# On the different Floquet Hamiltonians in a periodic-driven Bose-Josephson
junction

Xiaoshui Lin<EMAIL_ADDRESS>CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, 230026,
China  Zeyu Rao  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, 230026, China  Ming Gong
<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, 230026, China  Synergetic Innovation
Center of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Hefei National Laboratory,
University of Science and Technology of China, Hefei 230088, China

###### Abstract

The bosonic Josephson junction, one of the maximally simple models for
periodic-driven many-body systems, has been intensively studied in the past
two decades. Here, we revisit this problem with five different methods, all of
which have solid theoretical reasoning. We find that to the order of
$\omega^{-2}$ ($\omega$ is the modulating frequency), these approaches will
yield slightly different Floquet Hamiltonians. In particular, the parameters
in the Floquet Hamiltonians may be unchanged, increased, or decreased,
depending on the approximations used. Especially, some of the methods generate
new interactions, which still preserve the total number of particles; and the
others do not. The validity of these five effective models is verified using
dynamics of population imbalance and self-trapping phase transition. In all
results, we find the method by first performing a unitary rotation to the
Hamiltonian will have the highest accuracy. The difference between them will
become significate when the modulating frequency is comparable with the
driving amplitude. The results presented in this work indicate that the
analysis of the Floquet Hamiltonian has some kind of subjectivity, which will
become an important issue in future experiments with the increasing of
precision. We demonstrate this physics using a Bose-Josephson junction, and it
is to be hoped that the validity of these methods and their tiny differences
put forward in this work can be verified in realistic experiments in future
using quantum simulating platforms, including but not limited to ultracold
atoms.

##  I Introduction

Since the first observation of Bose-Einstein condensate (BEC) in ultracold
atom gas Anderson  _et al._ (1995); Davis  _et al._ (1995), ultracold atom has
become one of the most active research topics in the past two decades Leggett
(2001); Dalfovo  _et al._ (1999); Cornell and Wieman (2002); Ueda (2010). Due
to its long coherent time and the flexible tunability of interactions and
external potentials, this system offers a perfect platform for researching
macroscopic quantum phenomena, including interference of matter-wave Andrews
_et al._ (1997), quantized vortex lattice Fetter and Svidzinsky (2001),
Josephson oscillation and self-trapping effects Albiez  _et al._ (2005); Levy
_et al._ (2007), to name only a few of them. Furthermore, with Feshbach
resonance Chin  _et al._ (2010), the scattering length of the bosonic atomic
gas can be well-tuned, which opens an avenue for researching physics from the
weakly correlated physics to the strongly correlated physics with these
ultracold gases Ho (2004); Randeria and Taylor (2014); Makotyn  _et al._
(2014); Cowell  _et al._ (2002); Eigen  _et al._ (2017). This physics can also
be explored with ultracold Fermi gases.

Figure 1: (a) The schematic of the bosonic Josephson junction, which is
generated by confining the BEC in a double well. (b) The constant energy
surface of the nonrigid pendulum at $\Delta=0$ and $\nu=1$. When
$H_{\text{NP}}>\nu=1$, the system is in the MSTP with $z(t)\neq 0$ for any
time.

In these researches, the bosonic Josephson junction [see Fig. 1 (a)] plays a
unique role because of its simplicity and tunability. This model can be
written in the second quantization representation as

$H=\int
dx\hat{\Psi}^{\dagger}(x)[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(x)+\frac{g}{2}\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)]\hat{\Psi}(x),$
(1)

where $V(x)$ is the double well potential, $m$ is the atom mass, and
$g=4\pi\hbar^{2}a_{s}/m$ with $a_{s}$ the scattering length and $\hbar$ is the
reduced Planck constant. The scattering length can be tuned via Feshbach
resonance Chin  _et al._ (2010). When the potential depth is large, the lowest
two energy levels are well separated from the other excited energy levels. In
the two modes approximation, we can approximate the field operator using
$\hat{\Psi}(x)=\hat{a}\psi_{\text{L}}(x)+\hat{b}\psi_{\text{R}}(x)$, where
$\psi_{\text{L}}(x)$ and $\psi_{\text{R}}(x)$ are the ground state wave
functions of each well under harmonic approximation Milburn  _et al._ (1997).
The overlap between wave functions $\psi_{\text{L}}(x)$ and $\psi_{R}(x)$ is
small and its strength is characterized by $\nu$. Furthermore, the detuning
between the two wells is given by $\Delta$, and the on-site interaction
strength, assumed to be the same, is given by $\lambda$. hence, we should have
the following effective Hamiltonian

$H=\Delta(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b})-\nu(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})+\lambda(\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}+\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}\hat{b}),$
(2)

where $\hat{a}$ and $\hat{b}$ are the annihilation bosonic operators in each
well. By changing the parameters $\Delta$, $\nu$, and $\lambda$, this model
can undergo a transition from the oscillating phase to the macroscopic self-
trapping phase (MSTP) Raghavan  _et al._ (1999); Smerzi  _et al._ (1997);
Milburn  _et al._ (1997). We can use a mean field approximation to understand
this transition by replacing the operators $\hat{a}$ and $\hat{b}$ with some
complex numbers $a$ and $b$, which is expected to be valid in the large $N$
limit. In this way, $\hat{a}^{\dagger}$ and $\hat{b}^{\dagger}$ are
approximated by $a^{*}$ and $b^{*}$, respectively. Then we can map the mean
field Hamiltonian to a nonrigid pendulum Smerzi  _et al._ (1997) in phase
space with

$H_{\text{NP}}=\frac{\lambda}{2}z^{2}+\Delta z-\nu\sqrt{1-z^{2}}\cos(\phi),$
(3)

where $z=|a|^{2}-|b|^{2}$ and $2\cos(\phi)=(a^{*}b+b^{*}a)/|ab|$. The $\phi$
and $z$ are conjugate coordinates and momentum, respectively. The equation of
motion is given by

$\dot{\phi}={\partial H_{\text{NP}}\over\partial z},\quad\dot{z}=-{\partial
H_{\text{NP}}\over\partial\phi}.$ (4)

Furthermore, with energy conservation, it is found that the value of $z(t)=0$
becomes inaccessible at any time if

$\lambda>2\nu\frac{\sqrt{1-z(0)^{2}}\cos(\phi(0))+1}{z(0)^{2}},$ (5)

for $\Delta=0$ [see Fig. 1 (b)]. Especially, we find $\lambda_{c}=2\nu$ for
$z(0)=1$ and $\phi(0)=0$. This criterion for the MSTP works well for the mean
field dynamics. A detailed analysis of this problem in mean field theory in
terms of Josephson effect, $\pi$ oscillations, symmetry breaking, and
macroscopic quantum self-trapping have been presented in Ref. Raghavan  _et
al._ (1999). Intriguing generalization of this models to various systems
include dipole gases in Ref. Xiong  _et al._ (2009), chaos with single
impurity in Ref. Chen  _et al._ (2021), exciton-polaritons in Ref. Abbarchi
_et al._ (2013), Josephson oscillation in triple-well potentials in Ref.
Lahaye  _et al._ (2010), dynamical stability for BEC in a periodic potential
in Ref. Lellouch  _et al._ (2017), two coupled one-dimensional condensates in
Ref. Ji  _et al._ (2022), dissipative Josephson junctions coupling to a
bosonic bath in Refs. Saha  _et al._ (2023); Masuki  _et al._ (2022), and
spin-orbit coupled condensate in a single well in Ref. Zhuang  _et al._
(2021), to only mention a few of them. The self-trapping and tunneling in a
double well potential have been observed in experiments Albiez  _et al._
(2005). It can also be found in an array of coupled condensates Reinhard  _et
al._ (2013). This physics can also be explored using the other quantum
systems, including polaritons, superconductors, and trapped ions etc., which
are not discussed here.

