# Supplementary Material for “Gate-tunable phase transition in a bosonic Su-
Schrieffer-Heeger chain”

Lukas Johannes Splitthoff1,2<EMAIL_ADDRESS>Miguel Carrera Belo1,2
Guliuxin Jin2 Yu Li3 Eliska Greplova2 Christian Kraglund Andersen1,2 1QuTech,
Delft University of Technology, Delft 2628 CJ, The Netherlands  2Kavli
Institute for Nanoscience, Delft University of Technology, Delft 2628 CJ, The
Netherlands  3Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, 2100 Copenhagen, Denmark

(August 28, 2024)

††preprint: QuTech/AndersenLab

##  I SSH chain model

Our implementation has been derived from the SSH model presented in Sec. I.1.
We compare the model with the ideal implementation in Sec. I.2 and discuss its
symmetries and the localisation of the wavefunctions in Sec. I.3.2 and Sec.
I.3.3.

###  I.1 SSH Hamiltonian

The condensed matter system inspired tight-binding Hamiltonian of the SSH
chain with on-site potential $\epsilon$, intra-cell hopping $v$ and inter-cell
hopping $w$ yields a tri-diagonal real space Hamiltonian of dimension $2N$,
where $N$ is the number of unit cells in the system. The basis of this real
space Hamiltonian is formed by the sites along the chain. The Hamiltonian
reads

$\begin{split}H_{CM}&=\sum_{n=1}^{2N}\epsilon\ket{n}\bra{n}\\\
&+\sum_{n}^{N}\big{[}v\ket{2n-1}\bra{2n}\\\
&+w\ket{2n}\bra{2n+1}+h.c.\big{]}\\\ &=\begin{pmatrix}\epsilon&v&&&&&\\\
v&\epsilon&w&&&&\\\ &w&\epsilon&&&&\\\ &&&\ddots&&&\\\ &&&&\epsilon&w&\\\
&&&&w&\epsilon&v\\\ &&&&&v&\epsilon\\\ \end{pmatrix}\end{split}$ (1)

The spectral properties of this system and its eigenstates are displayed in
Fig. 1. The key feature is the symmetrically gaped spectrum spanned by the
bulk modes (black) with mid-gap modes in the topological insulator phase for
$v<w$. The appearance of the symmetric spectrum is linked to the chiral or
sublattice symmetry of the system, which can be tested by computing the anti-
commutator

$\\{\Gamma,H\\}=\Gamma H+H\Gamma=0$ (2)

with

$\Gamma=\mathbf{I}_{5}\otimes\sigma_{z}$ (3)

where the chiral symmetry operator $\Gamma$ is given by the outer product of
$N$ dimensional identity matrix $\mathbf{I_{5}}$ and the Pauli-z matrix
$\sigma_{z}$. For this tri-diagonal real space Hamiltonian described above the
anti-commutator is indeed zero. The spectral gap never closes due to the
finite size of the system, in contrast to the infinite limit in which the
spectral gap is given by $|v-w|$. Instead, the transition point at $v=w$ is
signaled by the equal free spectral range between the mid-gap modes and one of
the mid-gap modes and the first bulk mode. Beyond the transition, the system
enters a trivial phase where $v>w$ eventually leading to a full dimerization.
The corresponding eigenstates of the mid-gap modes are localized around the
edges of the chain, as exemplified for a specific configuration in Fig. 1d,
hence the name edge mode. An exponential fit to the wavefunction supported on
either sublattice A or B yields the localization length $\xi$ in Fig. 1b. As
long as the edge modes are quasi-degenerate, the localization length extracted
from the fit follows the theoretical limit $\log(w/v)^{-1}$, but the fit does
not capture the divergence of the localization length well. For a finite
system length, the inverse participation ratio appears to be a better measure
for the localization of the wave function.

