# Generating NOON states in circuit QED using multi-photon resonance in the
presence of counter-rotating interactions

Shi-fan Qi  Zhejiang Province Key Laboratory of Quantum Technology and Device,
Department of Physics, Zhejiang University, Hangzhou 310027, Zhejiang, China
Jun Jing  Email address<EMAIL_ADDRESS>Zhejiang Province Key Laboratory
of Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou 310027, Zhejiang, China

###### Abstract

The NOON states are valuable quantum resources, which have a wide range of
applications in quantum communication, quantum metrology, and quantum
information processing. Here we propose a fast, concise and reliable protocol
for deterministically generating the NOON states of two resonators coupled to
a single $\triangle$-type superconducting qutrit. In particular, we derive the
effective Hamiltonians at the multi-photon resonances by virtue of the strong
counter-rotating interaction between the resonator modes and the qutrit. Based
on these crucial effective Hamiltonians, our protocol simplifies the previous
ones using the single-photon resonance and consequently reduces the number of
operations for state preparation. To test the robustness of this protocol, we
analyze the effects from both the decoherence including dissipation and
dephasing and the crosstalk of resonator modes on the state fidelity through a
Lindblad master equation in the eigenstates of the full Hamiltonian.

##  I Introduction

Entanglement Horodecki  _et al._ (2009) plays a key role in quantum
technologies, such as quantum communication Gisin and Thew (2007), quantum
computing, and quantum information processing Xiu  _et al._ (2017). The
entangled states serve as valuable resources especially for the quantum
communication protocols, including quantum key distribution Ekert (1991),
quantum secret sharing Hillery  _et al._ (1999), and quantum secure direct
communication Long and Liu (2002); Deng  _et al._ (2003). Novel quantum
platforms as well as fast protocols for preparing and measuring the entangled
states have therefore been intensively pursued for a long time Wei  _et al._
(2006); Pan  _et al._ (2012a); Tashima  _et al._ (2016); Macrì  _et al._
(2018), and are still under active investigation.

Among various entangled states, the NOON states
$(|N0\rangle+|0N\rangle)/\sqrt{2}$ with $N$ integer consist of two orthogonal
component states in maximal superposition, which have been applied in quantum
optical lithography Boto  _et al._ (2000); D’Angelo  _et al._ (2001), quantum
metrology Kok  _et al._ (2002); Mitchell  _et al._ (2004) and quantum
information processing Gisin and Thew (2007); Bennett and DiVincenzo (2000).
To populate the Fock states with more photons in preparing the NOON state, it
is of great helpful to have adjustable atomic level splitting to be
selectively resonant with the desired process. Many protocols in the circuit-
QED systems have thus been proposed upon the ability of level-manipulation in
the artificial qubit, qutrit or qudit systems. The setup for NOON state-
preparation in Ref. Strauch  _et al._ (2010) consists of two superconducting
resonators and a tunable qubit under classical driving. The operation number
in that protocol is increased in a nonlinear way in terms of the number of the
photons $N$. Although in Ref. Merkel and Wilhelm (2010), only a linear number
of operations with respect to $N$ are required for NOON-state preparation, the
circuit on demand consists of more elements (three resonators and two qutrits)
in the experiment Wang  _et al._ (2011). The protocol in Ref. Su  _et al._
(2014) adopts a setup consisting of one superconducting transmon qutrit and
two resonators, which demands $2N$ operations to generate a NOON state with
$N$ photons. It has been shortly improved by Ref. Xiong  _et al._ (2015),
which demands only $N+1$ operations while using a four-level superconducting
flux device and two resonators. Recently, an efficient protocol Backens (2017)
for generating a “double” NOON state $(|NN00\rangle+|00NN\rangle)/\sqrt{2}$
was proposed, which consists of $N+2$ operations. However, it employs five
resonators and two superconducting flux qutrits (the latter are initialized as
a Bell state). Therefore, a trade-off has to be made between the complexity of
the devices (as well as their initial states) and the number of running steps
towards the NOON states.

In this work, we propose a fast protocol for generating the NOON states of two
resonators strongly coupled to a superconducting qutrit. The strong or
ultrastrong coupling opens a door to study the physics of virtual processes
governed by the interaction Hamiltonian and leads to many interesting
phenomena and applications Casanova  _et al._ (2010); Ai  _et al._ (2010);
Forn-Díaz  _et al._ (2010); Braak (2011); Ridolfo  _et al._ (2013); Zhao  _et
al._ (2015); Ashhab (2013); Garziano  _et al._ (2016); Pan  _et al._ (2012b);
Zhao  _et al._ (2017). With respect to the ratio of coupling strength and the
characteristic frequency in the Rabi model, $g/\omega\simeq 0.1$ is a rough
starting point of the strong-coupling regime Pan  _et al._ (2012b). In the
strongly coupled qubit-cavity systems, Rabi Hamiltonians Ma and Law (2015);
Garziano  _et al._ (2015) have been applied in the protocol Macrì  _et al._
(2018) to prepare the Bell states and the GHZ states. Without the counter-
rotating terms, it is impossible to achieve a desired effective Hamiltonian
connecting states with no conservation of the number of photons or excitons
from the full Hamiltonian. Here we extend the ideas in Ref. Ma and Law (2015);
Garziano  _et al._ (2015) to prepare the NOON states in light of the feasible
capabilities of manipulating the artificial atoms and atom-photon interactions
in the circuit-QED systems.

Around the specific multi-photon resonance points, we extract the effective
Hamiltonians for the interested system using the perturbation theory Garziano
_et al._ (2016); Macrì  _et al._ (2018); Stassi  _et al._ (2017). At such a
point, a selective pair of levels of the $\triangle$-type qutrit are multi-
photon resonant with the corresponding resonator, while the remaining level
and resonator do not interfere with the preceding resonance as well as the
relevant Hilbert subspace. Then our model of the $\triangle$-type qutrit
coupled to two resonators can be regarded as a “parallel” Rabi model. The
separability of the two subspaces and the valid regime of the effective
Hamiltonians have been verified by the standard numerical diagonalization of
the full Hamiltonian. The NOON states can be deterministically realized in the
effective model. Regarding the multi-photon resonance based on the strong or
ultrastrong atom-photon interaction and the controllability of the atomic
frequencies, our proposal can be implemented in the circuit-QED systems
Wallraff  _et al._ (2004); Schoelkopf and Girvin (2008); Peropadre  _et al._
(2010); You and Franco (2011).

The rest of the work is organized as follows. In Sec. II, we introduce the
full Hamiltonian. We numerically discuss and analytically explain the multi-
photon resonances arising from the counter-rotating interaction in the strong-
coupling regime. The double-photon and triple-photon resonances are discussed
in Sec. II.2 and Sec. II.3, respectively. The detailed derivation of the
effective Hamiltonians can be found in Appendices A and B. In Sec. III, we
analyze the decoherence effect from the external environment on the state
fidelity by a Lindblad master equation written in the eigenbases of the full
Hamiltonian. In Sec. IV, we provide a detailed procedure for the NOON state
preparation with an arbitrary $N$ and then consider the effect of the inter-
resonator coupling on their fidelities. In Sec. V, we conclude the work with a
brief discussion.

##  II Description of the model

###  II.1 Model Hamiltonian

The model we investigated is a general Rabi model describing a superconducting
$\triangle$-type qutrit coupled to two resonators (labelled $a$ and $b$) in
both longitudinal and transversal directions. The three levels of the
$\triangle$-type qutrit are labelled by $|g\rangle$, $|e\rangle$, and
$|f\rangle$, representing the ground state, the mediate state and the highest-
level state, respectively. The system Hamiltonian $(\hbar\equiv 1)$ can be
written as

$\displaystyle H$ $\displaystyle=H_{0}+V,$ (1) $\displaystyle H_{0}$
$\displaystyle=\omega_{a}a^{{\dagger}}a+\omega_{b}b^{{\dagger}}b+\omega_{eg}|e\rangle\langle
e|+\omega_{fg}|f\rangle\langle f|,$ $\displaystyle V$
$\displaystyle=[g_{eg}^{a}(a^{{\dagger}}+a)+g_{eg}^{b}(b+b^{{\dagger}})](\sigma_{x}^{eg}\cos\theta+\sigma_{z}^{eg}\sin\theta)$
$\displaystyle+[g_{fg}^{a}(a^{{\dagger}}+a)+g_{fg}^{b}(b+b^{{\dagger}})](\sigma_{x}^{fg}\cos\theta+\sigma_{z}^{fg}\sin\theta)$
$\displaystyle+[g_{fe}^{a}(a^{{\dagger}}+a)+g_{fe}^{b}(b+b^{{\dagger}})](\sigma_{x}^{fe}\cos\theta+\sigma_{z}^{fe}\sin\theta).$

Here $\omega_{a}$ and $\omega_{b}$ are the transition frequencies of the
resonators $a$ and $b$, respectively. $\omega_{jk}\equiv\omega_{j}-\omega_{k}$
with $j,k\in\\{g,e,f\\}$ represents the level spacing between $|j\rangle$ and
$|k\rangle$. The ground state energy is set as zero. $g^{n}_{jk}$ with
$n\in\\{a,b\\}$ and $j,k\in\\{g,e,f\\}$ is the coupling strength between the
resonator $n$ and the specific pair of levels $j$ and $k$. The coupling
operators of the qutrit are defined as $\sigma_{x}^{jk}\equiv|j\rangle\langle
k|+|k\rangle\langle j|$ and $\sigma_{z}^{jk}\equiv|j\rangle\langle
j|-|k\rangle\langle k|$, respectively. The angle $\theta$ parameterizes the
amount of the longitudinal and the transversal coupling between the qutrit and
the resonators, which is adjustable and independent from the choice of the
coupling strength $g^{n}_{jk}$. The arbitrary mixture of the longitudinal and
the transversal couplings has been realized in the circuit-QED experiments
Forn-Díaz  _et al._ (2010); Niemczyk  _et al._ (2010).

