# Passive superconducting circulator on a chip

Rohit Navarathna<EMAIL_ADDRESS>ARC Centre for Engineered Quantum
System, School of Mathematics and Physics, University of Queensland, Brisbane,
QLD 4072, Australia  Dat Thanh Le  ARC Centre for Engineered Quantum System,
School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia  Andrés Rosario Hamann  Current address: Department of
Physics, ETH Zürich, CH-8093 Zürich, Switzerland ARC Centre for Engineered
Quantum System, School of Mathematics and Physics, University of Queensland,
Brisbane, QLD 4072, Australia  Hien Duy Nguyen  School of Mathematics and
Physics, University of Queensland, Brisbane, QLD 4072, Australia  Thomas M.
Stace  ARC Centre for Engineered Quantum System, School of Mathematics and
Physics, University of Queensland, Brisbane, QLD 4072, Australia  Arkady
Fedorov<EMAIL_ADDRESS>ARC Centre for Engineered Quantum System, School
of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072,
Australia

###### Abstract

An on-chip microwave circulator that is compatible with superconducting
devices is a key element for scale-up of superconducting circuits. Previous
approaches to integrating circulators on chip involve either external driving
that requires extra microwave lines or a strong magnetic field that would
compromise superconductivity. Here we report the first proof-of-principle
realisation of a passive on-chip circulator which is made from a
superconducting loop interrupted by three notionally-identical Josephson
junctions and is tuned with only DC control fields. Our experimental results
shows evidence for nonreciprocal scattering, and excellent agreement with
theoretical simulations. We also present a detailed analysis of quasiparticle
tunneling in our device using a hidden Markov model. By reducing the junction
asymmetry and utilising the known methods of protection from quasiparticles,
we anticipate that Josephson-loop circulator will become ubiquitous in
superconducting circuits.

superconducting, passive, circulator, microwave

##  I Introduction

Circulators, nonreciprocal devices that route signals and isolate devices from
noise, are a crucial component in cryogenic environments such as
superconducting circuits [1, 2]. When used as isolators, they typically route
the low power signals from a quantum system to an amplifier, while redirecting
noise from the amplifier to a matched load, thus protecting the coherence of
the quantum system [3, 4]. Circulators can also function as duplexers that
separate the input and output signals to and from a device [5]. This is
particularly useful for reflection measurements, where the outgoing signal
needs to be separated from the incoming one.

Most commercial microwave circulators harness ferrite components and the
Faraday effect to achieve nonreciprocity [6, 1], which presents bottlenecks
for future development of large-scale superconducting quantum processors.
First, they are microwave-interference devices, so their size is set by the
microwave wavelength. Second, the ferrite materials make it impractical to
integrate them on a nano-fabricated microwave chip. Furthermore, losses at the
cable interconnections for cascaded circulators reduce the quantum efficiency
of measurement and control at the quantum noise limit.

Various approaches to miniaturising circulators have been proposed. Some
involve driven elements to induce symmetry-breaking fields that require active
microwave control lines and consume additional energy [7, 8, 9, 10, 11, 12,
13, 3, 14, 15, 16, 17]. Others nonreciprocal devices exploit the quantum Hall
effect in a two-dimensional gas that necessitates very large magnetic fields
[18, 19, 20] which are detrimental to superconducting circuits. In this paper,
we present the first implementation of an on-chip circulator which is both
passive and compatible with superconducting circuits. Our design is based on a
superconducting ring interrupted by three Josephson junctions, which forms
three superconducting islands that are capacitively coupled to external
input/output ports [21, 22]. The device is tuned by only DC control fields
that include three charge biases to control the charge distribution on the
superconducting islands and a small magnetic flux to break time-reversal
symmetry. Interference of transitions between the ground state and the excited
states of the superconducting ring results in nonreciprocal effects and signal
circulation.

We performed spectroscopic measurements on our device that confirm theoretical
predictions from Ref. [22] and allow us to extract the device parameters. We
also observed quasiparticle tunneling which gives rise to four quasiparticle
sectors, each with their own charge configuration and spectroscopic response
[23].

