# Interplay between topology and localization on superconducting circuits

Xin Guan  Department of Materials and Chemical Engineering, Taiyuan
University, Taiyuan, DC 030032, China<EMAIL_ADDRESS>Gang Chen  State
Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of
Laser Spectroscopy, Shanxi University, Taiyuan, DC 030006, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,
DC 030006, China  School of Physics, Zhengzhou University, Zhengzhou, DC
450001, China

###### Abstract

Topological insulator lie at the forefront of condensed matter physics.
However strong disorder can destroy the topological states and make all states
become localized. In this paper, we investigate the competition between
topology and localization in the one-dimensional Su-Schrieffer-Heeger (SSH)
model with controllable off-diagonal quasi-periodic modulations on
superconducting circuits. By utilizing external ac magnetic fluxes, each
transmon can be driven and all coupling strengths can be tuned independently.
Based on this model we construct phase diagrams that illustrate the extended
topologically nontrivial, critical localization, and coexisting topological
and critical localization phases. The dynamics of the qubits’ excitations are
also discussed in this paper, revealing distinct quantum state transfers
resulting from the interplay between topology and localization. Furthermore,
we propose a method for detecting different quantum phases using current
experimental setups.

##  I Introduction

Recent experimental breakthrough of superconducting circuits has opened up a
new avenue for processing quantum computingBuluta et al. (2011); Gong et al.
(2021); Liu et al. (2018) and quantum informationWendin (2017); Yang and Zheng
(2018); Blais et al. (2020), as well as implementing quantum
simulationGeorgescu et al. (2014); Zhang et al. (2023); Ying et al. (2023).
Due to the superconducting circuits can be fabricated into different lattice
structures, a wide range of condensed matter phenomena can be simulated, such
as chiral physics and spin clustersWang et al. (2019), topological
semimetalTan et al. (2018, 2019), quantum walksYan et al. (2019); Gong et al.
(2021), quantum phase transitionsKaur et al. (2021); Dborin et al. (2022), and
many-body quench dynamicsZhang et al. (2023). A recent experiment has
successfully engineered a chain of qubits on superconducting circuits with
tunable couplings through external drivingsLi et al. (2018). Utilizing this
experimental setup, a meticulous study has successfully observed the
topological magnon insulator and detected its winding numberCai et al. (2019).
These experiments provide a novel approach for investigating exotic
topological properties.

On the other hand, topological insulators(TIs)Hasan and Kane (2010); Qi and
Zhang (2011); Khanikaev et al. (2013); Rechtsman et al. (2013); Maciejko and
Fiete (2015); Asbóth et al. (2016); Rachel (2018); Schindler et al. (2018);
Tokura et al. (2019); Rudner and Lindner (2020); Biesenthal et al. (2022); Lei
et al. (2022) have received considerable attention in the past decades. As is
well known, in addition to possessing a bulk gap, topological insulators (TIs)
exhibit symmetry-protected gapless states localized at their boundaries. The
topological properties of TIs are robust against weak disorders such as
fluctuations, imperfections in device fabrications, and environmental noises.
Consequently, TIs have found numerous applications in quantum computation and
information processing. However, when subjected to strong enough disorder, the
topological state can be destroyed and replaced by a localization state.
Additionally, moderate levels of disorder can induce a topological Anderson
insulatorLi et al. (2009); Groth et al. (2009); Guo et al. (2010); Stützer et
al. (2018); Meier et al. (2018a); Yang et al. (2020); Zhang et al. (2021a). It
is intriguing to investigate the interplay between topology and localization.
Quasi-periodic disorder and dimer lattice structure have attracted much
attention as means to induce localization and topological states,
respectively. The famous Aubry-André(AA)Sokoloff (1985); Thouless (1988);
Roushan et al. (2017); An et al. (2021); Ye and Sun (2022); Yang and Cao
(2022) and Su-Schrieffer-Heeger(SSH)Su et al. (1979); Marchand et al. (2010);
Meier et al. (2016a); Xing et al. (2021); Zhang et al. (2021b); Kiczynski et
al. (2022); Youssefi et al. (2022); Wang et al. (2023); Wei et al. (2023)
models are abstracted, both of which have been realized in superconducting
circuitsRoushan et al. (2017); Youssefi et al. (2022). Therefore to design a
qubits chain with staggered coupling strength and controllable off-diagonal
quasi-periodic modulations is appealing.

In this paper, we propose a practical approach to realize the one-dimensional
SSH model on superconducting circuits with controllable off-diagonal quasi-
periodic modulations. Specifically each transmon is driven by an external ac
magnetic flux via the flux-bias line. All parameters can be independently
adjusted by manipulating the amplitudes and frequencies of the drivings.
Through this generalized SSH model, we have discovered extended topologically
nontrivial, critical localization, and co-existing topological and critical
localization phases transitions. We utilize the winding number and inverse
participation ratio (IPR) to characterize the topological and localization
properties, respectively. Additionally, we discuss the dynamics of qubit
excitations in this paper. We observe different quantum state transfers
induced by competition between topology and localization. Finally, we address
possible experimental observations of these quantum phases. Herein, we
introduce the mean chiral displacement (MCD) as a means to probe the winding
number. Our results provide a novel avenue for exploring exotic quantum phases
and dynamic behaviors arising from the interplay between dimerization and
quasi-periodic disorder strengths on superconducting circuits.

##  II Model and Hamiltonian

Figure 1:  Schematic diagram of transmon qubits chain with tunable coupling
strengths. All transmon qubits are coupled with their nearest-neighbor qubits
by capacitors. (a) Illustration of the detail circuits of two coupled qubits.
(b) Schematic diagram of the generalized SSH model with $t_{1}$ and $t_{2}$ as
the coupling strengths. $\Phi_{ext}(t)$ is the external ac magnetic flux.

