# Non-Abelian Aharonov-Bohm Caging in Photonic Lattices

Sheng Li  School of Physics, Huazhong University of Science and Technology,
Wuhan, 430074, China  Zheng-Yuan Xue  Guangdong Provincial Key Laboratory of
Quantum Engineering and Quantum Materials, GPETR Center for Quantum Precision
Measurement, and School of Physics and Telecommunication Engineering, South
China Normal University, Guangzhou 510006, China  Ming Gong  CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei 230026, China  Yong Hu<EMAIL_ADDRESS>School of
Physics, Huazhong University of Science and Technology, Wuhan, 430074, China

###### Abstract

Aharonov-Bohm (AB) caging is the localization effect in translational-
invariant lattices due to destructive interference induced by penetrated
magnetic fields. While current research focuses mainly on the case of Abelian
AB caging, here we go beyond and develop the non-Abelian AB caging concept by
considering the particle localization in a 1D multi-component rhombic lattice
with non-Abelian background gauge field. In contrast to its Abelian
counterpart, the non-Abelian AB cage depends on both the form of the nilpotent
interference matrix and the initial state of the lattice. This phenomena is
the consequence of the non-Abelian nature of the gauge potential and thus has
no Abelian analog. We further propose a circuit quantum electrodynamics
realization of the proposed physics, in which the required non-Abelian gauge
field can be synthesized by the parametric conversion method, and the non-
Abelian AB caging can be unambiguously demonstrated through the pumping and
the steady-state measurements of only a few sites on the lattice. Requiring
only currently available technique, our proposal can be readily tested in
experiment and may pave a new route towards the investigation of exotic
photonic quantum fluids.

##  I Introduction

Synthetic gauge fields in artificial atomic and photonic systems has been
studied extensively in the past decades Goldman  _et al._ (2014); Aidelsburger
_et al._ (2018). The motivation is to investigate exotic topological physics
Bernevig and Hughes (2013) in a well-controlled quantum simulator Cooper  _et
al._ (2019); Ozawa  _et al._ (2019). Inspired by the rapid progress in this
field, research attention has been recently devoted to the realization of
Aharonov-Bohm (AB) caging, where the interplay of the external magnetic field
and the lattice geometry leads to completely flat bands (FB) in the 1D rhombic
lattice Longhi (2014); Vidal  _et al._ (2000); Douçot and Vidal (2002);
Gladchenko  _et al._ (2009) and 2D dice lattice Vidal  _et al._ (1998) and
consequently exotic localization on the perfectly periodic lattices. Being
distinct from the localization due to disorder, this effect is interpreted by
the destructive interference induced by the Peierels phase of the penetrated
magnetic flux. The interaction mechanism then becomes dominant on the
dispersionless band, making the AB caging lattices ideal platforms of
exploring strongly correlated physics. Following the experimental realization
of AB caging early in solid state systems Abilio  _et al._ (1999a); Naud  _et
al._ (2001a) and recently in photonic systems Mukherjee  _et al._ (2018a);
Kremer  _et al._ (2018), a variety of related theoretical works has been
presented, including the effects of nonlinearity and disorder Vidal  _et al._
(2001); Tovmasyan  _et al._ (2018); Cartwright  _et al._ (2018); Gligorić  _et
al._ (2019); Di Liberto  _et al._ (2019); Kuno  _et al._ (2020), topological
pumping and edge states Pelegrí  _et al._ (2019a, b); Haug  _et al._ (2019),
FB laser Longhi (2019), and influence of non-Hermicity Jin (2019); Zhang and
Jin (2019).

Meanwhile, subtlety still exists in the sense that, discussions up to now are
mainly based on the assumption that an Abelian background gauge potential is
imposed. Few works have investigated models in which Rashba SOC can lead to
flat band localization in certain lattice geometries Bercioux  _et al._ (2004,
2005). On the other hand, non-Abelian gauge field Kogut (1983) has already
manifested its essence in condensed matter physics and quantum optics,
partially by the role of spin-orbital coupling (SOC) Galitski and Spielman
(2013) in the physics of topological quantum matters Bernevig and Hughes
(2013). In addition, synthetic SOC has been experimentally implemented in
various artificial system during the past few years, ranging from ultra-cold
atoms Wu  _et al._ (2016) and exciton-polariton microcavities Sala  _et al._
(2015) to coupled pendula chains Salerno  _et al._ (2017). These exciting
advances thus raise the curious questions that whether the AB caging concept
can be extended to the realm of non-Abelian gauge, and whether the non-Abelian
nature of the gauge field would bring any new physics that has no Abelian
correspondence.

In this manuscript, we propose the concept non-Abelian AB caging (i. e. AB
caging in the presence of non-Abelian background gauge field) in a 1D rhombic
lattice where the lattice sites contain multiple (pseudo)spin components and
the background gauge potential becomes matrix-valued. In particular, the non-
Abelian AB caging is defined by the emergence of nilpotent interference
matrix, which is the matrix generalization of the destructive interference
condition in the Abelian AB caging case. Detailed analysis further implies
that particle localization in this situation is drastically different from its
Abelian counterpart: The non-Abelian feature of the background gauge field
results in exotic spatial configuration of non-Abelian AB cage, which is
sensitive to both the nilpotent power of the interference matrix and the
initial state of the lattice.

In addition, we consider the circuit quantum electrodynamics (QED) system Gu
_et al._ (2017) as a promising candidate of implementing the proposed physics.
While the Abelian AB caging has been realized in a variety of physical systems
Mukherjee  _et al._ (2018a); Kremer  _et al._ (2018); Abilio  _et al._
(1999b); Vidal  _et al._ (2000); Naud  _et al._ (2001b), our proposal takes
the advantages of flexibility and tunability of superconducting quantum
circuit, which allow the future incorporation of photon-photon interaction and
disorder in a time- and site- resolved manner Houck  _et al._ (2012); Schmidt
and Koch (2013). The required link variables can be synthesized by the
parametric frequency conversion (PFC) approach Zakka-Bajjani  _et al._ (2011);
Wulschner  _et al._ (2016); Roushan  _et al._ (2017), which is feasible with
current technology and can lead to in situ tunability of the synthesized non-
Abelian gauge potential. Our numerical simulations pinpoint that localization-
in-continuum dynamics can be observed through the steady-state photon number
(SSPN) detection of only few sites on the lattice, which can serve as
unambiguous evidence of the non-Abelian AB caging.

