# Spectral signatures of non-trivial topology in a superconducting circuit

L. Peyruchat  R. H. Rodriguez  J.-L. Smirr  Ç. Ö. Girit<EMAIL_ADDRESS>Quantronics Group, Université Paris Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-
Yvette Cedex, France  $\Phi_{0}$, JEIP, USR 3573 CNRS, Collège de France, PSL
University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France  R.
Leone  Laboratoire de Physique et Chimie Théoriques, Université de Lorraine,
CNRS, F–54506 Vandœuvre lès Nancy Cedex, France

###### Abstract

Topology, like symmetry, is a fundamental concept in understanding general
properties of physical systems. In condensed matter systems, non-trivial
topology may manifest itself as singular features in the energy spectrum or
the quantization of observable quantities such as electrical conductance and
magnetic flux. Using microwave spectroscopy, we show that a superconducting
circuit with three Josephson tunnel junctions in parallel can possess energy
degeneracies indicative of _intrinsic_ non-trivial topology. We identify three
topological invariants, one of which is related to a hidden quantum mechanical
supersymmetry. Depending on fabrication parameters, devices are gapless or
not, and fall on a simple phase diagram which is shown to be robust to
perturbations including junction imperfections, asymmetry, and inductance.
Josephson tunnel junction circuits, which are readily fabricated with
conventional microlithography techniques, allow access to a wide range of
topological systems which have no condensed matter analog. Notable spectral
features of these circuits, such as degeneracies and flat bands, may be
leveraged for quantum information applications.

##  I Introduction

A physical system is topologically non-trivial if it has a discrete property
that is robust to perturbations [1]. For example a superconducting ring is
topologically non-trivial as the magnetic flux through it is quantized and
conserved no matter how the ring is deformed, as long as it is not broken. The
quantized transconductance of a semiconductor in the quantum Hall regime is
another topological invariant [2]. Some topological invariants may be related
to singular spectral features such as band crossings [3]. Many new invariants,
describing different kinds of topological order, have been discovered in
condensed matter systems [4, 5]. In electronic systems the topological
invariants often correspond to measurable, discrete, universal electromagnetic
quantities, such as flux ($h/e$), charge ($e$), or resistance ($h/e^{2}$) [6].
Both topological spectral features and quantized transport parameters find
applications in fields such as metrology [7] and quantum information [8].

Although topological phases in materials arise from symmetries of many-
particle Hamiltonians [9], simple single-particle systems with non-trivial
topology also exist, such as a spin-1/2 electron in a magnetic field $B$,
where the degeneracy at $B=0$ gives rise to a topological charge [3, 10].
Circuits with Josephson tunnel junctions, due to their versatility,
scalability, and reliability, have been used for the simulation of well known
topological models [11]. But these circuits, which can often be described by a
few bosonic degrees of freedom, also have _inherent_ topological properties.
These inherent properties are derived from the real Hamiltonian, a function of
the flux and charge variables [12], as opposed to a “simulated” one, and may
have no analog in real materials [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 23, 26]. We identify and measure three topological invariants in a
simple superconducting circuit with three Josephson junctions in parallel, the
BiSQUID. The invariants determine whether degeneracies exist (curvature
invariant $B$); characterizes the global structure of degeneracies (Witten
index $\operatorname{Tr}P$); and associates to each degeneracy a unique
winding number $N_{2}$.

Figure 1:  Hidden quantum mechanical supersymmetry in the spectrum of the
charging Hamiltonian. At charge offset $n_{g}=0$ (Witten index
$\operatorname{Tr}P=1$), supersymmetry is _unbroken_ : the ground state at
zero energy is not degenerate, whereas all excited states are doubly
degenerate. At $n_{g}=\pm 1/2$ ($\operatorname{Tr}P=0$), supersymmetry is
_broken_ : the ground state energy is greater than zero and all states are
doubly degenerate. “Bosonic” (even parity) and “fermionic” (odd parity)
superpartners are indicated by purple circles and green crosses.

##  II Supersymmetry of Charging Hamiltonian

The first invariant we consider is the Witten index as it has been largely
overlooked and pertains to a wide class of Josephson junction circuits with
degeneracies. Consider the spectrum of a quantum capacitor described by the
Hamiltonian $H_{C}=E_{C}(n-n_{g})^{2}$, where $E_{C}=(2e)^{2}/2C$ is the
charging energy and $2en_{g}$ is the offset charge [Fig. 1] [27, 28]. The
eigenstates $\ket{n}$, with $n$ integer, correspond to the number of excess
Cooper pairs on the ungrounded electrode, and span the Hilbert space of
$2\pi$-periodic wavefunctions in $\theta$, the phase difference across the
capacitor. The fundamental commutation relation is $[\theta,n]=i$. The
Hamiltonian $H_{C}$ can also describe the flux states of a superconducting
ring, a quantum rotor, or the angular momentum states of a charged particle
confined to a ring [29, 30, 31, 32, 33].

The energy diagram Fig. 1 has a peculiar degeneracy structure: at $n_{g}=0$
all states except the zero energy ground state are doubly degenerate, whereas
at $n_{g}=\pm 1/2$, all states are degenerate. This spectral signature is
reminiscent of supersymmetric quantum mechanics (SUSY QM), a toy model for
SUSY quantum field theories in which a bosonic Hamiltonian is augmented with
an additional fermionic “superpartner” degree-of-freedom [34, 35, 36, 37, 38].
The resulting generic supersymmetric Hamiltonian can either have a doubly
degenerate spectrum with a zero energy ground state singlet, called _unbroken_
SUSY ($n_{g}=0$), or in _broken_ SUSY, have a doublet non-zero energy ground
state ($n_{g}=\pm 1/2$).

Whereas $H_{C}$ has no explicit fermionic component, it has a hidden
supersymmetry which generates the same superalgebra that yields the
distinctive spectrum of SUSY QM models [Appendix A]. Hidden supersymmetry in
quantum bosonic systems, such as the free particle [38], bound state Aharonov-
Bohm problem [39, 40], and Jaynes-Cummings model [41] was discovered
previously. Supersymmetry also plays a role in models such as SYK [42], is
debated to exist in certain topological superconductors [43, 44] and has been
measured in artificial systems such as ion trap simulators [45]. A five
Josephson junction circuit with a configuration exhibiting hidden unbroken
supersymmetry was also proposed [46]. Here we consider the hidden
supersymmetry, both broken and unbroken, inherent in any superconducting
circuit where the Hamiltonian at degeneracy is the charging Hamiltonian
$H_{C}$.

The symmetry which partitions eigenstates into “bosons”, with even parity, and
“fermions”, with odd parity, is inversion about the point $n=n_{g}$. In the
charge basis, the parity operator is
$P(n_{g})=\sum_{n}\ket{-n+2n_{g}}\bra{n}$, defined only for $n_{g}=0,\pm 1/2$
when restricted to the Hilbert space. Since $[P,H_{C}]=0$ and $P^{2}=1$,
energy eigenstates can also be chosen to be parity eigenstates with
eigenvalues $\pm 1$. Combining $P$, referred to as a grading operator in
supersymmetry, with the “supercharge” operator,
$Q(n_{g})=\sqrt{E_{C}}(n-n_{g})$, we have the following algebra,

$\displaystyle P^{2}=1,\qquad\\{P,Q\\}=0,\qquad Q^{2}=H_{C},$ (1)

which is an equivalent definition of SUSY QM [47]. These relations imply that
all non-vanishing eigenvalues of $H_{C}$ are associated to degenerate
eigenvectors [Appendix A] and give rise to the energy level structure of Fig.
1. It can also be shown that the degenerate doublets are Kramers pairs
Appendix A. The singlet ground state at $n_{g}=0$ lacks a “fermionic” partner
because there is no odd parity eigenstate of zero momentum.

The Witten index, $\operatorname{Tr}P$, counts the difference between the
number of bosonic and fermionic states: $1$ for unbroken SUSY ($n_{g}=0$) and
$0$ for broken SUSY ($n_{g}=\pm 1/2$). $\operatorname{Tr}P$ is ill-defined for
non-integer $2n_{g}$ as the $2\pi$ periodicity of the wavefunctions restricts
$n$ to integer values. The curvature invariant, which dictates whether
supersymmetry occurs and the Witten index can be defined, as well as the
winding invariant will be defined below in the context of the design and
measurement of the BiSQUID.

##  III Superconducting topological circuit

In order to detect signatures of supersymmetry, we seek a quantum electrical
circuit described by the charging Hamiltonian. A small Josephson tunnel
junction, known as a Cooper pair box or charge qubit [27, 28], is quantum
coherent and has a capacitance with large $E_{C}\gg kT$. However the Josephson
coupling hybridizes the charge states, lifting the degeneracies at
$n_{g}=0,\pm 1/2$. By the Von Neumann-Wigner theorem [48], barring accidental
degeneracies, at least three parameters are necessary to obtain band crossings
in a generic Hamiltonian, whereas the Cooper pair box only has one, the gate
voltage.

Adding a junction in parallel yields a superconducting quantum interference
device (SQUID) or Cooper pair transistor, which has an additional variable,
the magnetic flux threading the SQUID loop. A SQUID can be considered a single
Josephson junction with a flux-tunable Josephson energy $E_{J}^{*}(\Phi)$.
Although a completely symmetric SQUID at half a quantum flux bias
($\Phi=\Phi_{0}/2=h/4e$) will have zero $E_{J}^{*}$, leaving only the charging
Hamiltonian $H_{C}$, this requires two junctions with exactly identical
critical currents which is impossible to fabricate.

Putting a third junction in parallel will result in a circuit with
topologically protected degeneracies: the BiSQUID [Fig. 2] [49, 25]. Such a
circuit has a flux parameter for each loop, $\Phi_{L}$ and $\Phi_{R}$, in
addition to the gate voltage $n_{g}$. For a large range of junction
parameters, by tuning both magnetic fluxes through the loops one can
effectively cancel out the overall Josephson coupling, leaving only the
capacitive term in the Hamiltonian.

