^{0}_{+-}=0$. With these
conditions, Eqs. (37) and (38) become

$\displaystyle\dot{\rho}_{-0}$
$\displaystyle=-[i(\Delta+\frac{\Omega_{c}}{2})+\Gamma]\rho_{-0}-\Lambda\rho_{+0}+\frac{i\Omega_{p}}{2\sqrt{2}},$
(39) $\displaystyle\dot{\rho}_{+0}$
$\displaystyle=-[i(\Delta-\frac{\Omega_{c}}{2})+\Gamma]\rho_{+0}-\Lambda\rho_{-0}+\frac{i\Omega_{p}}{2\sqrt{2}}.$
(40)

This set of equations can be solved by writing in the matrix form,

$\dot{R}=-MR+A$ (41)

with

$\displaystyle R$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\rho_{-0}\\\
\rho_{+0}\end{array}\right),$ (44) $\displaystyle M$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}i(\Delta+\frac{\Omega_{c}}{2})+\Gamma&\Lambda\\\
\Lambda&i(\Delta-\frac{\Omega_{c}}{2})+\Gamma\end{array}\right),$ (47)
$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\frac{i\Omega_{p}}{2\sqrt{2}}\\\
\frac{i\Omega_{p}}{2\sqrt{2}}\end{array}\right),$ (50)

and then integrating

$\displaystyle
R(t)=\int_{-\infty}^{t}e^{-M(t-t^{\prime})Adt^{\prime}}=M^{-1}A.$ (51)

This yields

$\displaystyle\rho_{-0}$ $\displaystyle=$
$\displaystyle\frac{i\Omega_{p}}{2\sqrt{2}}\frac{i(\Delta-\frac{\Omega_{c}}{2})+\Gamma-\Lambda}{[i(-\Delta-\frac{\Omega_{c}}{2})-\Gamma][i(-\Delta+\frac{\Omega_{c}}{2})-\Gamma]-\Lambda^{2}},$
$\displaystyle\rho_{+0}$ $\displaystyle=$
$\displaystyle\frac{i\Omega_{p}}{2\sqrt{2}}\frac{i(\Delta+\frac{\Omega_{c}}{2})+\Gamma-\Lambda}{[i(-\Delta-\frac{\Omega_{c}}{2})-\Gamma][i(-\Delta+\frac{\Omega_{c}}{2})-\Gamma]-\Lambda^{2}}.$

Then the absorption of the probe field is
Im$(\rho_{10})=\mathrm{Im}(\rho_{+0}+\rho_{-0})/\sqrt{2}$.

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