††thanks: corresponding author<EMAIL_ADDRESS>
# The time-domain Landau-Zener-Stückelberg-Majorana interference in optical
lattice clock

Wei-Xin Liu  Institute of Theoretical Physics and Department of Physics, State
Key Laboratory of Quantum Optics and Quantum Optics Devices, Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Tao Wang  Department of Physics, and Center of Quantum Materials and Devices,
Chongqing University, Chongqing, 401331, China  Chongqing Key Laboratory for
Strongly Coupled Physics, Chongqing, 401331, China  Xue-Feng Zhang  Department
of Physics, and Center of Quantum Materials and Devices, Chongqing University,
Chongqing, 401331, China  Chongqing Key Laboratory for Strongly Coupled
Physics, Chongqing, 401331, China  Wei-Dong Li  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

###### Abstract

The interference between a sequence of Landau-Zener (LZ) transitions can
produce Rabi oscillations (LZROs). This phenomenon is a kind of time-domain
Landau-Zener-Stückelberg-Majorana (LZSM) interference. However, experimental
demonstrations of this LZSM interference induced Rabi oscillation are
extremely hard due to the short coherence time of the driven quantum system.
Here, we study theoretically the time-domain LZSM interference between the
clock transition in one-dimensional (1D) 87Sr optical lattice clock (OLC)
system. With the help of both the adiabatic-impulse model and Floquet
numerical simulation method, the LZROs with special step-like structure are
clearly found both in fast- and slow-passage limit in the real experiment
parameter regions. In addition, the dephasing effect caused by the system
temperature can be suppressed with destructive interference in the slow-
passage limit. Finally, we discuss the possible Bloch-Siegert shift while the
pulse time is away from the half-integer and integer periods. Our work
provides a clear roadmap to observe the LZROs on the OLC platform.

###### pacs:

03.65Ud, 03.67. Hk, 42.50.Xa

##  I INTRODUCTION

The transition between two energy states at an avoided level crossing is known
as the Landau-Zener (LZ) transition. When the system is modulated to traverse
the avoided crossing points back and forth, the phase collected between two
levels during the transition may result in constructive or destructive
interference, which is often referred to Landau-Zener-Stückelberg-Majorana
(LZSM) interference Landau (1932); Zener and Fowler (1932); Stückelberg
(1932); Majorana (1932). The LZSM interference is analyzed in a variety of
physical systems Rotvig  _et al._ (1995); Oliver  _et al._ (2005); Sillanpää
_et al._ (2006); Childress and McIntyre (2010); Zhang  _et al._ (2008); van
Ditzhuijzen  _et al._ (2009); Sun  _et al._ (2011); Zhang  _et al._ (2014);
Forster  _et al._ (2014); Liu  _et al._ (2019); Lidal and Danon (2020), and it
provides a possible approach to control the quantum states and build quantum
gates for quantum computing Teranishi and Nakamura (1998); Saito and Kayanuma
(2004); Gaudreau  _et al._ (2012); Cao  _et al._ (2013); Gaitan (2003). In
particular, the LZSM interference process can be used for designing an
interferometer if it can be coupled to the environment Shytov  _et al._
(2003); Huang  _et al._ (2011).

On the other hand, the interference between multiple successive LZ transitions
can lead to periodic oscillations of the population in different energy levels
Shevchenko  _et al._ (2010); Ashhab  _et al._ (2007), which is similar to Rabi
oscillations Rabi (1937). Although this time-domain oscillations of LZSM
interferences are usually named as LZ-Rabi oscillations (LZROs), it presents
more rich and distinctive phenomena Zhou  _et al._ (2014); Neilinger  _et al._
(2016) in comparison with the conventional Rabi oscillations. Meanwhile, they
can also be used to analyze the decay rate of the quantum system Zhou  _et
al._ (2014); De Raedt  _et al._ (2012). In comparison with LZSM interference
which have just been observed in various experiments Oliver  _et al._ (2005);
Sillanpää  _et al._ (2006); Childress and McIntyre (2010); Forster  _et al._
(2014); Li  _et al._ (2013), the time-domain LZROs are hardly realized because
the coherence time of most related systems is practically shorter than
requirement for a long sequence of LZ processes. It has only been demonstrated
by making use of the nitrogen-vacancy (NV) center in diamond Zhou  _et al._
(2014), however, never observed in the atomic system.

The OLC system, based on the doubly forbidden clock transition between
1S0($|g\rangle$) and 3P0($|e\rangle$), is one of the most appropriate
candidates for observing LZROs. The long lifetime of the excited clock state
helps to keep the coherence and can also reduce the atom loss caused by the
spontaneous emission Ludlow  _et al._ (2015); Boyd  _et al._ (2006). On the
other hand, based on the state-insensitive optical trap with “magic”
wavelength, the fairly clean OLC system can finely tailor the target
Hamiltonian Wall  _et al._ (2016); Kolkowitz  _et al._ (2017). Then, the only
task left would be periodically modulating the transition frequency.

Recently, it has been realized in one dimensional 87Sr OLC system by
modulating the lattice frequency around the “magic” wavelength Yin  _et al._
(2021); Lu  _et al._ (2021). Due to the Doppler effect, the dynamics of
internal clock transition is governed by a photon-assisted LZSM Hamiltonian
Shevchenko  _et al._ (2010); Li  _et al._ (2013); Silveri  _et al._ (2017)

$\hat{H}_{\rm
LZSM}^{n_{z},n_{x}}(t)=\frac{\hbar}{2}[\delta+A\omega_{s}\cos(\omega_{s}t)]\hat{\sigma}_{z}+\frac{\hbar}{2}g_{n_{z},n_{x}}\hat{\sigma}_{x},$
(1)

where $\delta$ is the frequency difference between the two-level energy gap
and clock laser, $\omega_{s}$ is the driving frequency, $A$ is the
renormalized dimensionless driving amplitude, and $g_{n_{z},n_{x}}$ is the
clock laser coupling strength of the external energy level
$|n_{z},n_{x}\rangle$ Yin  _et al._ (2021); Blatt  _et al._ (2009a). As
demonstrated in previous work Yin  _et al._ (2021), more than ten Floquet
sidebands are clearly observed with stable spectroscopy sensitivity, and it
can be well understood under the resolved side-band approximation (RSBA) in
the parameter region $\omega_{s}\gg g_{n_{z},n_{x}}$. Meanwhile, the dynamics
of the Floquet sidebands show smooth sinusoidal behavior which follows the
Rabi oscillations with effective coupling strengths modified by Bessel
functions. However, the characteristics of LZROs are not found, such as the
population oscillations with step-like structure.

In this manuscript, we study theoretically the time-domain LZSM interference
between the clock states of the OLC system. Our proposal is based on the
experimental realized driven OLC system with Hamiltonian Eq.(1) Yin  _et al._
(2021). Different from previous paper Yin  _et al._ (2021) working on the
RSBA, we enter into the parameter regions beyond RSBA and make use of the
Floquet theory to solve the system and get accurate numerical results.
Meanwhile, we also give an analytic expression and qualitative explanation
with the analytic approach – adiabatic-impulse model (AIM) Damski and Zurek
(2006); Shevchenko  _et al._ (2010) to describe the time evolution of the
system. The LZSM interference between the clock states is studied in both fast
and slow passage limits, and the temperature effect is well treated. The AIM
results agree well with the numerical results from the Floquet approach, and
the time-domain LZROs are found in both limit.In the slow-passage limit, we
found the analytic expression of coarse-grained oscillation with effective
Rabi frequency is also suitable for the half-integer period in the adiabatic
basis. At last, the fixing of zero detuning in the real experiment parameter
region is analyzed, so that the time-domain LZSM interference can be possibly
experimental observed in the driven OLC platform under proper chosen parameter
region Yin  _et al._ (2021).