Figure 2:  (a) The BiSQUID, comprised of three Josephson tunnel junctions in
parallel, acts as a single junction with an effective Josephson energy
$E_{J}^{*}$ which can be represented geometrically as the sum of three vectors
of lengths $1,\alpha,\beta$ and angles $\varphi_{L},\varphi_{R}$. The junction
asymmetries $\alpha,\beta$ are the relative Josephson energies of the left and
right junctions and $\varphi_{L},\varphi_{R}$ are the reduced magnetic fluxes
threading each loop. The Hamiltonian reduces to the charging term $H_{C}$ when
$E_{J}^{*}=0$, or equivalently the vectors form a triangle, resulting in a
spectrum with a supersymmetric degeneracy structure (curvature invariant
$B=-1$). (b) The two lowest energy bands for $n_{g}=1/2$ (broken SUSY) are
plotted as a function of $\varphi_{L},\varphi_{R}$, showing ground state
degeneracy at two inequivalent points identified by their winding number
$N_{2}=\pm 1$ and corresponding to closing the triangle clockwise or counter-
clockwise. All excited states are also doubly degenerate at those points. If
the Josephson energies do not satisfy the triangle inequality (c,
$\alpha=\beta=0.4$) the system is always gapped (d) ($B=+1$). Neither the
ground nor excited states are degenerate. Energies are normalized by the
plasma energy $\hbar\omega_{p}=\sqrt{2E_{C}E_{J}(1+\alpha+\beta)}$. (e) Phase
diagram as a function of junction asymmetries. Critical line (dashed)
separates topologically trivial ($B=+1$, blue) and non-trivial ($B=-1$, red)
regimes. When $B=-1$, the Witten index $\operatorname{Tr}P=0$ if SUSY is
unbroken ($n_{g}=0$) and $\operatorname{Tr}P=1$ if it is broken ($n_{g}=1/2$).

The complex effective Josephson energy of the BiSQUID is

$E_{J}^{*}(\varphi_{L},\varphi_{R})=E_{J}\left(1+\alpha
e^{-i\varphi_{L}}+\beta e^{i\varphi_{R}}\right),$ (2)

where $E_{J}$ is the Josephson energy of the middle junction, $\alpha$ and
$\beta$ are the Josephson energies of the left and right junction scaled to
$E_{J}$, and $\varphi_{L,R}=2\pi\Phi_{L,R}/\Phi_{0}$ are the reduced magnetic
fluxes.

As expected for supersymmetry, the BiSQUID spectrum has a tower of
degeneracies when $E_{J}^{*}=0$ and $2n_{g}$ is integer. The condition
$E_{J}^{*}(\varphi_{L},\varphi_{R})=0$ is equivalent to the existence of a
closed triangle with relative angles $\varphi_{L},\varphi_{R}$ and side
lengths $1,\alpha$ and $\beta$. A symmetric BiSQUID ($\alpha=\beta=1$) has an
effective Josephson energy which is zero when $\varphi_{R}=-\varphi_{L}=\pm
2\pi/3$ [Fig. 2(a)], corresponding to an equilateral triangle oriented
clockwise ($\varphi_{R}>0$) or counterclockwise ($\varphi_{R}<0$). The
spectrum [Fig. 2(b)] for $n_{g}=1/2$ will then have two conical band crossings
where the ground state is degenerate (broken SUSY). All excited states will
also be doubly degenerate at the same diabolical points [3] in the
$(\varphi_{L},\varphi_{R})$ “Brillouin zone,” as shown in [Appendix Fig. 7].
When the junction asymmetry is too large, such as $\alpha+\beta<1$ [Fig.
2(c)], it is no longer possible to close the triangle. Then the ground state
is not degenerate and the spectrum has a gap [Fig. 2(d)]. For $n_{g}=0$
(unbroken SUSY), which is not shown in [Fig. 2], all excited states are
degenerate but the zero-energy ground state is not.

The phase diagram for the BiSQUID [Fig. 2(e)] is given by the triangle
inequality $|\alpha-\beta|<1<\alpha+\beta$, which determines whether the
triangle can be closed and there exists a solution to
$E_{J}^{*}(\varphi_{L},\varphi_{R})=0$ [50]. In the topologically non-trivial
phase, one can have either broken or unbroken SUSY depending on the value of
$n_{g}$. Unbroken SUSY ($n_{g}=0$) is a gapped topologically non-trivial state
whereas broken SUSY ($n_{g}=1/2$) is gapless. When the triangle condition
cannot be satisfied and therefore supersymmetry is not present, the system is
both gapped _and_ topologically trivial. The curvature invariant, described
below and measured for several devices, encodes the triangle inequality and
indicates whether a BiSQUID is topologically non-trivial.

Since the triangle condition can be satisfied for a range of $\alpha,\beta$,
the degeneracies at both $n_{g}=0$ and $n_{g}=1/2$ are robust to variations in
the charging energy $E_{C}$ and the three Josephson energies. The two
diabolical points will move in the Brillouin zone as the junction asymmetries
$\alpha$ and $\beta$ are varied, converging at high symmetry points where
$\varphi_{L}$ and $\varphi_{R}$ are either zero or $\pi$ as the parameters
approach the critical lines $|\alpha\pm\beta|=1$. On the critical lines the
greatest Josephson energy is equal to the sum of the two others and the
diabolical points merge to produce an energy dispersion described by a power
law.

In the analogy to a topological material, the reduced magnetic fluxes
$\varphi_{L,R}$ which are odd under time-reversal play the role of
$k_{x},k_{y}$ for a 2D crystal. The offset charge $n_{g}$ is related to an
electric field and therefore even under time-reversal and is considered a
control parameter [25]. In the condensed matter context, the BiSQUID at
$n_{g}=0$ or $1/2$ is a 2D topological semimetal with time-reversal and
inversion symmetry. The conical two-band crossings are referred to as either
2D Weyl points [51] or Dirac points of spinless graphene (class AI with
inversion) [52, 9].

##  IV Experimental Setup

Spectroscopy is a direct probe of energy transitions and is well suited to
detect the distinctive spectral signatures of supersymmetry. Circuit quantum
electro-dynamics [53] techniques are routinely used for superconducting qubit
characterization and manipulation. Spectroscopy has previously been used in an
ion trap quantum simulator to detect degeneracies due to supersymmetry [45].

We use a microwave cavity resonator to perform the spectroscopy of several
BiSQUIDs in both trivial and topological regimes (Fig. 3). A BiSQUID (purple)
is capacitively coupled to the voltage anti-node of a coplanar waveguide
lambda-mode resonator (blue) with $f_{r}$ approximately
$7\text{\,}\mathrm{GHz}9\text{\,}\mathrm{GHz}$ and loaded quality factor
$Q_{L}>10^{3}$. We first deposit $100\text{\,}\mathrm{nm}$ aluminum on a
silicon substrate then fabricate the resonator and control lines using photo-
lithography and wet etching. The Josephson junctions of the BiSQUID are made
of aluminum and are patterned using e-beam lithography and a double-angle
evaporation technique. The surface area of each JJ is approximately
$200\times$200\text{\,}{\mathrm{nm}}^{2}$$ such that $E_{J}\approx E_{C}$ (see
inset of Fig. 3(c)). All experiments are conducted in a dilution refrigerator
at $10\text{\,}\mathrm{mK}$. The transmission of the resonator at frequency
$f_{1}$ is probed using a vector network analyzer, and transitions between
energy levels of the BiSQUID are detected via changes in $f_{1}$ resulting
from a second tone of frequency $f_{2}$ exciting the BiSQUID via a local flux
line. The relevant transition matrix elements are $\braket{k}{\theta}{j}$
determining the type of spectral lines that can be resolved. Further details
of the setup are provided in [Appendix E].

Figure 3: Spectroscopy setup. (a) Electrical diagram of a BiSQUID (purple)
capacitively coupled to the voltage anti-node of a lambda-mode coplanar
waveguide resonator (blue). Gate voltage $V_{g}$ and microwaves at frequency
$f_{1}$ are applied to the resonator using a bias tee (left). Reduced loop
fluxes $\varphi_{L}$ and $\varphi_{R}$ are set by direct currents $I_{L}$ and
$I_{R}$ applied to local flux lines (red, green). A microwave drive at
frequency $f_{2}$ (left flux line, red) excites transitions of the BiSQUID.
(b) False-colored optical microscope image of device R18C. The resonator is
made of aluminum (gray) on a silicon substrate (black). (c) Optical image of
BiSQUID (yellow inset in (b)). Inset: SEM image of rightmost Josephson
junction (scale bar $100\text{\,}\mathrm{nm}$, junction surface area
$0.034\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$).

The charge offset $n_{g}$ is controlled by applying a gate voltage
$V_{g}=2en_{g}/C_{g}$ to the central conductor of the resonator. Even with
careful filtering of the DC gate voltage line, the value of $n_{g}$ slowly
drifts a few percent per hour and randomly jumps on a timescale of minutes. To
compensate we repeatedly perform quick calibrations approximately every
minute. We sweep $V_{g}$ over a few periods while probing the resonator to
find the value of $V_{g}$ corresponding to the desired $n_{g}$. This allows us
to correct for slow drift and detect charge jumps, in which case data is
discarded and remeasured.

Quasiparticle (QP) poisoning is a large source of error for many
superconducting quantum devices [54]. High-energy photons or ionizing
radiation [55, 56, 57, 58, 59] may create non-equilibrium quasiparticles which
can then tunnel to the superconducting island of the BiSQUID, shifting $n_{g}$
by 0.5 (1e). Despite broadband noise filtering and superconducting gap
engineering, we measured constant jumps between quasiparticle parities with a
typical switching time on the order of $300\text{\,}\mathrm{ms}$ and average
populations of $50\%$ for each parity [Appendix F]. Post-measurement
processing allows sorting the data for each QP parity.

Direct currents $I_{L}$ and $I_{R}$ are applied to two local flux lines (see
red and green in Fig. 3) to control loop fluxes $\varphi_{L}$ and
$\varphi_{R}$. The second microwave tone used to excite transitions of the
BiSQUID is applied via one of these flux lines. To find the conversion between
applied currents $\left(I_{L},I_{R}\right)$ and reduced fluxes
$\left(\varphi_{L},\varphi_{R}\right)$, we determine the unit cell for the
periodic resonator shift [Fig. 4(a,b)] and compensate accordingly.