##  II non-Abelian AB Caging: concept and examples

###  II.1 Definition of non-Abelian AB caging

In this section, the physics of non-Abelian AB caging is illustrated in the
context of a periodic 1-D rhombic lattice sketched in Fig. 1(a) Longhi (2014);
Mukherjee  _et al._ (2018a); Di Liberto  _et al._ (2019), with each site
consisting of $N$ (pseudo)spin modes. The Hamiltonian of the lattice in the
presence of an $U(N)$ background gauge field $\mathbf{A}$ takes the form

$H=-J\sum_{\left\langle
n\alpha,m\beta\right\rangle}\alpha_{n}^{\dagger}U_{{n}\alpha,{m}\beta}\beta_{m},$
(1)

where ${J}$ is the uniform positive hopping strength between the linked sites
shown in Fig. 1(a),
$\alpha_{n}=[\alpha_{n,1},\alpha_{n,2},\cdots,\alpha_{n,N}]^{T}$ with
$\alpha=A,B,C$ is the multi-component annihilation operator vector of the
$\alpha$th site in the $n$th unit-cell, and
$U_{{n}\alpha,{m}\beta}=\exp\left[{i}\int_{[m,\beta]}^{[n,\alpha]}\mathrm{d}\textbf{x}\cdot\textbf{A}(\textbf{x})\right]$
is the translational-invariant link variable describing the unitary
transformation experienced by a particle when it hops from site $[m,\beta]$ to
site $[n,\alpha]$ Goldman  _et al._ (2014); Kogut (1983).

We then define the non-Abelian AB caging by the condition that the
interference matrix

$I=\frac{1}{2}\left(U_{2}U_{1}+U_{4}U_{3}\right),$ (2)

is nilpotent, i.e.

$\exists\quad m\in\mathbb{N},\quad\mathrm{s.t.}\quad I^{m-1}\neq 0,\quad
I^{m}=0.$ (3)

Here $U_{1}$, $U_{2}$, $U_{3}$, and $U_{4}$ are the rightward link variables
labeled in Fig. 1(a). This definition can be explained intuitively as follow:
Imagine there is a particle initially populated in the $[n,A]$ site
highlighted in Fig. 1(a). Due to the geometry of the lattice, this particle
can move rightward to $[n+1,A]$ via only two paths. Along these two paths, it
will gain the unitary transformations $U_{\mathrm{up}}=U_{2}U_{1}$ and
$U_{\mathrm{down}}=U_{4}U_{3}$ respectively, and consequently an interference
described by the interference matrix $I$ in Eq. (2).

Figure 1: (Color Online) (a) Sketch of the 1D periodic rhombic lattice
composed of unit-cells with three sites labeled A, B, and C. (b) Circuit QED
implementation of the proposed two component rhombic lattice. The lattice is
built by the TLRs grounded at their common ends (the big dots) and connected
by the coupling SQUIDs (the crossed squares). The colors of the TLRs label
their different lengths and consequently different eigenfrequencies. (c)
Effective $[n,\alpha,\sigma_{1}]\Leftrightarrow[m,\beta,\sigma_{2}]$ mode
coupling mediated by parametric modulation of the coupling SQUID. (d) Typical
eigenfrequency level diagram of the neighboring $[n,\alpha]$ and $[m,\beta]$
sites, and the corresponding modulating tone configuration of the coupling
SQUID. Here $\delta=\omega_{\beta 1}-\omega_{\alpha 1}$ is the frequency
difference of the $\lambda/2$ modes of the two TLRs.

For the Abelian situation $N=1$, AB caging happens exactly when $\pi$ magnetic
flux is penetrated in each loop of the lattice. The two up and down paths
shown in Fig. 1(a) then interfere with each other destructively and result in
an vanishing $I=0$. Therefore, the particle initially in the site $[n,A]$
becomes localized as it cannot spread outward to sites further than $[n\pm
1,A]$. For the non-Abelian situation $N>1$, the localization of the particle
can still happen when $I=0$, just the same as the Abelian case Bercioux  _et
al._ (2004, 2005). However, the matrix feature of $\mathbf{A}$ now offers
possibilities of much more rich physics: A matrix-formed interference matrix
$I$ can be nilpotent, with c-number $I=0$ being its $N=1$ special case. This
point lies in the heart of our generalization of AB caging to the non-Abelian
gauge situation.

###  II.2 Two examples of nilpotent interference matrices

To validate the proposed non-Abelian AB caging concept, we consider a special
example

$\displaystyle I=$
$\displaystyle\sum_{n=1}^{N-1}\outerproduct{n}{n+1}=\frac{1}{2}\left[\sum_{n=1}^{N-1}\outerproduct{n}{n+1}+\outerproduct{1}{N}\right.$
(4)
$\displaystyle+\left.\sum_{n=1}^{N-1}\outerproduct{n}{n+1}-\outerproduct{1}{N}\right].$

with nilpotent power $m=N$. Here $\left|n\right\rangle$ is the $N$-column with
only its $n$th item being unity and others being zero, and a candidate
decomposition of $I$ into two $U(N)$ matrices (i. e. $U_{\mathrm{up}}$ and
$U_{\mathrm{down}}$) is implied. The evolution of particles on the lattice
still exhibits localization feature in this case. However, the non-Abelian AB
cage now has an enlarged size and a spatial location depending on the initial
state of the system: Let us assume that a particle is initially prepared in
the $[n,A,l]$ mode with $l\in[1,N]$. Due to the form of $I$ shown in Eq. (4),
the particle will hop into the $(l-1)$th mode when it arrives at the $[n+1,A]$
site. After it reaches the $[n+1,A]$ site, it can move rightward further. The
rightmost $A$ site that the particle can approach is $[n+l-1,A]$, and its
further moving towards $[n+l,A]$ is suppressed. The leftward moving of the
particle can be investigated in the similar way, with
$I^{\dagger}=(U_{\mathrm{up}}^{-1}+U_{\mathrm{down}}^{-1})/2$ being used as
the interference matrix due to the reversed moving direction. The particle can
thus reach as far as the $[n+l-N,A,N]$ mode. The above analysis therefore
indicates that the particle will experience a spin dependent asymmetric
directional breath dynamics. This is different from the Abelian case, in which
a particle initial prepared in $[n,A]$ site will breath in a symmetric manner
between $[n,A]$ and its four neighboring $B$ and $C$ sites Longhi (2014); Di
Liberto  _et al._ (2019). Such symmetry is manifestly broken in the considered
non-Abelian situation.