To simplify the discussion but with no loss of generality, we let
$g_{eg}^{n}=g_{fg}^{n}=g_{fe}^{n}=g_{n}$ with $n=a,b$. Thus, the interaction
Hamiltonian $V$ in Eq. (1) can be written as

$\displaystyle V$
$\displaystyle=[g_{a}(a^{{\dagger}}+a)+g_{b}(b+b^{{\dagger}})]\big{(}\sigma_{x}^{eg}\cos\theta+\sigma_{z}^{eg}\sin\theta$
(2)
$\displaystyle+\sigma_{x}^{fg}\cos\theta+\sigma_{z}^{fg}\sin\theta+\sigma_{x}^{fe}\cos\theta+\sigma_{z}^{fe}\sin\theta\big{)}.$

The rest part of this work is based on the Hamiltonian (2). We would apply the
standard perturbation theory to obtain the effective Hamiltonians in charge of
the desired Rabi oscillations, by which we can construct the NOON states. Also
for each effective Hamiltonian, a pair of effective coupling strength and
energy shift, which are consistent with each other to the leading-order of
$g_{a}$ or $g_{b}$, can be determined. The energy shifts as well as the
relevant level spacing for the multi-photon resonances are used in the
comparison between the numerical and analytical results.

###  II.2 Two-photon resonance

To show the basic mechanism for our proposal about the NOON-state generation
by the double-photon resonance, we plot in Fig. 1 the avoided level-crossings
of the system as a function of the qutrit frequencies. The eigenvalues
$\\{E_{n}\\}$ and the eigenstates of the full Hamiltonian (1) are obtained by
standard numerical diagonalization method in a truncated Hilbert space. A
sufficiently large number of energy eigenstates have been used to ensure that
the treatment here is not significantly affected by the truncation.

Figure 1: (Color online) (a) and (c) Energy-levels and avoided level-crossings
for the two-photon resonances, normalized by $\omega_{a}$ and plotted as a
function of the transition frequency $\omega_{eg}$ and $\omega_{fg}$,
respectively. The avoided level-crossings of the eigenstates are distinguished
in the dark circles. (b) and (d) Comparison between the numerically calculated
normalized effective coupling strengths $|g_{eff}|/\omega_{a}$ and
$|g_{eff}^{\prime}|/\omega_{a}$ (blue dots) and the corresponding analytical
results in Eqs. (4) and (7) from the second-order perturbation theory (orange
solid line), respectively. Here we fix $\omega_{b}=1.7\omega_{a}$,
$\theta=\pi/6$. For (a) and (b), $\omega_{fg}=2\omega_{b}$ and
$g_{b}=0.05\omega_{a}$; for (c) and (d), $\omega_{eg}=2\omega_{a}$ and
$g_{a}=0.05\omega_{a}$.

In Fig. 1(a), we see that an avoided level-crossing (distinguished in the dark
circle) occurs between two eigenstates of the full Hamiltonian when the level
spacing $\omega_{eg}$ approaches $2\omega_{a}$. It demonstrates a two-photon
resonance, in which two photons of mode-$a$ can be simultaneously created by
the atomic transition from level $|e\rangle$ to level $|g\rangle$, i.e.,
$|00e\rangle\rightarrow|20g\rangle$. And inversely, two photons of mode-$a$
can be simultaneously annihilated by the atomic transition from $|g\rangle$ to
$|e\rangle$, i.e., $|20g\rangle\rightarrow|00e\rangle$. In fact, when
$\omega_{eg}\approx 2\omega_{a}$, $|00e\rangle$ and $|20g\rangle$ are nearly
degenerate and they become the main component of the system eigenstates with
$E_{n}\approx 2\omega_{a}$. At the avoided-crossing point (the middle point in
the dark circle), $|E_{n}\rangle\approx(|00e\rangle\pm|20g\rangle)/\sqrt{2}$.
The lower eigenvector approaches $|00e\rangle$ ($|20g\rangle$) at the red
(blue) far-off-resonant end and the upper eigenvector does the other way
around. The existence of $V$ in either Eq. (1) or Eq. (2) lifts the degeneracy
of the two eigenstates and renders a strong Rabi-oscillation between them.
This phenomenon can also be well-explained by the effective Hamiltonian due to
the second-order process involving both the longitudinal and transversal
couplings in Eq. (1), provided that
$g_{a},g_{b}\ll\omega_{a},\omega_{b},|\omega_{b}-\omega_{a}|$. The detailed
derivation can be found in Appendix A. The effective Hamiltonian is found to
be

$H_{eff}=g_{eff}\left(|00e\rangle\langle 20g|+|20g\rangle\langle 00e|\right).$
(3)

Here the coupling strength

$g_{eff}=-\sqrt{2}g^{2}_{a}\left[\frac{\sin(2\theta)}{\omega_{a}}+\frac{\cos^{2}\theta}{\omega_{fg}-\omega_{a}}\right]$
(4)

is in the same order as the distinction $\delta$ of the avoided level-crossing
point from $2\omega_{a}$. At this point, $\omega_{eg}=2\omega_{a}+\delta$,
where $\delta$ is

$\displaystyle\delta$
$\displaystyle=-g^{2}_{a}\cos^{2}\theta\left(\frac{4}{\omega_{a}}+\frac{1}{\omega_{fg}-\omega_{a}}+\frac{3}{\omega_{fg}+\omega_{a}}\right)$
(5)
$\displaystyle-g^{2}_{b}\cos^{2}\theta\bigg{(}\frac{1}{2\omega_{a}+\omega_{b}}+\frac{1}{\omega_{fg}+\omega_{b}}+\frac{1}{2\omega_{a}-\omega_{b}}$
$\displaystyle+\frac{1}{2\omega_{a}-\omega_{b}-\omega_{fg}}\bigg{)}-\frac{4g^{2}_{a}\sin^{2}\theta}{\omega_{a}}-\frac{4g^{2}_{b}\sin^{2}\theta}{\omega_{b}}.$

The energy-splitting of the two eigenstates at the avoided level-crossing
point is $2|g_{eff}|$, which can be evaluated by the numerical simulation over
the whole Hilbert space. The comparison of $|g_{eff}|$ between the analytical
(4) and numerical (1) results are shown in Fig. 1(b) as a function of
normalized coupling strength $g_{a}/\omega_{a}$. The orange solid line is the
analytical result from the effective Hamiltonian (4) and the blue dots are the
results from the numerical diagonalization of the full Hamiltonian (1). One
can see that the effective Hamiltonian yields perfect results for normalized
interaction strengths $g_{a}/\omega_{a}\leq 0.095$. For an even larger
$g_{a}$, higher-orders contribution needs to capture all the effects from the
interaction Hamiltonian modifying the eigenstates of the bare system.

Figures 1(c) and 1(d) demonstrate the avoided level-crossing at the double-
photon resonance occurring in the subspace spanned by
$\\{|00f\rangle,|02g\rangle\\}$ (two hybridized states by the atomic levels
$|f\rangle$, $|g\rangle$ and mode-$b$) when the level spacing $\omega_{fg}$
approaches $2\omega_{b}$. Here we choose $\omega_{b}=1.7\omega_{a}$ to avoid
the unnecessary mixture of the demanding near-degenerate eigenstates or
subspaces. Followed by a similar perturbative derivation (see Appendix A), the
effective Hamiltonian is found to be

$H_{eff}^{\prime}=g_{eff}^{\prime}(|00f\rangle\langle 02g|+|02g\rangle\langle
00f|),$ (6)

where

$g_{eff}^{\prime}=-\sqrt{2}g^{2}_{b}\left[\frac{2\sin(2\theta)}{\omega_{b}}+\frac{\cos^{2}\theta}{\omega_{eg}-\omega_{b}}\right].$
(7)

To the second order of $g_{b}$, the energy shift is evaluated by

$\displaystyle\Delta$ $\displaystyle=\omega_{fg}-2\omega_{b}$ (8)
$\displaystyle=-g^{2}_{b}\cos^{2}\theta\left(\frac{4}{\omega_{b}}+\frac{1}{\omega_{eg}-\omega_{b}}+\frac{3}{\omega_{eg}+\omega_{b}}\right)$
$\displaystyle-g^{2}_{a}\cos^{2}\theta\bigg{(}\frac{1}{\omega_{eg}+\omega_{a}}+\frac{1}{2\omega_{b}+\omega_{a}}+\frac{1}{2\omega_{b}-\omega_{a}}$
$\displaystyle+\frac{1}{2\omega_{b}-\omega_{a}-\omega_{eg}}\bigg{)}.$

Here the normalized effective coupling strength for the double-photon
resonance is demonstrated in Fig. 1(d). Similar to Fig. 1(b), one can also
roughly estimate the valid range of the second-order perturbative Hamiltonian
through the comparison between the analytical result by Eq. (7) and the
numerical evaluation by the Hamiltonian (1). The two results match with each
other for normalized atom-photon interaction strength $g_{b}/\omega_{a}\leq
0.065$, the upper bound of which is still of a strong-coupling regime, and
depends on the choice of the other parameters, such as $g_{a}$, $\omega_{b}$,
and $\omega_{eg}$.

Figure 2: (Color online) Three groups of Rabi oscillations in one period. (1)
The blue dashed line and the blue dashed line with dots represent the
populations of state $|20g\rangle$ and $|00e\rangle$, respectively (the
initial state is $|00e\rangle$). (2) The orange dash-dotted line and the
orange dashed line with dots represent the state $|02g\rangle$ and
$|00f\rangle$, respectively (the initial state is $|00f\rangle$). (3) The dark
solid line and the dark solid line with dots represent the state
$(|20g\rangle+|02g\rangle)/\sqrt{2}$ and $(|00e\rangle+|00f\rangle)/\sqrt{2}$,
respectively (the initial state is $(|00e\rangle+|00f\rangle)/\sqrt{2}$). The
parameters are chosen as $\omega_{b}=1.7\omega_{a}$, $g_{a}=0.05\omega_{a}$
and $\theta=\frac{\pi}{6}$. The vertical grey dashed line represents the
analytical result.

The two effective Hamiltonians (3) and (6) are independent with each other and
can be simultaneously constructed provided $\omega_{eg}=2\omega_{a}+\delta$
and $\omega_{fg}=2\omega_{b}+\Delta$. When the whole system is initialled at
the $|00e\rangle$, i.e., the two resonators are in the vacuum state and the
qutrit is at $|e\rangle$, a completed Rabi oscillation between $|00e\rangle$
and $|20g\rangle$ can then be accurately realized with a period
$T_{R}=\pi/|g_{eff}|$. Similarly, when the whole system is initialled as
$|00f\rangle$, a completed Rabi oscillation between $|00f\rangle$ and
$|02g\rangle$ can also be observed with a period
$T^{\prime}_{R}=\pi/|g^{\prime}_{eff}|$. Thus if the system is initialled as
the symmetrical superposed state $(|00e\rangle+|00f\rangle)/\sqrt{2}$ and
$T_{R}=T^{\prime}_{R}$, it will evolve into
$(|20g\rangle+|02g\rangle)/\sqrt{2}=(|20\rangle+|02\rangle)/\sqrt{2}\otimes|g\rangle$
at $t=T_{R}/2$. For the resonator modes, they are now prepared as the two-
photon NOON state. In Fig. 2, we plot all these Rabi oscillations in one
period under the constrain $T_{R}=T^{\prime}_{R}$. With the parameters we
chosen, it is found that $\omega_{a}T_{R}=\pi/|g_{eff}|\approx 754$ by Eq.
(4), which is confirmed by the numerical simulation.