The method for implementing a one-dimensional SSH model with quasi-periodicity
on superconducting circuits is as follow. We consider a chain of transmon
qubits that are capacitively coupled to their next-nearest neighbor qubits, as
illustrated in Fig. 1(a). The corresponding Lagrangian is written as

$\mathcal{L}=\underset{j}{\sum}\left[\frac{C_{j}}{2}\dot{\phi}_{j}^{2}+E_{j}^{J0}\cos\left(\frac{\phi_{j}}{\phi_{0}}\right)\right]+\underset{\left\langle
i,j\right\rangle}{\sum}\left[\frac{C_{ij}}{2}\left(\dot{\phi}_{i}-\dot{\phi}_{j}\right)^{2}\right]$
(1)

Here, $C_{j}$ and $E_{j}^{J0}$ represent the effective capacitance and the
Josephson energy of the $j$th transmon qubit. $C_{ij}$ denotes the capacitor
used to couple the $i$th and $j$th transmon qubits with its nearest-neighbor
qubits. $\phi_{j}$ refers to node flux expressed in units of reduced flux
quantum $\phi_{0}=1/\left(2e\right)$. The symbol $\left\langle
i,j\right\rangle$ indicates that the summation runs over all pairs of the
nearest-neighbor sites. With the canonical coordinate $\hat{\phi}_{j}$, the
canonical momentum $\hat{q}_{j}=\partial L/\partial\dot{\hat{\phi}}_{j}$,
i.e.,

$\hat{q}_{j}=C_{j}\dot{\hat{\phi}}_{j}+\underset{j\left\langle
i\right\rangle}{\sum}C_{j}\left(\dot{\hat{\phi}}_{j}-\dot{\hat{\phi}}_{j\left\langle
i\right\rangle}\right)$ (2)

where $j\left\langle i\right\rangle$ is the summation of the nearest-neighbor
sites around the $j$th site. Considering the experimental condition that
$C_{j}\gg\\{C_{\left(j-1\right)j},C_{j\left(j+1\right)}\\}$, we have
$\dot{\hat{\phi}}_{j}\approx\hat{q}_{j}/C_{j}$. By applying the Legendre
transformation, we obtain the corresponding Hamiltonian

$\hat{H}=\hat{H}_{\text{0}}+\hat{H}_{\text{IJ}},$ (3)

where

$\hat{H}_{\text{0}}=\underset{j}{\sum}\left(\frac{\hat{q}_{j}^{2}}{2\tilde{C}_{j}}+\frac{\hat{\phi}_{j}^{2}}{2L_{j}^{J}}-\frac{E_{j}^{J0}}{24\hat{\phi}_{0}^{4}}\hat{\phi}_{j}^{4}\right),$
(4) $\hat{H}_{\text{IJ}}=\underset{\left\langle
i,j\right\rangle}{\sum}\frac{C_{ij}}{C_{i}C_{j}}\hat{q}_{i}\hat{q}_{j},$ (5)

with
$1/\tilde{C}_{j}=(C_{j}-C_{\left(j-1\right)j}-C_{j\left(j+1\right)})/C_{j}^{2}$
and $1/L_{j}^{J}=E_{j}^{J0}/\phi_{0}^{2}$.

By introducing bosonic annihilation and creation operators

$\displaystyle\hat{a}_{j}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\tilde{C}_{j}\omega_{j}^{0}}{2}}\hat{\phi}_{j}+i\sqrt{\frac{1}{2\tilde{C}_{j}\omega_{j}^{0}}}\hat{q}_{j},$
(6) $\displaystyle\hat{a}_{j}^{{\dagger}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\tilde{C}_{j}\omega_{j}^{0}}{2}}\hat{\phi}_{j}-i\sqrt{\frac{1}{2\tilde{C}_{j}\omega_{j}^{0}}}\hat{q}_{j},$
(7)

where $[\hat{a}_{j},\hat{a}_{j^{\prime}}^{{\dagger}}]=\delta_{j,j^{\prime}}$
and $\omega_{j}^{0}=\sqrt{8E_{j}^{C}E_{j}^{J0}}$ with
$E_{j}^{C}=e^{2}/(2\tilde{C}_{j})$, and after applying a rotating-wave
approximation, the Hamiltonian (3) is quantized as

$\displaystyle\hat{H}_{\text{0}}$ $\displaystyle=$
$\displaystyle\sum_{j}\omega_{j}^{0}\hat{a}_{j}^{{\dagger}}\hat{a}_{j}+\frac{1}{2}V_{j}\hat{n}_{j}(\hat{n}_{j}-1),$
(8) $\displaystyle\hat{H}_{\text{IJ}}$ $\displaystyle=$
$\displaystyle\sum_{j}g_{j}^{0}\hat{a}_{(j-1)}^{{\dagger}}\hat{a}_{j}+\text{H.c.},$
(9)

where H.c. is the Hermitian conjugate,
$\hat{n}_{j}=\hat{a}_{j}^{{\dagger}}\hat{a}_{j}$ is the particle number
operator, and

$\displaystyle V_{j}$ $\displaystyle=$ $\displaystyle-E_{j}^{C},$ (10)
$\displaystyle g_{j}^{0}$ $\displaystyle=$
$\displaystyle\frac{C_{\left(j-1\right)j}\sqrt{\omega_{\left(j-1\right)}^{0}\omega_{j}^{0}}}{2\sqrt{C_{\left(j-1\right)}C_{j}}}.$
(11)

In Eqs. (8) and (9), $V_{j}$ denotes the anharmonicity of each qubit,
$g_{j}^{0}$ is the coupling strength between the $j$th and ($j-1)$th qubits.
Since the requirement of $E_{j}^{C}\ll E_{j}^{J0}$ for transmon qubitsKoch et
al. (2007), we choose $E_{j}^{J0}/E_{j}^{C}=50$,
$\\{C_{\left(j-1\right)j},C_{j\left(j+1\right)}\\}\approx 0.5$ fF, and
$C_{j}\approx 100$ fF. Based on these parameters, we obtain
$V_{j}/\left(2\pi\right)\approx 200$ MHz and
$g_{j}^{0}/\left(2\pi\right)\approx 10$ MHzYe et al. (2019). This results in a
condition that $V_{j}\gg g_{j}^{0}$. In such case, the qubit exhibits strong
anharmonicity, allowing for only one boson to be excited per transmon qubit.
This results in $V_{j}\hat{n}_{j}(\hat{n}_{j}-1)/2\equiv 0$, and
$\hat{a}_{j}^{{\dagger}}\left|0\right\rangle_{j}=\left|1\right\rangle_{j}$,
$\hat{a}_{j}^{{\dagger}}\left|1\right\rangle_{j}=0$,
$\hat{a}_{j}\left|0\right\rangle_{j}=0$, and
$\hat{a}_{j}\left|1\right\rangle_{j}=\left|0\right\rangle_{j}$. As a result,
the Hamiltonian (3) becomes

$\hat{H}=\sum_{j}\omega_{j}^{0}\hat{a}_{j}^{{\dagger}}\hat{a}_{j}+\sum_{j}\left(g_{j}\hat{a}_{(j-1)}^{{\dagger}}\hat{a}_{j}+\text{H.c.}\right).$
(12)

The commutation relation can be reformulated as follows,

$[\hat{a}_{j},\hat{a}_{j^{\prime}}^{{\dagger}}]=(1-2\hat{n}_{j})\delta_{j,j^{\prime}}.$
(13)