We further study a more general situation in which $I$ has nilpotent power
smaller than $N$. Without loss of generality, we assume

$\displaystyle I=\sum_{n=1}^{m-1}\outerproduct{n}{n+N-m+1}$ (5)
$\displaystyle=\frac{1}{2}\left[\sum_{n=1}^{m-1}\outerproduct{n}{n+N-m+1}+\sum_{n=1}^{N-m+1}\outerproduct{n}{N+1-n}\right.$
$\displaystyle+\left.\sum_{n=1}^{m-1}\outerproduct{n}{n+N-m+1}-\sum_{n=1}^{N-m+1}\outerproduct{n}{N+1-n}\right],$

with nilpotent power $m\in(1,N)$. Re-performing the previous analysis, we find
that now the spatial configuration of the non-Abelian AB cage is not only
determined by the initial mode number $l$ of the particle, but also the
nilpotent power $m$ of $I$. Explicitly speaking, there are six situations,
with main results summarized in Tab.1. For each situation, the AB cage is
characterized by its size defined by the number of $A$ sites that can be
populated by the particle during its evolution, and its right and left edges,
defined by the rightmost and leftmost $A$ sites that can be populated by the
particle during its evolution. We should emphasize that Eqs. (4) and (5)
obviously do not exhaust the possible forms of the nilpotent interference
matrix $I$. Meanwhile, these two illuminating examples have already revealed
several properties of the non-Abelian AB caging which are significantly
different from its Abelian counterpart: The AB cage in the non-Abelian
situation becomes larger, with its size and edge location depending on both
the nilpotent power of the interference matrix and the initial state of the
particle. From the previous derivation, we can see that these exotic new
features stem from the matrix nature of the gauge field $\mathbf{A}$ and thus
have not Abelian analog.

Nilpotent power of $I$ | $m\in(1,[{N+1}/{2}])$ | $m\in[[{N+1}/{2}],N)$
---|---|---
Initial mode | $l\in[1,m]$ | $l\in(m,N-m]$ | $l\in(N-m,N]$ | $l\in[1,N-m]$ | $l\in(N-m,m]$ | $l\in(m,N]$
Size of the cage | $l$ | $1$ | $N-l+1$ | $l$ | $N$ | $N-l+1$
Right edge | $[n+l-1,A]$ | $[n,A]$ | $[n,A]$ | $[n+l-1,A]$ | $[n+l-1,A]$ | $[n,A]$
Left edge | $[n,A]$ | $[n,A]$ | $[n+l-N,A]$ | $[n,A]$ | $[n+l-N,A]$ | $[n+l-N,A]$
Table 1: Spatial configuration of the non-Abelian AB cage versus the nilpotent
power $m$ of the interference matrix $I$ and the mode number $l$ initially
populated by the particle. The interference matrix takes the form in Eq. (5).
For $m\in(1,[(N+1)/2])$ , the particle can move neither rightward nor leftward
if $l\in(m,N-m]$. The size of the AB cage in this situation is $1$, as the
same as the situation of Abelian AB caging Longhi (2014). On the other hand,
if $l\leq m$ (or $l>(N-m)$), the particle can move only rightward (or
leftward), and reach as far as the $[n+l-1,A,1]$ (or the $[n+l-N,A,N]$) mode.
The size of the AB cage in the situation is $l$ (or $N-l+1$). Meanwhile, for
$m\in[[(N+1)/2],N)$, the particle can move rightward to reach $[n-l+1,A,1]$
mode and leftward to arrive $[n+l-N,A,N]$ mode if $l\in(N-m,m]$. The size of
the AB cage in this situation is $N$. For $l\leq(N-m)$ (or $l>m$), the
particle can only move rightward (or leftward) and can reach as far as the
$[n-l+1,A,1]$ (or the $[n+l-N,A,N]$) mode. In this situation, the AB cage has
size $l$ (or $N-l+1$).

###  II.3 A minimal $U(2)$ model

We then turn to a $U(2)$ design which can be regarded as the minimal
realization of the proposed non-Abelian AB caging. Also, this is partially due
to recent advances of realizing two-component SOC in artificial systems Sala
_et al._ (2015); Wu  _et al._ (2016); Salerno  _et al._ (2017). The link
variables can be carefully set as

$U_{1}=U_{4}=\left[\begin{matrix}1&0\\\
0&1\end{matrix}\right],U_{2}=\left[\begin{matrix}0&1\\\
1&0\end{matrix}\right],U_{3}=\left[\begin{matrix}0&1\\\
-1&0\end{matrix}\right],$ (6)

and such that a nilpotent $I$ with $m=2$ can be achieved:

$I=\frac{1}{2}(U_{2}U_{1}+U_{4}U_{3})=\left[\begin{matrix}0&1\\\
0&0\end{matrix}\right].$ (7)

If the particle is initially prepared in the $[n,A,1]$ mode, it can move
rightward and arrive the $[n+1,A,2]$ mode, but it can reach neither the
$[n+2,A]$ nor the $[n-1,A]$ sites. Meanwhile, If the particle is initially
prepared in the $[n,A,2]$ mode, it can move leftward and arrive the
$[n-1,A,1]$ mode, but it cannot arrive the $[n-2,A]$ or the $[n+1,A]$ sites.

The proposed model is not the only possible $U(2)$ model exhibiting non-
Abelian AB caging. However, we can prove that other possible models should
take a very similar form to it upon a local basis transformation: Suppose we
want an $U(2)$ model exhibiting non-Abelian AB caging. The key point is to
find a nilpotent interference matrix $I$ and its decomposition into two
unitary matrices $I=(U_{\mathrm{up}}+U_{\mathrm{down}})/2$ . We recognize
that, any $2\times 2$ nilpotent matrix $I$ is equivalent to an upper
triangular matrix upon a certain unitary transformation $V$, i. e.

$V^{\dagger}IV=\left[\begin{matrix}0&\gamma\\\ 0&0\end{matrix}\right],$ (8)

where $V$ corresponds to the local basis choice on the $A$ site, and $\gamma$
is determined by $I$ upon a $U(1)$ phase and can be set real-valued through
the change of $V$. After figuring out the possible form of $I$, we further
notice that the only possible unitary decomposition of the RHS of Eq. (8)
takes the form:

$\left[\begin{matrix}0&\gamma\\\
0&0\end{matrix}\right]=\frac{1}{2}(U_{\mathrm{up}}+U_{\mathrm{down}})=\frac{1}{2}\left[\left(\begin{matrix}0&e^{\mathrm{i}\theta}\\\
e^{\mathrm{i}\psi}&0\end{matrix}\right)+\left(\begin{matrix}0&e^{\mathrm{i}\theta}\\\
-e^{\mathrm{i}\psi}&0\end{matrix}\right)\right],$ (9)

which further restrict the norm of $\gamma$ into $|\gamma|\leq 1$. Then the
final step of constructing the desired $U(2)$ model is to select the link
variables such that $U_{2}U_{1}=U_{\mathrm{up}}$ and
$U_{4}U_{3}=U_{\mathrm{down}}$. The explicit matrix forms of the link
variables depends on the basis choice on the $B$ and $C$ sites. Here we can
see that Eqs. (8) and (9) severely limit the choice of possible $U(2)$ models
to a very small region. Our choice in the manuscript corresponds to
$\gamma=1$. Other choice of models can only differ from our model upon a local
basis transformation by choosing a $\gamma$ with smaller norm.