It is noteworthy to point that if $T_{R}\neq T^{\prime}_{R}$, we can firstly
let the initial state to experience a half Rabi-oscillation driven by Eq. (3);
then use a microwave $\pi/2$ pulse to realize $|e\rangle\to|g\rangle$; and
finally control the system to undergo a half Rabi-oscillation driven by Eq.
(6). In the end, the NOON state $(|20\rangle+|02\rangle)/\sqrt{2}$ emerges,
consuming more time and resource than that in case of $T_{R}=T^{\prime}_{R}$.
The order of these two half-Rabi-oscillations can be exchanged if the $\pi/2$
pulse is performed between $|f\rangle$ and $|g\rangle$.

With the effective Hamiltonians (3) and (6), one can generate a double-photon
NOON state from the vacuum state of modes $a$ and $b$. This protocol can be
straightforwardly extended to generate a $(n+2)$-photon NOON state when the
system is in the state $(|n0e\rangle+|0nf\rangle)/\sqrt{2}$. In the subspace
spanned by $\\{|n0e\rangle,|(n+2)0g\rangle\\}$, one can construct an effective
Hamiltonian
$H_{eff}=g_{eff}(n)[|n0e\rangle\langle(n+2)0g|+|(n+2)0g\rangle\langle n0e|]$
through a derivation similar to Appendix A. Here the effective coupling
strength is $n$-dependent:

$\displaystyle g_{eff}(n)=\sqrt{\frac{(n+1)(n+2)}{2}}g_{eff},$ (9)

where $g_{eff}$ is found in Eq. (4) and the energy shift is also
$n$-dependent:

$\delta_{n}=\delta-
ng^{2}_{a}\cos^{2}\theta\left(\frac{8}{3\omega_{a}}+\frac{1}{\omega_{fg}+\omega_{a}}-\frac{1}{\omega_{fg}-3\omega_{a}}\right).$
(10)

While in the subspace spanned by $\\{|0nf\rangle,|0(n+2)g\rangle\\}$, one can
find

$\displaystyle
g_{eff}^{\prime}(n)=\sqrt{\frac{(n+1)(n+2)}{2}}g^{\prime}_{eff},$ (11)

and

$\Delta_{n}=\Delta-
ng^{2}_{b}\cos^{2}\theta\left(\frac{8}{3\omega_{b}}+\frac{1}{\omega_{eg}+\omega_{b}}-\frac{1}{\omega_{eg}-3\omega_{b}}\right).$
(12)

Note $\delta$ in Eq. (10) and $\Delta$ in Eq. (12) represent the energy shifts
in Eqs. (5) and (8), respectively. Roughly, the magnitude of the effective
coupling strength scales linearly with the initial photon number of the
resonators, which accordingly reduces the time for state-preparation.

###  II.3 Three-photon resonance

In this subsection, we provide a protocol to achieve the three-photon
resonances, which is based on Eq. (2) with $\theta=0$. The Hamiltonian with a
nonvanishing $\theta$ is of course available for the same target. Nevertheless
the relevant protocol raises more treatments in mathematics yet no extra
physics. Now the full Hamiltonian can be written as

$\displaystyle H$ $\displaystyle=H_{0}+V,$ (13) $\displaystyle H_{0}$
$\displaystyle=\omega_{eg}|e\rangle\langle e|+\omega_{fg}|f\rangle\langle
f|+\omega_{a}a^{{\dagger}}a+\omega_{b}b^{{\dagger}}b,$ $\displaystyle V$
$\displaystyle=[g_{a}(a^{{\dagger}}+a)+g_{b}(b+b^{{\dagger}})](\sigma_{x}^{eg}+\sigma_{x}^{fg}+\sigma_{x}^{fe}).$

Figure 3: (Color online) (a) and (c) Energy-levels and avoided level-crossings
for the three-photon resonances, normalized by $\omega_{a}$ and plotted as a
function of the transition frequency $\omega_{eg}$ and $\omega_{fg}$,
respectively. (b) and (d) Comparison between the numerically calculated
normalized effective coupling strengths $|g_{effs}|/\omega_{a}$ and
$|g_{effs}^{\prime}|/\omega_{a}$ (blue points) and the corresponding
analytical results (15) and (18) (orange solid line), respectively. Here we
fix $\omega_{b}=2.2\omega_{a}$. For (a) and (b), $\omega_{fg}=3\omega_{b}$ and
$g_{b}=0.1\omega_{a}$; for (c) and (d), $\omega_{eg}=3\omega_{a}$ and
$g_{a}=0.1\omega_{a}$.

Analog to Sec. II.2, here in Fig. 3(a) and Fig. 3(c), we describe two avoided
level-crossings required for three-photon resonances in the respective
subspaces. In Fig. 3(b) and Fig. 3(d), we justified the corresponding
effective Hamiltonians by comparing the effective coupling strengthes based on
analytical derivation to those obtained from the numerical results at the
avoided level-crossing points. Particularly, Fig. 3(a) demonstrates the
avoided level-crossing (distinguished in dark circle) when $\omega_{eg}$
approaches $3\omega_{a}$ in the subspace spanned by
$\\{|00e\rangle,|30g\rangle\\}$. Through a straightforward derivation in
Appendix B, the effective Hamiltonian is found to be

$H_{effs}=g_{effs}\left(|00e\rangle\langle 30g|+|30g\rangle\langle
00e|\right),$ (14)

where

$g_{effs}=-\frac{\sqrt{6}g^{3}_{a}}{2\omega_{a}}\left(\frac{1}{2\omega_{a}}-\frac{1}{\omega_{fg}-2\omega_{a}}+\frac{1}{\omega_{fg}-\omega_{a}}\right).$
(15)

The leading-order correction of $\omega_{eg}$ to $3\omega_{a}$ can be obtained
as

$\displaystyle\delta_{s}$ $\displaystyle=\omega_{eg}-3\omega_{a}$ (16)
$\displaystyle=-g^{2}_{a}\left(\frac{3}{\omega_{a}}+\frac{3}{\omega_{fg}-\omega_{a}}+\frac{4}{\omega_{fg}+\omega_{a}}+\frac{1}{2\omega_{a}-\omega_{fg}}\right)$
$\displaystyle-g^{2}_{b}\bigg{(}\frac{1}{3\omega_{a}+\omega_{b}}+\frac{1}{\omega_{b}+\omega_{fg}}+\frac{1}{3\omega_{a}-\omega_{b}}$
$\displaystyle+\frac{1}{3\omega_{a}-\omega_{b}-\omega_{fg}}\bigg{)}.$

In Fig 3(b), one can observe that the analytical result of the effective
energy splitting (15) remains perfect for the normalized interaction strengths
$g_{a}/\omega_{a}\leq 0.14$ in comparison with the numerical result obtained
from Eq. (13).

Figure 3(c) demonstrates the avoided level-crossing occurring in the subspace
spanned by $\\{|00f\rangle,|03g\rangle\\}$ when the level spacing
$\omega_{fg}$ approaches $3\omega_{b}$. Followed by a similar perturbative
derivation as the preceding case (see Appendix B), the effective Hamiltonian
is found to be

$H^{\prime}_{effs}=g_{effs}^{\prime}\left(|00f\rangle\langle
03g|+|03g\rangle\langle 00f|\right),$ (17)

where the effective coupling strength

$g_{effs}^{\prime}=-\frac{\sqrt{6}g^{3}_{b}}{2\omega_{b}}\left(\frac{1}{2\omega_{b}}-\frac{1}{\omega_{eg}-2\omega_{b}}+\frac{1}{\omega_{eg}-\omega_{b}}\right).$
(18)

The leading-order correction of the avoided level-crossing point, i.e.,
$\Delta_{s}=\omega_{fg}-3\omega_{b}$, can be obtained as

$\displaystyle\Delta_{s}$
$\displaystyle=-g^{2}_{a}\bigg{(}\frac{1}{3\omega_{b}+\omega_{a}}+\frac{1}{\omega_{a}+\omega_{eg}}+\frac{1}{3\omega_{b}-\omega_{a}-\omega_{eg}}$
(19)
$\displaystyle+\frac{1}{3\omega_{b}-\omega_{a}}\bigg{)}-g^{2}_{b}\bigg{(}\frac{1}{2\omega_{b}-\omega_{eg}}+\frac{3}{\omega_{b}}+\frac{3}{\omega_{eg}-\omega_{b}}$
$\displaystyle+\frac{4}{\omega_{eg}+\omega_{b}}\bigg{)}.$

And then we present both the analytical result in Eq. (18) and the numerical
result with full Hamiltonian in Fig. 3(d) to verify the regime in which they
can match with each other. It is found that when $g_{b}/\omega_{a}\leq 0.13$,
the third-order perturbation remains as a good approximation. Although the
upper bound of the coupling strength depends on the choice of parameters, it
enters the strong-coupling regime.

Figure 4: (Color online) Three groups of Rabi oscillations. (1) The blue
dashed line and the blue dashed line with dots represent the state
$|30g\rangle$ and $|00e\rangle$, respectively (the initial state is
$|00e\rangle$). (2) The orange dash-dotted line and the orange dashed line
with dots represent the state $|03g\rangle$ and $|00f\rangle$, respectively
(the initial state is $|00f\rangle$). (3) The dark solid line and the dark
solid line with dots represent the state $(|30g\rangle+|03g\rangle)/\sqrt{2}$
and $(|00e\rangle+|00f\rangle)/\sqrt{2}$, respectively (the initial state is
$(|00e\rangle+|00f\rangle)/\sqrt{2}$). Here $\omega_{b}=2.2\omega_{a}$ and
$g_{a}=0.1\omega_{a}$.