To achieve tunable coupling strength, an ac microwave driving is applied to
each transmon qubit, which can be implemented experimentally via the flux-bias
line lineLi et al. (2018). As illustrated in Fig. 1, we incorporate a
Josephson junction into a superconducting quantum interference device (SQUID).
Since the two Josephson junctions in SQUID possess equal energy, the effective
energy can be expressed as $E_{j}^{J}=2E_{j}^{J0}$. As a result of the
external magnetic flux $\Phi_{j}(t)$, a phase difference
$\Phi_{j}(t)/\phi_{0}$ arises between the two Josephson junctions, leading to
a modification in the potential energy of each qubit,

$\displaystyle
E_{j}^{J0}\left[\cos\left(\frac{\phi_{j}}{\phi_{0}}+\frac{\Phi_{j}(t)}{2\phi_{0}}\right)+\cos\left(\frac{\phi_{j}}{\phi_{0}}-\frac{\Phi_{j}(t)}{2\phi_{0}}\right)\right]$
(14) $\displaystyle=$ $\displaystyle
E_{j}^{J}(t)\cos\left(\frac{\phi_{j}}{\phi_{0}}\right),$

where $E_{j}^{J}(t)=E_{j}^{J}\cos[\Phi_{j}(t)/2\phi_{0}]$. The frequency of
the $j$th qubit becomes
$\omega_{j}(t)=\omega_{j}^{0}\sqrt{\cos[\Phi_{j}(t)/2\phi_{0}]}.$ Here we take
the external magnetic flux as,

$\Phi_{j}(t)=\alpha_{j}\cos(u_{j}^{0}t+\phi_{j}^{0}),$ (15)

where $\alpha_{j}\ll\omega_{j}^{0}$, $u_{j}^{0}$, and $\phi_{j}^{0}$ are
amplitude, frequency, and phase of the external magnetic flux respectively.
Next we conduct Taylor expansion of the frequency $\omega_{j}(t)$ at
$\Phi_{j}(t)=0$,

$\omega_{j}(t)\approx\omega_{j}^{0}+\varepsilon_{j}\sin(u_{j}t+\phi_{j}),$
(16)

where $\varepsilon_{j}=\omega_{j}^{0}\alpha_{j}^{2}/4\phi_{0}^{2}$,
$u_{j}=2u_{j}^{0}$, and $\phi_{j}=2\phi_{j}^{0}+3\pi/2.$ In experimentsLi et
al. (2018), $u_{j}$ is of the order $\sim 100$ MHZ, which is far larger than
$g_{j}^{0}$. Then an unitary operator $\hat{U}=\hat{U}_{A}\times\hat{U}_{B}$
is introduced to transform the Hamiltonian (12) with the frequency
$\omega_{j}(t),$ where,

$\displaystyle\hat{U}_{A}$ $\displaystyle=$
$\displaystyle\exp\left[-i\underset{j=1}{\overset{N}{\sum}}\omega_{j}^{0}\hat{n}_{j}t\right]$
$\displaystyle\hat{U}_{B}$ $\displaystyle=$
$\displaystyle\exp\left[i\underset{j=1}{\overset{N}{\sum}}f_{j}\cos(u_{j}t+\phi_{j})\right],$
(17)

with $f_{j}=\varepsilon_{j}/u_{j}$ and $N$ is the number of the qubits. The
transformed Hamiltonian can be obtained,

$\displaystyle\hat{H}_{t}=$
$\displaystyle\hat{U}^{{\dagger}}\hat{H}\hat{U}+i\frac{d\hat{U}^{{\dagger}}}{dt}\hat{U}$
$\displaystyle=$
$\displaystyle\overset{N-1}{\underset{j=1}{\sum}}g_{j}^{0}\left\\{e^{-i\Delta_{j}t}\exp[-if_{j}\cos(u_{j}t+\phi_{j})]\right.$
$\displaystyle\left.\exp[if_{j+1}\cos(u_{j+1}t+\phi_{j+1})]\hat{a}_{j}^{{\dagger}}\hat{a}_{j+1}+\text{H.c.}\right\\},$
(18)

where $\Delta_{j}=\omega_{j+1}^{0}-\omega_{j}^{0}.$ We choose
$\Delta_{j}=u_{j+1}(-u_{j+1})$ for odd (even) $j$. Next we utilize the Jacobi-
Anger identity,
$\exp[if_{j}\cos(u_{j}t+\phi_{j})]=\sum_{-\infty}^{\infty}i^{m}J_{m}(f_{j})\exp[im(u_{j}t+\phi_{j})],$
and apply the rotating-wave approximation to the Hamiltonian in Eq. (18). Here
$J_{m}(f_{j})$ represents the $m$th Bessel function of the first kind. By
neglecting the oscillating terms and setting $\phi_{j}=-\pi(0)$ for odd (even)
$j$, we can derive,

$\hat{H}_{eff}=\overset{N-1}{\underset{j=1}{\sum}}g_{j}\left(\hat{a}_{j}^{{\dagger}}\hat{a}_{j+1}+\text{H.c.}\right),$
(19)

where

$g_{j}=g_{j}^{0}J_{0}(f_{j})J_{1}(f_{j+1})$ (20)

As is evident, the coupling strength $g_{j}$ can be tuned independently. For
obtaining the SSH model with controllable off-diagonal quasi-periodic
modulations, we take $g_{j}=g[1+\lambda+\delta\cos(2\pi\beta j)]$ for odd $j$
and $g_{j}=g[1-\lambda+\delta\cos(2\pi\beta j)]$ for even $j$. For
constituting the quasi-periodic disorder, we take $\beta=(\sqrt{5}-1)/2$.
$\lambda$ and $\delta$ indicate the dimerization and quasi-periodic disorder
strengths respectively. In this paper we set $g=1$ as the unit of the energy.
In the following discussion, we denote the creation (annihilation) operator of
the qubit with odd and even $j$ as $\hat{a}_{i}^{{\dagger}}(\hat{a}_{i})$ and
$\hat{b}_{i}^{{\dagger}}(\hat{b}_{i})$ respectively, where $i$ is the unit
cell index in the case of $\delta=0$. Here we consider the single excitation
throughout our work.

##  III Phase diagram and quantum state transfer

###  III.1 Phase diagram

Figure 2:  Energy bands and wave functions of the $N/2$th and
$(N/2+1)$th$[l=N/2,(N/2+1)]$ eigenstates for the Hamiltonian in Eq. (19) with
open boundary condition. ($a_{1}$)-($c_{1}$) Energy bands as functions of
$\lambda$ with ($a_{1}$) $\delta=0$, ($b_{1}$) $\delta=1$, and ($c_{1}$)
$\delta=2$. ($a_{2}$)-($c_{2}$) Wave functions with ($a_{2}$) $\delta=0$ and
$\lambda=0.5$, ($b_{2}$) $\delta=1$ and $\lambda=0.5$ and ($c_{2}$) $\delta=2$
and $\lambda=0.5$.