An alternative perspective is to calculate the band structure of the lattice
with Eq. 6, we can get the corresponging K-space Hamiltonian

$H_{k}=\left[\begin{matrix}0&0&e^{ik}&1&1&e^{ik}\\\ 0&0&1&e^{ik}&e^{ik}&1\\\
e^{-ik}&1&0&0&0&0\\\ 1&e^{-ik}&0&0&0&0\\\ 1&e^{-ik}&0&0&0&0\\\
e^{-ik}&1&0&0&0&0\end{matrix}\right],$ (10)

As shown in Fig. 2(a), the six bands of the lattice become all completely
flat, implying that a particle on the lattice should move with velocity
$v=\partial E/\partial k=0$ Ashcroft and Mermin (1976). We also offer a brief
remark on the symmetry of the model Chiu  _et al._ (2016). The flat band
structure implies that the proposed $U(2)$ model does not preserve time
reversal symmetry(TRS) if the two components are spin-$\frac{1}{2}$
components, because the flat bands do not exhibit Kramers degeneracy at the
high symmetry points of the K-space. However, TRS is preserved if the two
components are pseudospin components, because in this situation TRS
corresponds to complex conjugation and all the link variables in our $U(2)$
model are real-valued. Moreover, the model has chiral symmetry(CS) where the
CS operator can be described as

$C=\left[\begin{matrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&-1&0&0&0\\\
0&0&0&-1&0&0\\\ 0&0&0&0&-1&0\\\ 0&0&0&0&0&-1\end{matrix}\right].$ (11)

The existence of CS is intuitive, as Fig. 2(a) indicates that the
eigenenergies $E(k)$ and $-E(k)$ come in pair. The particle-hole symmetry(PHS)
can then be defined by the combination of TRS and CS. PHS dose not existe if
the two components are spin-$\frac{1}{2}$ components, and the PHS does exist
with $PHS=1$ if the two components are pseudospin components. In addition, the
compact localized eigenstate (CLES) for each eigenenergy is calculated and
shown in Figs. 2(b)-(e) and Figs. 3(a)-(b). For the two-fold degenerate middle
bands with $E_{3}=E_{4}=0$, the CLES do not have $A$ site component (Figs.
2(b) and (c)). Also, it can be checked that they are the eigenstates of the CS
operator in Eq. (11). Meanwhile, for the other four bands with eigenenergies
$E_{1}=-E_{6}=\sqrt{6}J$ and $E_{2}=-E_{5}=\sqrt{2}J$, the corresponding CLES
containing the $[n,A,1]$ and $[n+1,A,2]$ mode components are explicitly shown
in Figs. 2(d)-(e) and 3(a)-(b), respectively.

Before proceeding, we offer another discussion on the difference between our
model in Eq. (6) and the Rashba caging investigated in Refs. Bercioux  _et
al._ (2004, 2005). As we have stated previously, flat-band localization of
particles can certainly happen with vanishing interference matrix $I=0$.
However, the matrix feature of $I$ can offer possibilities of much more rich
physics: A non-zero interference matrix $I$ can be nilpotent, and this
nilpotency leads to the flat band localization of particles. This point thus
helps us to classify these two models: As mentioned in Refs. Bercioux  _et
al._ (2004, 2005), flat-band localization can be induced in a two-component
rhombic lattice by Rashba SOC with specific SOC strength. The interference
matrix in that situation is completely zero, i. e. it belongs to the vanishing
interference matrix $I=0$ case. On the other hand, the interference matrix
calculated from Eq. (6) is non-zero with nilpotent power $2$, i. e. it belongs
to the nonzero nilpotent $I$ case.

Figure 2: (Color Online) $U(2)$ non-Abelian AB caging in a two-component
rhombic lattice with link variables defined in Eq. (6). The band structure of
the lattice is depicted in (a): All the six bands are completely flat,
indicating localization in the absence of disorder. Four of the six un-
normalized single-particle CLESs of these FBs are sketched in (b)-(e). The
two-fold degenerate $E_{3}=E_{4}=0$ bands correspond to (b) and (c), while the
$E_{6}=-\sqrt{6}J$, and $E_{5}=-\sqrt{2}J$ correspond to (d) and (e)
respectively. The colors of the circles indicate the involved spin component
(red for $\sigma=1$ and blue for $\sigma=2$), and the numbers in the circles
represent the relative ratio of superposition in the corresponding CLESs.

##  III A Candidate Circuit QED Implementation

###  III.1 The circuit QED lattice proposal

In this section, we consider the circuit QED lattice shown in Fig. 1(b) as the
candidate realization of the proposed non-Abelian AB caging described in Eqs.
(6) and (7). This circuit QED lattice consists of three types of
superconducting transmissionline resonators (TLRs) Wallraff  _et al._ (2004)
differed by their lengths and placed in an interlaced form. These TLRs play
the corresponding roles of the A, B, and C sites of the rhombic lattice
depicted in Fig. 1(a). In the bulk of the TLRs, the propagation of the voltage
and current fluctuations obey the wave equation Blais  _et al._ (2004), while
at their ends, the TLRs are grounded by inductors with inductances much
smaller than those of the TLRs, which impose the consequent low-voltage
shortcut boundary conditions for the TLRs Zakka-Bajjani  _et al._ (2011);
Felicetti  _et al._ (2014); Wang  _et al._ (2015, 2016); Yang  _et al._
(2016). Therefore, the eigenmodes of the resonators are approximately their
$\eta\lambda/2$ modes of the waveguide with $\eta$ being integers. For each
TLR, its lowest $\lambda/2$ and $\lambda$ modes are selected as the first and
the second pseudospin component, respectively. Through the parameter setting
of the circuit and the external pumping pulses described in what follows, we
expect that the excitation of the higher frequency modes is effectively
suppressed and thus the two-mode resonator approximation is valid. The lattice
can then be described by the Hamiltonian

$H_{\mathrm{S}}=\sum_{\alpha,n,\sigma}\omega_{\alpha\sigma}\alpha_{n\sigma}^{\dagger}\alpha_{n\sigma},$
(12)

where the eigenfrequencies of the cavity modes are specified as

$\displaystyle\left[\omega_{{A}1},\omega_{{B}1},\omega_{{C}1}\right]$
$\displaystyle=\left[\omega_{0}-\Delta,\omega_{{0}},\omega_{{0}}+\Delta\right],$
(13a) $\displaystyle\left[\omega_{{A}2},\omega_{{B}2},\omega_{{C}2}\right]$
$\displaystyle=2\left[\omega_{{A}1},\omega_{{B}1},\omega_{{C}1}\right],$ (13b)

with $\omega_{{0}}/2\pi\in\left[5,6\right]\,\mathrm{GHz}$ and
$\Delta/2\pi\in\left[1,2\right]\,\mathrm{GHz}$. Such configuration is for the
following application of the PFC method and can be precisely realized in
experiments through the length selection of the TLRs in the millimeter range
Zakka-Bajjani  _et al._ (2011); Underwood  _et al._ (2012).