The manuscript is organized as follows. In Sec.II, we briefly review the
Landau-Zener problem and related adiabatic-impulse model method. In Sec.IIIA,
we discuss the Landau Zener transition in the first half period. In Sec.IIIB,
we turn to the LZSM interference at one whole period with two avoided crossing
points. In Sec.IIIC, we consider the multi-periods in both slow- and fast-
passage limits, and the time-domain LZROs are observed with effective Rabi
frequency matches well with the analytic results. In Sec. IV, we discuss the
frequency shift during the time-domain LZROs. We make a conclusion in Sec.V. A
brief description of the model of the driven OLC system and the Floquet
approach are introduced in Appendix A and B, respectively.

Figure 1: The energy sketch of the driven two-level system. (a) Energy levels
versus the driving field $\varepsilon(t)=vt$. The two solid blue curves
represent the adiabatic energy levels $\varphi_{\pm}$, and the red dashed
lines show the crossing diabatic levels $|e\rangle$ and $|g\rangle$ . (b) Time
evolution of the energy levels during one period. At the avoided crossing
points, the system undergoes LZ transitions described by unitary transfer
matrix $X$, which are denoted by the orange lines with arrows. The colored
areas indicate the phases $\phi_{1,2,3}$ collected during the adiabatic
evolution. The diabatic energy levels $\pm\varepsilon(t)/2$ are shown by the
red dashed lines.

##  II LANDAU-ZENER PROBLEM

###  II.1 Model

The Landau-Zener problem can be described with generic LZSM-Hamiltonian

$\hat{H}_{\rm
LZSM}(t)=\frac{\hbar}{2}\varepsilon(t)\hat{\sigma}_{z}+\frac{\hbar}{2}g\hat{\sigma}_{x},$
(2)

where $\varepsilon(t)$ is the time-dependent longitudinal driving, $g$ is a
constant coupling, and the eigenstates of Pauli matrix $\hat{\sigma}_{z}$ are
also called diabatic states or basis ${|g\rangle,|e\rangle}$. On the other
hand, if we solve the instantaneous eigenfunction $\hat{H}_{\rm
LZSM}(t)|\varphi(t)\rangle=E(t)|\varphi(t)\rangle$, we can obtain the
instantaneous eigenvalues or eigen-energies
$E_{\pm}(t)=\pm\frac{\hbar}{2}\sqrt{\varepsilon^{2}(t)+g^{2}}$ with
instantaneous eigen-states:

$\displaystyle|\varphi_{+}(t)\rangle$
$\displaystyle=\sin\frac{\theta(t)}{2}|g\rangle+\cos\frac{\theta(t)}{2}|e\rangle,$
$\displaystyle|\varphi_{-}(t)\rangle$
$\displaystyle=\cos\frac{\theta(t)}{2}|g\rangle-\sin\frac{\theta(t)}{2}|e\rangle,$
(3)

where $\theta(t)=\arctan(g/\varepsilon(t))$, and $|\varphi_{\pm}(t)\rangle$
are also named as adiabatic basis. When $g=0$, the energy gap closes at the
zero points of $\varepsilon(t)$. After linearly expanding $\varepsilon(t)$
around the zero point, it can be approximately written as $vt$, where $v$ is
the sweep rate. As shown in Fig. 1(a) in negative $t$ region, the adiabatic
states $|\varphi_{+}(t)\rangle$ and $|\varphi_{-}(t)\rangle$ are same as
diabatic state $|g\rangle$ and$|e\rangle$. In contrast, they will exchange
when passing the energy crossing point (zero point) and entering the positive
$t$ region.

When $g$ is nonzero, the two adiabatic states display an avoided crossing with
the gap $\hbar g$ at $t=0$. When far away from the avoided crossing point
(means when $|\varepsilon(t)|\gg g$), the adiabatic states coincide
asymptotically with the diabatic states (see Fig. 1(a)). In the LZ model, one
can exactly get the final quantum state for arbitrary initial state by
expressing the Schrödinger equation in terms of the Weber functions Zener and
Fowler (1932); Shevchenko  _et al._ (2010). Assuming the system is in the
ground state $|g\rangle$ at initial time $t_{i}=-\infty$, then the probability
in $|g\rangle$ at final time $t_{f}=+\infty$ is exactly given by the famous LZ
formula Landau (1932); Zener and Fowler (1932); Stückelberg (1932)

$P_{\rm LZ}=\exp(-2\pi\chi)$ (4)

with $\chi=g^{2}/(4v)$ called the adiabatic parameter. In the fast-passage
limit ($\chi\ll 1$), $P_{\rm LZ}\approx 1$ means the sweep rate is so fast
that the system evolves along the diabatic state. In the opposite case – slow-
passage limit ($\chi\gg 1$), $P_{\rm LZ}\approx 0^{+}$ indicates the system
follows the adiabatic path.

###  II.2 Adiabatic-impulse model

In the LZ model, the adiabatic states change rather rapidly around the region
of the avoided crossing but approximately keep constant when far from that
region. Thus, the evolution of the system can be approximately treated as
experiencing a non-adiabatic transition just at the avoided crossing point
($t=0$), apart from which the system undergoes a free adiabatic time-
evolution, and it is called adiabatic-impulse model (AIM). The adiabatic
evolution can be described with help of the unitary evolution matrix in the
adiabatic basis

$U_{\phi}=e^{-i\phi(t_{f},t_{i})\hat{\sigma}_{z}},$ (5)

where $\phi(t_{f},t_{i})$ is the dynamical phase accumulated from initial time
$t_{i}$ to final time $t_{f}$

$\phi(t_{f},t_{i})=\frac{1}{2\hbar}\int_{t_{i}}^{t_{f}}[E_{+}(t)-E_{-}(t)]\,dt.$
(6)

Meanwhile, the instantaneous non-adiabatic transition is approximately
governed by the unitary time-independent matrix Shevchenko  _et al._ (2010);
Sillanpää  _et al._ (2006); Silveri  _et al._ (2017)

$X=\begin{pmatrix}\sqrt{1-P_{\rm LZ}}\exp(-i\varphi)&\sqrt{P_{\rm LZ}}\\\
-\sqrt{P_{\rm LZ}}&\sqrt{1-P_{\rm LZ}}\exp(i\varphi)\end{pmatrix},$ (7)

where $\varphi=-\pi/4+\chi(\ln\chi-1)+\arg[\Gamma(1-i\chi)]$, and $\Gamma$ is
the Gamma function. $\varphi$ approaches to $-\pi/4$ in fast-passage limit and
to $-\pi/2$ in slow-passage limit. Then, the dynamics of the Landau-Zener
problem from initial time $t_{i}<0$ to final time $t_{f}>0$ can be proximately
in a description of the AIM with the combined evolution matrices
$U_{\phi(t_{f},0)}XU_{\phi(0,t_{i})}$.