##  V Measured Spectra

Figure 4: Spectroscopy of topologically trivial (top row, R17A) and non-
trivial (bottom row, R17C) BiSQUIDs at $n_{g}=1/2$. (a,b) One-tone signal
shift of resonator as a function of currents $I_{L},I_{R}$ in the two flux
lines. Superposed are reduced flux axes $\varphi_{L}$ and $\varphi_{R}$ (solid
white), diagonal $\varphi_{\mathrm{diag}}=\varphi_{L}=\varphi_{R}$ (red), and
unit cell (dashed). (c,d) Two-tone phase shift of resonator along
$\varphi_{\mathrm{diag}}$. Overlaid on the data are the bare frequency of each
resonator (dash-dotted line) and the transition frequency $f_{01}$ obtained
from fitting to the BiSQUID Hamiltonian [Eq. 3] (dashed lines). Insets plot
first two energy levels of the Hamiltonian using fit parameters over the unit
cell. (e,f) Histograms of $\textrm{arg}\,S_{21}(f_{2})$ at
$\varphi_{\textrm{diag}}=\pi$ representative of data used to obtain spectra in
(c,d). The distribution is bimodal due to quasiparticle poisoning.

We fabricated several BiSQUIDs with different Josephson energy asymmetries
$\alpha$ and $\beta$ to explore the phase diagram [Fig. 2(e)]. BiSQUID R17A
was fabricated in the topologically trivial regime, with a middle junction
about three times larger than the outer ones ($\alpha\approx\beta\approx
1/3$). Two other BiSQUIDs, R17C and R18C, were fabricated with junctions of
similar area in order to be in the topologically non-trivial regime. For
simplicity only the middle junction area was varied, keeping
$\alpha\approx\beta$ such that degeneracies remain on the
$\varphi_{L}=\varphi_{R}\equiv\varphi_{\mathrm{diag}}$ axis. As $\alpha,\beta$
decreases from the equilateral configuration $\alpha=\beta=1$, degeneracies
should move toward each other along the $\varphi_{\mathrm{diag}}$ axis until
they merge at $\varphi_{\mathrm{diag}}=\pi$ for $\alpha=\beta=1/2$.

A first signature of the topological phase of the BiSQUID is found in the flux
maps [Fig. 4(a,b)]. The asymmetric BiSQUID R17A ($\alpha=\beta=0.34$) shows
only a single minimum (dark blue) in one unit cell [Fig. 4(a)]. On the
contrary for the more symmetric device R17C ($\alpha=\beta=0.67$) we observe
two maxima (yellow). The extrema in both maps correspond to local minima of
the BiSQUID transition energy. For a topologically trivial BiSQUID, such as
R17A (a), the gap is smallest at $\varphi_{\mathrm{diag}}=\pi$. For a BiSQUID
in the topologically non-trivial regime, there are always two degeneracies,
and thus two extrema.

We then performed two-tone spectroscopy by measuring the transmission $S_{21}$
of the resonator in the presence of a second tone of frequency $f_{2}$ at
different values of $\varphi_{\mathrm{diag}}$. The charge offset is set to
$n_{g}=0.5$, where degeneracies are expected, and post-processed to account
for quasiparticle poisoning. The two-tone phase shift is shown in [Fig.
4(c,d)], where the flux-dependent single-tone phase offsets have been removed.
We observe one main spectroscopy line corresponding to the transition $f_{01}$
from ground to first excited state. At $\varphi_{\mathrm{diag}}=2\pi$ the
transition has a maximum of $12-$13\text{\,}\mathrm{GHz}$$ for both BiSQUIDs.
However near $\varphi_{\mathrm{diag}}=\pi$ the transition strongly differs for
the two circuits. For the BiSQUID in the trivial regime, $f_{01}$ has a single
minimum of $3\text{\,}\mathrm{GHz}$ near $\varphi_{\mathrm{diag}}=\pi$ [Fig.
4(c)]. The BiSQUID in the topologically non-trivial regime shows two minima,
at $\varphi_{\mathrm{diag}}\approx 0.75\pi$ and
$\varphi_{\mathrm{diag}}\approx 1.25\pi$, and the transition has negative
curvature and a local maximum at $\varphi_{\mathrm{diag}}=1\pi$. The $f_{01}$
spectral line disappears below $1\text{\,}\mathrm{GHz}$ due to both thermal
effects and because the charge-coupled resonator shift goes to zero at the
degeneracy point [60].

The raw data leading to the maps [Fig. 4(c,d)] are shown for the respective
devices at a flux bias $\varphi_{\textrm{diag}}=\pi$ in Fig. 4(e,f). Each
horizontal cut is a histogram of the phase of the transmitted signal
$\textrm{arg}\,S_{21}$ at the resonator frequency obtained from multiple
measurements while driving at the second tone frequency $f_{2}$ (y-axis). The
color scale corresponds to the number of counts. Due to quasiparticle
poisoning the distribution is bimodal, with the second, spurious, mode
corresponding to a charge offset $n_{g}\pm 0.5=0$. A threshold is applied to
distinguish between the two QP states, and only data corresponding to
$n_{g}=0.5$ is shown in Fig. 4(c,d). The desired mode, $n_{g}=0.5$, shows
clear shifts when $f_{2}$ is resonant with a BiSQUID transition, whereas the
spurious mode is relatively constant. This is because the excitation gap near
$\varphi_{\textrm{diag}}=\pi$ for $n_{g}\equiv 0$ is above approximately
$8\text{\,}\mathrm{GHz}$.

The transition frequencies extracted from the raw data are plotted as a
function of $\varphi_{\mathrm{diag}}$ for four BiSQUID devices, R17A, R17B,
R17C, and R18C in Fig. 5 (circle markers). The error bars correspond to the
full-width half-max (FWHM) of the Gaussian fits.

Figure 5:  BiSQUID transitions $f_{01}$ (circles along solid lines) and
$f_{02}$ (circles along dashed lines) as a function of
$\varphi_{\textrm{diag}}$ extracted from two-tone spectroscopy measurements at
$n_{g}=1/2$ (a) and $n_{g}=0$ (b) for different devices. Error bars indicate
measured full-width half maximum of transitions, and lines are best fits to
the data using the BiSQUID Hamiltonian. Inset shows phase diagram with
placement of devices in topologically non-trivial (red) and trivial (blue)
regions, separated by the critical line (solid), which is shifted (dashed)
when taking into account geometric inductance, $\beta_{L}=0.1$. Labels at
degeneracies indicate the winding number $N_{2}$.

##  VI Discussion

The Hamiltonian is obtained via standard circuit quantization [12] in a form
incorporating the effective Josephson energy
$E_{J}^{*}(\varphi_{L},\varphi_{R})$ [Eq. 2],

$H=E_{C}(n-n_{g})^{2}-(E_{J}^{*}e^{i\theta}+\bar{E}_{J}^{*}e^{-i\theta})/2,$
(3)

and reduces to the charging Hamiltonian $H_{C}$ when $E_{J}^{*}=0$
($\bar{E}_{J}^{*}$ is the complex conjugate of $E_{J}^{*}$). Expanding out the
flux dependence in $E_{J}^{*}$ we have the more familiar form,

$\displaystyle H=$ $\displaystyle E_{C}(n-n_{g})^{2}-E_{J}\cos\theta$ (4)
$\displaystyle-
E_{J}\alpha\cos(\theta-\varphi_{L})-E_{J}\beta\cos(\theta+\varphi_{R}).$

We fit the transitions measured as a function of $\varphi_{\textrm{diag}}$ for
$n_{g}=1/2$ and $n_{g}=0$ to the eigenenergies of the BiSQUID Hamiltonian Eq.
4 using three free parameters: $E_{C}$, $E_{J}$, and outer junction symmetry
parameter $\alpha=\beta$. These fits are plotted as solid lines ($f_{01}$) and
dashed lines ($f_{02}$) in Fig. 5, and show good agreement with the data
(circles). Plots of the fit for $f_{01}$ are also overlaid on the spectra Fig.
4(c,d) as dashed lines.

The circuit parameters obtained from fits, along with properties of the
resonator, are given in Table 1 and are consistent with geometric estimates
based on scanning electron microscopy. As a control, we also performed
spectroscopy of a SQUID (device _R16_), which was designed to be symmetric,
and fit to a model in which $\beta=0$ [Appendix G]. Fitting yields a gap for
the BiSQUID in the topologically trivial regime and the control SQUID. In the
case of the SQUID the gap is due to a residual junction asymmetry of $2\%$
resulting from variability in fabrication.

| R17A | R17B | R17C | R18C | _R16_
---|---|---|---|---|---
$f_{r}$ (GHz) | 8.096 | $9.001$ | $8.171$ | $7.245$ | $9.178$
$Q_{L}$ | 37 520 | 4 200 | 65 460 | 17 520 | 5020
$Q_{\mathrm{ext}}$ | 185 000 | 75 000 | 185 000 | 50 000 | 7400
$E_{J}/h$ (GHz) | 9.37 | $6.24$ | $5.53$ | 2.15 | 7.56
$\alpha$ | 0.34 | 0.71 | 0.67 | 1.16 | 1.02
$\beta$ | 0.34 | 0.71 | 0.67 | 1.16 | 0
$E_{C}/h$ (GHz) | 7.93 | $7.61$ | $8.34$ | 16 | $9.05$
$\omega_{p}/2\pi$ (GHz) | 15.78 | $15.17$ | $14.72$ | 15 | $16.61$
$E_{01,\mathrm{min}}/h$ (GHz) | 3.0 | 0 | 0 | 0 | 0.2
Table 1:  Summary of parameters of BiSQUID devices and control device SQUID
_R16_ [Appendix G].

As in Fig. 4, the most indicative qualitative feature of a topologically non-
trivial regime at $n_{g}=1/2$ is the existence of negative curvature in
$f_{01}$ near $\varphi_{\mathrm{diag}}=\pi$, as observed in samples R17B,
R17C, and R18C [Fig. 5(a)]. This is in contrast to sample R17A (blue), which
has a spectrum with positive curvature at $\varphi_{\mathrm{diag}}=\pi$
reminiscent of an asymmetric SQUID. As shown in the phase diagram (inset), of
the topologically non-trivial devices, sample R17C with $\alpha=\beta\approx
0.67$ is closest to the critical line $\beta=1-\alpha$. In this case the
degeneracy triangle is more oblique, with angles $\varphi_{L,R}$ [Fig. 2]
closer to $\pi$. Hence the region of negative curvature is smaller, more
closely confined to a region around $\varphi_{\textrm{diag}}=\pi$.