We further consider the implementation of the required link variables on the
lattice, taking the form in Eq. (6) in the rotating frame of
${H}_{\mathrm{S}}$. Here we employ the dynamic modulation method Zakka-Bajjani
_et al._ (2011); Sirois  _et al._ (2015); Nguyen  _et al._ (2012); Wulschner
_et al._ (2016); Roushan  _et al._ (2017), which is different from the on-site
modulation method used in Refs. Longhi (2014); Mukherjee  _et al._ (2018b) in
the sense that the a.c. modulation here is imposed on the inter-site links.
This scheme has already been exploited in an experiment of generating
artificial Abelian gauge field in a three-qubit ring superconducting quantum
circuit Roushan  _et al._ (2017). The essential physics can be illustrated
intuitively through a toy model of two coupled cavities: Our aim is to
implement a controllable inter-cavity photon hopping process, described by the
effective Hamiltonian

${H}_{\mathrm{eff}}^{12}=Ja_{1}^{{\dagger}}a_{2}e^{i\theta_{12}}+\mathrm{h.c.},$
(14)

where $a_{i}/a_{i}^{{\dagger}}$ are the annihilation/creation operators of the
$i$th cavity for $i=1,2$, $J$ is the effective $1\leftrightarrow 2$ hopping
rates, and $\theta_{12}$ is the corresponding hopping phase. To achieve this
goal, we consider a two-cavity physical Hamiltonian

${H}_{TC}={H}_{0}+{H}^{12}_{\mathrm{a.c.}}(t),$ (15)

with

$\displaystyle{H}_{0}=\Sigma_{i=1}^{2}\omega_{i}a_{i}^{{\dagger}}a_{i},$ (16)
$\displaystyle{H}^{12}_{\mathrm{a.c.}}(t)=g_{12}(t)[a_{1}^{{\dagger}}+a_{1}][a_{2}^{{\dagger}}+a_{2}],$
(17)

where $\omega_{i}$ is the eigenfrequency of the $i$th cavity, $g_{12}(t)$ is
the tunable coupling constant between the cavities, and
${H}^{12}_{\mathrm{a.c.}}(t)$ takes the similar form to the inductive current-
current coupling of two TLRs which will be discussed later. Here we also
assume that $g_{12}(t)$ can be tuned harmonically. This corresponds to the
a.c. modulation of the coupling SQUID between the neighboring TLRs. Moreover,
we assume that the parameters in ${H}_{TC}$ satisfy the far off-resonance
condition:

$|\omega_{2}-\omega_{1}|\gg|g_{12}(t)|.$ (18)

If $g_{12}(t)$ is static, the $1\leftrightarrow 2$ photon hopping can hardly
happen because the two cavities are far off-resonant. Meanwhile, we can
implement the effective $1\leftrightarrow 2$ photon hopping by modulating
$g_{12}(t)$ dynamically as

$g_{12}(t)=2J\cos[(\omega_{1}-\omega_{2})t-\theta_{12}].$ (19)

Physically speaking, $g_{12}(t)$ carries energy quanta filling the gap between
the two cavity modes. For a photon initially placed in the 1st cavity, it can
absorb an energy quantum $|\omega_{2}-\omega_{1}|$ from the $1\leftrightarrow
2$ link, convert its frequency to $\omega_{2}$, and hop finally into the 2nd
cavity. We can further describe this process in a more rigorous way: in the
rotating frame with respect to ${H}_{0}$, ${H}^{12}_{\mathrm{a.c.}}(t)$
becomes

$\displaystyle{H}^{12}_{\mathrm{eff}}=e^{iH_{0}t}{H}^{12}_{\mathrm{a.c.}}(t)e^{-iH_{0}t}\approx
g_{12}a_{1}^{{\dagger}}a_{2}e^{i\theta_{12}}+\mathrm{h.c.},$ (20)

where the other $a_{1}^{\dagger}a_{2}^{\dagger}+\mathrm{h.c.}$ term are fast
oscillating in the rotating frame and thus are safely neglected.

From Eq. (20), we notice that both the effective hopping strength and the
hopping phase can be controlled by the modulating pulse $g_{12}(t)$. In
particular, the control of the hopping phase $\theta_{12}$ is important as we
are using this method to synthesize artificial gauge fields. Also, we notice
that the only approximation we exploit in getting the effective Hamiltonian is
the rotating wave approximation, which helps us to drop the off-resonant, fast
oscillating terms. Due to the fact that the a.c. modulation is now exerted on
the inter-site link. The Bessel function constants from the expansion of
oscillating exponential functions of the form $\exp[i\cos(\omega t)]$ in the
on-site modulation proposals Longhi (2014); Mukherjee  _et al._ (2018b) do not
appear in the effective Hamiltonian.

To facilitate the application of the described PFC method, the TLRs on the
lattice are connected at their ends by connecting superconducting quantum
interference devices (SQUIDs) (Fig. 1(b)) Peropadre  _et al._ (2013);
Wulschner  _et al._ (2016), which correspond to the linking bonds shown in
Fig. 1(a) one-to-one. It will be figured out in the following that each of the
bonds on the lattice can be independently controlled by the modulating pulse
threaded in the corresponding connecting SQUID, leading to the site-resolved
arbitrary control of the synthesized non-Abelian gauge field. In addition,
estimations in our previous works have shown that this method can result in
effective hopping strength in the range
$J/2\pi\in\left[5,15\right]\,\mathrm{MHz}$, which is robust against
unavoidably imperfection factors in realistic experiments including the
fabrication errors of the circuit and the background low-frequency noises Wang
_et al._ (2015, 2016); Yang  _et al._ (2016).