###  I.2 Lumped element, superconducting circuit Hamiltonian

The design of the lumped element, superconducting circuit implementation of
the SSH chain follows a simple guiding principle. We realize every site as
lumped element resonator with frequency $\omega$ and the coupling terms as
capacitors $C_{w}$ or inductors $L_{v}$. We then derive the circuit
Hamiltonian from a classical circuit Lagrangian formalism as part of a
standard circuit quantization [1]. The resulting Hamiltonian $H_{LC}$ indeed
takes the form of the tight-binding Hamiltonian

$\begin{split}H_{LC}&=\sum_{n=1}^{2N}\hbar\omega\ket{n}\bra{n}\\\
&+\hbar\omega\sum_{n}^{N}\bigg{[}\frac{L_{T}}{2L_{v}}\ket{2n-1}\bra{2n}\\\
&+\frac{C_{w}}{2C_{T}}\ket{2n}\bra{2n+1}+h.c.\bigg{]}\\\
&=\hbar\omega\begin{pmatrix}1&\frac{L_{T}}{2L_{v}}&&&&&\\\
\frac{L_{T}}{2L_{v}}&1&\frac{C_{w}}{2C_{T}}&&&&\\\
&\frac{C_{w}}{2C_{T}}&1&&&&\\\ &&&\ddots&&&\\\ &&&&1&\frac{C_{w}}{2C_{T}}&\\\
&&&&\frac{C_{w}}{2C_{T}}&1&\frac{L_{T}}{2L_{v}}\\\
&&&&&\frac{L_{T}}{2L_{v}}&1\\\ \end{pmatrix}\end{split}$ (4)

Based on the correspondence between the two systems, we can compare the
coefficients in the Hamiltonian and formalize the mapping between the on-site
energy $\epsilon$ and the resonator frequency $\omega$, between the coupling
strength $v$ and the Josephson inductance $L_{v}$, and between the the
coupling strength $w$ and the capacitive coupling $C_{w}$

$\displaystyle\begin{split}\epsilon&=\hbar\omega\\\
v&=\frac{\hbar\omega}{2}\frac{L_{T}}{L_{v}}\\\
w&=\frac{\hbar\omega}{2}\frac{C_{w}}{C_{T}}\end{split}$ (5)

It should be noted that $\omega=\omega(L_{v})$ which leads to a small increase
in $w$, but a significant shift in the on-site potential $\epsilon$ causing
the upwards shift of the spectrum as $L_{v}$ decreases. For further visual
comparison we also show the spectral properties and the eigenstates of the
ideal SSH chain implementation in Fig. 1.

Figure 1: Simulated spectrum and states of a tight-binding SSH model with on-
site potential $\epsilon=$6.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
hopping coefficient $w=$0.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ (a)
Energy spectrum of the bulk modes (black) and the edge modes (red) versus the
coupling strength $v$. (b) Free spectral range $\Delta$ between the edge
states and the bulk (blue, arrow in (a)) and between the two edge modes
(orange, arrow in (a)) versus hopping strength $v$. The dashed line indicates
the phase transition point between the topological and the trivial phase where
$v=w$. The spectral gap $|v-w|$ (grey) only closes in the infinite length
system (c) Localization length $\xi$ extracted as exponential decay fit to the
wave functions of the edge modes. The grey line shows the theoretical limit
$\log(w/v)^{-1}$. (c) Energy spectrum of the bulk modes (black) and the edge
modes (red) versus the coupling strength $v$. (d-g) Real space wave function
$|\psi|^{2}$ on the resonator lattice for the topological and the trivial
phase of the bulk and the edge states. The colors correspond to the markers in
(a).

###  I.3 Topological properties of SSH implementation

####  I.3.1 Real space winding number

Topological invariants characterize the global properties of topological
insulators. In case of the SSH model, the winding number is used as the
topological invariant, which takes a phase-dependent binary value: 0 in the
trivial insulator phase and 1 in the topological insulator phase [2, 3]. In
the finite-size SSH chain, the periodicity is absent, thus making momentum not
a good quantum number and the winding number becomes an ill-defined invariant.
Alternatively, the co-variant real space winding number (RSWN) serves as a
good approximation for the momentum space winding number given the large
enough system size, and the RSWN remains quantized in the presence of disorder
[4, 5, 6, 7]. In the following, we review the calculation of real space
winding number.