In this section we focus on the phase diagram resulting from the interplay
between quasi-periodic disorder and dimerization strengths. We employ 50
transmon qubits in this section.

When disorder is absent in the generalized SSH model (19), it is reduced to
the SSH model. Its energy bands are shown in Fig. 2(a), indicating the
presence of degenerate zero-energy modes that persist within the middle of the
energy gap, with a value of $\lambda>0$. The corresponding wave functions
exhibit exponential localization at the boundaries of the system, as shown in
Fig. 2(e). Subsequently, we activate the disorder and present the variation of
these energy bands in Figs. 2(b) and (c), demonstrating that as $\delta$
increases, the energy gap near zero-energy vanishes within the small $\lambda$
region. Figures 2 (e)-(g) demonstrate the transition of edge states into bulk
of the system as $\delta$ increases. These results demonstrate that strong
disorder can only destroy topological states in the case of small $\lambda$.
This implies that the computation between dimerization strength $\lambda$ and
quasi-periodic disorder strength $\delta$ can induce a transition from
topologically trivial to nontrivial phases. In the following, we will explain
these results analytically.

With $\delta=0,$ we can ansatz the edge states as
$\left|\Psi\right\rangle_{E=0}=\sum_{i}(\rho_{1}^{i}w_{1}\hat{a}_{i}^{{\dagger}}\pm\rho_{2}^{i}w_{2}\hat{b}_{i}^{{\dagger}})\left|G\right\rangle$,
where $\left|G\right\rangle$ is the vacuum state. According to the Schrödinger
equation, $\hat{H}\left|\Psi\right\rangle_{E=0}=0$, ie,

$(g_{1}\rho_{2}^{i}+g_{2}\rho_{2}^{i-1})w_{2}\hat{a}_{i}^{{\dagger}}\left|G\right\rangle+(g_{2}\rho_{1}^{i+1}+g_{1}\rho_{1}^{i})w_{1}\hat{b}_{i}^{{\dagger}}\left|G\right\rangle=0,$
(21)

the edge states can be obtained as

$\left|\Psi\right\rangle_{E=0}=\underset{i}{{\sum}}\left[\left(-\frac{g_{1}}{g_{2}}\right)^{i}w_{1}\hat{a}_{i}^{{\dagger}}\pm\left(-\frac{g_{2}}{g_{1}}\right)^{i}w_{2}\hat{b}_{i}^{{\dagger}}\right]\left|G\right\rangle.$
(22)

These results demonstrate that the state is localized in proximity to the
leftmost ($i=1$) $a-$type site and rightmost ($i=N/2$) $b-$type site.

Next, we introduce disorder and observe the changes in $\rho_{1}$ and
$\rho_{2}$,

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle-\frac{1-\lambda+\delta\cos\left[2\pi\beta\left(2i-1\right)\right]}{1+\lambda+\delta\cos\left[2\pi\beta\left(2i\right)\right]},$
(23) $\displaystyle\rho_{2}$ $\displaystyle=$
$\displaystyle-\frac{1+\lambda+\delta\cos\left[2\pi\beta\left(2i\right)\right]}{1-\lambda+\delta\cos\left[2\pi\beta\left(2i-1\right)\right]},$
(24)

which depend on the qubits’ indexes. Equations (23) and (24) indicates the
presence of non-zero $\delta$ prevents $\rho_{1}$ and $\rho_{2}$ from always
being larger or smaller than 1. This implies that the behavior of topological
edge states is influenced by $\delta$. Then for characterizing distinct
topological phases in the case of $\delta\neq 0$, we compute the topological
invariant. Since the Hamiltonian (19) possesses chiral, time-reversal and
particle-hole symmetries, it belongs to class BDI of the Altland–Zirnbauer
classification. The topological properties of the system can be characterized
by the winding number, which can be easily calculated in momentum space for
translationally invariant systems. However, in the presence of non-zero
disorder strength, translational symmetry is broken and thus the computation
of winding number must be performed in real space using a local
quantityMondragon-Shem et al. (2014); Meier et al. (2018b),

$\nu=\frac{1}{2}\text{Tr}^{\prime}(\hat{P}\hat{\Gamma}\hat{X}\hat{P}),$ (25)

where Tr′ indicate the trace over the single unit cell in the middle of the
qubits chian. $\hat{P}=\hat{P}_{+}-\hat{P}_{-}$ with
$\hat{P}_{+}(\hat{P}_{-})$ is the projection operator of the eigenstates with
positive (negative) eigenenergies.
$\hat{\Gamma}=\sum_{i=1}^{N}\left(\hat{\Gamma}_{a_{i}}^{e}-\hat{\Gamma}_{b_{i}}^{e}\right)$
where $\hat{\Gamma}_{di}^{e}=$ $\left|e\right\rangle_{di}\left\langle
e\right|$ with $\left|e\right\rangle_{di}$ ($d=a,b$) indicates the state with
exciting $d-$type qubit at the $i$th unit cell. $\hat{X}$ is the unit-cell
position operator. We numerically calculate the winding number in Fig. 3(a).
In region III ($\delta>1+\lambda$), $\nu=0$, indicating that the system is
topologically trivial. In regions I and II ($\delta<1+\lambda$), $\nu=1$,
indicating that the system is topologically nontrivial. This finding
accurately demonstrates that the coexistence of strong disorder and weak
dimerization strength can result in the annihilation of topological states.

Figure 3:  (a) Phase diagram in the $\lambda$-$\delta$ plane. I is extended
topologically nontrivial phase with winding number $\nu=1$. II is the
coexisting topological critical localization phase with winding number
$\nu=1$. III is the critical localization phase with winding number $\nu=0$.
(b) Energy bands denoted by the values of IPRn as a function of $\delta$ with
$\lambda=0.5$.