###  III.2 The PFC scheme of realizing non-Abelian gauge

The essential implementation of the PFC method can be illustrated step-by-step
by investigating the a.c. modulation of a particular
$[n,\alpha]\Leftrightarrow[m,\beta]$ coupling SQUID. The physical coupling
between these neighboring TLRs is established through the current dividing
mechanism recently implemented in experiment Roushan  _et al._ (2017):
Consider the two TLRs located at neighboring sites $[n,\alpha]$ and
$[m,\beta]$. The small grounding inductances of the two TLRs create low
voltage nodes at their ends, and their neighboring low voltage nodes are
connected by the $[n,\alpha]\leftrightarrow[m,\beta]$ coupling SQUID. The
connecting SQUID can be regarded as an inductance which can be tuned by the
penetrating flux bias in the coupling SQUID loop at very high frequencies
Wilson  _et al._ (2011). The inter-TLR coupling can be understood in an
intuitive way Geller  _et al._ (2015): An excitation current from the
$[n,\alpha]$ TLR, taking the form
$I_{n,\alpha}\simeq\sum_{\sigma}I_{\alpha,\sigma}(\alpha_{n,\sigma}^{\dagger}+\alpha_{n,\sigma})$,
will mostly flow through its grounding inductance to ground, with a small
fractions $I_{n\alpha,m\beta}$ flowing to its neighboring TLR $[m,\beta]$
through the tunable $[n,\alpha]\leftrightarrow[m,\beta]$ coupling SQUID.
Depending on the ratio of the Josephson inductance of the
$[n,\alpha]\leftrightarrow[m,\beta]$ coupling SQUID to the grounding
inductance of the $[n,\alpha]$ TLR, $I_{n\alpha,m\beta}$ will in turn flow
through the grounding inductance of the $[m,\beta]$ TLR. Therefore, the
$[n,\alpha]\leftrightarrow[m,\beta]$ inter-TLR coupling can be induced by such
inductive current-current coupling mechanism and can be written as

${H}_{\mathrm{a.c.}}^{n\alpha,m\beta}=g_{n\alpha,m\beta}(t)\sum_{\sigma,\sigma^{\prime}}(\alpha_{n\sigma}+\alpha_{n\sigma}^{\dagger})(\beta_{m\sigma^{\prime}}+\beta_{m\sigma^{\prime}}^{\dagger}).$
(21)

where $g_{n\alpha,m\beta}(t)$ is the inter-TLR coupling strength which can be
a.c. modulated through the flux bias penetrated in the coupling SQUID. Here we
assume that the parameters in Eq. (21) satisfy the condition

$|\omega_{\alpha\sigma}-\omega_{\beta\sigma^{\prime}}|\gg|g_{n\alpha,m\beta}(t)|.$
(22)

This assumption is needed for the future application of rotating wave
approximation in deriving the effective Hamiltonian.

In the first step, we aim at constructing a concrete
$[n,\alpha,\sigma_{1}]\Leftrightarrow[m,\beta,\sigma_{2}]$ photon hopping.
This task can be mapped to the discussed toy model by setting
$1\rightarrow[n,\alpha,\sigma_{1}]$ and $2\rightarrow[m,\beta,\sigma_{2}]$: As
the $[n,\alpha,\sigma_{1}]$ and $[m,\beta,\sigma_{2}]$ modes are far off-
resonant, the desired hopping can hardly be achieved with static
$g_{n\alpha,m\beta}(t)$. Meanwhile, we can complete this task by dynamically
modulating $g_{n\alpha,m\beta}(t)$ as

$g_{n\alpha,m\beta}(t)=2J\cos[(w_{\alpha\sigma_{1}}-w_{\beta\sigma_{2}})t-\theta_{n\alpha\sigma_{1},m\beta\sigma_{2}}].$
(23)

This is the process depicted in Fig. 1(c), where a photon initially in the
$[n,\alpha,\sigma_{1}]$ mode changes its energy by absorbing/emitting an
energy quanta $|w_{\beta\sigma_{2}}-w_{\alpha\sigma_{1}}|$ from/to the
oscillating $g_{n\alpha,m\beta}(t)$ and then hops into the
$[m,\beta,\sigma_{2}]$ mode.

More rigorously, we expand the $[n,\alpha]\leftrightarrow[m,\beta]$ coupling
in Eq. (22) in the rotating frame of $H_{\mathrm{S}}$ in Eq. (12) and get 32
terms: two from the positive and negative frequency choices of
$g_{n\alpha,m\beta}(t)=J\exp[-\mathrm{i}\theta_{n\alpha\sigma_{1},m\beta\sigma_{2}}]\exp[\mathrm{i}(w_{\alpha\sigma_{1}}-w_{\beta\sigma_{2}})t]+\mathrm{h.c.}$,
four hopping branches with two choices of $\sigma$ and two choices of
$\sigma^{\prime}$, and four terms for each hopping branch,taking the form

$[\alpha_{n\sigma}\exp{-\mathrm{i}\omega_{\alpha\sigma}t}+\mathrm{h.c.}][\beta_{m\sigma^{\prime}}\exp{-\mathrm{i}\omega_{\beta\sigma^{\prime}}t}+\mathrm{h.c.}].$

We then take a close look at the energy spectrum of the two TLRs. The typical
eigenfrequency level diagram of the neighboring $[n,\alpha]$ and $[m,\beta]$
sites is shown in Fig. 1(d). With out loss of generality, let us set the
paramter according to the Eq. (13). Therefore the frequencies of the eight
hopping branches are $\Delta,2\Delta,\omega_{0}+2\Delta,\omega_{0}-\Delta$,
respectively. If the modulating frequency of $g_{n\alpha,m\beta}(t)$ is
resonant with one of the four hopping branches, it is off-resonant with the
other three hopping branches with a detuning at least of the order $\Delta$.
Therefore we look back to the expansion: the $16$ Bogoliubov terms carries
frequency at least of the order $\omega_{0}$, they oscillate in the fastest
way and should be dropped in the very first step. Then, with this observation,
we can find out that only two terms are time independent, other terms carries
fast oscillating factors, six with frequencies of the order $\Delta$, and
eight with the frequencies of the order $\omega_{0}$. Since we have assumed
$\omega_{0}\gg\Delta\gg J$, we dropped these fast oscillating terms and
finally get the effective Hamiltonian.

$H_{\mathrm{eff}}^{n\alpha\sigma_{1},m\beta\sigma_{2}}=J\alpha_{n\sigma_{1}}^{\dagger}\beta_{m\sigma_{2}}\exp(-i\theta_{n\alpha\sigma_{1},m\beta\sigma_{2}})+\mathrm{h.c.}.$
(24)

Here we can see that both the phase and the amplitude of the effective
$[n,\alpha,\sigma_{1}]\leftrightarrow[m,\beta,\sigma_{2}]$ hopping can be
independently controlled by the $[n,\alpha]\Leftrightarrow[m,\beta]$
modulating pulse $g_{n\alpha,m\beta}(t)$.