In the AIM, the duration time of LZ transition, which is called the LZ jump
time, at the avoided crossing point is treated as zero. In order to check the
goodness of the approximation, the LZ jump time is defined as $t_{\rm
LZ}=P(\infty)/P^{\prime}(0)$, where $P(\infty)$ is the asymptotic value of the
LZ transition, and $P^{\prime}(0)$ is time derivative of the transition
probability at the crossing point $t=0$ Vitanov (1999). In the diabatic basis,
it proves the following relation as $t_{\rm LZ}\sim 2\sqrt{\pi/v}$ for
$\chi\ll 1$ and $t_{\rm LZ}\sim 2g/v$ for $\chi\gg 1$ Vitanov (1999);
Tayebirad  _et al._ (2010). Apparently, the AIM requires LZ jump time to be
sufficiently shorter than the time interval of successive two LZ transitions.

Figure 2: LZ transition probability in half driving period. The population
probability of state $|e\rangle$ is plotted as a function of time with zero
detuning in (a) fast-passage limit with $A=13.3$, $g/\omega_{s}=0.6$ and (b)
slow-passage limit with $A=21.8$, $g/\omega_{s}=5$. In both figures, the solid
lines are numerical results from the Floquet approach, and the dashed lines
are the asymptotic LZ transition probability $1-\langle P_{\rm
LZ}\rangle_{T}$. The red, blue, black (solid and dashed) lines corresponding
to the results in the temperature 1$\mu$K, 2$\mu$K, 3$\mu$K, respectively. The
time regions between the black dotted lines indicate the jump time of LZ
transition $\langle t_{\rm LZ}\rangle_{T}$, which almost remains unchanged in
the different temperatures above. The orange lines indicate the position of
non-adiabatic X-transition. The insets present the first derivatives of
$P_{e}$ versus time.

##  III LZSM interference in OLC

###  III.1 LZ Transition in Half Period

The OLC system under longitudinal periodical modulation in description of
Hamiltonian Eq.(1) can be taken as two LZ transitions in one period
$T=2\pi/\omega_{s}$ with $\varepsilon(t)=\delta+A\omega_{s}\cos(\omega_{s}t)$
and $g=g_{n_{z},n_{x}}$. When the driving frequency $\omega_{s}$ is larger
than $\delta/A$, the $\varepsilon(t)$ achieves two zero points at
$t_{1}=\arccos[-\frac{\delta}{A\omega_{s}}]/\omega_{s}+nT$ and
$t_{2}=T-t_{1}+nT$, where $n\in\mathbb{Z}$ (see Fig. 1(b)). Then, in the
vicinity of these avoided crossing points, the linear approximation can be
taken, i.e., $\varepsilon(t)\approx\mp vt$ with
$v=A\omega_{s}^{2}\sqrt{1-[\delta/(A\omega_{s})]^{2}}$, so that the AIM can be
utilized.

Different from LZ transition of the one-body problem, the cold atoms in the
OLC platform are trapped in each lattice site and follow the Boltzmann
distribution in the external motional energy levels. Then, the temperature
will cause dephasing because the atoms in different motional states follow
different LZ transitions. Therefore, the asymptotic population $P_{\rm LZ}$
should be modified by thermally averaging with the Boltzmann-weighted
superposition of all external states, that is

$\langle P_{\rm
LZ}\rangle_{T}=\sum_{n_{z},n_{x}}q(n_{z})q(n_{x})\exp(-\frac{\pi
g_{n_{z},n_{x}}^{2}}{2v}),$ (8)

where $q(n_{z,x})$ are Boltzmann factors in different directions, given in Eq.
(29). We also define
$\langle\mathcal{O}\rangle_{T}=\sum_{n_{z},n_{x}}\mathcal{O}(n_{z},n_{x})q(n_{z})q(n_{x})$
hereafter with $\mathcal{O}(n_{z},n_{x})$ is the observable of external state
$|n_{z},n_{x}\rangle$ (see Appendix A). In order to simulate the experiment,
we choose the parameters same as the 1D 87Sr OLC system Yin  _et al._ (2021),
i.e., the longitudinal (transverse) frequency is $\nu_{z}=65$ kHz
($\nu_{x}=250$ Hz), the number of state in longitudinal (transverse) direction
is $N_{z}=5$ ($N_{x}=1300$), and the misalignment angle is $\Delta\theta=10$
mrad. In order to quantitatively verify the temperature effect, we also adopt
the numerical Floquet method (see Appendix. B) Eckardt (2017).The population
probabilities of state $|e\rangle$ as the function of time $t$ are analyzed in
both (a) fast-passage limit at $A=13.3$, $g/\omega_{s}=0.6$ and (b) slow-
passage limit at $A=21.8$, $g/\omega_{s}=5$. As shown in Fig. 2, the numerical
results (solid lines) are approaching to the asymptotic LZ results $1-\langle
P_{\rm LZ}\rangle_{T}$ (dashed lines). In both cases, a clear jump can be
observed around the diabatic states crossing point $t_{1}=T/4$ with zero
detuning. Due to the time interval of the two successive LZ transitions at
$t_{1}$ and $t_{2}$ is $T/2$, the LZ jump time (the region between the black
dotted lines) $\langle t_{\rm LZ}\rangle_{T}\simeq 0.15T$, which is smaller
than $T/2$, is consistent with the step-like structures although it is
strictly not a instantaneous moment. Meanwhile, we can find increasing the
temperature can strongly depress the excitation, and it is because the Rabi
frequency in higher external energy levels is smaller (see Appendix A). On the
other hand, the different evolution behaviors are apparent in slow- and fast-
passage limits. The high probability of excitation in the slow-passage limit
demonstrates the adiabatic process, whereas near zero value in the fast-
passage limit reflects the non-adiabaticity. Meanwhile, the first derivatives
of $P_{e}$ in the insets of Fig. 2 present different behavior in different
limits, and it may result from state changing in the slow-passage limit.
Considering the temperature will not change the step-structure and the
transition point, it is set to be 1$\mu$K in the following sections.

Figure 3: Constructive versus destructive interference in one period. The
population probability of state $|e\rangle$ is plotted as a function of time
with zero detuning in (a) fast-passage limit at $g/\omega_{s}=0.6$, $A=13.3$
for constructive interference (red lines) and $A=11.75$ for destructive
interference (blue lines); (b) slow-passage limit at $g/\omega_{s}=5$,
$A=21.8$ for constructive interference (red lines) and $A=20.2$ for
destructive interference (blue lines). Meanwhile, the dashed lines are
$\langle 4P_{\rm LZ}(1-P_{\rm LZ})\rangle_{T}$. The insets in both figures are
the relationship between the interference phase $\langle\phi_{T}\rangle_{T}$
and $A$, where the red (blue) asterisks corresponding to the constructive
(destructive) interference.