Method |  $\Delta^{\prime}/\Delta$ |  $\lambda^{\prime}/\lambda$ |  $\nu^{\prime}/\nu$ |  $V_{\text{new}}$ ($\hat{V}_{\text{new}}$) |  $\lambda_{c}$ at $|A/\omega|\ll 1$
---|---|---|---|---|---
Q-I |  1 |  1 |  $1-\frac{A^{2}}{\omega^{2}}$ |  - |  $2\nu(1-\frac{A^{2}}{\omega^{2}})$
Q-II |  $1-\frac{\nu^{2}}{4\omega^{2}B}$ |  $1-\frac{8\nu^{2}B}{\omega^{2}}$ |  $J_{0}+\frac{2\nu^{2}D}{\omega^{2}}$ |  Eq. 29 |  $2\nu(1-\frac{A^{2}}{\omega^{2}})$
C-I |  1 |  $1+\frac{A^{2}}{2\omega^{2}}$ |  $1-\frac{A^{2}}{\omega^{2}}$ |  - |  $2\nu(1-\frac{3A^{2}}{2\omega^{2}})$
C-II |  $1-\frac{4\nu^{2}B}{\omega^{2}}$ |  $1-\frac{6\nu^{2}B}{\omega^{2}}$ |  $J_{0}+\frac{2\nu^{2}M}{\omega^{2}}$ |  Eq. LABEL:eq-method-4-new |  $2\nu(1-\frac{A^{2}}{\omega^{2}})$
C-III |  $1+\frac{A^{2}}{\omega^{2}}$ |  $1+\frac{A^{2}}{\omega^{2}}$ |  $1-\frac{A^{2}}{2\omega^{2}}$ |  - |  $2\nu(1-\frac{3A^{2}}{2\omega^{2}})$
Table 1: Summary of the parameter changes and new terms for the Floquet
Hamiltonian under five different methods. Here, the parameters $B$, $C$, $M$,
$D$, $\cdots$, can be found in the main text.

More intriguing physics can be found in this simple model in the presence of
periodic driving. For instance, periodic driving can induce coherent
destruction of tunneling Kayanuma and Saito (2008); Gong  _et al._ (2009);
Kierig  _et al._ (2008) for dynamic localization, Shapiro-like resonant effect
Raghavan  _et al._ (1999); Eckardt  _et al._ (2005), as well as the
Hamiltonian chaos Lee  _et al._ (2001); Weiss and Teichmann (2008); Salmond
_et al._ (2002); Zhang  _et al._ (2008); Jiang  _et al._ (2014). This system
with periodic modulation has been studied intensively in Refs. Holthaus and
Stenholm (2001); Holthaus (2001); Xie (2007); Xie and Hai (2009); Abdullaev
and Kraenkel (2000); Luo  _et al._ (2008); Watanabe and Mäkelä (2012);
Abdullaev and Kraenkel (2000). In all this research, the Floquet analysis
plays a unique role, and we expect a time-independent Floquet Hamiltonian
$\hat{H}_{\text{F}}$ to determine the stroboscopic dynamics. Then many aspects
of the periodic-driven systems can be understood with the help of this Floquet
Hamiltonian. However, it is very hard for us to find the exact expression of
the Floquet Hamiltonian in the most generic conditions. This Floquet
Hamiltonian is also not unique. From different starting points or different
approximations, one may yield different effective Hamiltonians. In this way,
some of the effective Hamiltonians may work well, and some of them may not.
Thus, it is crucial for us to identify which method can give the most precise
prediction. To date, this issue has not yet drawn much attention in theory and
has not been a concern in experiments, however, it will become a critical
issue in the future with the increasing of experimental precision, and it is
to be hoped that the different effective models can be verified in these
experiments.

###  I.1 Physical model

To address this problem, we consider the following maximally simple model
Milburn  _et al._ (1997); Ruostekoski and Walls (1998); Stroescu  _et al._
(2016); Kierig  _et al._ (2008); Shin  _et al._ (2004) (set $\hbar=1$)

$\displaystyle\hat{H}$ $\displaystyle=(\Delta+A\cos(\omega
t))(\hat{n}_{a}-\hat{n}_{b})-\nu(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})$
(6)
$\displaystyle+\frac{\lambda}{2}(\hat{n}_{a}-\hat{n}_{b})^{2}-\frac{\lambda}{2}(\hat{n}_{a}+\hat{n}_{b}).$

with the detuning being periodically modulated, and
$\hat{n}_{a}=\hat{a}^{\dagger}\hat{a}$ and
$\hat{n}_{b}=\hat{b}^{\dagger}\hat{b}$ the number operator in each well. When
$|\omega|$ is much larger than the magnitude of other parameters, we can
neglect the periodic driving, and the time-averaged Hamiltonian is equivalent
to Eq. 2. It will become important if $\omega$ and all the other parameters
are comparable. A widely used transformation to this model is
$\hat{U}=\exp(-i\frac{A}{\omega}\sin(\omega
t)(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}))$, which gives

$\displaystyle\hat{H}^{\prime}$
$\displaystyle=\hat{U}^{\dagger}\hat{H}\hat{U}-i\hat{U}^{\dagger}\partial_{t}\hat{U}$
(7) $\displaystyle=\Delta(\hat{n}_{a}-\hat{n}_{b})-(\nu
e^{i\frac{2A}{\omega}\sin(\omega t)}\hat{a}^{\dagger}\hat{b}+\mathrm{H.c.})$
$\displaystyle+\frac{\lambda}{2}(\hat{n}_{a}-\hat{n}_{b})^{2}-\frac{\lambda}{2}(\hat{n}_{a}+\hat{n}_{b}),$

with the coupling now being periodically modulated. Although it is expected
that the physics of Eqs. 6 and 7 should be exactly the same, the resulting
Floquet Hamiltonians up to a finite truncation are different in practice.

To clarify this difference, we use quantum and classical methods to derive the
corresponding Floquet Hamiltonians for Eqs. 6 and 7. For the quantum methods,
we treat these Hamiltonians as quantum operators using the approaches
developed in Refs. Rahav  _et al._ (2003); Goldman and Dalibard (2014);
Goldman  _et al._ (2014). For the classical methods Rahav  _et al._ (2003), we
directly employ the multiple scale analysis to the nonlinear mean field
equation and separate the slow part from the fast oscillating part. In all
these methods, the resulting Floquet Hamiltonian and mean field dynamics are
discussed up to the order of $\omega^{-2}$. The central results can be given
by (classical Hamiltonian)

$\displaystyle H_{\text{F}}$
$\displaystyle=\Delta^{\prime}(|a|^{2}-|b|^{2})-\nu^{\prime}(a^{*}b+b^{*}a)+\frac{\lambda^{\prime}}{2}(|a|^{2}-|b|^{2})^{2}$
$\displaystyle+V_{\text{new}},$ (8)

with $a$, $b\in\mathbb{C}$. The corresponding quantum Floquet Hamiltonian
reads as

$\displaystyle\hat{H}_{\text{F}}$
$\displaystyle=\Delta^{\prime}(\hat{n}_{a}-\hat{n}_{b})-\nu^{\prime}(\hat{a}^{\dagger}\hat{b}+\text{h.c.})+\frac{\lambda^{\prime}}{2}(\hat{n}_{a}-\hat{n}_{b})^{2}$
$\displaystyle-\frac{\lambda^{\prime}}{2}(\hat{n}_{a}+\hat{n}_{b})+\hat{V}_{\text{new}}.$
(9)

The changed parameters from different methods are summarized in Table 1. Due
to the preservation of the total number of particles, the new interaction only
admits terms like $\hat{n}_{a}\hat{n}_{b}$ and
$\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{b}\hat{b}+\text{h.c.}$; see below.

This manuscript is organized as follows. In Sec. II, we derive the Floquet
Hamiltonian using quantum methods (methods Q-I and Q-II) for Eqs. 6 and 7. In
Sec. III, we employ multiple scale analysis (methods C-I, C-II, and C-III) to
the mean field equation of motion for the Floquet Hamiltonian. In Sec. IV, we
present the numerical verification based on the short-time dynamics and
criterion of the MSTP. In Sec. V, we conclude and discuss the possible
application of our results. In Appendix A, we discuss the criterion for the
MSTP in the presence of a co-tunneling process. In Appendix B, an exact
solution of the Floquet Hamiltonian is presented. We hope these different
effective Hamiltonians put forward in this work can be directly verified in
future experiments.