As long as the energy levels of the artificial atom are adjusted to meet both
$\omega_{eg}=3\omega_{a}+\delta_{s}$ and $\omega_{fg}=3\omega_{b}+\Delta_{s}$,
the two effective Hamiltonians (14) and (17) can then be simultaneously
constructed. Consequently, when the whole system is initialled as an arbitrary
superposed state over $|00e\rangle$ and $|00f\rangle$ and
$|g_{effs}|=|g^{\prime}_{effs}|$, a completed Rabi oscillation occuring
between the initial state and the corresponding superposed state over
$|30g\rangle$ and $|03g\rangle$ with period $T_{R}=\pi/|g_{effs}|$. Thus if
the system is initialled as $(|00e\rangle+|00f\rangle)/\sqrt{2}$, it will
completely transform to
$(|30g\rangle+|03g\rangle)/\sqrt{2}=(|30\rangle+|03\rangle)/\sqrt{2}\otimes|g\rangle$
at $t=T_{R}/2$. At this moment, the cavity modes are prepared as the three-
photon NOON state. In Fig. 4, we plot the relevant Rabi oscillations in one
period under certain parameters by exact numerical evaluations. With the
chosen parameters, it is found that $\omega_{a}T_{R}=\pi/|g_{effs}|\approx
5562$ by Eq. (15), which is confirmed by numerical simulation (see the
vertical dashed grey line).

##  III Fidelity Analysis

As shown in Figs. 2 and 4, the initial state
$|00\rangle\otimes(|e\rangle+|f\rangle)/\sqrt{2}$ can be transformed to
$(|20\rangle+|02\rangle)/\sqrt{2}\otimes|g\rangle$ and
$(|30\rangle+|03\rangle)/\sqrt{2}\otimes|g\rangle$ at the respective multi-
photon resonances, by undergoing a half Rabi oscillation. The states of the
two modes are then the target NOON states with $N=2$ or $N=3$. Yet the whole
system cannot be isolated from the surrounding environment. The target NOON
state will be damaged by the influence from cavity mode damping and atomic
decay and dephasing. In this section, the fidelity of the prepared state is
studied based on the master equation approach. By applying the standard
Markovian approximation and tracing out the degrees of freedom of external
environment (assumed to be at the vacuum state), we arrive at the master
equation Ma and Law (2015); Beaudoin  _et al._ (2011); Ridolfo  _et al._
(2012) for the density-matrix operator $\rho(t)$ of the whole system
consisting of the two resonators and the artificial three-level atom,

$\displaystyle\dot{\rho}(t)$
$\displaystyle=-i[H_{diag},\rho(t)]+\kappa_{a}\mathcal{L}[X_{a}]\rho(t)+\kappa_{b}\mathcal{L}[X_{b}]\rho(t)$
(20)
$\displaystyle+\gamma_{eg}\mathcal{L}[S_{eg}]\rho(t)+\gamma_{fg}\mathcal{L}[S_{fg}]\rho(t)+\gamma_{fe}\mathcal{L}[S_{fe}]\rho(t)$
$\displaystyle+\gamma_{e}\mathcal{L}[S_{e}]\rho(t)+\gamma_{f}\mathcal{L}[S_{f}]\rho(t).$

Here $H_{diag}$ indicates that the full Hamiltonian $H$ is now expressed by
its eigenvectors $|E_{n}\rangle$’s. $\kappa_{a}$ ($\kappa_{b}$) is the decay
constant of the resonator mode $a$ ($b$). $\gamma_{eg}$, $\gamma_{fg}$, and
$\gamma_{fe}$ are respectively the energy relaxation constants associated with
the transitions $|e\rangle\to|g\rangle$, $|f\rangle\to|g\rangle$ and
$|f\rangle\to|e\rangle$. $\gamma_{e}$ ($\gamma_{f}$) is the dephasing constant
of the level $|e\rangle$ ($|f\rangle$). The superoperator $\mathcal{L}$ is
defined as

$\mathcal{L}[O]\rho=\frac{1}{2}(2O\rho O^{{\dagger}}-O^{{\dagger}}O\rho-\rho
O^{{\dagger}}).$ (21)

Here $O=X_{a},X_{b},S_{eg},S_{fg},S_{fe},S_{e},S_{f}$ are the dressed lowering
operators, defined respectively in terms of their bare counterparts
$o=a,b,\sigma^{-}_{eg},\sigma^{-}_{fg},\sigma^{-}_{fe},|e\rangle\langle
e|,|f\rangle\langle f|$ as

$O\equiv\sum_{E_{n}>E_{m}}\langle
E_{m}|(o+o^{{\dagger}})|E_{n}\rangle|E_{m}\rangle\langle E_{n}|.$ (22)

To simplify the discussion but with no loss of generality, we let
$\kappa_{a}=\kappa_{b}=\gamma_{fg}=\gamma_{fe}=\gamma_{eg}=\gamma_{e}=\gamma_{f}=\gamma$.

Figure 5: (Color online) (a) and (b) Fidelities of the NOON state
$(|20\rangle+|02\rangle)/\sqrt{2}$ and $(|30\rangle+|03\rangle)/\sqrt{2}$
prepared by our protocol given various decoherence rates, respectively. The
parameters of (a) are set the same as those in Fig. 2 and
$T_{R}=\pi/|g_{eff}|$ is evaluated by Eq. (4); the parameters of (b) are set
the same as those in Fig. 4 and $T_{R}=\pi/|g_{effs}|$ is evaluated by Eq.
(15).

The robustness of our protocols for generating
$|\phi\rangle=(|N0\rangle+|0N\rangle)/\sqrt{2}$ upon the multi-photon
resonance can be measured by the state-fidelity
$F=\sqrt{\langle\phi|\rho(t)|\phi\rangle}$. Here $\rho(t)$ is numerically
obtained by the master equation (20), whose parameters (the accurate value of
level-spacings and evolution time) are determined in the derivation of the
effective Hamiltonian.

In Fig. 5, we plot the fidelity under different decoherence rates $\gamma$.
One can observe from Fig. 5(a) that the protocol of the double-photon
resonance works rather well for $\gamma/\omega_{a}\leq 10^{-5}$ (Note in
recent experiments Yan  _et al._ (2015); You  _et al._ (2007); Peterer  _et
al._ (2015); Pop  _et al._ (2014), the relative magnitude of the decoherence
rates is about $10^{-6}\sim 10^{-5}$), producing the desired NOON state with a
fidelity of $96\%$, close to $97\%$ in the case with no decoherence. The
fidelity $F$ maintains above $80\%$ even when $\gamma$ is enhanced to
$10^{-4}\omega_{a}$. While for a larger decoherence rate
$\gamma/\omega_{a}\leq 10^{-3}$, the fidelity will drop below $50\%$. It is
consistent with the fact that now $\gamma$ becomes comparable with the
effective coupling strength of Eq. (4). Note $|g_{eff}|/\omega_{a}\approx
4\times 10^{-3}$.

The fidelities for the protocol of the triple-photon resonance are shown in
Fig. 5(b). It is clear that the protocol works well as long as
$\gamma/\omega_{a}\leq 10^{-6}$, producing the desired NOON state
$|\phi\rangle=(|30\rangle+|03\rangle)/\sqrt{2}$ with a fidelity over $95\%$.
The fidelity $F$ decreases to $82\%$ when $\gamma/\omega_{a}=10^{-5}$. And it
is below $50\%$ when $\gamma/\omega_{a}=10^{-4}$, which is the same order as
the effective coupling strength $|g_{effs}|/\omega_{a}\approx 6\times
10^{-4}$. It is also consistent with the fact that the effective coupling
strength for the triple-photon process is one order weaker than that for the
double-photon process.

##  IV Preparing NOON states with multiple photons

Suppose that the qutrit is initially in the state
$|\psi\rangle=\frac{1}{\sqrt{2}}(|e\rangle+|f\rangle)$ and the two resonators
are initially in the vacuum state $|00\rangle$. As shown in Fig. 6, the
procedure for generating the NOON states of the two resonators can be mainly
divided into two parts: in part $A$ ($B$), the subspace
$\\{|e\rangle,|g\rangle\\}$ ($\\{|f\rangle,|g\rangle\\}$) and the resonator
mode $a$ ($b$) are operated by $(N-1)$ steps to populate the $|2(N-1)\rangle$
Fock state of the mode $a$ ($b$). Throughout the part $A$ ($B$), the
irrelevant mode $b$ ($a$) is decoupled from the rest part of the whole system
by tuning its frequency to the far-off-resonant regime. In the end, an
operation by virtue of the effective dynamics involving both modes is
performed to complete the whole protocol.

Figure 6: (Color online) (a) and (c): Resonator $a$ ($b$) is tuned to be
multi-photon resonant with the $|e\rangle\leftrightarrow|g\rangle$
($|f\rangle\leftrightarrow|g\rangle$) transition and far-off multi-photon
resonant with the $|f\rangle\leftrightarrow|g\rangle$
($|e\rangle\leftrightarrow|g\rangle$) transition. Then the photon number of
the mode is added while the qutrit is transformed to the ground state. (b) and
(d): A driving pulse that is resonant with the
$|e\rangle\leftrightarrow|g\rangle$ ($|f\rangle\leftrightarrow|g\rangle$)
transition excites the qutrit back to the state $|e\rangle$ ($|f\rangle$).

For the target NOON state with $2N$ photons ($N\geq 1$), part-$A$ is divided
into $N-1$ steps as following.

Step-$A_{1}$: Let mode-$a$ and the qutrit resonate with the transition
$|00e\rangle\leftrightarrow|20g\rangle$, i.e.,
$\omega_{eg}=2\omega_{a}+\delta$, where $\delta$ is given by Eq. (5). Then as
shown in Fig. 6(a), the component $|00e\rangle$ in the initial state will
become $|20g\rangle$ through the Rabi oscillation under the effective
Hamiltonian (3) after an evolution time $t_{1}=\pi/|2g_{eff}|$. Hence, the
initial state $\Psi(0)=\frac{1}{\sqrt{2}}(|00e\rangle+|00f\rangle)$ of the
whole system becomes

$\Psi(t_{1})=\frac{1}{\sqrt{2}}(-i|20g\rangle+|00f\rangle).$ (23)

Then as shown in Fig. 6(b), a microwave pulse of
$\\{\omega_{eg},-\pi/2,\pi/(2\Omega_{eg})\\}$ is applied to transform the
state (23) to

$\Psi(t_{1}+\tau)=\frac{1}{\sqrt{2}}(-i|20e\rangle+|00f\rangle),$ (24)

where $\omega_{eg}$ is the pulse frequency, $-\pi/2$ is a time-independent
phase and $\tau\equiv\pi/(2\Omega_{eg})$ is the duration time of the pulse Su
_et al._ (2014). Here and below, we assume that Rabi frequency
$\Omega_{eg}\gg|g_{eff}(n)|$ (Note $g_{eff}(0)=g_{eff}$), so that ideally the
system evolution due to the qutrit-mode-$a$ interaction is negligible during
$\tau$.