###  III.2 LZSM Interference in One Period

Based on the AIM, the LZ transition can be approximately studied via combined
evolution matrices. Similarly, as shown in Fig. 1(b), the evolution matrices
at the avoided crossing points $t_{1}$ and $t_{2}$ are $X$. Meanwhile, in the
other time regions, the corresponding unitary evolution matrix $U_{\phi}$ is
with different accumulated phase $\phi$. Then, the whole dynamical process is
described by the following matrix

$U_{T}=U_{\phi_{3}}XU_{\phi_{2}}XU_{\phi_{1}}=\begin{pmatrix}\alpha&-\gamma^{*}\\\
\gamma&\alpha^{*}\end{pmatrix},$ (9)

where

$\displaystyle\alpha=(1-P_{\rm LZ})e^{-i\zeta_{+}}-P_{\rm LZ}e^{-i\zeta_{-}},$
(10) $\displaystyle\gamma=-2\sqrt{P_{\rm LZ}(1-P_{\rm
LZ})}\cos(\phi_{2}+\varphi)e^{-i(\phi_{1}-\phi_{3})},$ (11)

with $\zeta_{+}=2\varphi+(\phi_{1}+\phi_{3})+\phi_{2}$ and
$\zeta_{-}=(\phi_{1}+\phi_{3})-\phi_{2}$, $\phi_{i}=\phi(t_{i},t_{i-1})$,
$T=t_{3}-t_{0}$. If assuming the system is prepared in the ground state
$|g\rangle$ at $t_{0}=0$, the population probability of the excited state
$|e\rangle$ after one single period can be calculated as

$P_{T}=|\gamma|^{2}=4P_{\rm LZ}(1-P_{\rm LZ})\cos^{2}(\phi_{2}+\varphi).$ (12)

We can find $P_{T}$ is irrelevant to the $\phi_{1}$ and $\phi_{3}$ which means
only intermediate region between two avoided crossing points takes effect.

The population of excited state oscillates as a function of the interference
phase $\phi_{T}=\phi_{2}+\varphi$ and it can be interpreted physically as a
quantum interference between the two paths along which the system evolves. In
comparison with the Mach-Zehnder interferometer in which interference of paths
is located in real space, the paths of our system are in phase space.
Meanwhile, the avoided crossing acts as a beam splitter and the evolution path
acts as the two arms of the Mach-Zehnder interferometer Oliver  _et al._
(2005); Pezzé  _et al._ (2007). The destructive interference $P_{T}=0$ happens
at $\phi_{T}=(n+1/2)\pi$, while the constructive one with $P_{T}=4P_{\rm
LZ}(1-P_{\rm LZ})$ at $\phi_{T}=n\pi$. For the Hamiltonian Eq. (1), $\phi_{2}$
can be calculated analytically at $\delta=0$ from Eq. (6) Shevchenko and
Omelyanchouk (2006)

$\phi_{2}=\frac{\sqrt{A^{2}\omega_{s}^{2}+g_{n_{z},n_{x}}^{2}}}{\omega_{s}}\mathrm{EllipticE}(\frac{A\omega_{s}}{\sqrt{A^{2}\omega_{s}^{2}+g_{n_{z},n_{x}}^{2}}}),$
(13)

where $\mathrm{EllipticE}()$ is the complete elliptic integral of the second
kind, so the analytic value of $\phi_{T}=\phi_{2}+\varphi$ can be exactly
obtained. In Fig. 3, we show the numerical results of constructive and
destructive interference in one period at zero detuning in (a) fast-passage
limit with $\langle\phi_{T}\rangle_{T}\simeq 4\pi(7/2\pi)$ for constructive
(destructive) interference and (b) slow-passage limit with
$\langle\phi_{T}\rangle_{T}\simeq 7\pi(13/2\pi)$ for constructive
(destructive) interference. The analytic solution Eq. (12) denotes the maximum
of population is $\langle 4P_{\rm LZ}(1-P_{\rm LZ})\rangle_{T}$, and the
numerical results of constructive interference in both limits can be found
oscillating around it (red dashed lines) from Fig. 3. Therefore, with help of
Eq. (13), we can check the relation between the interference phase
$\langle\phi_{T}\rangle_{T}$ and renormalized driving amplitude $A$. As shown
in the inset of Fig. 3, it follows monotonic increase, but also presents nice
linearity in large parameter region. It indicates the LZSM interference in OLC
platform can be flexibly tuned.

###  III.3 Time-domain LZROs for Multi-Period

One of the most advantages of the ultra-cold 87Sr OLC platform is its long
coherence time, so it is straightforward to expect observing the time-domain
LZROs when considering the multiple LZ processes. Following the mathematics of
Appendix. B of Ref. Shevchenko  _et al._ (2010), the time-evolution over N
periods is governed by $N$th power of the single-period evolution operator

$U_{T}^{N}=(U_{\phi_{3}}XU_{\phi_{2}}XU_{\phi_{1}})^{N}.$ (14)

By diagonalizing the operator $U_{T}$, we can obtain

$U_{T}=ME_{T}M^{\dagger}$ (15)

with

$\displaystyle M=\frac{1}{Q}\begin{pmatrix}\alpha^{*}-e^{-i\Phi}&\gamma^{*}\\\
-\gamma&\alpha-e^{i\Phi}\end{pmatrix},$ (16) $\displaystyle
E_{T}=\begin{pmatrix}e^{-i\Phi}&0\\\ 0&e^{i\Phi}\end{pmatrix},$ (17)

where $Q=\sqrt{|r|^{2}+|\alpha-e^{-i\Phi}|^{2}}$, $\cos\Phi={\rm Re}\alpha$,
and thus

$\displaystyle U_{T}^{N}$
$\displaystyle=ME_{T}^{N}M^{\dagger}=\begin{pmatrix}u_{11}&-u_{21}^{*}\\\
u_{21}&u_{11}^{*}\end{pmatrix},$ (18) $\displaystyle u_{11}$
$\displaystyle=\cos N\Phi+i({\rm Im}\alpha)\frac{\sin N\Phi}{\sin\Phi},$ (19)
$\displaystyle u_{21}$ $\displaystyle=\gamma\frac{\sin N\Phi}{\sin\Phi},$ (20)

where $\cos\Phi={\rm Re}\alpha$ Shevchenko  _et al._ (2010). Then the
population probability of state $|e\rangle$ after N periods is

$P_{e}=|u_{21}|^{2}=\frac{|\gamma|^{2}}{|\gamma|^{2}+({\rm
Im}\alpha)^{2}}\sin^{2}N\Phi.$ (21)

We can correspond the integer periods time-dependent factor $\sin^{2}N\Phi$ in
Eq. (21) to a coarse-grained oscillation $\sin^{2}\frac{\Omega}{2}t$ with
frequency $\Omega=\frac{\omega_{s}}{\pi}\arccos|{\rm Re}\alpha|$, when
$\Omega$ is smaller than the driving frequency $\omega_{s}$ Ashhab  _et al._
(2007); Garraway and Vitanov (1997); Neilinger  _et al._ (2016). If ${\rm
Im}\alpha=0$ with solution

$P_{\rm LZ}\sin\zeta_{-}=(1-P_{\rm LZ})\sin\zeta_{+}$ (22)

which is called resonance condition, the amplitude of $P_{e}$ achieves its
maximum value.