##  II Floquet analysis based on quantum model

In the first two methods, we employ Floquet analysis for the quantum
Hamiltonian in Eq. 6 and Eq. 7, which we term as method Q-I and method Q-II.
Then we apply the mean field approximation by replacing all operators with
complex numbers. From the Floquet theorem Rahav  _et al._ (2003), the Floquet
eigenstates and eigenvalues can be solved by the following equation

$(\hat{H}-i\partial_{t})|u_{n}\rangle=\epsilon_{n}|u_{n}\rangle,$ (10)

with $|u_{n}(t)\rangle=|u(t+T)\rangle$. Thus, it is expected that there should
exist a family of time-independent Floquet Hamiltonians $\hat{H}_{\text{F}}$,
which has the same eigenstates and eigenvalues with $\hat{H}-i\partial_{t}$.
This Hamiltonian may be obtained from the unitary $\hat{U}=e^{i\hat{K}(t)}$ as

$\hat{G}=\hat{U}\hat{H}\hat{U}^{\dagger}-i\hat{U}\partial_{t}\hat{U}^{\dagger}.$
(11)

Once the transformation $\hat{U}$ makes $\hat{G}$ time-independent, we have
found the effective Floquet Hamiltonian $\hat{H}_{\text{F}}=\hat{G}$ exactly.
In practice, we assume $\hat{K}(t)$ and $\hat{H}_{\text{F}}$ can be expanded
in powers of $\omega^{-1}$ Rahav  _et al._ (2003); Goldman and Dalibard
(2014); Goldman  _et al._ (2014) and the resulting Floquet Hamiltonian up to
the order of $\omega^{-2}$ can be written as

$\hat{H}_{\text{F
}}=\hat{H}_{0}+\hat{H}_{\text{F}}^{(1)}+\hat{H}_{\text{F}}^{(2)}+\mathcal{O}(\omega^{-2}),$
(12)

with

$\displaystyle\hat{H}_{\text{F}}^{(1)}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{1}{\omega
j}[\hat{H}_{j},\hat{H}_{-j}],$ (13) $\displaystyle\hat{H}_{\text{F}}^{(2)}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{1}{2(\omega
j)^{2}}([[\hat{H}_{j},\hat{H}_{0}],\hat{H}_{-j}]+\text{H.c.})+\sum_{j,l=1}^{\infty}\frac{1}{3\omega^{2}jl}$
$\displaystyle([\hat{H}_{j},[\hat{H}_{l},\hat{H}_{-j-l}]]-[\hat{H}_{j},[\hat{H}_{-l},\hat{H}_{l-j}]]+\text{H.c.}),$
(14)

for the periodic driven Hamiltonian

$\hat{H}=\hat{H}_{0}+\sum_{j=1}^{\infty}(\hat{H}_{j}e^{ij\omega
t}+\hat{H}_{-j}e^{ij\omega t}).$ (15)

From these expressions, we can find that although Eqs. 6 and 7 have the same
eigenvalues, they would yield different Floquet Hamiltonian up to the order of
$\omega^{-2}$, as we discussed below. This method may be used to realize new
interactions for novel physics. For example, it has been used to engineer the
topological phase transitions Lindner  _et al._ (2011); Maczewsky  _et al._
(2017); Zhou  _et al._ (2023a); Weitenberg and Simonet (2021); Zhang  _et al._
(2023); Adiyatullin  _et al._ (2023); Wintersperger  _et al._ (2020) and to
realize the discrete time crystal Zhang  _et al._ (2017); Kyprianidis  _et
al._ (2021).

###  II.1 Method Q-I

In this method, we directly access the Hamiltonian of Eq. 6. We can write the
whole Hamiltonian as

$\hat{H}=\hat{H}_{0}+\hat{H}_{-1}e^{-i\omega t}+\hat{H}_{1}e^{i\omega t},$
(16)

with
$\hat{H}_{0}=\Delta(\hat{n}_{a}-\hat{n}_{b})-\nu(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})+\frac{\lambda}{2}(\hat{n}_{a}-\hat{n}_{b})^{2}-\frac{\lambda}{2}(\hat{n}_{a}+\hat{n}_{b})$
and $\hat{H}_{-1}=H_{1}=\frac{A}{2}(\hat{n}_{a}-\hat{n}_{b})$. By a direct
calculation, we find the new effective terms as

$\displaystyle\hat{H}^{(1)}_{\text{F}}$ $\displaystyle=\sum_{j}\frac{1}{\omega
j}[\hat{H}_{j},\hat{H}_{-j}]=0,$ (17) $\displaystyle\hat{H}^{(2)}_{\text{F}}$
$\displaystyle=\frac{A^{2}}{\omega^{2}}\nu(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a}).$
(18)

Combining the above results will yield the effective Floquet Hamiltonian in
Eq. 9 with modified parameters as

$\displaystyle\Delta^{\prime}=\Delta,\quad\lambda^{\prime}=\lambda,\quad\nu^{\prime}=(1-\frac{A^{2}}{\omega^{2}})\nu,\quad\hat{V}_{\text{new}}=0.$
(19)

In this method, the periodic driving can only weaken the strength of the
coupling. However, it would not change the parameters of $\lambda$ and
$\Delta$ at least up to the order of $\omega^{-2}$. Moreover, according to Eq.
5, we find this Floquet Hamiltonian predicts the criterion for MSTP as

$\lambda_{c}=2\nu(1-\frac{A^{2}}{\omega^{2}}),$ (20)

when $\Delta=0$, $a(0)=1$, and $b(0)=0$. This is a good approximation when
$|A|<\omega$; see the expression $J_{0}$ below.

###  II.2 Method Q-II

The Hamiltonian in Eq. 7 can be divided into

$\hat{H}^{\prime}=\hat{H}_{0}+\hat{H}_{1}(t),$ (21)

with

$\displaystyle\hat{H}_{0}$ $\displaystyle=\Delta(\hat{n}_{a}-\hat{n}_{b})-\nu
J_{0}(\hat{a}^{\dagger}\hat{b}+\mathrm{H.c.})$
$\displaystyle+\frac{\lambda}{2}[(\hat{n}_{a}-\hat{n}_{b})^{2}-(\hat{n}_{a}+\hat{n}_{b})],$
(22) $\displaystyle\hat{H}_{1}(t)$ $\displaystyle=\sum_{j=1}^{\infty}[\nu
J_{j}\hat{a}^{\dagger}\hat{b}+\nu J_{-j}\hat{b}^{\dagger}\hat{a}]e^{ij\omega
t}+\mathrm{H.c.}$ $\displaystyle=\sum_{j=1}^{\infty}H_{j}e^{ij\omega
t}+\mathrm{H.c.}.$ (23)

Here, we have employed the Jacobi-Anger expansion
$e^{iz\sin(x)}=\sum_{j=-\infty}^{\infty}J_{j}(z)e^{ijx}$ with $J_{j}(z)$ being
the Bessel function of the first kind. Hereafter for convenience we define
$J_{j}=J_{j}(\frac{2A}{\omega})$. It can be found that the time-averaged
Hamiltonian $\hat{H}_{0}$ can capture the physics predicted by the method in
Sec. II.1, since $J_{0}\sim 1-A^{2}/\omega^{2}$, which is a good approximation
when $|A|<\omega$. The resulting Floquet Hamiltonian considering the effect
from $\hat{H}_{1}(t)$ is give by