Step-$A_{j}$, $(j=2,3,\cdots,N-1)$: The level spacing $\omega_{eg}$ is tuned
to be resonant with the transition
$|(2j-2)0e\rangle\leftrightarrow|(2j)0g\rangle$, i.e.,
$\omega_{eg}=2\omega_{a}+\delta_{2j-2}$ according to Eq. (10). Through a
similar Rabi oscillation as in step-$A_{1}$, $|(2j-2)0e\rangle$ is transformed
to $-i|(2j)0g\rangle$ after $t_{j}=\pi/|2g_{eff}(2j-2)|$, where the effective
coupling strength is given by Eq. (9). $|(2j)0g\rangle$ is then further
transformed to $|(2j)0e\rangle$ by a microwave pulse of
$\\{\omega_{eg},-\pi/2,\pi/(2\Omega_{eg})\\}$ pumping the state $|g\rangle$
back to $|e\rangle$. Thus after these operations, the state (24) becomes

$\Psi(T_{A})=\frac{1}{\sqrt{2}}\left[(-i)^{N-1}|(2N-2)0e\rangle+|00f\rangle\right],$
(25)

where $T_{A}=\sum^{N-1}_{j=1}t_{j}+(N-1)\tau$. At this moment, part-$A$ is
completed. Then we focus on the $|00f\rangle\to|02g\rangle$ transition
involving mode-$b$ as shown in Fig. 6(c).

Step-$B_{1}$: Similar to step-$A_{1}$, the effective Hamiltonian (6) is used
to realize a half Rabi transition from $|00f\rangle$ to $|02g\rangle$. After
the interaction time $t^{\prime}_{1}=\pi/|2g^{\prime}_{eff}|$, where
$g^{\prime}_{eff}$ is given by Eq. (7), the state (25) becomes

$\Psi(T_{A}+t^{\prime}_{1})=\frac{1}{\sqrt{2}}\left[(-i)^{N-1}|(2N-2)0e\rangle-i|02g\rangle\right].$
(26)

Then as shown in Fig. 6(d), a microwave pulse of
$\\{\omega_{fg},-\pi/2,\tau^{\prime}\equiv\pi/(2\Omega_{fg})\\}$ pumping
$|g\rangle$ to $|f\rangle$, transforms the state (26) to

$\Psi(T_{A}+t^{\prime}_{1}+\tau^{\prime})=\frac{1}{\sqrt{2}}\left[(-i)^{N-1}|(2N-2)0e\rangle-i|02f\rangle\right].$
(27)

Here and below, we assume $\Omega_{fg}\gg|g_{eff}^{\prime}(n)|$ so that the
interaction between the qutrit and mode-$b$ is negligible in the pulse
duration time.

Step-$B_{k}$, $(k=2,3,...,N-1)$: Tune the level difference $\omega_{fg}$ to be
resonant with the transition $|0(2k-2)f\rangle\leftrightarrow|0(2k)g\rangle$,
namely, $\omega_{fg}=2\omega_{b}+\Delta_{2k-2}$ according to Eq. (12). The
system undergoes a similar Rabi oscillation as in step-$B_{1}$ with a
evolution time set as $t_{k}^{\prime}=\pi/|2g_{eff}^{\prime}(2k-2)|$ due to
Eq. (11), after which the basis $|0(2k-2)f\rangle$ moves to
$-i|0(2k)g\rangle$. It then moves to $-i|0(2k)f\rangle$ by a microwave pulse
of $\\{\omega_{fg},-\pi/2,\pi/(2\Omega_{fg})\\}$ pumping $|g\rangle$ to
$|f\rangle$. Thus after these steps, the state (27) becomes

$\displaystyle\psi(T_{A}+T_{B})$
$\displaystyle=\frac{1}{\sqrt{2}}\big{[}(-i)^{N-1}|(2N-2)0e\rangle$ (28)
$\displaystyle+(-i)^{N-1}|0(2N-2)f\rangle\big{]},$

where $T_{B}=\sum^{N-1}_{k=1}t_{k}^{\prime}+(N-1)\tau^{\prime}$. At this
moment, part-$B$ is completed. The order of part-$A$ and part-$B$ can be
mutually exchanged.

The final step is started by simultaneously tuning the level splittings
$\omega_{eg}$ and $\omega_{fg}$ to be resonant with
$|(2N-2)0e\rangle\leftrightarrow|(2N)0g\rangle$ and
$|0(2N-2)f\rangle\leftrightarrow|0(2N)g\rangle$, respectively. Namely,
$\omega_{eg}=2\omega_{a}+\delta_{2N-2}$ and
$\omega_{fg}=2\omega_{b}+\Delta_{2N-2}$, and now two effective Rabi
oscillations are simultaneously switched on. Under the coupling strength
$g_{eff}=g^{\prime}_{eff}$ and the operation time
$t_{N}=\pi/|2g_{eff}(2N-2)|$, the state (28) eventually becomes

$\displaystyle\Psi(T)$
$\displaystyle=\frac{1}{\sqrt{2}}\left[(-i)^{N}|(2N)0g\rangle+(-i)^{N}|0(2N)g\rangle\right]$
(29)
$\displaystyle=\frac{(-i)^{N}}{\sqrt{2}}[|(2N)0\rangle+|0(2N)\rangle]\otimes|g\rangle.$

The total time for completing our protocol using two-photon resonance is found
to be

$\displaystyle T$ $\displaystyle=T_{A}+T_{B}+t_{N}$ (30)
$\displaystyle=\sum^{N-1}_{j=1}\frac{\pi}{|g_{eff}(2j-2)|}+(N-1)\left(\frac{\pi}{2\Omega_{fg}}+\frac{\pi}{2\Omega_{eg}}\right)$
$\displaystyle+\frac{\pi}{2|g_{eff}(2N-2)|}.$

Here the time for all the frequency adjustments has been omitted. When $N=1$,
the whole procedure of preparation is reduced to the final step, which is
exactly the same as illustrated in Sec. (II.2). Note $\delta_{0}=\delta$ and
$\Delta_{0}=\Delta$.

The NOON state (29) can be also constructed when $g_{eff}\neq
g_{eff}^{\prime}$. In this case, we first arrive at $|\Psi(T_{A})\rangle$
(25). Then we perform an extra Step-$A_{N}$, whose definition is analog to
Step-$A_{j}$ ($j<N$), so that $|\Psi(T_{A})\rangle$ evolves to

$\Psi(\tilde{T}_{A})=\frac{1}{\sqrt{2}}\left[(-i)^{N}|(2N)0e\rangle+|00f\rangle\right],$
(31)

where $\tilde{T}_{A}=\sum^{N}_{j=1}t_{j}+N\tau$. Then we continue the
operations in part-$B$. After that, the state (31) becomes

$\Psi(\tilde{T}_{A}+T_{B})=\frac{(-i)^{N-1}}{\sqrt{2}}\left[-i|(2N)0e\rangle+|0(2N-2)f\rangle\right].$
(32)

Next a pulse of $\\{\omega_{eg},\pi/2,\pi/(2\Omega_{eg})\\}$ is used to
realize $|e\rangle\to|g\rangle$. Finally, the level spacing $\omega_{fg}$ is
tuned to be resonant with $|0(2N-2)f\rangle\leftrightarrow|0(2N)g\rangle$.
After an evolution time $t_{N}=\pi/|2g_{eff}^{\prime}(2N-2)|$, we obtain

$\Psi(T)=\frac{(-i)^{N}}{\sqrt{2}}\left[|(2N)0\rangle+|0(2N)\rangle\right]\otimes|g\rangle.$
(33)

The time of the whole procedure in total becomes

$\displaystyle T$
$\displaystyle=\sum^{N}_{j=1}\left[\frac{\pi}{2|g_{eff}(2j-2)|}+\frac{\pi}{2|g_{eff}^{\prime}(2j-2)|}\right]+\frac{N\pi}{2\Omega_{fg}}$
(34) $\displaystyle+\frac{(N-1)\pi}{2\Omega_{eg}}.$

To prepare the NOON states with $2N-1$ (odd number, $N\geq 1$) photons, one
needs to revise merely the final step after the system arrives at the state in
Eq. (28). At that moment, we set $\omega_{eg}=\omega_{a}$,
$\omega_{fg}=\omega_{b}$ and $g_{a}=g_{b}$. Then two mutually-independent Rabi
oscillations can simultaneously occur in their respective subspaces by virtue
of the first-order/single-photon process determined by the interaction
Hamiltonian (2). In particular, after the evolution time
$t_{N}=\pi/[2\sqrt{2N-1}g_{a}\cos\theta]$, the state (28) becomes

$\Psi(T)=\frac{(-i)^{N}}{\sqrt{2}}\left[|(2N-1)0\rangle+|0(2N-1)\rangle\right]\otimes|g\rangle,$
(35)

where

$\displaystyle T$ $\displaystyle=T_{A}+T_{B}+t_{N}$ (36)
$\displaystyle=\sum^{N-1}_{j=1}\left[\frac{\pi}{2|g_{eff}(2j-2)|}+\frac{\pi}{2|g_{eff}^{\prime}(2j-2)|}\right]+(N-1)$
$\displaystyle\times\left(\frac{\pi}{2\Omega_{fg}}+\frac{\pi}{2\Omega_{eg}}\right)+\frac{\pi}{2\sqrt{2N-1}g_{a}\cos\theta}.$

The results (29) and (35) show that the two resonators $a$ and $b$ are
eventually prepared in a NOON state.

Figure 7: (Color online) The fidelity of the NOON state
$(|0N\rangle+|N0\rangle)/\sqrt{2}$ under different inter-cavity coupling
strengths $g_{ab}$. When $g_{ab}=0.1g_{a}$, the fidelities are $96.7\%$
($N=2$), $88.4\%$ ($N=4$), $80.1\%$ ($N=6$) for an even $N$; and $85.3\%$
($N=3$), $73.5\%$ ($N=5$) for an odd $N$, respectively.