The unbroken SUSY regime, $n_{g}=0$, is shown in Fig. 5(b). Data of the
topologically non-trivial device R17B (red) shows negative curvature at
$\varphi_{\textrm{diag}}\approx\pi$ for both transitions $f_{01}$ and
$f_{02}$. This is in agreement with the model, plotted as a red solid line
($f_{01}$) and dashed line ($f_{02}$). The negative curvature in the
topologically non-trivial state is in contrast to the expectation for device
R17A, where the fit lines for $f_{01}$ and $f_{02}$ do not intersect and both
transitions would have positive curvature near $\pi$. For this sample there is
no data at $\varphi_{\textrm{diag}}\approx\pi$ because the resonator frequency
is coincident with the transition frequency. In the unbroken SUSY state
($n_{g}=0$), the lack of data where $f_{01}$ is degenerate with $f_{02}$ at
$\varphi_{\textrm{diag}}\approx 0.75\pi$ (indicated by $\pm 2$ winding
number), is due to the vanishing of the resonator frequency shift, similarly
to what occurs for the ground state degeneracy in broken SUSY ($n_{g}=1/2$)
[60].

As with topological insulators, negative band curvature at a high-symmetry
point can be indicative of an inverted gap. Indeed the phase diagram is
described by a topological invariant which relates the curvature of the ground
state band at high-symmetry points to the validity of the triangle inequality.
At the four inequivalent high symmetry points
$(\varphi_{L},\varphi_{R})=(0,0),(0,\pi),(\pi,0),(\pi,\pi)$ of the 2D
Brillouin zone, there is time-reversal symmetry and
$E_{J}^{*}(\varphi_{L},\varphi_{R})$ is real. As shown in Appendix B, the
triangle inequality can be fulfilled only if one of the three points
$[(0,\pi),(\pi,0),(\pi,\pi)]$ has negative $E_{J}^{*}$.The $\mathbb{Z}_{2}$
curvature invariant $B$ is defined as the product of the signs of $E_{J}^{*}$
evaluated at these points. From the Hamiltonian Eq. 3, the sign of the
expectation value of $\cos\theta$ in the ground state,
$\operatorname{sgn}\braket{\cos\theta}_{0}$, must be equal to the sign of
$E_{J}^{*}$ at each of the symmetry points in order to minimize the energy.
Therefore $B$ can also be calculated from the product of the signs of the
ground state expectation values $\braket{\cos\theta}_{0}$ over the symmetry
points.

Using the Hellmann-Feynman theorem, it can be shown that for $\alpha=\beta$,
the curvature of an energy band in the $\varphi_{\textrm{diag}}$ direction at
$\varphi_{\textrm{diag}}=\varphi_{L}=\varphi_{R}=\pi$ is proportional to the
expectation value of $\braket{\cos\theta}$ [Appendix B]. The sign of the
curvature measured for the transition energy $f_{01}$ at
$\varphi_{\textrm{diag}}=\pi,n_{g}=1/2$ is then related to
$\operatorname{sgn}\braket{\cos\theta}_{0}$. Furthermore, for the
experimentally realized devices where $\alpha,\beta<1$ (R17A, R17B, R17C), it
suffices to determine the sign of $\braket{\cos\theta}_{0}$ at the unique
point $\varphi_{\textrm{diag}}=\pi$ to determine $B$. Hence the sign of the
curvature at $\varphi_{\textrm{diag}}=\pi$ of the measured spectra Fig. 5(a)
is a direct indication of whether the device is topologically non-trivial.

In topological insulators with inversion and time-reversal symmetry, there is
a similar $\mathbb{Z}_{2}$ invariant calculated from the product of parity
eigenvalues over the high symmetry points [61]. For the BiSQUID, $B$ is not a
band invariant since it is only defined for the ground state, but it encodes
information about the junction asymmetries and hence the triangle inequality.

In the topologically non-trivial regime one can also assign a topological
charge to each degeneracy, or diabolical point, calculated from integrating
the Berry curvature on a surface enclosing the degeneracy [3, 10, 49]. This
local topological charge, or winding number $N_{2}$ [62], is distinct from the
Witten index, which characterizes the global structure of the degeneracy
“ladder” [Fig. 1], and the band inversion invariant which determines the
global phase diagram [Fig. 2(e)]. In the broken SUSY configuration, Fig. 5(a),
the topological charge of the ground state degeneracy is either $+1$ or $-1$,
as indicated by black circles. In unbroken SUSY (b), the topological charges
of the degeneracies between states $\ket{1}$ and $\ket{2}$ are $\pm 2$, as
indicated by black circles. Higher order degeneracies have correspondingly
higher charges, but are odd for $n_{g}=1/2$ and even for $n_{g}=0$ [23]. Along
the critical line of the phase diagram separating trivial and non-trivial
states, the degeneracies merge and the integral of the Berry curvature around
the band intersection is zero, equal to the sum of the topological charges.

Higher experimental sensitivity would allow the measurement of transitions
closer to the degeneracy points at $n_{g}=0$ and $n_{g}=1/2$. However
spectroscopic techniques can only give an upper bound to a possible energy
gap. Thermal effects, sample disorder, and instrument limitations set bounds
on the emission linewidth and detector resolution, ultimately determining the
smallest transition that can be measured. Controversy over the existence of a
gap occurred previously in angular resolved photoemission studies of epitaxial
graphene [63, 64, 65].

Despite the fact that the data is consistent with a Hamiltonian [Eq. 3] which
has a non-trivial phase, one cannot exclude that unknown, perturbative, terms
open a gap in the region of the spectrum that we cannot resolve
experimentally. Although many possible mechanisms could introduce such terms,
including capacitive non-linearities [66], we only discuss the role of
Josephson harmonics and the loop inductance.

Higher order Josephson effects are important in short, transparent
superconducting weak links and add harmonic terms of the type $\cos k\theta$
for $k>1$, corresponding to the simultaneous tunneling of multiple Cooper
pairs. These processes, which are related to multiple Andreev reflections, are
always weaker than the single Cooper pair tunneling process. Although they are
strongly suppressed in typical tunnel junctions, they can occur when there are
defects [67]. Increased junction transparency can be incorporated in the
effective Josephson energy [Eq. 2] by adding exponential terms with arguments
$ik\varphi_{L,R}$ with $k>1$. The triangle condition for degeneracy will be
modified but can still be satisfied. Considering the symmetry point
$\varphi_{L}=\varphi_{R}=\pi$, the even and odd harmonic terms in the
effective Josephson energy will sum with opposite signs, in effect making it
harder to “close” the triangle when even harmonics are present. However one
can compensate by increasing the total Josephson energy of the outer
junctions. The size of the non-trivial region of the phase diagram will
therefore shrink as the junction transparency increases and higher order
Josephson terms become more important, but will not disappear completely.

The superconducting loops of the BiSQUID will have a small inductance, both
geometric and kinetic, which can be accounted for by adding terms of the form
$E_{L}\gamma^{2}=E_{J}/\beta_{L}\gamma^{2}$, where $\beta_{L}$ is the ratio of
the loop inductance to the Josephson inductance and $\gamma$ the phase drop
across the inductance. In the perturbative limit $\beta_{L}\ll 1$, inductances
effectively reduce the Josephson energy of the junctions by a factor
$\cos\rho\gamma\approx 1-(\rho\gamma)^{2}/2$ where $\rho$ is a coefficient of
order unity depending on the geometry [Appendix C]. Replacing $\gamma^{2}$ by
its expectation value for an $l$-photon oscillator state, the renormalization
factor is approximately $1-\rho^{2}\frac{2l+1}{4}\sqrt{E_{C}/E_{L}}$.
Therefore when accounting for inductance, the low-lying degeneracies of a
BiSQUID will not be lifted if the BiSQUID is deep in the topologically non-
trivial regime, far from the critical lines. However for highly excited
BiSQUID states, the oscillator degrees of freedom may be in a large number
state $m\gg 1$ and it will be harder to satisfy the triangle inequality.

Solving numerically for the energy levels in the presence of arbitrary loop
inductance [Appendix C], we find that although the lowest lying degeneracies
persist, both for integer and half-integer $n_{g}$, higher order degeneracies
may split according to their topological charge or even merge and open a gap
[68]. The deformation of the phase diagram for the ground state degeneracy at
$n_{g}=1/2$ due to an artificially large series inductances $\beta_{L}=0.1$ on
each branch of a BiSQUID with
$E_{J}=E_{C}=\sqrt{E_{JL}E_{CL}}=\sqrt{E_{JR}E_{CR}}$ is shown in the inset of
Fig. 5(a). For small $\beta_{L}$, the prediction of the perturbative model for
the shift in the phase diagram of the position of the critical line near the
point $\alpha=\beta=1/2$ agrees with numerical results [Appendix C]. Similarly
to the effect of higher order Josephson terms, the inductive perturbation
reduces the size of the topologically non-trivial region. However, whereas the
phase diagram accounting for junction transparency is identical for all
excited states, leading to a supersymmetric spectrum [Fig. 1] in the
topologically non-trivial state, the inductive terms will gap out states at
high enough energy. Accounting for inductance, supersymmetry is effectively
destroyed in the ultra-violet limit. The role of additional bosonic degrees of
freedom in lifting supersymmetry at the superconducting critical point of
fermionic Dirac/Weyl systems has been considered previously [44].

##  VII Conclusion

The BiSQUID is a model topological system which has at least three different
topological invariants: the curvature invariant $B$, the Witten index
$\operatorname{Tr}P$, and the winding number $N_{2}$. The curvature invariant,
which is a function of the relative Josephson junction energies, determines
whether a given BiSQUID sample can have degeneracies. For negative $B$, the
structure of degeneracies is characterized by the Witten index, and the
winding number classifies individual degeneracies. We have measured the
curvature invariant directly via microwave spectroscopy of the ground state
transition. The spectra are fit to a model with a supersymmetric degeneracy
structure from which we infer the winding numbers of the degeneracies.