In fast-passage limit, $1-P_{\rm LZ}\approx 0^{+}$, the resonance condition
Eq. (22) is proximate $\zeta_{-}=k\pi$. In the large driving amplitude limit
$A\omega_{s}\gg g_{n_{z},n_{x}}$, we can approximately derive an analytic
result Shevchenko  _et al._ (2010); Ashhab  _et al._ (2007)

$\zeta_{-}\simeq\frac{1}{2}\int_{0}^{T}\delta+A\omega_{s}\cos(\omega_{s}t)\,dt=\frac{\delta\pi}{\omega_{s}},$
(23)

so the resonance condition changes into $\delta=k\omega_{s}$ which is same as
the $k$th Floquet sideband in Ref. Yin  _et al._ (2021) with resolved Floquet
sideband approximation (RFSA) in the condition $\omega_{s}\gg
g_{n_{z},n_{x}}$. Meanwhile, the oscillation frequency can be expressed as

$\Omega=g_{n_{z},n_{x}}\sqrt{\frac{2}{A\pi}}\cos\left[A-\frac{\pi}{4}(2k+1)\right]$
(24)

and it will approach to the effective Rabi frequency of $k$th Floquet sideband
$g_{k}^{\rm eff}=g_{n_{z},n_{x}}J_{k}(A)$ Yin  _et al._ (2021) because of the
asymptotic behavior of the Bessel functions:
$J_{k}(A)\sim\sqrt{\frac{2}{A\pi}}\cos\left[A-\frac{\pi}{4}(2k+1)\right]$ at
$A\gg|k|$.

In the slow-passage limit $P_{\rm LZ}\approx 0$, resonance condition is
proximate $\zeta_{+}=k\pi$. Assuming $A\omega_{s}\gg g_{n_{z},n_{x}}$ and
$\delta=0$, we get $\zeta_{+}\simeq 2A-\pi$ Shevchenko  _et al._ (2010);
Neilinger  _et al._ (2016).Then the resonance condition for constructive
interference takes the form $A=\frac{\pi}{2}(2k+1)$ with integer k and the
corresponding oscillation frequency is
$\Omega=\frac{2\omega_{s}}{\pi}\sqrt{P_{\rm LZ}}$. Moreover, the resonance
condition for destructive interference is $A=k\pi$, which corresponds to
$P_{T}=0$ in Eq. (12).

Figure 4: The multiple driving periods in fast-passage limit. The population
probability of state $|e\rangle$ is plotted as a function of time with zero
detuning. The parameters for red and blue lines are the same as those in Fig.
3(a). The black dashed line shows the coarse-grained oscillation with the Rabi
frequency in Eq. (24) for constructive interference. Destructive interference
corresponding to CDT is shown with blue solid lines. All the solid lines are
numerical results from the Floquet method.

In order to check the analysis above based on the AIM, we also implement the
numerical simulation of long-time evolution. In the fast-passage limit, as
shown in Fig. 4, we plot the population probability of excited state $P_{e}$
as the function of time t. We can see that $P_{e}$ exhibits a clear
oscillation with step-like structure, and it matches well with the Rabi
oscillation with effective frequency given in Eq. (24) (black dashed line).
Meanwhile, because of the dephasing effect caused by temperature, the
magnitude of $P_{e}$ is decreasing over time. On the other hand, the blue
solid line indicates the destructive interference between diabatic states,
which also corresponds to the so-called coherent destruction of tunneling
(CDT) Grossmann  _et al._ (1991). The CDT phenomena can also be understood
with the help of the Rabi frequency Eq. (24). The destructive interference
with $A=11.75,k=0$ gives $J_{0}(11.75)\simeq 0$ and $\Omega\simeq 0$. So the
CDT can also be interpreted as the disappearance of the effective Rabi
frequency of 0th Floquet sideband Kayanuma (1994).

Figure 5: The time-domain LZSM interference fringes in the fast-passage limit.
The excited-state population probability $P_{e}$ is plotted as a function of
driving amplitude $A$ and the detuning $\delta$ at (a) $t=1.5T$ and (b) $t=3T$
and $g/\omega_{s}=0.6$. The red dashed lines denote the boundary of the LZSM
interference: the region above them indicates the system can experience
avoided crossing.

Meanwhile, the time-domain interference fringes in the fast-passage limit at a
certain driving time are shown in Fig. 5. Compared with the Fig. 7 of Ref.
Shevchenko  _et al._ (2010), which is the time-averaged LZSM interference,
here we show the interference pattern at different evolution times. We can see
that along with the increase of driving amplitude $A$, more interference
patterns can be observed, which correspond to more Floquet sidebands being
visible. Meanwhile, because of the coarse-grained Rabi oscillation, the
probability $P_{e}$ is monotonic increasing in the first half Rabi period
$T_{\Omega}=2\pi/\Omega$. Thus, stronger interference requires longer
evolution time but shorter than $T_{\Omega}$, as demonstrated in Fig. 5(b).

In addition, we find the interference pattern is asymmetry when measuring time
is set at integer period ($NT$), but symmetry at half-integer period
($(N+1/2)T$), where $N$ is an integer, as shown in Fig. 5. This symmetry can
be interpreted with the difference of collected dynamic phases during the time
evolution. From Fig. 6, one could see that, when the measuring time is set at
$NT$, the picked up effective dynamical phase is
$N\phi_{2}+(\phi_{1}+\phi_{3})(N-1)$ (Fig. 6(a)), while at the measuring time
$(N+1/2)T$, the picked up effective dynamical phase is
$N\phi_{2}+N(\phi_{1}+\phi_{3})$ (Fig. 6(b)). From the Eq. (6) we can see
that, when $\delta$ is replaced by $-\delta$, the dynamic phase $\phi_{2}$ and
$\phi_{1}+\phi_{3}$ are exchanged with each other. Thus the total effective
picked up dynamical phase at measuring time $(N+1/2)T$ will be the same, while
at $NT$ it will be different, which accounts for the asymmetry.

Figure 6: Time evolution of the energy levels with (a) $2T$ and (b) $2.5T$.
The colored areas indicate the effective dynamic phases $\phi_{1,2,3}$
collected during the adiabatic evolution.

Figure 7: The multiple driving periods in slow-passage limit. The population
probability of excited state $P_{e}$ in diabatic basis (blue solid lines) and
$P_{+}$ in adiabatic basis (red solid lines) are plotted as a function of time
with zero detuning. The parameters in (a) and (b) correspond to red and blue
line in Fig. 3(b), respectively. The black dashed line shows the coarse-
grained oscillation with the effective Rabi frequency
$\Omega=\frac{\omega_{s}}{\pi}\arccos|{\rm Re}\alpha|$ for constructive
interference. All the solid lines are numerical results from the Floquet
method.