On the other hand, localization always emerges in a quasi-periodic system. We
can capture the localization property of the $n$th eigenstate
$\left|\Psi_{n}\right\rangle=\sum_{j}\psi_{n}(j)\left|e\right\rangle_{j}$
($\psi_{n}(j)$ is the wave function) in our system by the inverse
participation ratio(IPR), which is defined as
IPR${}_{n}=\sum_{j}\left|\psi_{n}(j)\right|^{4}$. For an extended (a fully
localized) state, IPR${}_{n}\approx(2N)^{-1}$ (IPR${}_{n}\approx$1). For a
critically localized state, IPR${}_{n}\approx(2N)^{-\theta},$where
$\theta\in(0,1)$ depends on the multifractal nature of the wave function. We
present the phase diagram in Fig. 3(a) and find our system exhibits
localization transition boundaries, where the values of IPRn shift from zero
in region I ($\delta<1-\lambda$ $\cup$ $\delta<\lambda-1$) to finite values
between 0 and 1 in regions II and III
($\delta>1-\lambda\cap\delta>\lambda-1$). This result suggests that the
localization properties are also influenced by the strength of dimerization.

In conclusion, the competition between $\delta$ and $\lambda$ induces four
different quantum phases which are extended topologically nontrivial [region I
in Fig. 3(a)], coexisting topological and critical localization [region IIin
Fig. 3(a)], and critical localization [region III in Fig. 3(a)] phases. In
addition, the energy bands with open boundary condition marked by the values
of IPRn are shown in Fig. 3(b), which indicates there are no mobility edges
except few edge states.

###  III.2 Quantum state transfer

Figure 4:  (a)Time evolution of qubits’ excitations with $\delta=0$ and
$L=11$. (b) The fidelity of the state transfer between $\left|L\right\rangle$
and $\left|R\right\rangle$. (c) Time evolution of qubits’ excited states with
$\lambda=0.5$ and $L=10$

In this section, we demonstrate the transfer of quantum states both with and
without quasi-periodic disorder.

Firstly, considering $\delta=0$, the edge states are presented in Eq.(22).
When the number of qubits is even, the edge states will localize at both
boundaries of the system. Therefore, regardless of how $\lambda$ changes, the
quantum state localized at leftmost boundary cannot transfer to the rightmost.
While in the case of odd qubits, both boundaries are composed of $a$-type
qubits which result in the localization of edge states at the left- and
rightmost boundaries with $\lambda>0$ and $\lambda<0$, respectively. This
implies the quantum state transfer can occur when $\lambda$ changes from
negative to positive values. To achieve this change, we set
$\lambda=\cos(\gamma)$ with $\gamma\in[0,\pi]$. In order to verify this
quantum state transfer, the periodic parameter $\gamma$ should be a function
of time $t$. Here we choose $\gamma=\Omega t$ with $\Omega$ as the rate of
change and the initial state is chosen as the leftmost edge state, which can
be written as $\left|L\right\rangle=(egg...g)$. We illustrate the temporal
evolution of the qubits’ excitations in Fig.4(a). This result indicates that
the quantum state is transferred from the leftmost boundary to the rightmost
boundary of the system through bulk with $\gamma$ varying from 0 to 2$\pi$.
Subsequently, we introduce quasi-periodic disorder $\delta$ and present the
fidelity of this quantum state transfer $F=\left\langle
R\right|\psi_{f}\rangle$ in Fig.4(b). $\left|R\right\rangle=(gg..ge)$ is the
rightmost edge state. $\left|\psi_{f}\right\rangle$ is the final state. When
$\delta$ is sufficiently small, the quantum state transfer persists. While as
$\delta$ increases beyond a certain threshold, the transfer becomes
unfeasible.

We then observe the quantum state transfer by varying the quasi-periodic
disorder strength $\delta$. Similarly to the aforementioned discussion, here
we set $\delta=\Omega t$ and choose $\left|L\right\rangle$ as the initial
state. The corresponding temporal evolutions of the qubits’ excitations are
illustrated in Fig. 4(c). We find the qubits’ excitations localize at the
leftmost boundary when $\Omega t$ is smaller than approximately 0.5, which can
be attributed to the system being in a topological extended phase as depicted
in Fig. 3(a). As $\Omega t$ varies from 0.5 to 1.5, these excitations depart
from the leftmost boundary and diffuse throughout the bulk of the system. In
this case, the system is in coexisting topological and critical localization
phase [Fig. 3(a)]. Therefore this diffusion arises due to competition between
topology and localization. In the region where $\Omega t>1.5$, the qubits’
excitations are localized at one or some bulk states of the system due to
critical localization phase with topologically trivial properties[Fig. 3(a)].
These results suggest that the quantum state can transfer from a boundary
state to certain states within the bulk, which is induced by the interplay
between topology and localization.

##  IV Possible experimental observation

In this section, we propose an experimental scheme for detecting the
aforementioned phases. Furthermore, it is worth noting that the quantum state
transfer can be observed directly.

Figure 5:  Time evolution of all qubits’ excited states in two dimensional
representation with (a) $\delta=2$ and (b) $\delta=0.2$. (c) Time evolution of
chiral displacement with $\delta=2$ and $\delta=0.2$ denoted by blue and
orange curves, respectively. In all figures $\lambda=0.5$

In superconducting circuits, the winding number (25) can be dynamically
detected though the mean chiral displacement (MCD) $\overline{C}$ Meier et al.
(2016b); Cardano et al. (2017); Meier et al. (2018b) with
$\overline{C}=\nu/2$. The MCD is defined as,

$\overline{C}=\frac{1}{T}\int_{0}^{T}Cdt,$ (26)

with
$C=\left\langle\psi(t)\right|\hat{\Gamma}\hat{X}\left|\psi(t)\right\rangle$.
Here $\left|\psi(t)\right\rangle=e^{-i\hat{H}t}\left|\psi(0)\right\rangle$ is
the state of the system at time $t.$ $\left|\psi(0)\right\rangle$ is the
initial state. Here we utilize four qubits to detect the MCD. Firstly we
prepare an initial state $\left|\psi(0)\right\rangle=\left|gegg\right\rangle,$
which excite one of the middle qubits to a excited state and leave the others
in ground states. The time evolution of this excitation can be measured
directly, which is shown in Fig. 5(a) and (b). We can see, in the topological
trivial phase, this excitation can propagate to the boundaries of the qubit
chain[Fig. 5(a)], whereas in the topologically nontrivial phase, it is
confined within the bulk[Fig. 5(b)]. Subsequently, the time evolution of the
CD can be derived, which as depicted in Fig. 5(c). The blue and orange curves
denote topological trivial and nontrival phases respectively. The CD exhibits
oscillations around -0.01 and 0.42 in topological trivial and nontrivial
phases, respectively, indicating distinct winding numbers of -0.02 and 0.84.
This finding is in close proximity to the ideal values.