In future experiments, both the resonator-BiSQUID coupling and BiSQUID
excitation should be modified to optimize the dispersive shifts near level
crossings. It is also desirable to excite high-order level crossings,
demonstrating that they occur at the same bias point, establishing the
defining characteristic of SUSY. This requires pumping transitions at higher
frequencies, pumping with multiple drive tones, using alternative techniques
such as amplitude spectroscopy [69], and optimizing the Josephson junction
plasma frequency and resonator frequency. Single tone measurements with a low-
frequency resonator should directly give the ground state band curvature and
could be a powerful probe of circuit topological properties. Increasing the
loop inductance $\beta_{L}$ with granular aluminum [70] or other high-kinetic
inductance materials may lead to the observation of gapping of high order
degeneracies and the lifting of supersymmetry. With three-tone spectroscopy
one may also be able to probe transitions from excited states, such as
$f_{12}$.

By measuring the transition energies along multiple flux axes and determining
the Gaussian curvature at $\varphi_{\textrm{diag}}=\pi$, it may be possible to
obtain the topological invariant $B$ for the general case $\alpha\neq\beta$.
The numerically calculated energy bands [Fig. 4(c,d)] indicate a saddle point
or negative Gaussian curvature for topologically non-trivial devices with
$\alpha=\beta$, but this may be true in general. Other pump-probe
spectroscopic techniques [71, 72, 73] should be able to directly determine the
winding number of degeneracies.

We have shown how the topological properties are sensitive to realistic
perturbations. Whereas the higher order Josephson effect does not affect
supersymmetry, it is lifted at high energies by inductances. Neglect of the
role of perturbations can lead to the misinterpretation of experimental
results and tenuous claims of topological non-triviality [74, 75]. Further
theoretical work should elucidate how the supersymmetric spectrum at $n_{g}=0$
and $n_{g}=1/2$ is modified as $\beta_{L}$ is increased and predict the
splitting and merging of high order topological charges.

Experiments on devices with an additional junction and one more loop, the
TriSQUID, would effectively allow in-situ tuning of the junction asymmetry
$\alpha$ [76, 25]. One could then traverse the phase diagram, switching
between topologically trivial and non-trivial regimes, with an additional
local flux line. Crossing the critical line would allow observing the merging
of Weyl points and the opening of a gap [77]. With more sophisticated
circuits, one could perform Weyl point teleportation [78] or probe higher
order topological invariants [79].

It would also be interesting to investigate, both theoretically and
experimentally, the existence of supersymmetry in circuits with many junctions
[80] or circuits with non-tunnel junctions, which harbor Andreev states [81].
Single [82, 83, 84, 85] or multi-terminal [86, 87, 26] circuits with Andreev
states may have degeneracies described by supersymmetry [88].

With respect to topological order in condensed matter systems, the material
analog of a BiSQUID is 2D spinless graphene. It remains to establish a general
mapping between the topological classes of materials [89, 9, 90] and Josephson
junction based circuits. Inversion symmetry plays an important role in the
BiSQUID and it may be possible to establish the link between the “weak”
BiSQUID invariants, $\operatorname{sgn}\braket{\cos\theta}$ at high symmetry
points, the “strong” invariant $B$, and the corresponding topological
invariants of inversion symmetric fermionic topological insulators [61]. Sub-
lattice symmetries [Appendix A] can also lead to degeneracies, as in Weyl
semimetals and Josephson junction circuits with higher order terms [91].
Although one can define a Brillouin zone for the BiSQUID, it does not have a
boundary in real space and therefore no edge states. On the other hand, it is
possible to use Josephson junctions to produce high-dimensional topological
systems without any real-world analog [25]. A full classification of the
topological properties of arbitrary Josephson junction circuits is lacking
[14, 76].

The dual circuit of the BiSQUID, the Cooper pair pump, and its variants have
been in development for current metrology [92]. In addition several proposals
exist for transconductance quantization exploiting topological properties of
Josephson tunnel junction circuits [24, 93]. Similar spectroscopic techniques
as employed here for BiSQUIDs could help evaluate the suitability of the
proposed circuits for metrology applications. Circuits similar to the BiSQUID
can also serve for quantum information, where its topological properties can
help mitigate decoherence or decay [94, 95, 96, 97, 8, 98], or offer platforms
for geometric and holonomic computation [99, 100, 101].

##  Appendix A Supersymmetry of Charging Hamiltonian

Figure 6:  Quantum mechanical supersymmetry of charging Hamiltonian.
Supercharge operator $Q$ exchanges eigenstates of even parity $\ket{m_{+}}$
(bosonic) and odd parity $\ket{m_{-}}$ (fermionic) for $m_{\pm}>0$. The
fermion number operator $c^{{\dagger}}c$ annihilates bosonic states while
leaving fermionic states invariant. Eigenstates are written in charge basis
$\ket{m}$ satisfying $Q\ket{m}=\sqrt{E_{C}}(m-n_{g})\ket{m}$.

Fig. 6 is a diagram of the quantum mechanical supersymmetry properties of the
charging Hamiltonian $H_{C}$ at $n_{g}=0$ and $n_{g}=+1/2$. The simultaneous
parity and energy eigenstates are given by
$\ket{m_{\pm}}=\frac{1}{\sqrt{2}}(\ket{m}\pm\ket{-m+2n_{g}})$ where $m>0$,
except for the unique ground state at $n_{g}=0$ which is given by $\ket{0}$.
Note that charge eigenstates $\ket{\pm m}$ are distinct from energy
eigenstates $\ket{m_{\pm}}$. For unbroken SUSY, $n_{g}=0$, the real
wavefunctions are sines and cosines with wavevector $m$, whereas for broken
SUSY, $n_{g}=1/2$, the complex wavefunctions have a magnitude given by sines
and cosines with wavevector $m/2$.

For $n_{g}=0$ and $n_{g}=+1/2$ we therefore have $P\ket{m\pm}=\pm\ket{m\pm}$
and $H_{C}\ket{m\pm}=E_{m}\ket{m\pm}$ with $E_{m}=E_{C}(m-n_{g})^{2}$. The
positive sign corresponds to even parity, or “bosonic” states, and the
negative sign to odd parity, or “fermionic” states.

The supersymmetry operator $Q=\sqrt{E_{C}}(n-n_{g})$ swaps parity, or
exchanges the bosonic and fermionic states, up to a multiplicative factor,
$Q\ket{m_{\pm}}=\sqrt{E_{C}}(m-n_{g})\ket{m_{\mp}}$. By construction we obtain
the fundamental spectral property of supersymmetry: for an eigenstate
$\ket{m_{\pm}}$ with non-zero energy, $Q\ket{m_{\pm}}$ is a _distinct_
degenerate eigenstate. The uniqueness of the two states ultimately arises from
the anti-commutator $\\{P,Q\\}=0$ (Eq. 1). When a symmetry operator $S$
commutes with the Hamiltonian, unless the operator is anti-unitary, the
eigenstates $\ket{\psi}$ and $S\ket{\psi}$ may be equivalent up to a phase
factor. But since both $P$ and $Q$ commute with the Hamiltonian _and_ the
anti-commutator $\\{P,Q\\}$ is zero, one can show that for any non-zero energy
eigenstate $\ket{\psi}$, either $P\ket{\psi}$ or $Q\ket{\psi}$ is also an
eigenstate and differs from $\ket{\psi}$.

For the bosonic charging Hamiltonian, the supersymmetry is hidden, in the
sense that there is no explicit fermionic degree of freedom. However we can
identify a “fermionic” operator $c=\frac{1+P}{2}Q$, which swaps parity and
then projects states to the even, or bosonic sector. Since $\\{P,Q\\}=0$, the
adjoint $c^{\dagger}=\frac{1-P}{2}Q$ also swaps parity but projects to the odd
sector. The operators $c,c^{{\dagger}}$ are “fermionic” in the sense that they
satisfy $c^{2}=c^{{\dagger}2}=0$ and $\\{c,c^{{\dagger}}\\}/E_{m}=1$ for all
eigenstates with non-zero energy. When applied to a “fermionic” state
$\ket{m_{-}}$, the fermionic number operator $c^{\dagger}c$ is the identity
(up to a constant), but $c^{\dagger}c$ annihilates a “bosonic” state
$\ket{m_{+}}$, hence the particle naming convention for the different
parities.

Using the fermion number operator or parity operator, one can partition the
Hilbert space in two, and rewrite the Hamiltonian in terms of Pauli matrices
[47]. This process yields a supersymmetric Hamiltonian in the same way that in
conventional SUSY QM, a Hamiltonian with a generic potential but no explicit
fermionic component is supplemented by its superpartner [37, 38].

A normalized analogue to the superexchange operator can be defined as
$q=Q/\sqrt{H_{C}}=(n-n_{g})/|n-n_{g}|=\operatorname{sgn}(n-n_{g})$ which is
unitary if we take the convention $\operatorname{sgn}(0)=+1$ and not $0$. Like
$P$, $q$ is a grading operator which squares to the identity and partitions
states by the sign of the charge relative to $n_{g}$. Note that with unitary
$q$, since $q\ket{0}=\ket{0}$, the fundamental SUSY anti-commutator
$\\{q,P\\}$ is zero for all eigenstates except the ground state at $n_{g}=0$.
For $n_{g}=1/2$, since the eigenstates $\ket{m_{\pm}}$ mix even and odd charge
states, the superexchange operator can also be written $q=e^{i\pi n}$, which
in the position basis is a half-lattice translation.

Degeneracies in the spectrum can also be understood as arising from Kramers
theorem. The anti-unitary symmetry $PqK$, where $K$ is time reversal (complex
conjugation in the charge basis), squares to negative one for all eigenstates
except the ground state at $n_{g}=0$. Therefore all non-zero energy
eigenstates are Kramers pairs. This statement is equivalent to the SUSY
degeneracy theorem. At $n_{g}=1/2$, $Pq$ can be considered a glide symmetry,
since $P$ is a reflection and $q=e^{i\pi n}$ is a half-lattice translation.
Such sub-lattice symmetry also arises in Weyl/Dirac systems [102].

For the five-junction unbroken supersymmetric circuit proposed by Ulrich et
al. [46], the grading operator is the same as for $n_{g}=0$, but the
supercharge has an additional “non-trivial” term proportional to $\sin\theta$.
At a special value of the proportionality constant, all states except the
ground state are doubly degenerate. Unfortunately the supersymmetry is not
robust to imperfections in the fabrication process and the circuit is more
difficult to realize experimentally.