$\displaystyle\hat{H}^{(1)}_{\text{F}}$ $\displaystyle=0,$ (24)
$\displaystyle\hat{H}^{(2)}_{\text{F}}$
$\displaystyle=-\frac{4\nu^{2}B\Delta}{\omega^{2}}(\hat{n}_{a}-\hat{n}_{b})-\frac{2\nu^{3}D}{\omega^{2}}(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a})$
$\displaystyle-\frac{4\nu^{2}B\lambda}{\omega^{2}}[(\hat{n}_{a}-\hat{n}_{b})^{2}-(\hat{n}_{a}+\hat{n}_{b})]$
$\displaystyle-\frac{4\nu^{2}C\lambda}{\omega^{2}}(\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{b}\hat{b}+\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{a}\hat{a})+\frac{8\nu^{2}B\lambda}{\omega^{2}}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}.$
(25)

with $B=\sum_{j=1}^{\infty}\frac{1}{j^{2}}J_{j}J_{j}$,
$C=\sum_{j=1}^{\infty}\frac{1}{j^{2}}J_{j}J_{-j}$, and
$D=(C-B)J_{0}+\sum_{jl}\frac{2}{3jl}[(-1)^{l}-(-1)^{j+l}](J_{-l}J_{l+j}J_{-j}-J_{-l}J_{l-j}J_{j})$.
As a result, in Eq. 9, we have

$\displaystyle\Delta^{\prime}$
$\displaystyle=(1-\frac{4\nu^{2}B}{\omega^{2}})\Delta,$ (26)
$\displaystyle\lambda^{\prime}$
$\displaystyle=(1-\frac{8\nu^{2}B}{\omega^{2}})\lambda,$ (27)
$\displaystyle\nu^{\prime}$
$\displaystyle=(J_{0}+\frac{2\nu^{2}D}{\omega^{2}})\nu,$ (28)
$\displaystyle\hat{V}_{\text{new}}=-\frac{4\nu^{2}C\lambda}{\omega^{2}}$
$\displaystyle[\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{b}\hat{b}+\mathrm{H.c.}]+\frac{8\nu^{2}B\lambda}{\omega^{2}}\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b}.$
(29)

This approximation generates a new term for co-tunneling between the two wells
Folling  _et al._ (2007), which still preserves the total particle number
$\hat{n}_{a}+\hat{n}_{b}$. The appearance of this term can be understood from
the two-photon process of the periodic-driven Hamiltonian in Eq. 21 and it
would be the dominant tunneling mechanism when the direct process is
suppressed Lignier  _et al._ (2007). For instance, when we set $2A/\omega$ at
the zero points of the Bessel function $J_{0}$, it is expected the tunneling
between the two wells is completely suppressed from the zero-order Floquet
Hamiltonian in Eq. 22. However, the tunneling of the whole system persists due
to the co-tunneling process.

From this result, we find that in this effective model, all parameters are
changed. In particular, new interaction $V_{\text{new}}$ emerges, which is
completely different from the result in Sec. II.1. As discussed above, this
term may lead to observable features. When $A<\omega$, following the criterion
presented in Appendix A, we find a similar criterion for MSTP at

$\lambda_{c}=2\nu\frac{\omega^{2}J_{0}+2\nu^{2}D}{\omega^{2}-12\nu^{2}B+4\nu^{2}C}.$
(30)

When $|\omega|\gg|A|$ and $|\nu|$, the Bessel functions have the following
asymptotic forms
$J_{j}=J_{j}(\frac{2A}{\omega})\sim\frac{1}{\Gamma(j+1)}(\frac{A}{\omega})^{j}$,
with $j\geq 1$, from which we expect

$\displaystyle B\sim\frac{A^{2}}{\omega^{2}},\quad
C\sim-\frac{A^{2}}{\omega^{2}},\quad D\sim-2\frac{A^{2}}{\omega^{2}}J_{0}.$
(31)

Thus, the criterion for MSTP becomes $\lambda_{c}\sim
2\nu(1-\frac{A^{2}}{\omega^{2}})$, which is consistent with method Q-I. This
result demonstrate that to the same order of $\omega^{-2}$, the different
approximations can yield different effective Hamiltonians. The similar
features will be observed in the mean field models.

##  III Floquet analysis based on mean field model

The Floquet Hamiltonian can also be obtained from the mean field Hamiltonian
and its associated equation of motion. In this way, the operators are repalced
by $\mathbb{C}$-numbers. The general equation of motion can be written as Wu
and Niu (2000); Choi and Niu (1999); Ananikian and Bergeman (2006);
Ruostekoski and Walls (1998); Lyu  _et al._ (2020)

$\displaystyle i\frac{da}{dt}=\gamma(t)a+\lambda(|a|^{2}-|b|^{2})a-\nu(t)b,$
(32) $\displaystyle
i\frac{db}{dt}=-\gamma(t)b-\lambda(|a|^{2}-|b|^{2})b-\nu(t)a.$ (33)

The equation of motion of $a^{*}$ and $b^{*}$ can be obtained by making a
complex conjugate to the above equations, thus is not shown. The same
preterition will be made below. When $\gamma(t)=\Delta+A\cos(\omega t)$ and
$\nu(t)=\nu$, the equation captures the dynamics of Eq. 6. When
$\gamma(t)=\Delta$ and $\nu(t)=\nu e^{i\frac{2A}{\omega}\sin(\omega t)}$, the
equation captures the dynamics of Eq. 7. In the following, we use multiple
scale analysis to derivate the Floquet Hamiltonian and term the two methods as
methods C-I and C-II. In the last part of this section, we have also presented
a more simple approach for the Floquet Hamiltonian, termed method C-III. A
general treatment of this approach can be found in the mechanics textbook
Landau and Lifshitz (1976).

###  III.1 Method C-I

Following the approach developed by Rahav et. al Rahav  _et al._ (2003), we
separate the dynamic variables into slow and fast modes as

$\displaystyle a$
$\displaystyle=a_{0}+\xi_{a}=a_{0}+\sum_{j=1}^{\infty}\frac{1}{\omega^{j}}\xi_{a,j},$
(34) $\displaystyle b$
$\displaystyle=b_{0}+\xi_{b}=b_{0}+\sum_{j=1}^{\infty}\frac{1}{\omega^{j}}\xi_{b,j},$
(35)

with $\xi_{a}$ and $\xi_{b}$ (neglecting the $j$ index) being time-dependent
as a function of $\tau=\omega t$ for fast modes, $a_{0}$ and $b_{0}$ for the
slow modes, respectively. Obviously, $\xi_{a}$ and $\xi_{b}$ also depend on
$a_{0}$ and $b_{0}$; see discussion of Karpitz inverted pendulum in Chap 5 of
Ref. Landau and Lifshitz (1976). Recently, a quantum version Karpitz pendulum
was realized with ultracold atoms in Ref. Jiang  _et al._ (2023). We can
assume that $\int_{0}^{T}dt\xi_{a}=\int_{0}^{T}dt\xi_{b}=0$, with
$T=2\pi/\omega$ the period. In this way, all the fast oscillating terms are
averaged out, leaving only the slow variation terms for the effective
dynamics. Then we have

$\displaystyle\frac{d\xi_{a}}{dt}$
$\displaystyle=\omega\frac{\partial\xi_{a}}{\partial\tau}+\frac{\partial\xi_{a}}{\partial
a_{0}}\dot{a}_{0}+\frac{\partial\xi_{a}}{\partial
b_{0}}\dot{b}_{0}+\frac{\partial\xi_{a}}{\partial
a_{0}^{*}}\dot{a}_{0}^{*}+\frac{\partial\xi_{a}}{\partial
b_{0}^{*}}\dot{b}_{0}^{*},$ (36) $\displaystyle\frac{d\xi_{b}}{dt}$
$\displaystyle=\omega\frac{\partial\xi_{b}}{\partial\tau}+\frac{\partial\xi_{b}}{\partial
a_{0}}\dot{a}_{0}+\frac{\partial\xi_{b}}{\partial
b_{0}}\dot{b}_{0}+\frac{\partial\xi_{b}}{\partial
a_{0}^{*}}\dot{a}_{0}^{*}+\frac{\partial\xi_{b}}{\partial
b_{0}^{*}}\dot{b}_{0}^{*}.$ (37)

The complete conjugate of these results can give the equation of motion of
$\xi_{a}^{*}$ and $\xi_{b}^{*}$. Then we can solve the nonlinear equation in
Eq. 33 according to the order of $\omega^{-1}$. In this method, we choose
$\gamma(t)=\Delta+A\cos(\omega t)$ and $\nu(t)=\nu$. In practice, we gather
all the terms according to $\omega^{-n}$ and then choose the solution of
$\xi_{n+1}$ to cancel the dependence of $\tau$. The constant terms in the next
order $\omega^{-(n+1)}$ give rise to the effective contribution with order
$\omega^{-(n+1)}$ for the slow part $a_{0}$ and $b_{0}$.