For a 1D chiral symmetric system, recall the anti-commutation relation between
the Hamiltonian $H$ and the chiral operator $\Gamma$ in Eq. (2). This relation
results in the fact that the Bloch Hamiltonian $H(k)$ takes the off diagonal
form [8]

$H(k)=\begin{pmatrix}0&h(k)\\\ h(k)^{\dagger}&0\\\ \end{pmatrix}.$ (6)

The winding number is given by [8]

$\nu=\frac{1}{2\pi
i}\int_{BZ}\text{Tr}\\{[h(k)^{-1}\partial_{k}h(k)]\\}\in\mathbb{Z}$ (7)

Eq. (7) has a direct covariant form in the real space [4]. Starting from the
real space SSH Hamiltonian $H$, the homotopically equivalent flatband version
of $H$ is given by

$Q=\frac{H}{|H|}=\begin{pmatrix}0&Q_{0}\\\ Q_{0}^{\dagger}&0\\\
\end{pmatrix},$ (8)

where the off-diagonal part $Q_{0}$ enters the calculation of real space
winding number. The co-variant real-space form of the winding number can be
obtained by applying the Bloch-Floquet transformation of Eq. (7) written in
terms of flatband Hamiltonian [4]

$\nu=-\text{Tr}_{\text{volume}}\\{Q_{0}^{-1}[X,Q_{0}]\\},$ (9)

where $X$ is the position operator and $\text{Tr}_{\text{volume}}$ denotes the
trace per volume. In the case of SSH model, the off-diagonal term $Q_{0}$ can
be obtained as the following. The position eigenstates are $|x,c\rangle$,
where $x=1,\ldots,L$ denote the unit cells and $c=A,B$ are the sublattice
degree of freedom. The position operator therefore takes the form
$X=\sum_{x\in\mathbb{Z}}\sum_{c=A,B}x|x,c\rangle\langle x,c|$. In order to
obtain $Q_{0}$, we define spectral projectors as

$\begin{split}P_{A}=&\sum_{x\in\mathbb{Z}}\sum_{c=A}|x,c\rangle\langle
x,c|,\\\ P_{B}=&\sum_{x\in\mathbb{Z}}\sum_{c=B}|x,c\rangle\langle
x,c|=\hat{I}-P_{A}.\end{split}$ (10)

The chiral operator $\Gamma$ is given by $\Gamma=P_{A}-P_{B}$. In parallel, we
define the projectors onto the upper and lower half of the energy spectrum

$\begin{split}P_{-}&=\sum_{E_{n}\leq 0}|n\rangle\langle n|,\\\
P_{+}&=\sum_{E_{n}\geq 0}|n\rangle\langle n|=\Gamma P_{-}\Gamma.\end{split}$
(11)

Then the flatband Hamiltonian in Eq. (8) can be obtained by $Q=P_{+}-P_{-}$.
As a chiral-symmetric operator, $Q$ satisfies the condition
$Q=P_{A}QP_{B}+P_{B}QP_{A}$. The off-diagonal terms in Eq. (8) can be written
in terms of the above defined operators as

$Q_{0}=P_{A}QP_{B},\quad(Q_{0})^{-1}=P_{B}QP_{A}.$ (12)

Finally, one can express real space winding number Eq. (9) as

$\nu=-\frac{1}{L}\text{Tr}\left\\{P_{B}QP_{A}\left[X,P_{A}QP_{B}\right]\right\\}.$
(13)

In our study, we compute the RSWN versus the coupling strength $v/w$ for
various chain lengths $N$ and for $w=0.5$, see Fig. 2 (a). On either end of
the coupling strength range ($v\in(0,1)$), the system approaches the the
topological and trivial state with RSWN=1 and RSWN=0, respectively, regardless
of the chain length. In-between $v\in(0,1)$, the RSWN indicates a cross-over
between the two different insulator states, but only in the thermodynamic
limit $N\rightarrow\infty$ the RSWN correctly predicts the phase transition
point as it approximates the k-space winding number. For small $N$, the RSWN
does not capture the phase transition, which is spectroscopically still
defined, due to finite size effects. Additionally, we investigate the
robustness of the topological state by examining the RSWN under varying levels
of disorder in the coupling parameter $v$ and $w$ for three different intra-
cell couplings $v=\\{0.01,0.2,1\\}$, see Fig. 2 (b-d). While the RSWN does not
aim at benchmarking the phase transition for our finite size system, we see
that in the limiting cases we have distinct topological phases which providing
valuable insights into the topological phase transition that we observe
experimentally.