Figure 7:  Plot of the first few energy levels of a fully symmetric
topologically non-trivial BiSQUID with $E_{C}=E_{J}/3$ and $\alpha=\beta=1$.
When $\varphi_{\textrm{diag}}=\pm 2\pi/3\approx\pm 0.67\pi$, the triangle
condition is satisfied and the effective Josephson energy $E_{J}^{*}$ is zero.
In the unbroken SUSY case ($n_{g}=0$, top) all states except the ground state
at zero energy are doubly degenerate. For broken SUSY ($n_{g}=1/2$, bottom),
all states are doubly degenerate. Line colors determined by wavefunction
continuity [103].

##  Appendix B Curvature Invariant

We show that the curvature invariant $B$, the product of
$\operatorname{sgn}E_{J}*$ over the high symmetry points, encodes the triangle
condition in Fig. 2 by enumerating its value for all possible configurations
of $\alpha,\beta$ [Table 2]. At the four high symmetry points
$(\varphi_{L},\varphi_{R})=(p\pi,q\pi)$ where $p,q=0$ or $1$ (first column),
the effective normalized Josephson energy $E_{J}^{*}(\varphi_{L},\varphi_{R})$
Eq. 2 can be written $1+(-1)^{p}\alpha+(-1)^{q}\beta$ (second column).
Assuming that $\alpha\geq\beta$ without loss of generality, there are three
possible orderings of the Josephson energies relative to $E_{J}$:
$1\leq\beta\leq\alpha$, $\beta<1\leq\alpha$, and $\beta\leq\alpha<1$ (top
row). Except for the last row, the cells with $+,-,\pm,\mp$ gives the sign of
$E_{J}^{*}$ for the corresponding high symmetry point and Josephson energy
ordering. For the cells with two possibilities, the sign of $E_{J}^{*}$ needs
to be resolved by using the triangle inequality: whether the sum of the
smallest of $1,\alpha,\beta$ is greater than the largest one. For the
orderings in columns three and four, the triangle inequality is
$1+\beta\geq\alpha$ and for the last column, it is $\alpha+\beta\geq 1$. For
the double entries indicated by $\pm$ or $\mp$, the top sign corresponds to
the Josephson energy configuration that satisfies the triangle inequality.

The last row gives the product of the signs along a column, which is
equivalent to the topological curvature invariant $B$. In all columns, the
product is negative if and only if the upper sign in $\pm$ or $\mp$ is taken,
or equivalently when the triangle inequality is satisfied. Therefore,
independent of the Josephson energy ordering, $B=-1$ implies that the triangle
inequality can be satisfied.

|  | $\operatorname{sgn}E_{J}^{*}$
---|---|---
$\varphi_{L},\varphi_{R}$ | $E_{J}^{*}$ | $1\leq\beta\leq\alpha$ | $\beta<1\leq\alpha$ | $\beta\leq\alpha<1$
$(0,0)$ | $1+\alpha+\beta$ | $+$ | $+$ | $+$
$(\pi,0)$ | $1-\alpha+\beta$ | $\pm$ | $\pm$ | $+$
$(0,\pi)$ | $1+\alpha-\beta$ | $+$ | $+$ | $+$
$(\pi,\pi)$ | $1-\alpha-\beta$ | $-$ | $-$ | $\mp$
$\Pi\operatorname{sgn}E_{J}^{*}$ | $\mp$ | $\mp$ | $\mp$
Table 2:  Calculation of curvature invariant $B$.

To relate $B$ to the curvature $E_{j}^{\prime\prime}$ of the $j$-th energy
band along the direction $\varphi_{\textrm{diag}}$ for the mirror symmetric
configuration $\alpha=\beta$, we first rewrite the potential energy term in
the Hamiltonian Eq. 4 as a function of flux variables
$\sigma=(\varphi_{R}+\varphi_{L})/2$ and $\delta=(\varphi_{R}-\varphi_{L})/2$,
$-E_{J}\left(1+2\alpha\cos\sigma\cos(\theta+\delta)\right)$. With this
transformation, the diagonal $\varphi_{\textrm{diag}}=\varphi_{R}=\varphi_{L}$
in the Brillouin zone Fig. 4(a) is equivalent to
$\sigma=\varphi_{\textrm{diag}},\delta=0$. Therefore the prime notation ′
refers to derivatives with respect to $\sigma$.

Taking the derivative of the Hellmann-Feynman theorem
$E_{j}^{\prime}=\bra{\psi_{j}}H^{\prime}\ket{\psi_{j}}$, where
$\ket{\psi_{j}}$ is the $j$-th energy eigenstate, we obtain an expression for
the curvature,

$E_{j}^{\prime\prime}=\bra{\psi_{j}}H^{\prime\prime}\ket{\psi_{j}}+\bra{\psi_{j}^{\prime}}H^{\prime}\ket{\psi_{j}}+\bra{\psi_{j}}H^{\prime}\ket{\psi_{j}^{\prime}}.$

At the high symmetry point $\varphi_{\textrm{diag}}=\pi$, or
$\sigma=\pi,\delta=0$, the last two terms are zero as long as $\alpha=\beta$.
Hence $E_{j}^{\prime\prime}/E_{J}=-2\alpha\braket{\cos\theta}_{j}$ and the
curvature is proportional to $\braket{\cos\theta}_{j}$, the expectation value
of $\cos\theta$ in state $\ket{\psi_{j}}$.

All measured devices except R18C have Josephson energies satisfying the
inequality $\alpha\approx\beta<1$. According to the last column of the table
above, for $\alpha,\beta<1$ the invariant $B$ will be given by the sign of the
effective Josephson energy at $\varphi_{\mathrm{diag}}=\pi$ (last row), or
equivalently the sign of $\braket{\cos\theta}$ in the ground state,
$\braket{\cos\theta}_{0}$. From the numerical surface plots of the first two
energy manifolds, insets Fig. 4(c,d), one sees that $E_{0}^{\prime\prime}$ is
expected to be positive for the topologically non-trivial configuration
($B=-1$) and negative in the trivial configuration ($B=1$). This concurs with
$\operatorname{sgn}E_{0}^{\prime\prime}=-\operatorname{sgn}\braket{\cos\theta}_{0}=-B$
as inferred from the formula
$E_{j}^{\prime\prime}/E_{J}=-2\alpha\braket{\cos\theta}_{j}$.

Since Fig. 5 shows the transition energy $f_{01}$, the curvature of the data
at $\varphi_{\textrm{diag}}=\pi$ should be compared to
$E_{1}^{\prime\prime}-E_{0}^{\prime\prime}=-2\alpha
E_{J}(\braket{\cos\theta}_{1}-\braket{\cos\theta}_{0})$. At
$\varphi_{\mathrm{diag}}=\pi$ and for $n_{g}=0$ or $n_{g}=1/2$, the parity
operator $P$ commutes with the Hamiltonian and energy eigenstates are also
parity eigenstates. Wavefunctions constructed from the $n_{g}=1/2$ parity
basis, $\ket{m\pm}=1/\sqrt{2}\left(\ket{-m}\pm\ket{m+1}\right)$ [Appendix A],
always have a node at either $\theta=\pi$ (positive parity) or $\theta=0$
(negative parity). Therefore the expectation value $\braket{\cos\theta}_{1}$
of the first excited state, which has opposite parity than the ground state,
cannot be larger in absolute value than $\braket{\cos\theta}_{0}$. This
implies that
$\operatorname{sgn}(E_{1}^{\prime\prime}-E_{0}^{\prime\prime})=-\operatorname{sgn}(\braket{\cos\theta}_{1}-\braket{\cos\theta}_{0})=\operatorname{sgn}\braket{\cos\theta}_{0}=B$.

As stated in the main text, the sign of the curvature measured at the high
symmetry point $\varphi_{\mathrm{diag}}=\pi,n_{g}=1/2$ is a direct indication
of $B$ for all samples except R18C, which has $\alpha\approx\beta>1$. For
R18C, looking at [Table 2], the curvature is always negative at
$\varphi_{\mathrm{diag}}=\pi$ and a second measurement of the curvature at
$\varphi_{L}=\pi,\varphi_{R}=0$ would be needed to resolve the triangle
inequality. However, in practice since we know that $\alpha\approx\beta\approx
1.16$ for R18C [Table 1], we expect the triangle inequality
$1+\beta\geq\alpha$ to be satisfied and that a measurement of the curvature at
$\varphi_{L}=\pi,\varphi_{R}=0$ would have positive sign. The blue curve in
Fig. 5(a) ($n_{g}=1/2$) for the gapped device R17A has positive curvature
($B=1$) at $\varphi_{\mathrm{diag}}=\pi$ whereas the two gapped devices (red)
have negative curvature ($B=-1$).

##  Appendix C Role of Inductance

The BiSQUID Hamiltonian Eq. 4 can be modified to incorporate inductances on
each branch,

$\displaystyle H_{l}=H_{1}$
$\displaystyle(n_{1},\theta_{1})+H_{2}(n_{2},\theta_{2})+E_{C\Sigma}(n-n_{g})^{2}+$
$\displaystyle-E_{J}[\,$
$\displaystyle\cos(\theta+\gamma_{1}\theta_{1}+\gamma_{2}\theta_{2})+$
$\displaystyle\alpha$
$\displaystyle\cos(\theta+\gamma_{3}\theta_{1}+\gamma_{4}\theta_{2}-\varphi_{L})+$
$\displaystyle\beta$
$\displaystyle\cos(\theta+\gamma_{5}\theta_{1}+\gamma_{6}\theta_{2}+\varphi_{R})],$

where $H_{i}(n_{i},\theta_{i})$ are harmonic oscillator Hamiltonians
describing the two normal modes of the linear part of the circuit composed of
branch capacitors and inductors. The coordinates of the normal modes are the
conjugate quantum variables $n_{i},\theta_{i}$. Each mode has frequency
$\omega_{i}$ and dimensionless impedance $z_{i}$ which is the characteristic
impedance scaled by a resistance $\hbar/(2e)^{2}$. The constants
$\gamma_{i}<1$ are participation ratios [104] and describe the coupling of the
harmonic modes to the Josephson junction phases. The effective Josephson
degrees of freedom $n,\theta$ are associated to a common mode with charging
energy $E_{C\Sigma}=(2e)^{2}/(C_{J}+C_{L}+C_{R})$ where $C_{L,R}$ are the
capacitances on the left and right branch.