This parametric formalism can then be generalized to establish the matrix-form
non-Abelian link variables required in Eq. (6). In this situation, the
coupling strength $g_{n\alpha,m\beta}(t)$ in Eq. (21) contains multiple
frequencies, with each tone controlling one hopping branch in the linking
variable. This is schematically sketched in Fig. 1(d), where the red and blue
lines label the eigenfrequencies of the $[n,\alpha]$ and $[m,\beta]$ sites,
and the dashed and the solid arrows represent the component-preserving $U_{1}$
and $U_{4}$ and the compoent-mixing $U_{2}$ and $U_{3}$, respectively. The
major obstacle of the generalization is the cross talk effect, i.e. a
particular tone in $g_{n\alpha,m\beta}(t)$ may induce other unwanted hopping
process which corresponds to the fast oscillating terms which are dropped in
the derivation of the effective Hamiltonian. For instance, if we drive
$g_{n\alpha,m\beta}(t)$ with frequency $2\Delta$ in order to establish the
$[n,\alpha,2]\leftrightarrow[m,\beta,2]$ hopping, this driving can also induce
the $[n,\alpha,1]\leftrightarrow[m,\beta,1]$ hopping in an off-resonant manner
with detuning $\Delta$. This process is fast oscillating and dropped in the
derivation of the effective Hamiltonian. However, when we calculate the Dyson
series up to the second order, this process can give a second order a.c. Stark
energy shift of the involved modes. Physically, we can imagine that a particle
initially populated in the $[n,\alpha,1]$ mode can absorb a $2\Delta$
frequency photon from $g_{n\alpha,m\beta}(t)$ and hop virtually to
$[m,\beta,1]$. However this population is not stable because it is not energy-
conserving. Therefore the only fate of the particle is that it emits the
absorbed $2\Delta$ photon back to $g_{n\alpha,m\beta}(t)$ and jump back to
$[n,\alpha,1]$. This is a second order perturbation process and it results in
an a.c. Stark shift of the eigenenergy of the modes $\sim J^{2}/\Delta$.
Meanwhile, this effect can be effectively suppressed as follow. As shown in
Eq. 6, the link variables have been selected to take relatively simple forms.
Each link variable matrix contains at most $2$ non-zero items, implying that
any coupling strength $g_{n\alpha,m\beta}$ consists of at most $2$ tones. For
instance, the a.c. modulation of the $[n,A]\Leftrightarrow[n,B]$ connecting
SQUID have two tones with frequencies $2\Delta$ and $\Delta$, which induce the
$[n,A,1]\Leftrightarrow[n,B,1]$ and the $[n,A,2]\Leftrightarrow[n,B,2]$ PFC
bonds by bridging their frequency gaps, respectively. Moreover, the
eigenfrequencies of the cavity modes are set such that the two tones are
significantly different. This has already been indicated in Eqs. (13) and (22)
and shown in Fig. 1(d). In this situation, the controlling tone of one hopping
branch can hardly influence the other, resulting in arbitrary control of the
linking variables. With these strategies exploited, the leading effect is the
a.c. Stark shifts of the cavity mode frequencies induced by the off-resonant
pumping, which is of second-order and can be further compensated by the
adjustment of the frequencies of $g_{n\alpha,m\beta}(t)$ which corresponds to
the renormalization of $H_{\mathrm{S}}$ in Eq. (12), as implied in the
derivation of Eq. (24).

###  III.3 Measurement of the non-Abelian AB caging

The proposed non-Abelian AB caging physics can be observed through the
coherent monochromatic pumping of a particular $[n,A,\sigma]$ mode on the
lattice, which can be described by

$H_{\mathrm{pump}}=P^{\dagger}\alpha e^{-i\Omega_{P}t}+\mathrm{h.c.},$ (25)

where $\alpha$ is the vector of the annihilation operators of the whole
lattice, $P$ is the corresponding pumping strength vector, and $\Omega_{P}$ is
the detuning of the pumping frequency with respect to $\omega_{A\sigma}$. The
pumping can inject photons into the lattice which will then experience the
exotic localization dynamics analyzed in the previous section. Therefore, we
expect that information about the proposed non-Abelian AB caging can be
extracted from the driven-dissipation steady state of the lattice.

Figure 3: CLESs and the corresponding SSPN distribution of the lattice with
the $[0,A]$ site being pumped. The single-particle CLESs of these FBs in (a)
and (b) correspond to $E_{1}=\sqrt{6}J$ and $E_{2}=\sqrt{2}J$, respectively.
The detunings and the pumped modes are selected as: (c).
$\Omega_{P}=\sqrt{6}J$, $[0,A,1]$/$[0,A,2]$; (d). $\Omega_{P}=\sqrt{2}J$,
$[0,A,1]$/$[0,A,2]$. For each subfigure, the left and right panel corresponds
to the SSPN calculated by using the full time-dependent physical Hamiltonian
and the effective Hamiltonian.

We then numerically calculate the SSPN distribution of this lattice in the
presence of driving and dissipation, with results shown in Fig. 3(c)-(d).
Roughly speaking, there are two methods of getting the SSPN. The first
corresponds to the solution of the equation

$i\frac{\mathrm{d}\langle\mathrm{\alpha}\rangle}{\mathrm{d}t}=\left[\mathcal{B}-\left(\Omega_{P}+\frac{1}{2}i\kappa\right)\mathcal{I}\right]\langle\mathrm{\alpha}\rangle+P=0,$
(26)

where the matrix $\mathcal{B}$ is defined by
$\alpha^{\dagger}\mathcal{B}\alpha=H$ in Eq. (1) and $\kappa$ is the assumed
uniform decay rate of the cavity modes, and the second one is solving the full
time-dependent Schrodinger equation in the driven-dissipative setting for a
sufficiently long time based on the physical time dependent Hamiltonian. Both
method are exploited with very similar results obtained. These results thus
partially validate the rotating wave approximation we have employed in getting
the effective Hamiltonian in the previous section. In both simulation we
consider a chain of 11 unit-cells with the pumped $A$ site at the central
$x=0$ position. The parameters are chosen as
$J/2\pi=|P|/2\pi=10\quad\mathrm{MHz}$ and $\kappa=0.1J$. To maximize the
injection of photons into the lattice, we set $\Omega_{P}$ to be resonant with
one of the the eigenenergies shown in Fig. 2(a). Explicitly, Figs. 3(c)
correspond to $\Omega_{P}=\sqrt{6}J$ and pumping the$[0,A,1]$/$[0,A,2]$ modes,
and Figs. 3(d) correspond to $\Omega_{P}=\sqrt{2}J$ and pumping the
$[0,A,1]$/$[0,A,2]$ modes. For each subfigure, the up and lower panel denote
the SSPN distribution in the first and second components of the sites,
respectively. It can be clearly observed that the calculated SSPN
distributions exhibit several features of the discussed non-Abelian AB caging.
In particular, the injected photons become localized with asymmetric SSPN
distribution with respect to the pumped $A$ site. The SSPN distribution extend
more rightward(leftward) if the pumped mode is the first(second) component, as
predicted in Sec. II.