Figure 2: Real space winding number. (a) RSWN versus coupling strength ratio.
The dashed vertical lines correspond to the $v/w$ ratio used in the disorder
simulations shown in (b-d). (b-d) RSWN versus disorder strength for three
different coupling strength.

####  I.3.2 Symmetries

To further strengthen the claim that the chiral symmetry persists in the ideal
SSH chain implementation, we present the spectrum of Fig. 1c normalized by the
instantaneous resonator frequency $\omega(L_{v})$ in Fig. 3(a). This
normalized spectrum is symmetric around the on-site energy. Moreover, we can
compute the anti-commutator and find that it vanishes $\\{c,H_{LC}\\}=0$.

Figure 3: (a) Normalized simulated spectrum of the bosonic SSH chain with
ideal the on-site and coupling energies as in Fig. 1. (b,c) Inverse
participation ratio for different states for the tight-binding model (b) and
the ideal SSH chain implementation (c) versus the coupling strength $v$. The
dashed line indicates the $1/2N$ limit. The colors correspond to the
respective wave functions in Fig. 1.

####  I.3.3 Localization

The localization of the wavefunctions of the edge states is a characteristic
of the SSH chain. For finite chain lengths however, the localization length
defined as exponential decay $\exp(-x/\xi)$ does not capture the
delocalization of the edge modes as the system undergoes the phase transition,
as seen in Fig. 1(b). Instead, for finite chain length we can define an
inverse participation ratio (IPR) [9], which measures the support of a wave
function on a specific site and is defined as

$IPR=\frac{\sum_{x}|\psi(x)|^{4}}{(\sum_{x}|\psi(x)|^{2})^{2}}$ (14)

We observe that the IPR of the edge modes is finite and larger than for the
bulk modes in the topological phase. As the system undergoes the phase
transition, the IPR decreases and approaches the delocalization limit
$1/2N=1/10$ of the bulk modes at $v=w$, see Fig. 3b,c. The IPR of the
localized modes is of value one only in the fully dimerized case $v/w=0$,
hence we observe a kink in the IPR in Fig. 3b close to $v=0$.

##  II Circuit parameter estimation

From the presence of the bulk and gap modes in the spectrum for a given
$L_{v}$ configuration, we can extract the circuit parameters $C_{0}$, $L_{0}$,
$C_{w}$ and $L_{v}$. First, we identify and list the eigenmodes of the system.
Then, we fit the 10 dimensional Hamiltonian $H_{LC}$ to the list of eigenmodes
while allowing for free variation of all circuit parameters along the chain.
For completeness, we also take the two $C_{w}$ couplings to the coupling sites
$C$ into account, which expands the $C_{w}$ list from four to six elements.
Since the eigenfrequencies of the coupling sites are far detuned from the SSH
spectrum, we do not consider them in the system Hamiltonian. The circuit
parameters obtained from this optimization routine in the topological phase
are shown in Fig. 4. We assume the design parameters as start parameters for
the Nelder–Mead optimization. The variation of the obtained parameters is
smaller than $1\text{\,}\mathrm{\char 37\relax}$. Finally, we input the
obtained circuit parameters into the full Hamiltonian to compute the spectrum
and the eigenstates to simulate the data in Fig. 2 and Fig. 3d. The agreement
of this optimization with the measurement results is on the order of a few
$\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which is also the spectral
resolution of the measurement. This parameter estimation yields the static
parameters $C_{0}$, $L_{0}$, and $C_{w}$. The optimization routine can then be
repeated for different $L_{v}$ configurations to the trivial phase.
While any on-site disorder and second-nearest neighbor interaction due to
residual capacitive coupling breaks the chiral symmetry, strictly speaking,
the mid-gap modes remain localized and can therefore be understood as chiral
for practical purposes [9].

Figure 4: Extracted circuit parameters in the topological phase. (a,b)
Resonator capacitance $C_{0}$ and inductance $L_{0}$ per site. (c,d) Coupling
capacitance $C_{w}$ and coupling inductance $L_{v}$ per coupling site.

##  III Single gate dependence

A transmission measurement through the five unit cell SSH chain allows for a
site specific tune-up of every tunable coupling element. We run single gate
scans for every nanowire Josephson junction over a wide gate voltage range,
while we keep all other gates at the pinch-off voltage $V_{p}$, see Fig.
5(a-e). Hence, all scans begin deep in the topological regime. For the bulk
gates $NW_{i}$ with $i\in[2,3,4]$ we zoom on the low band in the topological
state, while we focus on the gap for the edge gates $NW_{i}$ with $i\in[1,5]$,
as we expect the biggest change in frequency in those spectral ranges.