(1) Zero order. We have the following equations

$\displaystyle i(\frac{da_{0}}{dt}+\frac{\partial\xi_{a,1}}{\partial\tau})=$
$\displaystyle\gamma(\tau)a_{0}+\lambda za_{0}-\nu b_{0},$ (38) $\displaystyle
i(\frac{db_{0}}{dt}+\frac{\partial\xi_{b,1}}{\partial\tau})=$
$\displaystyle-\gamma(\tau)b_{0}-\lambda zb_{0}-\nu a_{0},$ (39)

with $z=|a_{0}|^{2}-|b_{0}|^{2}$. To cancel the dependence of $\tau$, we chose
$i\frac{\partial\xi_{a,1}}{\partial\tau}=A\cos(\tau)a_{0}$ and
$i\frac{\partial\xi_{b,1}}{\partial\tau}=-A\cos(\tau)b_{0}$, which yield the
solutions

$\displaystyle\xi_{a,1}(\tau)=-iA\sin(\tau)a_{0},\quad\xi_{b,1}(\tau)=iA\sin(\tau)b_{0}.$
(40)

Substitute these solutions back to Eqs. 38 and 39, we got the zero-order
effective equation for $a_{0}$ and $b_{0}$, which consistent with time-
averaged mean field Hamiltonian

(2) First order. The equations in this order are

$\displaystyle
i\frac{\partial\xi_{a,2}}{\partial\tau}+i\dot{a}_{0}\frac{\partial\xi_{a,1}}{\partial
a_{0}}=(\gamma(\tau)+2\lambda|a_{0}|^{2}-\lambda|b_{0}|^{2})\xi_{a,1}$
$\displaystyle-(\nu+\lambda
b_{0}^{*}a_{0})\xi_{b,1}+\lambda(a_{0})^{2}\xi_{a,1}^{*}-\lambda
a_{0}b_{0}\xi_{b,1}^{*},$ (41) $\displaystyle
i\frac{\partial\xi_{b,2}}{\partial\tau}+i\dot{b}_{0}\frac{\partial\xi_{b,1}}{\partial
b_{0}}=-(\gamma(\tau)+\lambda|a_{0}|^{2}-2\lambda|b_{0}|^{2})\xi_{b,1}$
$\displaystyle-(\nu+\lambda
a_{0}^{*}b_{0})\xi_{a,1}+\lambda(b_{0})^{2}\xi_{b,1}^{*}-\lambda
a_{0}b_{0}\xi_{a,1}^{*}.$ (42)

It can be found that the solution in this order should be

$\displaystyle\xi_{a,2}(\tau)$
$\displaystyle=2A\nu\cos(\tau)b_{0}-\frac{A^{2}}{4}\cos(2\tau)a_{0},$ (43)
$\displaystyle\xi_{b,2}(\tau)$
$\displaystyle=-2A\nu\cos(\tau)a_{0}+\frac{A^{2}}{4}\cos(2\tau)b_{0}.$ (44)

In this order, we can not find any constant terms for $a_{0}$, $b_{0}$, and
their conjugate. Thus, we find $H_{\text{F}}^{(1)}=0$.

(3) Second order. In this order, the equations are much longer than the
previous equations. Thus, we focus on the equation for $a$, while the
derivation for $b$ is also straightforward. The equation for $a$ reads as

$\displaystyle i\frac{\partial\xi_{a,3}}{\partial\tau}$
$\displaystyle+i\dot{a}_{0}\frac{\partial\xi_{a,2}}{\partial
a_{0}}+i\dot{b}_{0}\frac{\partial\xi_{a,2}}{\partial b_{0}}=$
$\displaystyle(\gamma(\tau)+2\lambda|a_{0}|^{2}-\lambda|b_{0}|^{2})\xi_{a,2}-(\nu+\lambda
b_{0}^{*}a_{0})\xi_{b,2}$
$\displaystyle+\lambda(a_{0})^{2}\xi_{a,2}^{*}-\lambda
a_{0}b_{0}\xi_{b,2}^{*}+\lambda a_{0}^{*}\xi_{a,1}^{2}+2\lambda
a_{0}|\xi_{a,1}|^{2}$ $\displaystyle-\lambda
b_{0}^{*}\xi_{b,1}\xi_{a,1}-\lambda b_{0}\xi_{b,1}^{*}\xi_{a,1}-\lambda
a_{0}\xi_{b,1}^{*}\xi_{b,1}.$ (45)

Gathering all the constant terms, we find the whole effective equation of
motion for $a_{0}$ and $b_{0}$ up to the order of $\omega^{-2}$ is

$\displaystyle i\frac{da_{0}}{dt}=\Delta
a_{0}+\lambda^{\prime}(|a_{0}|^{2}-|b_{0}|^{2})a_{0}-\nu^{\prime}b_{0},$ (46)
$\displaystyle i\frac{db_{0}}{dt}=-\Delta
b_{0}-\lambda^{\prime}(|a_{0}|^{2}-|b_{0}|^{2})b_{0}-\nu^{\prime}a_{0},$ (47)

with modified parameters in the Floquet Hamiltonian as

$\displaystyle\Delta^{\prime}=\Delta,$
$\displaystyle\quad\lambda^{\prime}=(1+\frac{A^{2}}{2\omega^{2}})\lambda,$
(48) $\displaystyle V_{\text{new}}=0,$
$\displaystyle\quad\nu^{\prime}=(1-\frac{A^{2}}{\omega^{2}})\nu.$

Similar to the results we obtained in Sec. II.1, we find the periodic driving
suppresses the coupling strength. However, it also enhances the interaction
strength $\lambda$. Therefore, with this Floquet Hamiltonian, it is expected
that the criterion for MSTP become

$\lambda_{c}=4\nu\frac{\omega^{2}-A^{2}}{2\omega^{2}+A^{2}},$ (49)

when $\Delta=0$, $a(0)=1$, and $b(0)=0$. When $|A/\omega|\ll 1$, we have
$\lambda_{c}=2\nu(1-3A^{2}/2\omega^{2})$.

###  III.2 Method C-II

Here, we choose $\gamma(t)=\Delta$ and $\nu(t)=\nu
e^{i\frac{2A}{\omega}\sin(\omega t)}$. Then, we employ a similar method that
we have used in Sec. III.1. we can obtain the effective equations for $a_{0}$,
$b_{0}$, $a_{0}^{*}$, and $b_{0}^{*}$ order by order.

(1) Zero order. The effective equations in this order are

$\displaystyle i\frac{da_{0}}{dt}+i\frac{\partial\xi_{a,1}}{\partial\tau}$
$\displaystyle=\Delta a_{0}+\lambda za_{0}-\nu
J_{0}b_{0}-\mathcal{B}(\tau)b_{0},$ (50) $\displaystyle
i\frac{db_{0}}{dt}+i\frac{\partial\xi_{b,1}}{\partial\tau}$
$\displaystyle=-\Delta b_{0}-\lambda zb_{0}-\nu
J_{0}a_{0}-\mathcal{B}^{*}(\tau)a_{0},$ (51)

with
$\mathcal{B}(\tau)=\nu\sum_{j=1}^{\infty}[J_{j}e^{ij\tau}+J_{-j}e^{-ij\tau}]$
and $z$ same to Eq. 38. We chose the solution of $\xi_{a,1}$ and $\xi_{b,1}$
as

$\displaystyle\xi_{a,1}=\mathcal{C}(\tau)b_{0},$
$\displaystyle\quad\xi_{b,1}=-\mathcal{C}^{*}(\tau)a_{0},$ (52)

with
$\mathcal{C}(\tau)=\sum_{j=1}^{\infty}\frac{\nu}{j}(J_{j}e^{ij\tau}-J_{-j}e^{-ij\tau})$.