To identify the critical localization, the detection of IPRn is necessary. The
IPRn can be obtained from the eigenstates of Hamiltonian (19). Since each
qubit can be represented as a point on the Bloch sphere, the $j$th qubit can
be denoted by
$\left|\psi_{j}\right\rangle=\cos(\theta_{j})\left|0\right\rangle_{j}+e^{i\chi_{j}}\sin(\theta_{j}/2)\left|1\right\rangle_{j}$,
where $\theta_{j}$ ($\chi_{j}$) is the angle between the qubit
$\left|\psi_{j}\right\rangle$ and the $z(x)$ axis in the Bloch sphere. The
eigenstates can be described as
$\left|\psi\right\rangle_{n}=\prod_{j}\left|\psi_{j}\right\rangle.$ To
determine the specific values of the angles $\theta_{j}^{n}$ and
$\chi_{j}^{n}$ for the $n$th eigenstate, it is necessary to numerically
calculate the amplitude of the single-qubit excitation $\lambda_{j}^{n}$ for
each qubit in the $n$th eigenstate. Then angles $\theta_{j}^{n}\ $and
$\chi_{j}^{n}$ can be obtained as
$2\arccos(\sqrt{1-\left|\lambda_{j}^{n}\right|^{2}})$ and
$-i\ln[\lambda_{j}^{n}/\sin(\theta_{j}/2)]$ respectively. The $n$th eigenstate
can be obtained after applying a rotation operator
$\hat{U}_{j}^{n}=e^{i\theta_{j}^{n}\hat{\sigma}_{z}+i\chi_{j}^{n}\hat{\sigma}_{x}}$
for each qubit. This rotation operation can be realized by a microwave pules.
Finally the IPRn can be derived directly.

##  V Conclusion

In summary, we have proposed a feasible experimental method to realize the one
dimensional SSH model with controllable off-diagonal quasi-periodic
modulations on superconducting circuits. For realizing independently tunable
coupling strengths, we apply an external ac magnetic flux to drive each
transmon qubit. Then the generalized SSH model with both dimerization and
quasi-periodic can be constructed. Based on this model, extended topologically
nontrivial, critical localization, and coexisting topological and critical
localization phases transitions are discovered. We use the winding number and
IPR to characterize the topological and localization properties respectively.
The dynamics of the qubits’ excitations are also discussed in this paper,
revealing distinct quantum state transfers resulting from the interplay
between topology and localization. Furthermore, we propose a method for
detecting different quantum phases using current experimental setups. Notably,
we introduce the mean chiral displacement (MCD) as a means to probe the
winding number. Our results provide a novel avenue for exploring exotic
quantum phases and dynamic behaviors arising from the interplay between
dimerization and quasi-periodic disorder strengths on superconducting
circuits.