We notice the non-monotonic single gate dependence of the modes in the
respective test spectrum due to microscopic properties, most likely spurious
quantum dots in the $100\text{\,}\mathrm{n}\mathrm{m}$ long junction. In fact,
we observe discontinuities due to jumps in the microscopic properties of the
junctions. Repetitions of these scans reveal that the slopes change over time.
Hence, even these single gate dependencies cannot be used to calibrate for a
joint gate scan across the phase transition point with identical nanowire
inductances along the SSH chain.

The measured single gate dependencies are in qualitative agreement with the
simulated spectra on the right of Fig. 5. From the single gate dependencies we
can extract the voltage $V_{o}$, which corresponds to the maximal
supercurrent, hence minimal $L_{J}$ or, expressed in Josephson energy, maximal
$E_{J}$. This voltage point seem robust enough and constant over time and over
several gate scans.

Figure 5: Single gate dependence. (a-e) Measured spectrum per gate while the
other gates are set to $V_{p}$. Zoom-in on low band for bulk gates and zoom-in
on gap for edge gates. The data is normalized on the background to enhance the
visibility of the change. (f-j) Corresponding simulated spectrum.

##  IV Additional joint gate dependence

In Fig. 6, we repeat the measurement shown in Fig. 4 over the gate settings
$V_{p}$ and $V_{o}$ defined as minimal and maximal value of the single gate
dependencies shown in Fig. 5 at $-30\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$
signal power. The splitting of the modes with increasing gate voltage, thus
higher gate setting index, is in qualitative agreement with the equivalent
scan from an earlier cooldown presented in Fig. 4, which uses similar gate
settings $V_{p}=[400,150,430,400,1900]$\text{\,}\mathrm{m}\mathrm{V}$$ and
$V_{o}=[1800,1800,1800,1800,4000]$\text{\,}\mathrm{m}\mathrm{V}$$ for the
nanowire gates 1 to 5.

Figure 6: Additional joint gate dependence with $V_{p}$ and $V_{o}$ taken as
minimal and maximal value of the single gate dependencies shown in Fig. 5. The
dashed lines indicate the gate settings used for the power dependence shown in
Fig. 7

##  V Power-induced phase transition

Fig. 7 shows the mid-gap spectrum versus signal power at the signal generator
output for three gate voltage settings corresponding to the three gate
settings [50, 100, 150] in Fig. 6, (a) close to the topological regime, (b)
close to the normal regime and (c) deep in the trivial regime. The dark lines
on the light background indicate the mid-gap modes. We observe that the two
modes are quasi-degenerate at high signal powers and the spectrum is nearly
indistinguishable from the topological regime regardless of the exact gate
setting. As we reduce the signal power, the two modes split and approach the
low power spectrum shown in Fig. 6. In the low power regime, the spectral
evolution exhibits a gate dependence. For a gate set-point deep in the
topological regime, the modes do not disperse with signal power (not shown
here). We understand this power-induced phase transition as a result of the
power-dependent change in the kinetic inductance of the proximitized
nanowires. The inductance in these nanowires follows the equation

$\displaystyle
L_{NW}=L_{0}(V_{g})\bigg{[}1+\bigg{(}\frac{I_{s}}{I_{\ast}(V_{g})}\bigg{)}^{2}\bigg{]}$
(15)

where $L_{0}(V_{g})$ is the gate-dependent low-power inductance, $I_{s}$ is
the signal current through the chain, and $I_{\ast}(V_{g})$ is the gate-
dependent critical current. The signal power enters the equation via the
signal current to which it is proportional $P_{s}\leavevmode\nobreak\
I_{s}^{2}$. Hence, an increasing signal power increases the inductance, which
in turn transitions the SSH chain into the topological regime. The non-
monotonic power dependence as well as the different transition points most
likely arise from the different critical currents $I_{\ast}(V_{g})$ per
nanowire. Consequently, in order to recover the SSH spectrum versus gate
voltage, the measurements must be acquired with a signal power below
$-30\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$.