When the Josephson energies are smaller than the inductive energies one can
neglect the last term in $H_{l}$ and the eigenstates of the normal modes are
harmonic oscillator number states $\ket{l_{1},l_{2}}$. In this perturbative
limit, $\beta_{L}\lesssim 1$ and $\alpha,\beta\lesssim 1/\beta_{L}$, we can
trace out the oscillator degrees of freedom. Integrating $H_{l}$ over the
harmonic oscillator wavefunctions, the effective BiSQUID Hamiltonian in the
presence of inductance is

$\displaystyle H_{l}\approx\hbar\omega_{1}$
$\displaystyle(l_{1}+1/2)+\hbar\omega_{2}(l_{2}+1/2)+E_{C\Sigma}(n-n_{g})^{2}+$
$\displaystyle-E_{J}[\,$
$\displaystyle\rho_{0}\cos\theta+\alpha\rho_{L}\cos(\theta-\varphi_{L})+\beta\rho_{R}\cos(\theta+\varphi_{R})].$

The Josephson energies of the junctions are renormalized by exponential
factors

$\displaystyle\rho_{0}$
$\displaystyle=e^{-\gamma_{1}^{2}(2l_{1}+1)z_{1}/4-\gamma_{2}^{2}(2l_{2}+1)z_{2}/4}$
$\displaystyle\rho_{L}$
$\displaystyle=e^{-\gamma_{3}^{2}(2l_{1}+1)z_{1}/4-\gamma_{4}^{2}(2l_{2}+1)z_{2}/4}$
$\displaystyle\rho_{R}$
$\displaystyle=e^{-\gamma_{5}^{2}(2l_{1}+1)z_{1}/4-\gamma_{6}^{2}(2l_{2}+1)z_{2}/4}.$

They depend not only on the participation ratios and impedances of the
harmonic modes, but also on the photon occupation, resulting in a smaller
topologically non-trivial phase as $l_{1},l_{2}$ are increased.

We can now solve for the Josephson degrees of freedom $n,\theta$ with the only
difference being the renormalization of the Josephson energies. The triangle
inequality
$|\alpha\rho_{L}-\beta\rho_{R}|<\rho_{0}<\alpha\rho_{L}+\beta\rho_{R}$ can
still be satisfied but the position of the critical lines in the phase diagram
will be shifted. Since $\varphi_{L,R}$ is either $0$ or $\pi$ on the critical
lines, one can solve the transcendental equations
$\rho_{0}\pm\alpha\rho_{L}\pm\beta\rho_{R}=0$ to determine their positions.
The solutions of these equations for the ground state $l_{1}=l_{2}=0$ and
$\beta_{L}=0.1,1$ (solid lines) shows excellent agreement with numerically
obtained critical lines in the region $\alpha,\beta\lesssim 1/\beta_{L}$ [Fig.
8]. We use the python package scQubits [105, 106] to numerically diagonalize
the BiSQUID Hamiltonian with inductance, determining the boundary between
topologically trivial and non-trivial regions (circle markers) by computing
the energy gap $f_{01}$ for $n_{g}=1/2$.

Figure 8:  BiSQUID phase diagram in the presence of inductance
$E_{J}/\beta_{L}$ on each branch. The critical line separating the
topologically trivial and non-trivial phases is shown for $\beta_{L}=0$ (zero
inductance, black), $\beta_{L}=0.1$ (blue) and $\beta_{L}=1$ (orange). The
solid lines for $\beta_{L}>0$ are determined from the perturbative model
[Appendix C]. The circle markers are obtained by numerical diagonalization of
the BiSQUID Hamiltonian including inductance. The plasma frequency of each
junction is kept constant for all $\alpha$ and $\beta$. The capacitance of the
middle junction is fixed to $C_{J}=(8e^{2}L_{J})/(\varphi_{0}^{2})$ with
$L_{J}$ the Josephson inductance of the middle junction and
$\varphi_{0}=\hbar/2e$ the reduced flux quantum.

##  Appendix D Fabrication

The devices are fabricated on $350\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thick wafers of high-resistivity intrinsic silicon capped with a
$150\text{\,}\mathrm{nm}$ Si02 surface oxide. We begin by cleaning the
substrate using a soft oxygen plasma etching step in a reactive ion etcher.
Next, we deposit a layer of aluminum on the entire wafer. This aluminum is
deposited at a rate of 1 nm/s using a high-vacuum Plassys electron gun
evaporator. Before exposing the wafer to ambient atmosphere, the aluminum is
passivated in the Plassys in 200 mbar of pure dioxygen during 10 minutes.

In a first optical lithography step, we define the resonator, control lines,
and alignment structures. To make the negative mask, we spin coat a layer of
S1813 resist, bake it at 115°C for one minute, expose it using a LW405B laser
writer with a dose of about 200 mJ/cm2, and develop it in MF319 developer for
one minute. To remove the aluminum in exposed areas, we perform a wet etch.
This involves first hard baking the resist at 150°C for 3 minutes, followed by
immersing the wafer in MF319 for 5 to 7 minutes and rinsing it with water.
Then we dice the wafer in individual 10x10 mm2 dies.

We then fabricate BiSQUID’s Josephson junctions and test structures using
electron-beam lithography. We spin coat a bilayer of MMA EL11 and PMMA 950 A4
at 3000 rpm for 60 seconds each. The lithography mask is patterned using a 30
kV FEI Magellan scanning electron microscope equipped with Raith ELPHY quantum
software. The typical Josephson junction area is 200x200 nm2 with a critical
current density of
150-$200\text{\,}\mathrm{nA}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-2}$.
The mask is developed in a 1:3 MIBK/IPA mixture for 55 seconds and rinsed in
IPA for 60 seconds. The Josephson junctions are deposited using a Plassys
electron-beam evaporator and a double-angle Dolan evaporation technique.
Residual resist is removed by a soft oxygen plasma etch. Then a first layer of
12 nm of aluminum is deposited at a rate of 0.1 nm/s and an angle of 30°. The
tunnel barrier is formed by exposing the junctions to 200 mbar of pure
dioxygen for 10 minutes. A second layer of 80 nm of aluminum is deposited at a
rate of 1 nm/s and an angle of $-30$°. The difference in thickness between the
first and second electrodes leads to a superconducting gap difference across
the junctions, which can prevent quasiparticles from tunneling across the
junction [Appendix F]. The junctions are finally passivated in 200 mbar of
pure dioxygen for 10 minutes. The resist is stripped using NMP 1165 for a few
hours at 80°C.

The BiSQUID leads are capacitively coupled to the ground plane over an area of
approximately $800\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$. We do not
remove the aluminum oxide of the ground plane, so the connection is capacitive
with a high capacitance. However for some devices (SQUID _R16_ and BiSQUID
R17C), an extra step has been added to connect test JJs to measurement pads.
We use the same resist as for the first lithography and open small patching
areas around test JJs leads and measurement pads. The resist is then hard-
baked at 150°C for 3 minutes. In a Plassys electron-beam evaporator, we mill
the aluminum oxide in two 4-minute argon milling steps with an acceleration
voltage of 500 V and a current of 30 mA. We then deposit 100 nm of aluminum at
a rate of 1 nm/s and passivate the aluminum by 10 min oxidation in 200 mbar of
pure dioxygen.

##  Appendix E Experimental setup

Figure 9:  Full wiring of the cryostat for two-tone spectroscopy experiments.
NbTi twisted pair wires are used for DC currents $I_{L}$ and $I_{R}$. An in-
house flexible polyimide cable is used for the gate voltage $V_{g}$. It
supports both distributed and lumped low-pass RC filtering at all stages of
the fridge.

The simplified wiring used to perform two-tone spectroscopy is shown in Fig.
9. A vector network analyzer (VNA) is used to generate a signal at a frequency
$f_{1}$ to probe the resonator. This signal is combined at the mixing chamber
in a bias tee with a DC gate voltage $V_{g}$ generated by a Yokogawa 7651
voltage source. The DC gate voltage is filtered by custom-made flexible cables
with both lumped and distributed filtering, resulting in a low-pass cutoff of
around 50 Hz and a voltage attenuation factor of 17. A second tone at a
frequency $f_{2}$ is produced by an Agilent signal generator and combined with
the DC current $I_{X}$ in a second bias tee which is connected to the flux
line. We use NbTi twisted pairs for the DC currents, with attenuation at the
4K stage and additional low-pass filtering at the mixing chamber stage.
Microwave input signals are attenuated at different stages of the cryostat,
with a total of around -75 dB for the $f_{1}$ line and -55 dB for the $f_{2}$
line. Low-pass and high-pass microwave filters are used on the DC and
microwave lines respectively. Later in the experiment, we also added eccosorb
filters on all DC and RF lines at the mixing chamber stage. However, we
observe no significant change, and quasiparticle poisoning rates were
unaffected by this additional filtering.

The output signal is amplified first at 4K with a LNF cryo-HEMT with a gain of
42 dB and again at room temperature with a Mini-Circuits ZVA amplifier with a
gain of 26 dB. A Josephson traveling-wave parametric amplifier was also used
for some measurements [107]. The BiSQUID device is isolated from amplifier
noise by a Microtronics band-pass filter, a single isolator at the mixing
chamber and a double circulator at 4K. The output signal is recombined with
the reference signal in the VNA, enabling the transmission parameter $S_{21}$
to be measured.

The measured resonator and BiSQUID parameters are given in Table 1. To
characterize the resonators, we fit the transmission data to extract the
loaded quality factor $Q_{L}$. We use the FastCap tool in LayoutEditor to
estimate the external coupling capacitances from the CPW resonator to the
feedlines. From these capacitances, we estimate the external quality factor
$Q_{\mathrm{ext}}$. The output capacitance is designed to be larger than the
input capacitance such that most of the photons from the resonator are emitted
into the output feedline. The CPW resonator is designed to have an impedance
of 75 Ohm to increase its coupling with BiSQUIDs. The coupling of a BiSQUID
and CPW resonator is estimated from vacuum Rabi splitting measurements. We
find typical values of $g=2\pi\,20$ MHz. Further details are given in [76].