In the slow-passage limit, the population of the excitation $P_{e}$ follows
the coarse-grained oscillation at the integer-period time, but not the half
integer ones, as demonstration of Fig. 7(a). It is because the evolution of
the system is almost adiabatic so that the population changes drastically
between diabatic states at every avoiding crossing point. Considering the
corresponding probability of excited state $P_{+}$ in the adiabatic basis is
small in the first half period, we expect it can follow the AIM result in the
whole region. As pointed in the Fig. 5 of Ref. Shevchenko  _et al._ (2010) in
the slow-passage limit, here we also plot the excited probability $P_{+}$ of
the state $|\varphi_{+}(t)\rangle$ versus the time t. In Fig. 7, we compare
the population probability in both diabatic and adiabatic basis for both
constructive and destructive interference. For the constructive case shown in
Fig. 7(a), we can see the step-like structure associated with multiple LZ
transitions, but the long-time dynamics are irregular in diabatic basis. In
contrast, after transferring to the adiabatic basis, we see clearly that all
the steps follow the LZ Rabi oscillations (black dashed line) including the
half period. In fact, considering the constructive interference under
resonance condition Eq. (22) and zero detuning, the population of adiabatic
state $P_{+}$ at N $+1/2$ periods can be expressed as
$P_{+}^{\prime}=\sin^{2}N\Phi+P_{\rm LZ}\cos 2N\Phi+\sqrt{P_{\rm LZ}(1-P_{\rm
LZ})}\sin 2N\Phi$ with the combined evolution operator
$U_{\phi_{2}/2}XU_{\phi_{1}}U_{T}^{N}$. Considering the small $P_{\rm LZ}$ in
the slow-passage limit, we can approximate $P_{+}^{\prime}$ as
$\sin^{2}N\Phi$, which gives the same Rabi oscillation frequency as the N
periods results.

Figure 8: Destructive interference in long evolution time for slow-passage
limit compared with the non-driven time evolution. The population probability
of excited state $|e\rangle$ are plotted with time. The parameters in (a) are
the same as those of blue line in Fig. 7(b) except $A=0$ for red solid line.
And the parameters for blue (red) solid line in (b) are chosen with
$g/\omega_{s}=8$, $A=28.62$ ($A=0$), and temperature $T_{z}=T_{x}=1\mu$K. The
parameters for blue (red) solid line in (c) are chosen with $g/\omega_{s}=8$,
$A=19.56$ ($A=0$), and temperature $T_{z}=T_{x}=3\mu$K. All the solid lines
are numerical results from the Floquet method.

For the destructive case shown in Fig. 7(b), the effect CDT is visible in the
adiabatic basis. In comparison, the population probability of excited state in
diabatic basis $P_{e}$ also shows CDT at integer period, but has high
amplitude and also the time-domain step-like structures at half-integer
period. Meanwhile, interestingly, no apparent dephasing is observed. In order
to verify it, we prolong the evolution time up to fifty-period shown in Fig.
8(a). We can see the magnitude of $P_{e}$ (see blue solid line in Fig. 8(a))
is nearly one without any decay even after fifty driving periods. For
comparison, we also plot the population probability of excited state
$|e\rangle$ (red solid line in Fig. 8(a)) under the non-driven situation (that
is, $A=0$), which decay to $0.5$ very fast. We also show this non-dephasing
effect by enlarging the coupling strength $g$ and elevating the system
temperature. In Fig. 8(b) we plot the population probability of excited state
$|e\rangle$ with blue (red) solid line with $g/\omega=8$, $A=28.62$ ($A=0$) at
the system temperature $T_{z}=T_{x}=1\mu$K, corresponding to the destructive
interference (blue solid line) with $\langle\phi_{T}\rangle_{T}\simeq
19/2\pi$; in Fig. 8(c) we plot the population probability of state $|e\rangle$
with blue (red) solid line with $g/\omega=8$, $A=19.56$ ($A=0$) at the system
temperature $T_{z}=T_{x}=3\mu$K, corresponding to
$\langle\phi_{T}\rangle_{T}\simeq 13/2\pi$ (blue solid line). From both of
them we still can not observe any signature of dephasing effect for
destructive interference. These results imply that the destructive
interference in slow-passage limit may suppress the dephasing effect of clock
transition caused by the temperature.

##  IV Frequency shift

Based on the simulation and analysis above, the time-domain LZRO could be well
observed in real experiment parameter regions. However, the step-like
structure may cause the frequency shift, so that the zero detuning can not be
fixed. In order to check it, we simulated the excitation probability as a
function of detuning at (a) $t=1.5T$ and (b) $t=3T$ with $A=13.3$,
$g/\omega_{s}=0.6$ and $1\mu$K. In Fig. 9, we can see the excitation
probabilities for zero detuning are well recognized and thus could be easily
chosen to fix the detuning of clock laser.

Figure 9: The population probability of $|e\rangle$ as the function of
detuning at $t=1.5T$ (red line) and $t=3T$ (blue line) with $A=13.3$,
$g/\omega_{s}=0.6$ at 1$\mu$K. The solid lines are the numerical results from
Floquet approach, and the colored diamonds indicate the results from AIM.

However, if we consider the pulse duration away from the integer and half-
integer period, as shown in Fig. 10(a) near one-period driving time, there
exist obvious shifts in the locations of the peaks corresponding to the
generalized Bloch-Siegert shift Tuorila  _et al._ (2010). In contrast, the
spectra near four-period driving time in Fig. 10(b) do not show any deviations
of the peak positions. Thus, the step-like structure of time-domain LZRO
should be with distortion at short evolution time, but more obvious at longer
evolution time in the fast-passage limit. In addition, here we only consider
the region where the driving amplitude is much larger than the coupling
strength, that is, $A\omega_{s}\gg g_{n_{z},n_{x}}$ due to these reasons, (i)
small $A\omega_{s}$ will give large LZ jump time $t_{\rm LZ}$, and then can
deteriorate the validity of AIM; (ii) when $A\omega_{s}$ is large enough, the
diabatic and adiabatic states can be treated coincide except in a small region
around the avoided crossing. This is not only consist with AIM but also
convenient for the further experimental treatment, because one can transform
the state populations from the diabatic basis (the measurement basis) to the
adiabatic basis more straightforwardly Zhou  _et al._ (2014).

Figure 10: The population probability of $|e\rangle$ as the function of
detuning near one-period driving time (a) and four-period driving time (b)
with $A=13.3$, $g/\omega_{s}=0.6$ at $1\mu$K, calculated with numerical
Floquet approach.

##  V CONCLUSION AND DISCUSSION

With the help of the AIM analytic approach and the numerical Floquet theory
method, we study theoretically the periodically modulated one dimensional
optical lattice clock system. In both slow- and fast-passage limits, the
analytic method matches well with the numerical approach, so that we can
locate the proper parameter regions to observe the time-domain LZROs in the
real experimental region. In the fast-passage limit with zero detuning, if the
constructive condition is fulfilled, the time-domain LZROs follow the coarse-
grained oscillation with effective frequency which can be directly obtained
from the analytic formula. Meanwhile, the CDT effect can be clearly observed
if the destructive interference condition satisfies. On the other hand, in the
slow-passage limit, the coarse-grained oscillation curve is proved to be more
in accord with the population probability of excited state in the adiabatic
basis, which is quite different from the fast-passage limit case. In
particular, the long time evolution in slow-passage limit demonstrates the
dephasing effect is suppressed with destructive interference.