During the state preparation, the whole system is not only subject to the
external decoherence channels for each constituent described by the master
equation (20), but also under the influence of intrinsic disturbance, such as
the crosstalk between the two resonators $a$ and $b$ Peropadre  _et al._
(2013). Namely, the interaction Hamiltonian $V$ in Eq. (2) can be generalized
to

$\tilde{V}=V+g_{ab}(a+a^{{\dagger}})(b+b^{{\dagger}}),$ (37)

where $g_{ab}$ is the inter-cavity coupling strength. And the full Hamiltonian
in Eq. (20) is modified accordingly.

In a typical system consisting of a flux qutrit and two resonators Yan  _et
al._ (2015); You  _et al._ (2007); Peterer  _et al._ (2015); Pop  _et al._
(2014); Leek  _et al._ (2010), the transition frequencies among the three
levels of the qutrit can be manipulated within the range $\sim[1,20]$ GHz. In
the numerical evaluation of the fidelity under both decoherence and crosstalk,
the frequencies of the resonators $a$ and $b$ are set as
$\omega_{a}/(2\pi)\sim 4$GHz and $\omega_{b}/(2\pi)\sim 6.8$GHz, respectively.
The decay rates are set as $\gamma_{e}^{-1}=5\mu$s, $\gamma_{f}^{-1}=5\mu$s,
$\gamma_{eg}^{-1}=10\mu$s, $\gamma_{fg}^{-1}=20\mu$s,
$\gamma_{fe}^{-1}=5\mu$s, and $\kappa_{a}^{-1}=\kappa_{b}^{-1}=20\mu$s. The
Rabi frequencies of the microwave pulses are set as
$\Omega_{eg}=\Omega_{fg}=300$MHz. The effect of the inter-cavity coupling on
the NOON-state fidelity is demonstrated in Fig. 7 for different $N$.

We fix the coupling strength of resonator $a$ and the qutrit to be
$g_{a}/(2\pi)=120$MHz, which is about $0.03\omega_{a}$ according to the
previous setting. Note this value is both feasible in experiments and valid
for the obtained effective Hamiltonian as shown in Fig. 1. Due to the fact
that the preparation protocol is parity-dependent, so that for an even $N$,
$g_{b}$ is found to be $\sim 69.4$MHz to fulfill the condition
$g_{eff}=g^{\prime}_{eff}$ and for an odd $N$, $g_{b}=g_{a}=120$MHz.

$N$ | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---
$F(\delta_{n},\Delta_{n})$ | 92.2% | 82.7% | 81.6% | 70.8% | 70.0%
$F(\tilde{\delta}_{n},\tilde{\Delta}_{n})$ | 96.5% | 84.2% | 86.6% | 72.8% | 75.7%
Table 1: The fidelities of the NOON state $(|0N\rangle+|N0\rangle)/\sqrt{2}$
under $g_{ab}=0.5g_{a}$ using the energy shifts $\delta_{n}$ and $\Delta_{n}$
in Eqs. (10) and (12), respectively or the modified $\tilde{\delta}_{n}$ and
$\tilde{\Delta}_{n}$ in Eq. (38).

As expected, the fidelities in Fig. 7 decline with the increasing $N$ for
either group of parity, i.e., $F(N=2)>F(N=4)>F(N=6)$ and $F(N=3)>F(N=5)$. It
turns out that the effect of the crosstalk of cavities could be negligible as
long as $g_{ab}\leq 0.1g_{a}$ (it is feasible in experiments Yang  _et al._
(2012)), by which a high fidelity $\geq 73\%$ can be still obtained for $N\leq
5$. When $N=6$, it is interesting to find that
$F(g_{ab}=0.1g_{a})>F(g_{ab}=0)>F(g_{ab}=0.05g_{a})>F(g_{ab}=0.5g_{a})$, which
indicates the deficiency of the preceding effective Hamiltonian. To
accommodate a larger $N$ and a stronger inter-cavity coupling, we could
revisit the perturbative treatment in Appendix A using the updated full
Hamiltonian with the interaction Hamiltonian (37). It gives rises to the
effective Hamiltonians in the same formation as in Eqs. (3) and (6), but with
modified energy shifts:

$\displaystyle\tilde{\delta}_{n}=\delta_{n}+2g^{2}_{ab}\left(\frac{1}{\omega_{a}-\omega_{b}}-\frac{1}{\omega_{a}+\omega_{b}}\right),$
(38)
$\displaystyle\tilde{\Delta}_{n}=\Delta_{n}+2g^{2}_{ab}\left(\frac{1}{\omega_{b}-\omega_{a}}-\frac{1}{\omega_{a}+\omega_{b}}\right),$

which have taken account the effect of inter-cavity crosstalk into the double-
photon resonant points of $\omega_{eg}$ and $\omega_{fg}$, respectively. As
demonstrated in Table 1, the application of the modified $\tilde{\delta}_{n}$
and $\tilde{\Delta}_{n}$ can really improve the fidelities with different $N$
even under $g_{ab}=0.5g_{a}$.

##  V Discussion and Conclusion

The effective Hamiltonians obtained in Sec. II respectively in charge of the
two-photon resonance (see Sec. II.2) and the three-photon resonance (see Sec.
II.3) capture the effects from the relevant transitions in the whole Hilbert
space and have been justified by the numerical evaluation. The energy shifts
in the leading order $\delta$, $\Delta$, $\delta_{s}$ and $\Delta_{s}$
respectively presented in Eqs. (5), (8), (16) and (19) determine the points
for the intrinsic multi-photon resonances. Subsequently the effective coupling
strengthes $g_{eff}$, $g_{eff}^{\prime}$, $g_{effs}$ and $g_{effs}^{\prime}$
respectively presented in Eqs. (4), (7), (15) and (18) are used to evaluate
the period of the desired Rabi oscillations.

More than the protocols for constructing the NOON states with $N=2$ and $N=3$,
the effective Hamiltonians are the basic elements in generating the NOON
states with more photons in Sec. IV. Comparing to the previous protocol Su
_et al._ (2014) to the NOON state with the same number of photons, ours in
Sec. IV reduces the number of operations to about a half. The state fidelity
under preparation has been estimated by a Lindblad master equation in the
eigenbases of the original Hamiltonian of Sec. III.

The protocol we proposed is essentially based on two independent and parallel
Rabi oscillations in the $\triangle$-type qutrit, which can adapt to the other
types of three-level systems, such as the $V$-type and the ladder-type qutrit
Su  _et al._ (2014), via a properly modified interaction Hamiltonian $V$. For
example, the interaction Hamiltonian for the $V$-type three-level system has
no terms containing $\sigma^{fe}_{x,z}$. While a more complex $\triangle$-type
qutrit model established in the circuit-QED system is a compromise with
respect to the practical realization by the state-of-art quantum technology.
The first reason for this choice is that the excited states of the atomic
system are able to be adjusted on demand, which greatly reduces the
possibility of choosing the natural atoms. The second one is that to realize
the target NOON state as fast as possible (otherwise the accumulated
decoherence effect will destroy the target state), the coupling strength
between the atomic system and the resonators should be of the strong regime.
The circuit-QED system satisfies these two conditions; and in many situations
the three-level system of the superconduting circuit is of a $\triangle$-type
qutrit.

In conclusion, we have presented a concise protocol for the deterministic
preparation of the NOON states. We do this in a setup consisting of two
resonator modes strongly coupled to a single $\triangle$-type superconducting
qutrit within the framework of a general Rabi model. The protocol relies on
the effective Hamiltonians at the avoided level-crossing points, which
reserves the effects of the counter-rotating terms and the leading-order
contributions of the state transitions. By properly using the effective
Hamiltonian, the population at the excited states of the qutrit pumped by the
microwave driving pulses can be transferred to multiple photons in the
corresponding resonator modes. Moreover, our protocol is found to be robust
against the external decoherence with a typical magnitude of decay rate and
the internal crosstalk of the resonators. Hence, our study is of interested in
pursuit of the entangled state with the counter-rotating interaction and of
important to control the quantum state in the circuit-QED system with fewer
steps and fewer devices.

## Acknowledgments

We acknowledge grant support from the National Science Foundation of China
(Grants No. 11974311 and No. U1801661), and Zhejiang Provincial Natural
Science Foundation of China under Grant No. LD18A040001.

##  Appendix A The effective Hamiltonian for the two-photon resonances

To extract the effective Hamiltonian (3) in the subspace spanned by
$\\{|00e\rangle\equiv|0\rangle_{a}|0\rangle_{b}|e\rangle,|20g\rangle\equiv|2\rangle_{a}|0\rangle_{b}|g\rangle\\}$
from the full Hamiltonian (1) for the two-photon resonance shown in Fig. 1(a),
one can apply the standard perturbation theory with respect to the atom-photon
coupling strengths $g_{a}$ and $g_{b}$. In Fig. 8, all the second-order
processes involving with these two bases will be considered in the following
construction of the effective Hamiltonian.

Figure 8: (Color online) All the second-order (leading-order) paths involving
the bases $|00e\rangle$ and $|20g\rangle$. Blue solid (dashed) lines mark the
transitions mediated by $\sigma_{x}$ ($\sigma_{z}$) relative to the resonator
$a$. Red dotted (long-dashed) lines mark the transitions mediated by
$\sigma_{x}$ ($\sigma_{z}$) relative to the resonator $b$.

The interaction Hamiltonian $V$ in Eq. (2) can be regarded as a perturbation
provided that $g_{a},g_{b}\ll\omega_{a},\omega_{b},|\omega_{b}-\omega_{a}|$.
To the second order, the effective coupling strength or energy shift between
any eigenstates $|i\rangle$ and $|j\rangle$ of the unperturbed Hamiltonian
$H_{0}$ in Eq. (1) is given by Garziano  _et al._ (2015); Macrì  _et al._
(2018)

$g=\sum_{n\neq i,j}\frac{V_{jn}V_{ni}}{\omega_{i}-\omega_{n}}$ (39)

where $V_{nm}\equiv\langle n|V|m\rangle$ and $\omega_{n}$ is the eigenenergy
of $|n\rangle$.