(2) First order. The equations in this order are

$\displaystyle
i\frac{\partial\xi_{a,2}}{\partial\tau}+i\frac{db_{0}}{dt}\frac{\partial\xi_{a,1}}{\partial
b_{0}}=(\Delta+2\lambda|a_{0}|^{2}-\lambda|b_{0}|^{2})\xi_{a,1}-$
$\displaystyle(\nu J_{0}+\mathcal{B}(\tau)+\lambda
b_{0}^{*}a_{0})\xi_{b,1}+\lambda(a_{0})^{2}\xi_{a,1}^{*}-\lambda
a_{0}b_{0}\xi_{b,1}^{*},$ (53) $\displaystyle
i\frac{\partial\xi_{b,2}}{\partial\tau}+i\frac{da_{0}}{dt}\frac{\partial\xi_{b,1}}{\partial
a_{0}}=-(\Delta+\lambda|a_{0}|^{2}-2\lambda|b_{0}|^{2})\xi_{b,1}-$
$\displaystyle(\nu J_{0}+\mathcal{B}^{*}(\tau)+\lambda
a_{0}^{*}b_{0})\xi_{a,1}+\lambda(b_{0})^{2}\xi_{b,1}^{*}-\lambda
a_{0}b_{0}\xi_{a,1}^{*}.$ (54)

With these equations, we can find the solutions for $\xi_{a,2}$ and
$\xi_{b,2}$ are

$\displaystyle\xi_{a,2}=-\mathcal{D}(\tau)(2\Delta
b_{0}+4\lambda|a_{0}|^{2}b_{0}-2\lambda|b_{0}|^{2}b_{0}+\nu J_{0}a_{0})$
$\displaystyle+\mathcal{D}^{*}(\tau)(\nu J_{0}a_{0}+2\lambda
a_{0}^{2}b_{0}^{*})+\mathcal{F}(\tau)a_{0},$ (55)
$\displaystyle\xi_{b,2}=\mathcal{D}^{*}(\tau)(2\Delta
a_{0}+2\lambda|a_{0}|^{2}a_{0}-4\lambda|b_{0}|^{2}a_{0}-\nu J_{0}b_{0})$
$\displaystyle+\mathcal{D}(\tau)(\nu J_{0}b_{0}+2\lambda
b_{0}^{2}a_{0}^{*})+\mathcal{F}^{*}(\tau)b_{0},$ (56)

with
$\mathcal{D}(\tau)=\sum_{j=1}^{\infty}\frac{\nu}{j^{2}}(J_{j}e^{ij\tau}+J_{-j}e^{-ij\tau})$
and $\mathcal{F}(\tau)=\sum_{j\neq 0,l\neq 0,j\neq
l}=\frac{J_{j}J_{l}}{(j-l)l}e^{i(j-l)\tau}$. Here, $j$ and $l$ run from
negative infinity to positive infinity. The solution for $\xi_{a,1}^{*}$ and
$\xi_{b,1}^{*}$ can also be obtained by applying conjugation.

(3) Second order. The constant terms in this order will contribute to the
effective equations. The resulting Floquet Hamiltonian at this order is

$\displaystyle H_{\text{F}}^{(2)}$
$\displaystyle=-\frac{4\nu^{2}B\Delta}{\omega^{2}}(|a_{0}|^{2}-|b_{0}|^{2})-\frac{3\nu^{2}B\lambda}{\omega^{2}}(|a_{0}|^{2}-|b_{0}|^{2})^{2}$
(57)
$\displaystyle-\frac{2\nu^{3}M}{\omega^{2}}(a_{0}^{*}b_{0}+b_{0}^{*}\tilde{a}_{0})+\frac{8\nu^{2}B\lambda}{\omega^{2}}|a_{0}|^{2}|b_{0}|^{2}$
$\displaystyle-\frac{4\nu^{2}C\lambda}{\omega^{2}}(a_{0}^{*}a_{0}^{*}b_{0}b_{0}+b_{0}^{*}b_{0}^{*}a_{0}a_{0}),$

with $B$ and $C$ being given by Eq. 25, M being given by
$M=(C-B)J_{0}+\sum_{j,l>0,j\neq
l}\frac{1}{jl}(J_{-j}J_{j+l}J_{l}-J_{-j}J_{j-l}J_{-l})$. Compared with Eq. 8,
we find

$\displaystyle\Delta^{\prime}$
$\displaystyle=(1-\frac{4\nu^{2}B}{\omega^{2}})\Delta,$ (58)
$\displaystyle\lambda^{\prime}$
$\displaystyle=(1-\frac{6\nu^{2}B}{\omega^{2}})\lambda,$ (59)
$\displaystyle\nu^{\prime}$
$\displaystyle=(J_{0}+\frac{2\nu^{2}M}{\omega^{2}})\nu,$ (60) $\displaystyle
V_{\text{new}}$
$\displaystyle=-\frac{4\nu^{2}C\lambda}{\omega^{2}}(a^{*}a^{*}bb+b^{*}b^{*}aa)+\frac{8\nu^{2}B\lambda}{\omega^{2}}|a|^{2}|b|^{2}.$

In this method, we also find the co-tunneling term as we have discussed in
method Q-II. Similarly, the criterion for MSTP becomes (see Appendix A)

$\lambda_{c}=2\nu\frac{\omega^{2}J_{0}+2\nu^{2}M}{\omega^{2}-10\nu^{2}B+4\nu^{2}C}.$
(62)

Using a similar analysis in method Q-II, we find $\lambda_{c}\sim
2\nu(1-\frac{A^{2}}{\omega^{2}})$, at the limit of $|\omega|\gg|A|$ and
$|\nu|$.

###  III.3 Method C-III

In this method, we use a much simpler but useful approach for the derivation
of the Floquet Hamiltonian, following the essential idea in the Karpitz
pendulum Landau and Lifshitz (1976). We expand $a$ and $b$ into the slow and
fast modes up to the order of $\omega^{-2}$ as

$a=a_{0}+a_{1}+a_{2},\quad b=b_{0}+b_{1}+b_{2},$ (63)

with $a_{j}$ and $b_{j}$ of the $\omega^{-j}$ order. Using this expansion, we
can rewrite the nonlinear equations as

$\displaystyle
i\frac{d}{dt}(a_{0}+a_{1}+a_{2})=\Delta(a_{0}+a_{1}+a_{2})+\lambda(|a_{0}+a_{1}+a_{2}|^{2}$
$\displaystyle-|b_{0}+b_{1}+b_{2}|^{2})(a_{0}+a_{1}+b_{2})-\nu(b_{0}+b_{1}+b_{2})$
$\displaystyle+A\cos(\omega t)(a_{0}+a_{1}+a_{2}),$ (64) $\displaystyle
i\frac{d}{dt}(b_{0}+b_{1}+b_{2})=-\Delta(b_{0}+b_{1}+b_{2})-\lambda(|a_{0}+a_{1}+a_{2}|^{2}$
$\displaystyle-|b_{0}+b_{1}+b_{2}|^{2})(b_{0}+b_{1}+b_{2})-\nu(a_{0}+a_{1}+a_{2})$
$\displaystyle-A\cos(\omega t)(b_{0}+b_{1}+b_{2}).$ (65)

Similarly, we can gather all terms according to the power of $\omega$. We have
the equations of $a_{1}$ and $b_{1}$, as follows

$i\dot{a}_{1}=A\cos(\omega t)a_{0},\quad i\dot{b}_{1}=-A\cos(\omega t)b_{0},$
(66)

with solution

$a_{1}=-i{A\sin(\omega t)\over\omega}a_{0},\quad b_{1}=i{A\sin(\omega
t)\over\omega}b_{0}.$ (67)