All the time-domain LZROs we simulated theoretically here are feasible in the
real experiment parameter region Yin  _et al._ (2021), so it can be directly
used for guiding the experiment. Based on our discussion of the frequency
shift, the time-domain LZROs is more easily detectable in the fast-passage
limit with long time evolution. Our work not only pave the way for observation
of LZROs in the OLC platform, but also shed a light on the quantum metrology
based on the LZSM interference in the atomic system. Additionally, the
frequency shift discussed in Sec.V opened an interesting issue to be explored
about the generalized BS shifts in the OLC system in the future. Moreover, our
theory can build a fundamental theoretical framework, serve as the latter for
adapting those methods to other situations of interest, such as more
complicated optical lattice geometries, or different forms of modulation.

##  VI ACKNOWLEDGMENTS

This work is supported by Special Foundation for theoretical physics Research
Program of China (No. 11647165) and China Postdoctoral Science Foundation
Funded Project (Project No. 2020M673118). X.-F. Z. acknowledges funding from
the National Science Foundation of China under Grants No. 11804034, No.
11874094 and No.12047564, Fundamental Research Funds for the Central
Universities Grant No. 2020CDJQY-Z003 and 2021CDJZYJH-003. W.-D. L.
acknowledges the funding from the National Natural Science Foundation of China
under Grant No. 11874247, the National Key Research and Development Program of
China, Grant No. 2017YFA0304500, and the Program of State Key Laboratory of
Quantum Optics and Quantum Optics Devices, China, Grant No. KF201703, and the
support from Guangdong Provincial Key Laboratory, Grant No. 2019B121203002.

##  Appendix A Hamiltonian of the driven OLC system

Here we will give a brief introduction to how we treat the driven OLC with
both the internal and external degrees of freedom. One can also get a more
detailed derivation in the Supplemental material of Ref. Yin  _et al._ (2021).

Under the lattice laser modulation
$\omega_{L}(t)=\bar{\omega}_{L}+\omega_{a}\sin(\omega_{s}t)$, the intensity of
the lattice laser along the $z$ direction could be described as
$I=I_{0}\sin^{2}[\bar{\omega}_{L}(z+\int v(t)dt/c)]$, where
$v(t)\simeq\omega_{a}\omega_{s}L\cos(\omega_{s}t)/\bar{\omega}_{L}$ is the
effect lattice velocity, $L=0.3$ m denotes the distance between the center of
the MOT and the HR mirror. So in the lattice co-moving frame, the frequency of
the clock laser that the atom feels needs to shift to
$\omega_{p}^{\prime}\simeq[1-v(t)/c]\omega_{p}$ duo to the relativistic
Doppler effect. This modulating pattern does not change the external trapping
potential. Considering a small misalignment angle $\Delta\theta$ along the
transverse $x$-direction (we can choose the $\Delta\theta$ along $x$-axis
because the transverse trap is isotropic) between the clock laser axis and the
lattice axis in real experiments, the external Hamiltonian with lattice
potential having a Gaussian profile in the $x$ direction can be expressed as

$\hat{H}_{\rm
ext}=\frac{\hat{p}^{2}}{2m}-U_{0}\sin^{2}\left(\frac{\bar{\omega}_{L}z}{c}\right)e^{-\frac{2x^{2}}{w_{0}^{2}}},$
(25)

where $U_{0}=4\alpha_{0}P_{0}/(\pi c\varepsilon_{0}w_{0}^{2})$ is the lattice
depth, $x$ denotes the transverse distance from the lattice axis. Due to the
large lattice depth $U_{0}$, we can neglect the intersite tunneling. With the
harmonic approximation, the external Hamiltonian can be expressed as

$\hat{H}_{\rm
ext}\simeq\frac{\hat{p}^{2}}{2m}-\left(\frac{U_{0}\bar{\omega}_{L}^{2}}{c^{2}}z^{2}+\frac{2U_{0}}{w_{0}^{2}}x^{2}\right).$
(26)

This external Hamiltonian has the harmonic oscillator eigenstates
$|n_{z},n_{x}\rangle$ with the corresponding eigenenergies

$E_{n_{z},n_{x}}=h\nu_{z}(n_{z}+1/2)+h\nu_{x}(n_{x}+1),$ (27)

where $\nu_{z}=\sqrt{U_{0}\bar{\omega}_{L}^{2}/(2m\pi^{2}w_{0}^{2})}$,
$\nu_{x}=\sqrt{U_{0}/(m\pi^{2}w_{0}^{2})}$ are the longitudinal and transverse
trap frequencies, respectively. The number of the harmonic oscillator states
in the trap is approximately given by $N_{z}N_{x}^{2}$, where $N_{z,x}\simeq
U_{0}/(h\nu_{z,x})$ is the number of states in the longitudinal (transverse)
direction. Meanwhile, the internal Hamiltonian (after the rotating wave
approximation) can be written as the LZSM Hamiltonian Eq. (1) Yin  _et al._
(2021), in which $\delta=\omega_{0}-\omega_{p}$ is the frequency detuning of
the clock laser, $A=\hbar\omega_{a}\omega_{p}L/(c\bar{\omega}_{L})$ is the
renormalized driving amplitude, which is dimensionless and independent of
driving frequency $\omega_{s}$.
$g_{n_{z},n_{x}}=ge^{-\eta_{z}^{2}/2}e^{-\eta_{x}^{2}/2}L_{n_{z}}(\eta_{z}^{2})L_{n_{x}}(\eta_{x}^{2})$
is the coupling strength in the external state $|n_{z},n_{x}\rangle$ Blatt
_et al._ (2009b), where $L_{n}$ is the Laguerre polynomial.
$\eta_{z}=\sqrt{h/(2m\nu_{z})}/\lambda_{p}$,
$\eta_{x}=\sqrt{h/(2m\nu_{x})}\Delta\theta/\lambda_{p}$ are the Lamb-Dicke
parameters.

Due to the temperature being a few $\mu$K in the system, we can consider a
normalized Boltzmann distribution to describe the cold atoms in the external
states $|n_{z},n_{x}\rangle$. Then the clock transition probability can be
expressed by

$\langle P_{e}\rangle_{T}=\sum_{n_{z},n_{x}}q(n_{z})q(n_{x})P_{e}$ (28)

as a Boltzmann-weighted superposition of single external state transition
probability $P_{e}$ obtained from the Hamiltonian Eq. (1), where $q(n_{z})$
($q(n_{x})$) are the Boltzmann-weights corresponding to the longitudinal
(transverse) temperature $T_{z}$ ($T_{x}$)

$q(n_{z,x})=\frac{e^{-E_{n_{z},x}/(k_{B}T_{z,x})}}{\sum_{n_{z,x}}e^{-E_{n_{z},x}/(k_{B}T_{z,x})}}$
(29)

with $E_{n_{z,x}}=(n_{z,x}+1/2)h\nu_{z,x}$ is the energy of motional state
$|n_{z,x}\rangle$, $k_{B}$ is the Boltzmann constant.

##  Appendix B Floquet approach

The dynamics of internal clock transition governed by the time-periodic LZSM
Hamiltonian Eq. (1) can also be solved by the Floquet theory, which provides
an accurate approach to deal with the periodically driven quantum system
Shirley (1965); Eckardt (2017); Bukov  _et al._ (2015).