We write the avoid level-crossing point $\omega_{eg}=2\omega_{a}+\delta$,
where $\delta$ is an undetermined energy shift consistent with the second-
order effective Hamiltonian. It is worthy to emphasis this shift should not be
omitted as in some literatures, since it is in the same order of the effective
coupling strength. We first consider the contribution to the effective
coupling strength from the two paths connecting $|00e\rangle$ and
$|20g\rangle$, i.e., $|00e\rangle\to|10f\rangle\to|20g\rangle$ and
$|00e\rangle\to|10g\rangle\to|20g\rangle$ as shown in Fig. 8. By virtue of Eq.
(39), one can get

$\displaystyle g_{eff}$
$\displaystyle=-\sqrt{2}g^{2}_{a}\left[\frac{\sin(2\theta)}{\omega_{a}+\delta}+\frac{\cos^{2}\theta}{\omega_{fg}-\omega_{a}-\delta}\right]$
(40)
$\displaystyle=-\sqrt{2}g^{2}_{a}\left[\frac{\sin(2\theta)}{\omega_{a}}+\frac{\cos^{2}\theta}{\omega_{fg}-\omega_{a}}\right]+\mathcal{O}(\delta),$

where $\mathcal{O}(\delta)$ means all the other orders of $\delta$ from the
zeroth order (the first term of the second line) in terms of Taylor expansion.
The other paths in Fig. 8 (For example,
$|00e\rangle\to|01f\rangle\to|00e\rangle$) are in charge of the energy shifts
for $|00e\rangle$ or $|20g\rangle$.

Summarizing all the four paths from $|00e\rangle$ and back to $|00e\rangle$
through a mediate state, i.e., $|00e\rangle\to|10g\rangle\to|00e\rangle$,
$|00e\rangle\to|10f\rangle\to|00e\rangle$,
$|00e\rangle\to|01g\rangle\to|00e\rangle$, and
$|00e\rangle\to|01f\rangle\to|00e\rangle$, one can obtain the second-order
energy correction (shift) $\epsilon_{1}$ for the state $|00e\rangle$ according
to Eq. (39)

$\displaystyle\epsilon_{1}$
$\displaystyle=g^{2}_{a}\cos^{2}\theta\left(\frac{1}{\omega_{a}}-\frac{1}{\omega_{fg}-\omega_{a}}\right)$
(41)
$\displaystyle+g^{2}_{b}\cos^{2}\theta\left(\frac{1}{2\omega_{a}-\omega_{b}}-\frac{1}{\omega_{fg}+\omega_{b}-2\omega_{a}}\right)+\mathcal{O}(\delta).$

And in the same way, the energy shift $\epsilon_{2}$ for the state
$|20g\rangle$ is found to be

$\displaystyle\epsilon_{2}$
$\displaystyle=-g^{2}_{a}\cos^{2}\theta\left(\frac{3}{\omega_{a}}+\frac{2}{\omega_{fg}-\omega_{a}}+\frac{3}{\omega_{fg}+\omega_{a}}\right)$
(42)
$\displaystyle-g^{2}_{b}\cos^{2}\theta\left(\frac{1}{2\omega_{a}+\omega_{b}}+\frac{1}{\omega_{fg}+\omega_{b}}\right)-\frac{4g^{2}_{a}\sin^{2}\theta}{\omega_{a}}$
$\displaystyle-\frac{4g^{2}_{b}\sin^{2}\theta}{\omega_{b}}+\mathcal{O}(\delta).$

The collection of the results in Eqs. (40), (41), and (42) gives rise to the
second-order effective Hamiltonian:

$\displaystyle H_{eff}$
$\displaystyle=(\omega_{eg}+\epsilon_{1})|00e\rangle\langle
00e|+(2\omega_{a}+\epsilon_{2})|20g\rangle\langle 20g|$ (43)
$\displaystyle+g_{eff}(|00e\rangle\langle 20g|+|20g\rangle\langle 00e|).$

An exact double-resonance facilitated by Eq. (43) allows a completed Rabi
oscillation between $|00e\rangle$ and $|20g\rangle$, which requires that the
first line of Eq. (43) becomes an identity operator in the very subspace. Thus
$\omega_{eg}+\epsilon_{1}=2\omega_{a}+\epsilon_{2}$. Recalling the assumption
that $\omega_{eg}=2\omega_{a}+\delta$, one can then obtain

$\displaystyle\delta$ $\displaystyle=\epsilon_{2}-\epsilon_{1}$ (44)
$\displaystyle=-g^{2}_{a}\cos^{2}\theta\left(\frac{4}{\omega_{a}}+\frac{1}{\omega_{fg}-\omega_{a}}+\frac{3}{\omega_{fg}+\omega_{a}}\right)$
$\displaystyle-g^{2}_{b}\cos^{2}\theta\bigg{(}\frac{1}{2\omega_{a}+\omega_{b}}+\frac{1}{\omega_{fg}+\omega_{b}}+\frac{1}{2\omega_{a}-\omega_{b}}$
$\displaystyle-\frac{1}{\omega_{fg}+\omega_{b}-2\omega_{a}}\bigg{)}-\frac{4g^{2}_{a}\sin^{2}\theta}{\omega_{a}}-\frac{4g^{2}_{b}\sin^{2}\theta}{\omega_{b}}.$

Eventually the effective Hamiltonian (43) can be written as

$H_{eff}=g_{eff}(|00e\rangle\langle 20g|+|20g\rangle\langle 00e|),$ (45)

which is Eq. (3) in the main text. The foregoing perturbative calculation
prohibits $\omega_{b}\approx k\omega_{a}$ and $\omega_{fg}\approx k\omega_{a}$
with $k$ integer. Otherwise, the effect from the other degenerate states can
not be omitted. For example, when $\omega_{a}\approx\omega_{b}$, the states
$|20g\rangle$ and $|02g\rangle$ can not be differentiated by the qutrit.

Figure 9: (Color online) (a) and (b) Comparison between the numerically
evaluated normalized energy shifts from the full Hamiltonian (blue points) and
the corresponding analytical results from the second-order effective
Hamiltonian (red solid line) in Eqs. (44) and (50), respectively. Here
$\omega_{b}=1.7\omega_{a}$ and $\theta=\frac{\pi}{6}$. For (a)
$g_{b}=0.05\omega_{a}$, $\omega_{fg}=2\omega_{b}$; for (b)
$g_{a}=0.05\omega_{a}$, $\omega_{eg}=2\omega_{a}$.

A similar derivation yields the effective Hamiltonian (6) for the two-photon
resonance shown in Fig. 1(c), which lives in the subspace spanned by
$\\{|00f\rangle,|02g\rangle\\}$. Again the avoid level-crossing point
$\omega_{fg}$ is expressed by $\omega_{fg}=2\omega_{b}+\Delta$ with
$|\Delta|\ll\omega_{a},\omega_{b},|\omega_{b}-\omega_{a}|$ the undetermined
energy shift. The effective Hamiltonian can be written as

$\displaystyle H_{eff}^{\prime}$
$\displaystyle=(\omega_{fg}+\epsilon_{3})|00f\rangle\langle
00f|+(2\omega_{b}+\epsilon_{4})|02g\rangle\langle 02g|$ (46)
$\displaystyle+g_{eff}^{\prime}(|00f\rangle\langle 02g|+|02g\rangle\langle
00f|),$

where the effective coupling strength connecting $|00f\rangle$ and
$|02g\rangle$ is

$g_{eff}^{\prime}=-\sqrt{2}g^{2}_{b}\left[\frac{2\sin(2\theta)}{\omega_{b}}+\frac{\cos^{2}\theta}{\omega_{eg}-\omega_{b}}\right]+\mathcal{O}(\Delta),$
(47)

the energy shift for the state $|00f\rangle$ is

$\displaystyle\epsilon_{3}$
$\displaystyle=g^{2}_{a}\cos^{2}\theta\left(\frac{1}{2\omega_{b}-\omega_{a}-\omega_{eg}}+\frac{1}{2\omega_{b}-\omega_{a}}\right)$
(48)
$\displaystyle+g^{2}_{b}\cos^{2}\theta\left(\frac{1}{\omega_{b}-\omega_{eg}}+\frac{1}{\omega_{b}}\right)-\frac{4g^{2}_{a}\sin^{2}\theta}{\omega_{a}}$
$\displaystyle-\frac{4g^{2}_{b}\sin^{2}\theta}{\omega_{b}}+\mathcal{O}(\Delta),$

and the energy shift for the state $|02g\rangle$ is

$\displaystyle\epsilon_{4}$
$\displaystyle=-g^{2}_{a}\cos^{2}\theta\left(\frac{1}{\omega_{eg}+\omega_{a}}+\frac{1}{2\omega_{b}+\omega_{a}}\right)$
(49)
$\displaystyle-g^{2}_{b}\cos^{2}\theta\left(\frac{3}{\omega_{b}}+\frac{2}{\omega_{eg}-\omega_{b}}+\frac{3}{\omega_{eg}+\omega_{b}}\right)$
$\displaystyle-\frac{4g^{2}_{a}\sin^{2}\theta}{\omega_{a}}-\frac{4g^{2}_{b}\sin^{2}\theta}{\omega_{b}}+\mathcal{O}(\Delta).$

Again, to realize a completed Rabi oscillation between $|00f\rangle$ and
$|02g\rangle$, $\omega_{fg}+\epsilon_{3}$ should be equivalent to
$2\omega_{b}+\epsilon_{4}$. It turns out that

$\displaystyle\Delta$ $\displaystyle=\epsilon_{4}-\epsilon_{3}$ (50)
$\displaystyle=-g^{2}_{a}\cos^{2}\theta\bigg{(}\frac{1}{\omega_{eg}+\omega_{a}}+\frac{1}{2\omega_{b}+\omega_{a}}+\frac{1}{2\omega_{b}-\omega_{a}}$
$\displaystyle+\frac{1}{2\omega_{b}-\omega_{a}-\omega_{eg}}\bigg{)}-g^{2}_{b}\cos^{2}\theta\Big{(}\frac{4}{\omega_{b}}+\frac{1}{\omega_{eg}-\omega_{b}}$
$\displaystyle+\frac{3}{\omega_{eg}+\omega_{b}}\Big{)}.$

Then the effective Hamiltonian (46) is written as

$H_{eff}^{\prime}=g_{eff}^{\prime}\left(|00f\rangle\langle
02g|+|02g\rangle\langle 00f|\right).$ (51)

The two energy shifts in Eqs. (44) and (50) can be justified by Fig. 9. The
normalized $\delta$ ($\Delta$) as a function of $g_{a}$ ($g_{b}$) is compared
with that from the standard diagonalization of the full Hamiltonian (1). It is
shown that the analytical results do match with the numerical ones at least
for normalized atom-photon interaction strength $g_{a},g_{b}/\omega_{a}\leq
0.1$.