Strictly speaking, the above solution is not accurate if we consider the time-
dependence of $a_{0}$ and $b_{0}$, making $a_{1}(\tau,t)$, with $\tau=\omega
t$. In the multiple scale analysis as discussed in methods C-I and C-II, the
time-dependence of $a_{0}$ and $b_{0}$ has been taken into account. Ignoring
this time dependence leads to results with much less accuracy; however, it is
still instructive and has pedagogical value Landau and Lifshitz (1976). In
this way, there are no constant terms that can be separated in $\omega^{-1}$
order. Then we focus on the solution to the order of $\omega^{-2}$, with

$\displaystyle a_{2}$ $\displaystyle=\frac{A}{\omega^{2}}\cos(\omega t)[\Delta
a+\lambda(|a_{0}|^{2}-|b_{0}|^{2})a_{0}+\nu b_{0}]$
$\displaystyle+\frac{A^{2}}{(2\omega)^{2}}\cos(2\omega t)a_{0},$ (68)
$\displaystyle b_{2}$ $\displaystyle=-\frac{A}{\omega^{2}}\cos(\omega
t)[-\Delta b-\lambda(|a_{0}|^{2}-|b_{0}|^{2})b_{0}+\nu a_{0}]$
$\displaystyle+\frac{A^{2}}{(2\omega)^{2}}\cos(2\omega t)b_{0}.$ (69)

In this way, the effective term for $a_{0}$ comes from

$\displaystyle\frac{1}{T}\int_{0}^{T}dtA\cos(\omega t)a_{2}(t)$
$\displaystyle=\frac{A^{2}}{2\omega^{2}}[\Delta
a_{0}+\lambda(|a_{0}|^{2}-|b_{0}|^{2})a_{0}+\nu b_{0}],$ (70)
$\displaystyle\frac{1}{T}\int_{0}^{T}dt\lambda(2a_{0}|a_{1}|^{2}+a_{0}^{*}a_{1}^{2}-a_{0}|b_{1}|^{2}-b_{0}^{*}b_{1}a_{1}-b_{0}b_{1}^{*}a_{1})$
$\displaystyle=\frac{A^{2}}{2\omega^{2}}\lambda(|a_{0}|^{2}-|b_{0}|^{2})a_{0},$
(71)

and the same analysis can be employed to $b_{0}$. Combining all these terms,
we have the effective equation of motion for $a_{0}$ and $b_{0}$ as follows

$\displaystyle i\frac{da_{0}}{dt}=$
$\displaystyle+\Delta^{\prime}a_{0}+\lambda^{\prime}(|a_{0}|^{2}-|b_{0}|^{2})a_{0}-\nu^{\prime}b_{0},$
(72) $\displaystyle i\frac{db_{0}}{dt}=$
$\displaystyle-\Delta^{\prime}b_{0}-\lambda^{\prime}(|a_{0}|^{2}-|b_{0}|^{2})b_{0}-\nu^{\prime}a_{0},$
(73)

with

$\displaystyle\Delta^{\prime}$
$\displaystyle=(1+\frac{A^{2}}{2\omega^{2}})\Delta,$ (74)
$\displaystyle\lambda^{\prime}$
$\displaystyle=(1+\frac{A^{2}}{\omega^{2}})\lambda,$ (75)
$\displaystyle\nu^{\prime}$
$\displaystyle=(1-\frac{A^{2}}{2\omega^{2}})\nu,\quad V_{\text{new}}=0.$ (76)

In this result, the criterion for the MSTP is

$\lambda_{c}=\nu\frac{2\omega^{2}-A^{2}}{\omega^{2}+A^{2}},$ (77)

for $\Delta=0$, $a(0)=1$ and $b(0)=0$. When $A$ is much smaller than $\omega$,
it yields $\lambda_{c}=2\nu(1-\frac{3A^{2}}{2\omega^{2}})$, which is the same
as that from Eq. 49; please see a direct comparison between different
approaches in Table 1.

##  IV Numerical verification

Figure 2:  The left column denotes the population $P(t)=|a|^{2}-|b|^{2}$
against evolution time $t$ using Eq. 33 and other methods. The black solid
lines are the results obtained by Eq. 33, which we call $P_{0}(t)$. The right
column denotes the difference $\Delta P(t)=P(t)-P_{0}(t)$ of population
dynamics between Eq. 33 and other methods. We have used $\Delta=1$,
$\lambda=1.9$, $\nu=1$, and $\omega=50$ in all figures and we have used
$A=0.1$ in (a) and (b); $A=1$ in (c) and (d); $A=10$ in (e) and (f); and
$A=30$ in (g) and (h).  Figure 3:  The time-averaged population difference
with different methods. In the calculation of $\overline{P}$, we set
$\Delta=0$, $\nu=1$, $A=10$, $\omega=20$, $a(0)=1$, $b(0)=0$, and $M=500$. It
is expected $\lambda_{c}=1.5$ for method Q-I, $\lambda_{c}\sim 1.543$ for
method Q-II, $\lambda_{c}=4/3$ for method C-I, $\lambda_{c}\sim 1.538$ for
method C-II, and $\lambda_{c}\sim 1.4$ for method C-III. The accurate
numerical result gives $\lambda_{c}=1.54$.

With the above five methods and their different Floquet Hamiltonian, it is
crucial to ask whether all these Floquet Hamiltonian give rise to a good
description of the periodic driven BEC or only one yields the correct physics.
To answer this question, we numerically solve the mean field equation of
motion that is obtained by Eq. 33 and other effective Hamiltonians. Then we
compare the population dynamics $P(t)=|a|^{2}-|b|^{2}$ from these methods in
Fig. 2. It is found that all methods give rise to an excellent agreement with
the exact dynamics in the short time for $A=0.1$ and $A=1$ with $\Delta
P(t)\sim 10^{-4}-10^{-3}$. However, when $A=10$ and $A=30$, the difference
between the dynamics in Eq. 33 and other methods becomes much larger. Of all
the methods, we find the Q-II and C-II are the most accurate, whose $\Delta
P\sim 10^{-2}-10^{-1}$ even for strong periodic driving. However, methods C-I
and C-III, where interaction strength is enhanced by periodic driving, give
rise to the worst approximation. This can also be understood by the example we
discussed in Appendix B.

Then we should verify our prediction for the criterion of the MSTP based on
these different methods. To identify the oscillating phase and MSTP, we use
the time average of the population

$\overline{P}=\frac{1}{MT}\int_{0}^{MT}dtP(t),$ (78)

with $T$ the periodicity. It approaches zero for the oscillating phase while
its value will remain finite for the MSTP. We present the results in Fig. 3.
It can be found that $\overline{P}$ exhibits a transition from zero to finite
for all methods. However, the transition point is different for each method.
It is found that the transition point predicted by methods Q-II and C-II is
much closer to the accurate numerical results while the other methods predict
much smaller values. This may be understood from the Bessel function in Eq. 7,
which includes the contribution higher than $\omega^{-2}$. On the other hand,
we can find the quantum method Q-I is more accurate than C-I. This difference
should come from the nonlinear feature of method C-I. Among all methods,
method C-III works the poorest in all methods due to the classical
approximation and the lack of time-dependent terms in $a_{0}$ and $b_{0}$.
This result may be observed in experiments with increasing of resolution in
experiments in the future.