##  Appendix F Quasiparticle poisoning

Figure 10:  Quasiparticle poisoning in BiSQUID R18. (a) Measured magnitude of
the resonator transmission at fixed frequency as a function of gate voltage
$V_{g}$ for fixed $\varphi_{L}=\varphi_{R}=0$. For each value of $V_{g}$, we
collect 50 data points with an individual averaging time of
$5\text{\,}\mathrm{ms}$ and indicate counts with the color scale. The two
roughly sinusoidal lines shifted by approximately $0.68\text{\,}\mathrm{V}$
correspond to the two QP parities. Values of $V_{g}$ are given after
attenuation in the measurement lines. (b) First 10 seconds of an 800-second
measurement of the resonator with an averaging time of $5\text{\,}\mathrm{ms}$
per point. The reduced flux threading the SQUID is set to
$\varphi_{\mathrm{diag}}\approx\pi$ and the charge offset to $n_{g}\approx
0.5$. Data points are separated in two parity states (blue, orange), then
parity switching events are identified (black dashed lines). Outlier data
(green markers) are not included in the analysis. (c) The histogram of the
phase of the full time trace is fitted to two Gaussian peaks. We find two
states with occupation probabilities of 53$\%$ for state 0 (blue) and 47$\%$
for state 1 (orange). (d, e) The histograms of occupation times $t_{0}$ of
state 0 (d) and $t_{1}$ of state 1 (e) are fitted to exponential distributions
with time constants $\tau_{0}$ and $\tau_{1}$. The vertical axes indicate
counts in logarithmic scale.

The presence of quasiparticles (QPs), or unpaired electrons, is harmful to
many superconducting devices [54, 108]. The origin of these non-equilibrium
QPs is still unclear, but possible explanations include stray infrared photons
and ionizing radiation from environmental radioactive sources and cosmic rays
[109, 55, 57, 58].

In the absence of QPs, the energy of a Josephson junction is periodic with the
charge offset $2en_{g}$. However the tunneling of a QP through the junction
changes the parity of the superconducting island and shifts the charge offset
by $1e$. If QPs are tunneling to the island faster than they recombine or
tunnel out, the parity of the island will be constantly changing. By measuring
faster than the parity switching time, we can observe the typical "eye-
pattern" consisting of two states with different quasiparticle parity and $1e$
charge offset periodicity [Fig. 10(a)].

To extract the typical QP parity switching time, we measure the state of
BiSQUID device R18 at $\varphi_{\mathrm{diag}}\approx\pi$ and $n_{g}=0.5$
during 800 seconds. The first 10 seconds of the time trace are shown in Fig.
10(b). We identify a parity switching event if the state of the system changes
for at least three data points. From the histogram of the full time trace, we
can clearly identify two states with similar average occupation of 50$\%$
[Fig. 10(c)]. We then extract all the occupation times $t_{i}$ of state $i$
and fit their histogram to an exponential distribution
$f(t_{i})=A\exp(-t_{i}/\tau_{i})$ which yields time constants of about
$0.4\text{\,}\mathrm{s}$. We measure QP switching times at other values of
$\varphi_{L},\varphi_{R}$ and $n_{g}$ and found consistent switching times
between 0.1 and 1 seconds.

We increased the superconducting gap $\Delta$ of the BiSQUID island by
reducing the thickness of the aluminum to $12\text{\,}\mathrm{nm}$, leading to
an increase of about $40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ of $\Delta$
compared to the $80\text{\,}\mathrm{nm}$ thick aluminum layer used for the
rest of the BiSQUID [Fig. 11]. This creates an energy barrier, potentially
preventing QP from tunneling to the island [110, 111, 112, 113, 114, 115].
Gap-engineering worked well in some devices, with typical QP tunneling time of
several seconds and average population of poisoned parity of only few
percents. However, the effectiveness of gap-engineering seems to be highly
fabrication dependent and in most cases did not function properly. One
possible explanation of the failure of gap-engineering is the presence of
defects in the oxide barrier of the junctions which may form spatially
localized subgap states [116]. We tried adding eccosorb filters to all DC and
RF lines but did not observe any impact on QP poisoning. Increasing the
attenuation of eccosorb filtering and improving the sample shielding with
tight indium seals may help suppress the high-frequency photon flux and reduce
quasiparticle poisoning [59].

Figure 11:  Superconducting gap engineering using varying aluminum
thicknesses. (a) The $12\text{\,}\mathrm{nm}$ thick island has a larger
superconducting gap than the rest of the circuit. The extra energy of
$\delta\Delta=$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ can partially
prevent QP from entering the island. (b) Optical microscope image of a SQUID
with a $12\text{\,}\mathrm{nm}$ thick island (faint dark blue) and
$80\text{\,}\mathrm{nm}$ thick leads (white). The thin island can only be seen
with a large brightness. The mirror image of the island (white) is due to the
angle evaporation technique, but is not connected to the rest of the circuit.
(c) Resonator transmission measured as in Fig. 10 (a) but with the gap-
engineered SQUID _R16_. Only a single quasiparticle state is observed, meaning
that QP poisoning events are strongly suppressed. However, for several other
fabrications of gap-engineered SQUIDs and BiSQUIDs with similar recipes and
designs, the gap-engineering technique did not work and we observed typical QP
parity switching times $\tau_{i}$ of 0.1-0.5 s.

In addition to quasiparticle parity switching, BiSQUIDs are also sensitive to
charge noise which may be attributed to fluctuating two-level systems in
dielectrics or surface defects [117]. By monitoring the charge offset over
time, we observe random amplitude jumps approximately every ten minutes. We
also observe a slow drift of the charge offset of order of $0.1\%$ of a $2e$
period per minute [76]. To maintain a constant value of charge offset over a
long measurement, we regularly perform quick calibration measurements. We
probe the resonator while sweeping the charge offset of the BiSQUID over 1.5
periods. We set the reduced fluxes $\varphi_{L},\varphi_{R}$ of the BiSQUID
such that its energy is close to the resonator without crossing it. This
allows an easier fitting procedure to extract the correspondence between
$V_{g}$ and $n_{g}$. This calibration measurement takes about 5 seconds and we
perform it every 1-2 minutes to keep track of the charge offset. If we detect
a small drift during a measurement, we adjust the charge offset accordingly,
and if we detect a jump we discard the last data and measure it again.

Figure 12:  SQUID _R16_ energy transitions as a function of reduced flux
$\varphi_{X}$ extracted from two-tone spectroscopy measurements. (a) The
transition $f_{01}$ is measured at charge offset $n_{g}=0.5$ (blue circles
along solide lines) and $n_{g}=0$ (orange circles along dashed lines). The
transition $f_{02}$ is measured at $n_{g}=0$ (orange squares along dotted
line) and is out of the measurement range at $n_{g}=0.5$. Error bars indicate
measured full-width half maximum of transitions, and lines are best fits to
the data using the BiSQUID Hamiltonian with $\beta=0$. The resonator bare
frequency $f_{r}=9.178$ GHz is indicated by a gray dash-dot line. (b) Zoom on
the $f_{01}$ transition at $n_{g}=0.5$ around $\varphi_{X}=\pi$. The best fit
of the data yields the asymmetry parameter $d=(\alpha-1)/(\alpha+1)=0.9\,\%$
(solid line). Other two lines show simulations for an upper bound of the
asymmetry of $d=2.5\,\%$ (dotted line) and a fully symmetric SQUID with d=0
(dashed line).

##  Appendix G SQUID data

We also performed the spectroscopy of a SQUID as a control experiment. Two
external parameters control a SQUID energy levels: one reduced flux
$\varphi_{X}$ and one charge offset $n_{g}$. At half-flux quantum
$\varphi_{X}=\pi$, a perfectly symmetric SQUID reduces to the charging
Hamiltonian $H_{C}$ which has a hidden quantum mechanical supersymmetry [Fig.
1]. However any asymmetry between left and right junctions prevents the
cancellation of the effective Josephson energy $E_{j}^{*}$ which lifts all the
degeneracies of the energy levels.

The SQUID circuit _R16_ was designed to be symmetric with equal Josephson
energies, however the fabrication process introduces a small asymmetry between
left and right junctions. The spectroscopy of SQUID _R16_ is shown in Fig. 12
as a function of $\varphi_{X}$ at two values of $n_{g}$. At $n_{g}=0$, we
measured the first two transitions $f_{01}$ (orange circles) and $f_{02}$
(orange squares). They converge to a similar value at $\varphi_{X}=\pi$.
However we cannot resolve the transitions near this point because it is almost
resonant with the readout resonator frequency $f_{r}$ (gray dash-dot line).
The first transition $f_{01}$ is also measured at $n_{g}=0.5$, and a zoom down
to $0.5\text{\,}\mathrm{GHz}$ is shown in Fig. 12(b). Similarly to the
BiSQUIDs, the spectral line of $f_{01}$ disappears near $\varphi_{X}=\pi$
because of thermal effects and because of the decrease of the charge-coupled
resonator shift.

We fit the transitions to the eigenenergies of the BiSQUID Hamiltonian Eq. 3
with $\beta=0$. We use three free parameters: the Josephson energy $E_{j}$,
the charging energy $E_{C}$ and the asymmetry parameter
$d=(\alpha-1)/(\alpha+1)$. The best fit parameters are given in Table 1 and
the simulated energy levels are shown in Fig. 12. We find an asymmetry
parameter of $d=0.9\,\%$. Additional simulated energy levels are shown in Fig.
12(b) for an upper bound of the asymmetry of $d=2.5\,\%$ and for a fully
symmetric SQUID with $d=0$. Measuring $f_{01}$ down to lower frequencies would
be required to further constrain the asymmetry parameter.

###### Acknowledgements.

We thank Jean-Noël Fuchs for elucidating the analogy to topological
semimetals. We thank Jeffrey Teo, Roman Riwar, Valla Fatemi, Joël Griesmar,
Hugues Pothier, Landry Bretheau, Benoît Douçot, and Benjamin Wieder for
discussions. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement 636744). The research was also supported by
Japanese Society for the Promotion of Science International Research
Fellowship L22559, IDEX Grant No. ANR-10-IDEX-0001-02 PSL, a Paris “Programme
Emergence(s)” Grant, the Office of Naval Research under Award No.
N00014-20-1-2356, and the Thomas Jefferson Fund, a program of FACE Foundation
launched in collaboration with the French Embassy. We thank IARPA and MIT
Lincoln Labs for providing a TWPA and ESPCI Paris for providing SEM services.

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