Figure 7: Power-induced phase transition. (a-c) Mid-gap spectrum versus signal
power at the signal generator output for the three gate voltage settings [50,
100, 150] in Fig. 4.

##  VI Fabrication

We fabricate the SSH chain circuit and the gate lines from a
$40\text{\,}\mathrm{nm}$-thick sputtered NbTiN film (kinetic inductance
$4\text{\,}\mathrm{p}\mathrm{H}\,\Box^{-1}$) on high resistivity n-doped Si.
We pattern the NbTiN film using e-beam lithography and SF6/O2 based reactive
ion etching. $30\text{\,}\mathrm{nm}$-thick plasma enhanced chemical vapour
deposition (PECVD) SiN defined by a buffered oxide etch serves as bottom gate
dielectric. We transfer the two-facet InAs/Al nanowire on top of the SiN
bottom gate using a nano-manipulator. The InAs nanowires were grown by
vapor–liquid–solid (VLS) growth with a diameter of $110\pm
5.\text{\,}\mathrm{nm}$, and nominal thickness of the Al of
$6\text{\,}\mathrm{nm}$ [10]. We selectively etch the
$110\text{\,}\mathrm{n}\mathrm{m}$ long Josephson junctions into Al film.
Then, we electrically contact the nanowires to the circuit via lift-off
defined $150\text{\,}\mathrm{nm}$-thick sputtered NbTiN leads after prior Ar
milling to minimize the contact resistance.

##  VII SSH chain environment

We measure the transmission spectrum of the SSH chain in the topological state
over the accessible frequency range
($4-$8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$), see Fig. 8. We observe a
peaked background transmission with a finer modulation and with a maximal
transmission around $6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. We attribute
the resonance-like peak at around $6\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$
to a spurious feature hosted by the PCB enclosure spanning between the two
test ports S and D, which acts as a band pass filter in parallel to the SSH
chain, which in turn affects the overall transmission. We also observe that
the transmission spectrum recorded in several experimental runs and on
different samples, but within the same enclosure differs from the expected
transmission spectrum. We highlight the resemblance of the transmission
spectrum with a broad band resonator with a fit to a Lorentzian (orange). The
finer modulation of the transmission probably arises from impedance mismatches
along the lines connecting to the measurement electronics. To simplify the
data analysis in the presence of the broad spectral feature, we obtain a
linearly interpolated background transmission spectrum to which we normalize
the measurement data to better identify the transmission peaks and dips of the
SSH chain spectrum.

While we cannot extract the exact effective inductance and capacitance leading
to the spurious feature formed by the enclosure, we can approximate its effect
in a lumped element simulation in which we assume a resonator in parallel to
the SSH chain. The results of these lumped element simulations are presented
in Fig. 9(a) for the ideal case and in Fig. 9(b) for the case in presence of a
spurious resonance. In the ideal case, we expect two, or three well defined
bands with in total 10 modes and otherwise suppressed background transmission.
However, a spurious mode yields a higher overall transmission due its band
pass feature and distorts the measured line shape of the modes due to is
capacitive contribution on the rising edge and its inductive contribution on
the falling edge. Consequently, the bulk modes appear as dips and peaks and
the mid gap modes in the topological state appear as deep dip with a shallow
peak in the center. The feature, which we modelled for simplicity as a single
mode, does not affect the SSH chain, but it does complicate the interpretation
of the overall transmission spectrum. To overcome the effect, a new sample
enclose should be carefully designed [11]. Going beyond the current
experiment, one could also intentionally coupled a SSH chain to a resonator as
described in Ref. [12].

Figure 8: Box mode. $S_{21}$ transmission spectrum over the accessible
frequency range $4-$8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ with SSH
spectrum removed (blue). A Lorentzian fit highlights the presence of a box
mode (orange). The dashed grey line indicates the expected transmission given
the input attenuation and output amplification. Figure 9: Lumped element
simulation of the three regimes topological, normal, trivial for different
tuning scenarios. (a) Expected, ideal SSH chain spectrum for a joint
synchronous sweep of $L_{v}$. (b) Same device tuning as in (a), but accounting
for the presence of a box mode modelled as transmission resonator in parallel
to the SSH chain. Note that all spectra per panel are offset for better
visibility.

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