## References

  * Buluta et al. [2011] Iulia Buluta, Sahel Ashhab, and Franco Nori.  Natural and artificial atoms for quantum computation.  _Reports on Progress in Physics_ , 74(10):104401, 2011.
  * Gong et al. [2021] Ming Gong, Shiyu Wang, Chen Zha, Ming-Cheng Chen, He-Liang Huang, Yulin Wu, Qingling Zhu, Youwei Zhao, Shaowei Li, Shaojun Guo, Qian Haoran, Ye Yangsen, Chen Fusheng, Ying Chong, Yu Jiale, Fan Daojin, Wu Dachao, Su Hong, Deng Hui, Rong Hao, Zhang Kaili, Cao Sirui, Jin Lin, Yu Xu, Lihua Sun, Cheng Guo, Na Li, Futian Liang, M. Bastidas, V., Nemoto Kae, J. Munro, W., Yong-Heng Huo, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan.  Quantum walks on a programmable two-dimensional 62-qubit superconducting processor.  _Science_ , 372(6545):948–952, 2021.
  * Liu et al. [2018] Tong Liu, Bao-Qing Guo, Chang-Shui Yu, and Wei-Ning Zhang.  One-step implementation of a hybrid fredkin gate with quantum memories and single superconducting qubit in circuit qed and its applications.  _Optics express_ , 26(4):4498–4511, 2018.
  * Wendin [2017] Göran Wendin.  Quantum information processing with superconducting circuits: a review.  _Reports on Progress in Physics_ , 80(10):106001, 2017.
  * Yang and Zheng [2018] Chui-Ping Yang and Zhen-Fei Zheng.  Deterministic generation of greenberger–horne–zeilinger entangled states of cat-state qubits in circuit qed.  _Optics Letters_ , 43(20):5126–5129, 2018.
  * Blais et al. [2020] Alexandre Blais, Steven M Girvin, and William D Oliver.  Quantum information processing and quantum optics with circuit quantum electrodynamics.  _Nature Physics_ , 16(3):247–256, 2020.
  * Georgescu et al. [2014] Iulia M Georgescu, Sahel Ashhab, and Franco Nori.  Quantum simulation.  _Reviews of Modern Physics_ , 86(1):153, 2014\.
  * Zhang et al. [2023] Xueyue Zhang, Eunjong Kim, Daniel K Mark, Soonwon Choi, and Oskar Painter.  A superconducting quantum simulator based on a photonic-bandgap metamaterial.  _Science_ , 379(6629):278–283, 2023.
  * Ying et al. [2023] Chong Ying, Bin Cheng, Youwei Zhao, He-Liang Huang, Yu-Ning Zhang, Ming Gong, Yulin Wu, Shiyu Wang, Futian Liang, Jin Lin, Xu Yu, Deng Hui, Rong Hao, Peng Cheng-Zhi, Yung Man-Hong, Zhu Xiaobo, and Jian-Wei Pan.  Experimental simulation of larger quantum circuits with fewer superconducting qubits.  _Physical review letters_ , 130(11):110601, 2023\.
  * Wang et al. [2019] Da-Wei Wang, Chao Song, Wei Feng, Han Cai, Da Xu, Hui Deng, Hekang Li, Dongning Zheng, Xiaobo Zhu, H Wang, Shi-Yao Zhu, and O Scully.  Synthesis of antisymmetric spin exchange interaction and chiral spin clusters in superconducting circuits.  _Nature Physics_ , 15(4):382–386, 2019.
  * Tan et al. [2018] Xinsheng Tan, Dan-Wei Zhang, Qiang Liu, Guangming Xue, Hai-Feng Yu, Yan-Qing Zhu, Hui Yan, Shi-Liang Zhu, and Yang Yu.  Topological maxwell metal bands in a superconducting qutrit.  _Physical review letters_ , 120(13):130503, 2018\.
  * Tan et al. [2019] Xinsheng Tan, YX Zhao, Qiang Liu, Guangming Xue, Hai-Feng Yu, ZD Wang, and Yang Yu.  Simulation and manipulation of tunable weyl-semimetal bands using superconducting quantum circuits.  _Physical review letters_ , 122(1):010501, 2019\.
  * Yan et al. [2019] Zhiguang Yan, Yu-Ran Zhang, Ming Gong, Yulin Wu, Yarui Zheng, Shaowei Li, Can Wang, Futian Liang, Jin Lin, Yu Xu, CHeng Guo, Lihua Sun, Zhi-Cheng Peng, Keyu Xia, Hui Deng, Hao Rong, Q You, J, Franco Nori, Heng Fan, Xiaobo Zhu, and Jian-Wei Pan.  Strongly correlated quantum walks with a 12-qubit superconducting processor.  _Science_ , 364(6442):753–756, 2019.
  * Kaur et al. [2021] Kuljeet Kaur, Théo Sépulcre, Nicolas Roch, Izak Snyman, Serge Florens, and Soumya Bera.  Spin-boson quantum phase transition in multilevel superconducting qubits.  _Physical Review Letters_ , 127(23):237702, 2021\.
  * Dborin et al. [2022] James Dborin, Vinul Wimalaweera, Fergus Barratt, Eric Ostby, Thomas E O’Brien, and Andrew G Green.  Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer.  _Nature Communications_ , 13(1):5977, 2022.
  * Li et al. [2018] Xuegang Li, Y Ma, J Han, Tao Chen, Y Xu, W Cai, H Wang, YP Song, Zheng-Yuan Xue, Zhang-qi Yin, and Luyan Sun.  Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings.  _Physical Review Applied_ , 10(5):054009, 2018\.
  * Cai et al. [2019] Weizhou Cai, Jiaxiu Han, Feng Mei, Yuan Xu, Yuwei Ma, Xuegang Li, Haiyan Wang, YP Song, Zheng-Yuan Xue, Zhang-qi Yin, Suotang Jia, and luyan Sun.  Observation of topological magnon insulator states in a superconducting circuit.  _Physical review letters_ , 123(8):080501, 2019\.
  * Hasan and Kane [2010] M Zahid Hasan and Charles L Kane.  Colloquium: topological insulators.  _Reviews of modern physics_ , 82(4):3045, 2010\.
  * Qi and Zhang [2011] Xiao-Liang Qi and Shou-Cheng Zhang.  Topological insulators and superconductors.  _Reviews of Modern Physics_ , 83(4):1057, 2011\.
  * Khanikaev et al. [2013] Alexander B Khanikaev, S Hossein Mousavi, Wang-Kong Tse, Mehdi Kargarian, Allan H MacDonald, and Gennady Shvets.  Photonic topological insulators.  _Nature materials_ , 12(3):233–239, 2013.
  * Rechtsman et al. [2013] Mikael C Rechtsman, Julia M Zeuner, Yonatan Plotnik, Yaakov Lumer, Daniel Podolsky, Felix Dreisow, Stefan Nolte, Mordechai Segev, and Alexander Szameit.  Photonic floquet topological insulators.  _Nature_ , 496(7444):196–200, 2013.
  * Maciejko and Fiete [2015] Joseph Maciejko and Gregory A Fiete.  Fractionalized topological insulators.  _Nature Physics_ , 11(5):385–388, 2015.
  * Asbóth et al. [2016] János K Asbóth, László Oroszlány, and András Pályi.  A short course on topological insulators.  _Lecture notes in physics_ , 919:166, 2016.
  * Rachel [2018] Stephan Rachel.  Interacting topological insulators: a review.  _Reports on Progress in Physics_ , 81(11):116501, 2018.
  * Schindler et al. [2018] Frank Schindler, Ashley M Cook, Maia G Vergniory, Zhijun Wang, Stuart SP Parkin, B Andrei Bernevig, and Titus Neupert.  Higher-order topological insulators.  _Science advances_ , 4(6):eaat0346, 2018.
  * Tokura et al. [2019] Yoshinori Tokura, Kenji Yasuda, and Atsushi Tsukazaki.  Magnetic topological insulators.  _Nature Reviews Physics_ , 1(2):126–143, 2019\.
  * Rudner and Lindner [2020] Mark S Rudner and Netanel H Lindner.  Band structure engineering and non-equilibrium dynamics in floquet topological insulators.  _Nature reviews physics_ , 2(5):229–244, 2020\.
  * Biesenthal et al. [2022] Tobias Biesenthal, Lukas J Maczewsky, Zhaoju Yang, Mark Kremer, Mordechai Segev, Alexander Szameit, and Matthias Heinrich.  Fractal photonic topological insulators.  _Science_ , 376(6597):1114–1119, 2022.
  * Lei et al. [2022] Ying Lei, Xi-Wang Luo, and Shaoliang Zhang.  Second-order topological insulator in periodically driven optical lattices.  _Optics Express_ , 30(13):24048–24061, 2022.