##  Appendix B The effective Hamiltonian for the three-photon resonances

To construct the effective Hamiltonian (14) from the full Hamiltonian (13) in
the subspace spanned by $\\{|00e\rangle,|30g\rangle\\}$ for the three-photon
resonance remarked in Fig. 3(a). The leading-order paths plotted in Fig. 10
are either the paths of the three-order processes connecting $|00e\rangle$ and
$|30g\rangle$ or those of the two-order processes in charge of the energy
shifts for these two bases. Due to the standard perturbation theory, the
effective coupling strength between any eigenstates $|i\rangle$ and
$|j\rangle$ of the unperturbed Hamiltonian $H_{0}$ (13) is given by Garziano
_et al._ (2015); Macrì  _et al._ (2018)

$g_{s}=\sum_{n,m\neq
i,j}\frac{V_{jn}V_{nm}V_{mi}}{(\omega_{i}-\omega_{n})(\omega_{i}-\omega_{m})}.$
(52)

Figure 10: (Color online) All the leading-order paths involving the bases
$|00e\rangle$ and $|30g\rangle$. Blue solid lines mark the transitions
mediated by the coupling to the resonator $a$, which are charge of the
connections between $|00e\rangle$ and $|30g\rangle$. Red dashed lines mark the
transition mediated by the coupling to the resonator $b$, which are charge of
the energy shift for $|00e\rangle$ or $|30g\rangle$.

The avoid level crossing point can be written as
$\omega_{eg}=3\omega_{a}+\delta_{s}$, where $\delta_{s}$ is a to-be-determined
energy shift. Considering all the three paths (shown in Fig. 10) connecting
the basis $|00e\rangle$ and the basis $|30g\rangle$, i.e.,
$|00e\rangle\to|10g\rangle\to|20f\rangle\to|30g\rangle$,
$|00e\rangle\to|10g\rangle\to|20e\rangle\to|30g\rangle$ and
$|00e\rangle\to|10f\rangle\to|20e\rangle\to|30g\rangle$, we can find the
effective coupling strength by using Eq. (52),

$\displaystyle g_{effs}$
$\displaystyle=-\sqrt{6}g^{3}_{a}\bigg{[}\frac{1}{2\omega_{a}(2\omega_{a}+\delta_{s})}-\frac{1}{2\omega_{a}(\omega_{fg}-2\omega_{a}-\delta_{s})}$
(53)
$\displaystyle+\frac{1}{(2\omega_{a}+\delta_{s})(\omega_{fg}-\omega_{a}-\delta_{s})}\bigg{]}$
$\displaystyle=-\frac{\sqrt{6}g^{3}_{a}}{2\omega_{a}}\left(\frac{1}{2\omega_{a}}-\frac{1}{\omega_{fg}-2\omega_{a}}+\frac{1}{\omega_{fg}-\omega_{a}}\right)$
$\displaystyle+\mathcal{O}(\delta_{s}).$

Summarizing all the paths from the state $|00e\rangle$ and back to itself
through a mediate state, one can obtain its second-order energy shift
$\epsilon_{s1}$ according to Eq. (39)

$\displaystyle\epsilon_{s1}$
$\displaystyle=g^{2}_{a}\left(\frac{1}{2\omega_{a}}+\frac{1}{2\omega_{a}-\omega_{fg}}\right)$
(54)
$\displaystyle+g^{2}_{b}\left(\frac{1}{3\omega_{a}-\omega_{b}}+\frac{1}{3\omega_{a}-\omega_{b}-\omega_{fg}}\right)+\mathcal{O}(\delta_{s}).$

In the same way, the energy shift $\epsilon_{s2}$ for the basis $|30g\rangle$
is found to be

$\displaystyle\epsilon_{s2}$
$\displaystyle=-g^{2}_{a}\left(\frac{5}{2\omega_{a}}+\frac{3}{\omega_{fg}-\omega_{a}}+\frac{4}{\omega_{fg}+\omega_{a}}\right)$
(55)
$\displaystyle-g^{2}_{b}\left(\frac{1}{3\omega_{a}+\omega_{b}}+\frac{1}{\omega_{b}+\omega_{fg}}\right)+\mathcal{O}(\delta_{s}).$

With Eqs. (53), (54) and (55), the effective Hamiltonian in the subspace
spanned by $\\{|00e\rangle,|30g\rangle\\}$ is

$\displaystyle H_{effs}$
$\displaystyle=(\omega_{eg}+\epsilon_{s1})|00e\rangle\langle
00e|+(3\omega_{a}+\epsilon_{s2})|30g\rangle\langle 30g|$ (56)
$\displaystyle+g_{effs}(|00e\rangle\langle 30g|+|30g\rangle\langle 00e|).$

Similar to the double-photon resonance treatment in Appendix A, the first line
of Eq. (56) should be an effective identity operator in the very subspace to
facilitate a completed Rabi oscillation driven by the second line. It is
achieved by equating the diagonal elements of (56), which gives

$\displaystyle\delta_{s}$ $\displaystyle=\epsilon_{s2}-\epsilon_{s1}$ (57)
$\displaystyle=-g^{2}_{a}\left(\frac{3}{\omega_{a}}+\frac{3}{\omega_{fg}-\omega_{a}}+\frac{4}{\omega_{fg}+\omega_{a}}+\frac{1}{2\omega_{a}-\omega_{fg}}\right)$
$\displaystyle-g^{2}_{b}\bigg{(}\frac{1}{3\omega_{a}+\omega_{b}}+\frac{1}{\omega_{b}+\omega_{fg}}+\frac{1}{3\omega_{a}-\omega_{b}}$
$\displaystyle+\frac{1}{3\omega_{a}-\omega_{b}-\omega_{fg}}\bigg{)}.$

Eventually the effective Hamiltonian (56) becomes

$H_{effs}=g_{effs}(|00e\rangle\langle 30g|+|30g\rangle\langle 00e|)$ (58)

which is Eq. (14) in the main text.

Figure 11: (Color online) (a) and (b) Comparison between the numerically
calculated normalized energy shifts from the full Hamiltonian (blue points)
and the corresponding analytical results from the leading-order effective
Hamiltonian (red solid line) in Eqs. (57) and (62), respectively. Here
$\omega_{b}=2.2\omega_{a}$. For (a) $g_{b}=0.05\omega_{a}$,
$\omega_{fg}=3\omega_{b}$; for (b) $g_{a}=0.05\omega_{a}$,
$\omega_{eg}=3\omega_{a}$.

In parallel, one can derive the effective Hamiltonian (17) in the subspace
spanned by $\\{|00f\rangle,|03g\rangle\\}$ at the avoided level crossing point
$\omega_{fg}=3\omega_{b}+\Delta_{s}$ as shown in Fig. 3(c). The effective
Hamiltonian is found to be

$\displaystyle H_{effs}^{\prime}$
$\displaystyle=(\omega_{fg}+\epsilon_{s3})|00f\rangle\langle
00f|+(3\omega_{b}+\epsilon_{s4})|03g\rangle\langle 03g|$ (59)
$\displaystyle+g_{effs}^{\prime}(|00f\rangle\langle 03g|+|03g\rangle\langle
00f|),$

where the effective coupling strength is

$\displaystyle g_{effs}^{\prime}$
$\displaystyle=-\frac{\sqrt{6}g^{3}_{b}}{2\omega_{b}}\left(\frac{1}{2\omega_{b}}-\frac{1}{\omega_{eg}-2\omega_{b}}+\frac{1}{\omega_{eg}-\omega_{b}}\right)$
(60) $\displaystyle+\mathcal{O}(\Delta_{s}),$

and the energy shifts for $|00f\rangle$ and $|03g\rangle$ are

$\displaystyle\epsilon_{s3}$
$\displaystyle=g^{2}_{a}\left(\frac{1}{3\omega_{b}-\omega_{a}-\omega_{eg}}+\frac{1}{3\omega_{b}-\omega_{a}}\right)$
(61)
$\displaystyle+g^{2}_{b}\left(\frac{1}{2\omega_{b}-\omega_{eg}}+\frac{1}{2\omega_{b}}\right)+O(\Delta_{s}),$
$\displaystyle\epsilon_{s4}$
$\displaystyle=-g^{2}_{a}\left(\frac{1}{3\omega_{b}+\omega_{a}}+\frac{1}{\omega_{a}+\omega_{eg}}\right)$
$\displaystyle-g^{2}_{b}\left(\frac{5}{2\omega_{b}}+\frac{3}{\omega_{eg}-\omega_{b}}+\frac{4}{\omega_{eg}+\omega_{b}}\right)+\mathcal{O}(\Delta_{s}),$

respectively.

Again, to realize a completed Rabi oscillation between $|00f\rangle$ and
$|02g\rangle$, we let $\omega_{fg}+\epsilon_{3}=3\omega_{b}+\epsilon_{4}$. It
turns out that

$\displaystyle\Delta_{s}$ $\displaystyle=\epsilon_{s4}-\epsilon_{s3}$ (62)
$\displaystyle=-g^{2}_{a}\bigg{(}\frac{1}{3\omega_{b}+\omega_{a}}+\frac{1}{\omega_{a}+\omega_{eg}}+\frac{1}{3\omega_{b}-\omega_{a}-\omega_{eg}}$
$\displaystyle+\frac{1}{3\omega_{b}-\omega_{a}}\bigg{)}-g^{2}_{b}\bigg{(}\frac{1}{2\omega_{b}-\omega_{eg}}+\frac{3}{\omega_{b}}+\frac{3}{\omega_{eg}-\omega_{b}}$
$\displaystyle+\frac{4}{\omega_{eg}+\omega_{b}}\bigg{)}.$

Then we get the effective Hamiltonian (17) in the main text:

$H_{effs}^{\prime}=g_{effs}^{\prime}(|00f\rangle\langle
03g|+|03g\rangle\langle 00f|).$ (63)

In Fig. 11, we demonstrate the analytical results in Eqs. (57) and (62). The
normalized $\delta_{s}$ ($\Delta_{s}$) as a function of $g_{a}$ ($g_{b}$) is
compared to that from the standard diagonalization of the full Hamiltonian in
Eq. (13). It is shown that the analytical results do match with the numerical
ones at least for normalized atom-photon interaction strength
$g_{a}/\omega_{a}\leq 0.14$ and $g_{b}/\omega_{a}\leq 0.11$. They are in the
strong-coupling regime.

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