We present several remarks on the above results. (I) The most striking feature
of these different methods is that $\lambda^{\prime}$, $\Delta^{\prime}$, and
$\nu^{\prime}$ can be unchanged, increased, or decreased, depending strongly
on the approximations employed. However, all of them seem to be reasonable and
accurate when $\omega$ is large enough. (II) In all the classical methods, the
variable is divided into fast and slow modes. Thus, the normalization of
$|a|^{2}+|b|^{2}=1$ is not guaranteed in the multiple scale analysis. In
contrast, the Hamiltonian is transformed according to unitary transformation
in quantum methods. This may account for the inaccuracy of the classical
methods when the frequency of the driving becomes small. (III) On the other
hand, quantum methods also have their problems. From the quantum Hamiltonian
in Eq. 15 and perturbative expansion of the Floquet Hamiltonian in Eq. 12, we
can find the Floquet Hamiltonian is highly sensitive to the expression of
$\hat{H}_{0}$ and $\hat{H}_{j}$. Thus the unitary transformation would affect
$\hat{H}_{0}$ and $\hat{H}_{j}$, leading to different Floquet Hamiltonian in
the presence of truncation. In this way, some of the approximations will yield
much better results than the other methods. Thus, the choice of the
$\hat{H}_{0}$ and $\hat{H}_{j}$ by a proper unitary transformation is a rather
tricky issue in practice. In Appendix B, we show that Q-II and C-II based on
the unitary transformation will yield the most accurate result.

##  V Conclusion and discussion

In this work, we have systematically revisited the Floquet Hamiltonian of the
BEC in a double well with periodic driving using five different methods, all
of which have solid theoretical grounds and seem to be reasonable. In all
methods, the Floquet Hamiltonian and associated mean field dynamics are
discussed up to the order of $\omega^{-2}$. In method Q-I, we find that
periodic driving can only weaken the coupling, while it can not affect other
parameters. In method Q-II, all parameters are modified by periodic driving
and we find that both the coupling and the interaction strength are weakened.
Moreover, new interaction terms for co-tunneling, which still preserves the
total number of particles, emerge from the second-order process. In methods
C-I, C-II, and C-III, we use multiple scale analysis methods to obtain the
Floquet Hamiltonian. It is found that the terms, coming from the nonlinear
interaction (i.e. $|\xi_{a,1}|^{2}$, $|\xi_{b,1}|^{2}$, etc.), also contribute
to the Floquet Hamiltonian. In method C-III, we have employed the same method
used in Karpitz pendulum Landau and Lifshitz (1976) to derive the Floquet
Hamiltonian, yielding results that all parameters are modified. Finally, we
compare all these results on equal footing from the dynamics of the condensate
and the associated macroscopic self-trapping process. We find that while all
methods can give a good approximation to the physics in the high frequency
limit, significant and observable differences can emerge when the strength of
the driving becomes comparable with the modulating frequency. It is found that
the quantum methods give much better approximation than the classical methods.
Especially, the last method (C-III) yields the poorest approximation to all
physics. This justifies that the Floquet Hamiltonian based on Eq. 7 is more
accurate than that based on Eq. 6.

These results may indicate that the analysis based on the Floquet Hamiltonian
may contain some subjectivity. If the calculation is performed to infinity
orders, one may expect that different starting points may yield the same
conclusion. However, due to the presence of truncation, the different methods
can yield different effective Floquet Hamiltonians. More strikingly, the
parameters even get an opposite modification in different methods. Since all
these results give rise to reasonable descriptions at the high frequency
limit, it is hard for us to claim that periodic driving can strengthen or
weaken some particular parameters. A similar problem also arise in the other
periodic-driven systems. Therefore, we have to be careful in employing the
Floquet Hamiltonian approach to analyze the modulated systems in the future if
high precision measurement can be achieved, in which the validity of these
different approaches from different approximations can be readily verified
using various simulation platforms. In the current work, we consider the
spinless models; and in the presence of spin degrees of freedom, the Floquet
engineering can be used to realize new interactions that do not exist in the
static Hamiltonian Jotzu  _et al._ (2014); Eckardt (2017); Hainaut  _et al._
(2018); Oka and Kitamura (2019); Zhou  _et al._ (2023b); Zhao  _et al._
(2023), thus the modification of the parameters in the Hamiltonian and the
emergence of new terms $\hat{V}_{\text{new}}$ from $\hat{H}_{\text{F}}^{(n)}$
is reasonable and meaningful. In the low modulating limit, some of the above
approaches may become invalid, yielding new band structures Lyu  _et al._
(2020); Vogl  _et al._ (2020). This is a common problem in all driven systems
based on various methods Goldman and Dalibard (2014); Goldman  _et al._
(2014); Vajna  _et al._ (2018); Venkatraman  _et al._ (2022); Vogl  _et al._
(2019), and the question that which method is more accurate is then important
and needs to be concerned in much more careful in future.

###### Acknowledgements.

This work is supported by the Strategic Priority Research Program of the
Chinese Academy of Sciences with grant No. XDB0500000, and the Innovation
Program for Quantum Science and Technology (No. 2021ZD0301200 and No.
2021ZD0301500).

##  Appendix A The criterion for MSTP

In this appendix, we present a complementary derivation for the criterion of
MSTP. It is found that all the mean field Floquet Hamiltonian in this
manuscript can be summarized as

$\displaystyle H$
$\displaystyle=\Delta(|a|^{2}-|b|^{2})-\nu(a^{*}b+b^{*}a)+\lambda_{2}|a|^{2}|b|^{2}$
(79)
$\displaystyle+\frac{\lambda}{2}(|a|^{2}-|b|^{2})^{2}-\lambda_{1}(a^{*}a^{*}bb+b^{*}b^{*}aa).$

It can be mapped to the following pendulum model

$\displaystyle H_{\text{NP}}$
$\displaystyle=\frac{2\lambda-\lambda_{2}}{4}z^{2}+\Delta
z-\nu\sqrt{1-z^{2}}\cos(\phi)$ (80)
$\displaystyle-\frac{\lambda_{1}}{2}(1-z^{2})\cos(2\phi).$

When $\lambda_{1}=\lambda_{2}=0$, this model reduces to the model in the main
text. Because this model is not chaos, the particle can only move along the
constant energy surface. In the following, we set $\Delta=0$ while the
discussion with $\Delta\neq 0$ is also straightforward. When $z=0$, we find

$E_{\text{NP}}=-\nu\cos(\phi)-\frac{\lambda_{1}}{2}\cos(2\phi).$ (81)

In this work, we mainly focus on the high frequency limit, where the value of
$\lambda_{1}$ is much smaller than $\nu$. Thus, the maximum of $E$ is at
$\phi=\pi$ with $\max(E)=\nu-\frac{\lambda_{1}}{2}$ and the criterion for MSTP
is

$\displaystyle\frac{2\lambda-\lambda_{2}}{4}z(0)^{2}-\nu\sqrt{1-z(0)^{2}}\cos(\phi(0))$
(82)
$\displaystyle-\frac{\lambda_{1}}{2}(1-z(0)^{2})\cos(2\phi(0))>\nu-\frac{\lambda_{1}}{2}$

Substitute the expressions of $\lambda^{\prime}$, $\lambda_{1}$,
$\nu^{\prime}$, and $\lambda_{2}$ into the above equation will yields
$\lambda_{c}$, which is used in Eq. 62. We see that in method Q-II, we also
have the similar new term (in Eq. 29), with the same constant $B$ and $C$.
Yet, the parameters of $\nu^{\prime}$, $\lambda^{\prime}$ and $\nu^{\prime}$
in these two approaches (Q-II and C-II) are slightly different.

##  Appendix B Exact solution

For the sake of self-contained, here we present an analysis of the nonlinear
model without tunneling (set $\nu=0$). In this case, the two equations are
totally decoupled. By a proper phase transformation $a\rightarrow
a\exp(iK(t))$ and $b\rightarrow b\exp(-iK(t))$, where
$\dot{K}=\Delta+A\cos(\omega t)$, the above two equation becomes

$i\dot{a}=\lambda(|a|^{2}-|b|^{2}),\quad i\dot{b}=-\lambda(|a|^{2}-|b|^{2}).$
(83)

This result is exact, from which we find $i\frac{d}{dt}(a+b)=0$. If we use the
pendulum analogy, it can be found that

$\displaystyle\frac{d}{dt}z=0,\quad\frac{d}{dt}\phi=\lambda z.$ (84)

The solution is $z=c$, and $\phi=\lambda ct$, with $c$ being a constant. We
find that this solution is fulfilled only by methods Q- II, and C-II, which
are realized by first performing the unitary transformation.

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