  * Li et al. [2009] Jian Li, Rui-Lin Chu, Jainendra K Jain, and Shun-Qing Shen.  Topological anderson insulator.  _Physical review letters_ , 102(13):136806, 2009\.
  * Groth et al. [2009] CW Groth, M Wimmer, AR Akhmerov, J Tworzydło, and CWJ Beenakker.  Theory of the topological anderson insulator.  _Physical review letters_ , 103(19):196805, 2009\.
  * Guo et al. [2010] H-M Guo, G Rosenberg, G Refael, and M Franz.  Topological anderson insulator in three dimensions.  _Physical review letters_ , 105(21):216601, 2010\.
  * Stützer et al. [2018] Simon Stützer, Yonatan Plotnik, Yaakov Lumer, Paraj Titum, Netanel H Lindner, Mordechai Segev, Mikael C Rechtsman, and Alexander Szameit.  Photonic topological anderson insulators.  _Nature_ , 560(7719):461–465, 2018.
  * Meier et al. [2018a] Eric J Meier, Fangzhao Alex An, Alexandre Dauphin, Maria Maffei, Pietro Massignan, Taylor L Hughes, and Bryce Gadway.  Observation of the topological anderson insulator in disordered atomic wires.  _Science_ , 362(6417):929–933, 2018a.
  * Yang et al. [2020] Bing Yang, Hongfang Zhang, Qiang Shi, Tong Wu, Yong Ma, Zengtao Lv, Xia Xiao, Ruixin Dong, Xunling Yan, and Xiangdong Zhang.  Details of the topological state transition induced by gradually increased disorder in photonic chern insulators.  _Optics Express_ , 28(21):31487–31498, 2020.
  * Zhang et al. [2021a] Weixuan Zhang, Deyuan Zou, Qingsong Pei, Wenjing He, Jiacheng Bao, Houjun Sun, and Xiangdong Zhang.  Experimental observation of higher-order topological anderson insulators.  _Physical Review Letters_ , 126(14):146802, 2021a.
  * Sokoloff [1985] JB Sokoloff.  Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials.  _Physics Reports_ , 126(4):189–244, 1985.
  * Thouless [1988] DJ Thouless.  Localization by a potential with slowly varying period.  _Physical review letters_ , 61(18):2141, 1988\.
  * Roushan et al. [2017] Pedram Roushan, Charles Neill, J Tangpanitanon, Victor M Bastidas, A Megrant, Rami Barends, Yu Chen, Z Chen, B Chiaro, A Dunsworth, A Fowler, B Foxen, M Giustina, E Jeffrey, J Kelly, E Lucero, J Mutus, M Neeley, C Quintana, D Sank, A Vainsencher, J Wenner, T White, H Neven, G Angelakis, D, and J Martinis.  Spectroscopic signatures of localization with interacting photons in superconducting qubits.  _Science_ , 358(6367):1175–1179, 2017.
  * An et al. [2021] Fangzhao Alex An, Karmela Padavić, Eric J Meier, Suraj Hegde, Sriram Ganeshan, JH Pixley, Smitha Vishveshwara, and Bryce Gadway.  Interactions and mobility edges: Observing the generalized aubry-andré model.  _Physical review letters_ , 126(4):040603, 2021\.
  * Ye and Sun [2022] Fan Ye and Xiankai Sun.  Hofstadter butterfly and topological edge states in a quasiperiodic photonic crystal cavity array.  _Optics Express_ , 30(15):26620–26627, 2022.
  * Yang and Cao [2022] Ying Yang and Dawei Cao.  Observation of the topological phase transition from the spatial correlation of a biphoton in a one-dimensional topological photonic waveguide array.  _Optics Express_ , 30(21):37899–37909, 2022.
  * Su et al. [1979] W_P Su, JR Schrieffer, and Ao J Heeger.  Solitons in polyacetylene.  _Physical review letters_ , 42(25):1698, 1979\.
  * Marchand et al. [2010] DJJ Marchand, G De Filippis, V Cataudella, M Berciu, N Nagaosa, NV Prokof’Ev, AS Mishchenko, and PCE Stamp.  Sharp transition for single polarons in the one-dimensional su-schrieffer-heeger model.  _Physical review letters_ , 105(26):266605, 2010\.
  * Meier et al. [2016a] Eric J Meier, Fangzhao Alex An, and Bryce Gadway.  Observation of the topological soliton state in the su–schrieffer–heeger model.  _Nature communications_ , 7(1):13986, 2016a.
  * Xing et al. [2021] Bo Xing, Wei-Ting Chiu, Dario Poletti, Richard T Scalettar, and George Batrouni.  Quantum monte carlo simulations of the 2d su-schrieffer-heeger model.  _Physical Review Letters_ , 126(1):017601, 2021\.
  * Zhang et al. [2021b] Yiqi Zhang, Boquan Ren, Yongdong Li, and Fangwei Ye.  Topological states in the super-ssh model.  _Optics Express_ , 29(26):42827–42836, 2021b.
  * Kiczynski et al. [2022] Mitchell Kiczynski, SK Gorman, Helen Geng, MB Donnelly, Yousun Chung, Yu He, JG Keizer, and MY Simmons.  Engineering topological states in atom-based semiconductor quantum dots.  _Nature_ , 606(7915):694–699, 2022.
  * Youssefi et al. [2022] Amir Youssefi, Shingo Kono, Andrea Bancora, Mahdi Chegnizadeh, Jiahe Pan, Tatiana Vovk, and Tobias J Kippenberg.  Topological lattices realized in superconducting circuit optomechanics.  _Nature_ , 612(7941):666–672, 2022.
  * Wang et al. [2023] Ziteng Wang, Xiangdong Wang, Zhichan Hu, Domenico Bongiovanni, Dario Jukić, Liqin Tang, Daohong Song, Roberto Morandotti, Zhigang Chen, and Hrvoje Buljan.  Sub-symmetry-protected topological states.  _Nature Physics_ , pages 1–7, 2023.
  * Wei et al. [2023] Mei-Song Wei, Ming-Jie Liao, Ce Wang, Chengjie Zhu, Yaping Yang, and Jingping Xu.  Topological laser with higher-order corner states in the 2-dimensional su-schrieffer-heeger model.  _Optics Express_ , 31(3):3427–3440, 2023.
  * Koch et al. [2007] Jens Koch, M Yu Terri, Jay Gambetta, Andrew A Houck, David I Schuster, Johannes Majer, Alexandre Blais, Michel H Devoret, Steven M Girvin, and Robert J Schoelkopf.  Charge-insensitive qubit design derived from the cooper pair box.  _Physical Review A_ , 76(4):042319, 2007.
  * Ye et al. [2019] Yangsen Ye, Zi-Yong Ge, Yulin Wu, Shiyu Wang, Ming Gong, Yu-Ran Zhang, Qingling Zhu, Rui Yang, Shaowei Li, Futian Liang, Jin Lin, Yu Xu, Cheng Guo, Lihua Sun, Chen Cheng, Nvsen Ma, Yang Meng, Zi, Hui Deng, Hao Rong, Chao-Yang Lu, Cheng-Zhi Peng, Heng Fan, Xiaobo Zhu, and Jian-Wei Pan.  Propagation and localization of collective excitations on a 24-qubit superconducting processor.  _Physical review letters_ , 123(5):050502, 2019\.
  * Mondragon-Shem et al. [2014] Ian Mondragon-Shem, Taylor L Hughes, Juntao Song, and Emil Prodan.  Topological criticality in the chiral-symmetric aiii class at strong disorder.  _Physical review letters_ , 113(4):046802, 2014\.
  * Meier et al. [2018b] Eric J Meier, Fangzhao Alex An, Alexandre Dauphin, Maria Maffei, Pietro Massignan, Taylor L Hughes, and Bryce Gadway.  Observation of the topological anderson insulator in disordered atomic wires.  _Science_ , 362(6417):929–933, 2018b.
  * Meier et al. [2016b] Eric J Meier, Fangzhao Alex An, and Bryce Gadway.  Observation of the topological soliton state in the su–schrieffer–heeger model.  _Nature communications_ , 7(1):13986, 2016b.
  * Cardano et al. [2017] Filippo Cardano, Alessio D’Errico, Alexandre Dauphin, Maria Maffei, Bruno Piccirillo, Corrado de Lisio, Giulio De Filippis, Vittorio Cataudella, Enrico Santamato, Lorenzo Marrucci, Maciej Lewenstein, and Pietro Massignan.  Detection of zak phases and topological invariants in a chiral quantum walk of twisted photons.  _Nature communications_ , 8(1):15516, 2017.