From another perspective, the calculated SSPN distributions reflect to some
extent the spatial configuration of the corresponding CLESs. This can be
intuitively understood because pumping a particular cavity mode with a
particular eigenfrequency corresponds to exciting the corresponding CLES
containing the component of the pumped cavity mode. In this sense, the pumping
of the $[0,A]$ site can them be decomposed into the pumping of the
superposition of the four relevant CLESs. If the pumping is resonant with one
of the four CLESs and if the dissipation is sufficiently small compared with
the energy gaps, only one CLES is effectively excited. For instance, the
superposition of the up and lower panel in Fig. 3(c) coincides exactly with
the CLES shown in Fig. 3(a). Fig. 3(d) also coincide with CLES in Fig. 3(b) in
this sense. In addition, it should be noticed that the SSPN patterns shown in
Figs. 3(c) are different from those in Figs. 3(d). This difference can also be
attributed to the excitation of different CLESs by using different pumping
frequencies.

The calculated SSPN distribution can be experimentally detected by the
following simple measurement scheme. Each of the lattice sites sufficiently
involved in the SSPN calculation is capacitively connected to an external coil
with input/output ports for pumping/measurement. The steady state of the
lattice can be prepared by injecting microwave pulses through the input port
for a sufficiently long time. During the steady-state period, energy will leak
out from the coupling capacitance, which is proportional
$\omega_{\alpha\sigma}\langle\alpha_{n\sigma}^{\dagger}\alpha_{n\sigma}\rangle$
with the proportional constant determined by the coupling capacitance. The
target observable $\langle\alpha_{n\sigma}^{\dagger}\alpha_{n\sigma}\rangle$
can therefore be measured by simply integrating the energy flowing to the
output port in a given time duration. This measurement method has already been
used in experiment with both the amplitude and the phase of a coherent state
of a superconducting 3D cavity were measured Sirois  _et al._ (2015).

Figure 4:  Fidelity $F(t)$ with the $[0,A,1]$ mode being pumped. The detunings
are selected as: (a). $\Omega_{P}=\sqrt{6}J$; (b). $\Omega_{P}=\sqrt{2}J$. In
each subfigure, the time-evolving coherent state vector $\langle\alpha\rangle$
is calculated by using both the effective Hamiltonian and the full time-
dependent physical Hamiltonian, and the corresponding fidelity obtained are
shown by the red lines and blue lines, respectively. Moreover, the green line
sketches the abstract value of normalized inner product of the two calculated
coherent state vectors.

As the proposed circuit QED lattice is linear, its dynamics and steady state
properties can be well described in the framework of multi-mode coherent
state. Previous theoretical and experimental papers Longhi (2014); Mukherjee
_et al._ (2018b) focused mainly on the intensity property (i. e. the photon
number) because the density population of particles can visualize the
localization of particles directly. Meanwhile, the measurement of the phase
information of the coherent state can be implemented in superconducting
circuit experiments through standard microwave manipulation technique Sirois
_et al._ (2015). we then use a fidelity to characterize the difference between
the calculated multimode coherent state of the lattice and the target CLES of
the effective Hamitonian we aim to excite. This fidelity takes the form

$F(t)=\frac{|\innerproduct{a(t)}{E_{n}}|}{\norm{a(t)}\norm{E_{n}}},$ (27)

where $\ket{a(t)}$ is the expectation value of the annihilation operator
vector of the whole lattice, and $\ket{E_{n}}$ is the vector describing the
CLES of $H_{\mathrm{eff}}$ in the $n$th band we aim to excite. Both the time
evolution of $F(t)$ under the physical time dependent Hamiltonian and the
effective Hamiltonian are calculated and compared, with results shown in the
Fig. 4. We see that $F(t)$ calculated from these two methods closely coincide
with each other and both approach unity as $t\rightarrow\infty$. Therefore we
come to the conclusion that both the amplitude and the phase information of
the steady state can be effectively described by the corresponding CLES of
$H_{\mathrm{eff}}$ (otherwise the calculated fidelities would deviate
significantly). In addition, the coincidence of the two calculated fidelities
indicates that the effective Hamiltonian is indeed correctly working.

##  IV Discussion and Conclusion

In conclusion, we have proposed in this manuscript the non-Abelian AB caging
concept which is the multi-component matrix generalization of the existing
Abelian AB caging concept. Distinct from its Abelian counterpart, the AB
caging in this situation become sensitive to both the explicit form of the
gauge potential and the initial state of the lattice. These features are the
consequence of the matrix nature of our theory and thus have no Abelian
analog. Moreover, we suggest a superconducting quantum circuit implementation
of the proposed physics, in which the lattice sites are built by
superconducting TLRs and the required non-Abelian gauge is constructed by the
PFC method. The unambiguous verification of the non-Abelian AB caging can
further be achieved through the steady-state manipulation of only few sites on
the lattice.

While the localized steady states of the CLES excitations considered in this
manuscript can be thoroughly understood in the single particle picture, what
is more important is that the dispersionless flat band is an ideal platform of
investigating correlated many-body states. The introduction of interaction
will lead us to the realm where rich but less explored physics locates. On the
other hand, as the strong coupling has already been achieved in circuit QED Gu
_et al._ (2017), the Bose-Hubbard type Hu  _et al._ (2011) and Jaynes-
Cummings-Hubbard type photon-photon interaction Houck  _et al._ (2012);
Schmidt and Koch (2013) can be incorporated by coupling the TLRs with
superconducting qubits. Therefore, our further direction should be the
characterization of nonequilibrium strongly-correlated photonic quantum fluids
in the proposed architecture. Also, with the advances of technology, we expect
the implementation of our ideas in other atomic and photonic quantum simulator
platforms Noguchi  _et al._ (2014).

###### Acknowledgements.

We thank Y. H. Wu and J. H. Gao for helpful discussions. This work was
supported in part by the National Science Foundation of China (Grants No.
11774114 and No. 11874156).

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