According to the Floquet theorem, the quantum system described by the time-
periodic Hamiltonian $\hat{H}(t)=\hat{H}(t+T)$, gives rise to generalized
stationary states called Floquet states. These Floquet states are the
solutions of the periodically time-dependent Schrödinger equation

$i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle$
(30)

and have the form

$|\psi_{\alpha}(t)\rangle=|u_{\alpha}(t)\rangle
e^{-\frac{i}{\hbar}\varepsilon_{\alpha}t},$ (31)

where $\varepsilon_{\alpha}$ are the quasienergies, which are the eigenvalues
of the so-called Floquet Hamiltonian
$\hat{H}_{F}=\hat{H}(t)-i\hbar\partial/\partial t$:

$\hat{H}_{F}|u_{\alpha}(t)\rangle=\varepsilon_{\alpha}|u_{\alpha}(t)\rangle$
(32)

and $|u_{\alpha}(t)\rangle$ are the Floquet modes, which have the same
periodicity with the Hamiltonian, i.e.
$|u_{\alpha}(t+T)\rangle=|u_{\alpha}(t)\rangle$.

The Floquet states are also the eigenstates of the time-evolution operator
over integer driving periods with the eigenvalues
$\exp(-i\varepsilon_{\alpha}nT/\hbar)$ ($n$ is an arbitrary integer):

$\hat{U}(t_{0}+nT,t_{0})|\psi_{\alpha}(t_{0})\rangle=e^{-\frac{i}{\hbar}\varepsilon_{\alpha}nT}|\psi_{\alpha}(t_{0})\rangle,$
(33)

$\hat{U}(t_{2},t_{1})$ is the time evolution operator from $t_{1}$ to $t_{2}$.
The Floquet states can be computed with the time-evolution operator,
$|\psi_{\alpha}(t)\rangle=\hat{U}(t,t_{0})|\psi_{\alpha}(t_{0})\rangle$. Note
that the eigenvalue $\exp(-i\varepsilon_{\alpha}T/\hbar)$ does not depend on
the initial time $t_{0}$. So the time-evolution operator can be composed by
these complete and orthogonal Floquet states at any fixed time $t$

$\hat{U}(t,t_{0})=\sum_{\alpha}e^{-\frac{i}{\hbar}\varepsilon_{\alpha}(t-t_{0})}|u_{\alpha}(t)\rangle\langle
u_{\alpha}(t_{0})|.$ (34)

Now consider the state

$|u_{\alpha,n}(t)\rangle=|u_{\alpha}(t)\rangle e^{in\omega_{s}t},$ (35)

which gives physically equivalent state to $|u_{\alpha}(t)\rangle$, it is also
a solution to Eq.(32) with the shifted quasienergy
$\varepsilon_{\alpha,n}=\varepsilon_{\alpha}+n\hbar\omega_{s}$, but the
corresponding Floquet state is not altered

$|\psi_{\alpha}(t)\rangle=|u_{\alpha,n}(t)\rangle
e^{-\frac{i}{\hbar}\varepsilon_{\alpha,n}t}=|u_{\alpha}(t)\rangle
e^{-\frac{i}{\hbar}\varepsilon_{\alpha}t}.$ (36)

Now considering the system is initially in the ground state $|g\rangle$, which
means $|\psi_{\alpha}(t_{0})\rangle=|g\rangle$, then the population
probability of the state $|e\rangle$ at time $t$ can be obtained by

$P_{e}=|\langle e|\hat{U}(t,t_{0})|g\rangle|^{2}.$ (37)

Then the problem turns to finding out the Floquet modes
$|u_{\alpha}(t)\rangle$, and the corresponding quasienergy
$\varepsilon_{\alpha}$. The Eq. (34) constitutes an eigenvalue problem in an
extended Hilbert space Shirley (1965); Sambe (1973), which is given by the
product space of the original Hilbert space and the time-dependent Fourier
space. Here this extended Hilbert space can be constructed by:

$[|g\rangle,|e\rangle]\otimes[1,e^{\pm i\omega_{s}t},e^{\pm
2i\omega_{s}t}\ldots]$ (38)

In this extended Hilbert space,

$|u(t)\rangle=\sum_{n=-\infty}^{+\infty}|u^{n}\rangle e^{in\omega_{s}t},$ (39)

where each $|u^{n}\rangle=(u_{p}^{n},u_{s}^{n})^{\rm T}$ (the ${\rm T}$ here
denotes a transpose) is a two-component vector, and the component of the
Floquet Hamiltonian $\hat{H}_{F}$ can be represented as

$\displaystyle\hat{H}_{F}^{m-n}$
$\displaystyle=\frac{\omega_{s}}{2\pi}\int_{0}^{\frac{2\pi}{\omega_{s}}}e^{-im\omega_{s}t}\left[\hat{H}_{\rm
LZSM}^{n_{z},n_{x}}(t)-i\hbar\frac{\partial}{\partial
t}\right]e^{in\omega_{s}t}\,dt$
$\displaystyle=\hat{H}_{m-n}+n\hbar\omega_{s}\delta_{m,n},$ (40)

where $\hat{H}_{m-n}+n\hbar\omega_{s}\delta_{m,n}\hat{I}$ ($\hat{I}$ is a
2$\times 2$ identity matrix) forms the $m$th row and $n$th column of the
Floquet block. For clarity, the Floquet Hamiltonian can be visualized in this
extended Hilbert space as the following matrix form

$\left(\begin{array}[]{ccccc}\ddots&\vdots&\vdots&\vdots&\\\
\cdots&\hat{H}_{0}-\hbar\omega_{s}&\hat{H}_{-1}&0&\cdots\\\
\cdots&\hat{H}_{+1}&\hat{H}_{0}&\hat{H}_{-1}&\cdots\\\
\cdots&0&\hat{H}_{+1}&\hat{H}_{0}+\hbar\omega_{s}&\cdots\\\
&\vdots&\vdots&\vdots&\ddots\end{array}\right),$ (41)

where

$\hat{H}_{0}=\frac{\hbar}{2}\begin{pmatrix}\delta&g_{n_{z},n_{x}}\\\
g_{n_{z},n_{x}}&-\delta\end{pmatrix},$
$\hat{H}_{+1}=\hat{H}_{-1}=\frac{\hbar}{4}\begin{pmatrix}A\omega_{s}&0\\\
0&-A\omega_{s}\end{pmatrix},$

and the blocks with $|m-n|$ larger than 1 are zero.

Now, we can see that the time-dependent Schrödinger equation is transformed
into an eigenvalue equation of an infinite dimensional but time-independent
Hamiltonian with infinitely repeating block structure. Obviously, this
eigenvalue problem has an infinite number of solutions, but we can see that
those infinite solutions can be divided into two groups, each of which are
related to each other, and the same group of the solutions generate the
identical Floquet state by Eq. (36). So we can truncate the number of the
Floquet blocks that converged sufficiently to get the eigenvalues of this
infinite dimensional Hamiltonian, of which just two solutions are needed. In
order to get the numerical solutions in Sec.III and IV, $161$ Floquet blocks
($n=-80$ to $80$) are truncated, and we choose the two solutions with $n=0$ to
get the transition probability with Eq. (34) and Eq. (37). It is worthwhile to
note that we have not made any approximation to solve the original time-
periodic Hamiltonian. This should not be confused with the Floquet-Magnus
expansion method which is only applicable in high frequency regime Eckardt
(2017). In contrast, the Floquet approach used in our manuscript can be
applied to any parameter regime.

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