With three readout chains in the dilution refrigerator, we were able to
perform three-port device characterisation. Based on our experimental
observations, we show that our device exhibits weak nonreciprocity. However,
the excellent quantitative and qualitative agreement with theoretical
simulations points to systematic improvements that will produce high fidelity
circulation. Furthermore, using fast three-port measurements and hidden Markov
model analysis [24, 25, 26] we deduced the tunnelling rates for each of the
quasiparticle sectors and determined their average lifetime. We additionally
found the presence of a single charge fluctuator as well as charge drift at
slower time scales.

##  II Device description

Figure 1: a) Lumped-element circuit of the passive on-chip superconducting
circulator. The device is a superconducting loop that is interrupted by three
Josephson junctions to form three superconducting islands. The islands are
biased by three gate charges and coupled to the external ports by
interdigitated capacitors. The superconducting loop is threaded by an external
flux. b) False coloured optical microscope image of a fabricated loop. The
three islands are red, blue and green. The Josephson junctions are formed by
overlapping two layers of aluminium with a layer of aluminium oxide in
between.

Our circulator is schematically displayed as a lumped-element circuit in Fig.
1a, consisting of a superconducting loop interrupted by three Josephson
junctions. This creates three superconducting islands that are capacitively
coupled to external $50~{}\Omega$ microwave lines. These lines are the
input/output ports, carrying microwave signals and supplying DC bias voltages
$V_{i}$ ($i=1,2,3$) to each superconducting island. The superconducting loop
is also threaded by an external flux $\Phi$ provided by a coil. The false-
colour optical microscope image in Fig. 1b shows a fabricated sample
consistent with the lumped-element model. The sample is made of aluminum
deposited on a high-resistivity silicon substrate. It is mounted to the mixing
chamber plate of the dilution refrigerator and connected to three readout
chains (see Fig. 6 for details). This facilitates performing simultaneous
three-port characterisation.

The device is described by a Hamiltonian given by [22, 23]

$\begin{split}\hat{H}=&(2e)^{2}(\hat{\textbf{n}}-\textbf{n}_{g})\mathbb{C}^{-1}(\hat{\textbf{n}}-\textbf{n}_{g})\\\
&-\sum_{i=1}^{3}E_{Ji}\cos{(\hat{\phi}_{i}-\hat{\phi}_{i+1}-\tfrac{1}{3}\phi)},\end{split}$
(1)

where $\hat{\textbf{n}}=\left\\{\hat{n}_{1},\hat{n}_{2},\hat{n}_{3}\right\\}$
and $\textbf{n}_{g}=\left\\{n_{g1},n_{g2},n_{g3}\right\\}$ are respectively
the charge operators and the bias charges (normalised to Cooper-pair charge
$2e$) on the three superconducting islands, $\mathbb{C}$ is the capacitance
matrix, $E_{Ji}$ are the Josephson energies of the three Josephson junctions,
$\hat{\phi}_{i}$ are the superconducting phases of the islands, and
$\phi=2\pi\Phi/\Phi_{0}$ is the reduced flux threading through the
superconducting loop. In our experiment, $n_{gi}$ are controlled by applying
DC voltages $V_{i}$ to the coupling ports through bias tees at room
temperature with $n_{gi}=C_{c}V_{i}/(2e)$, whereas $\Phi$ is tuned by a
current in a superconducting coil in the same chip with the device as shown in
Fig. 6.

In the above Hamiltonian, there are two primary energy scales: the charging
energy $E_{C}$ (which is implicit in the first term of Eq. 1 but appears
explicitly after a coordinate transformation [23]) and the mean Josephson
energy $\bar{E}_{J}$. We chose the values of $E_{C}$ and $\bar{E}_{J}$ such
that $\bar{E}_{J}/E_{C}\sim 2$, which is in between the Cooper-pair-box [27]
and transmon regimes [28]. This, as pointed out in Ref. [23], helps increase
the bandwidth and relax constraints on junction fabrication, while introducing
charge noise to the device. We also selected the working frequency of the
device to be around 6 GHz, which is within the bandwidth of the amplifiers
used in our experiment.

Signal circulation in the device is mediated by transitions between the ground
and excited states of the Hamiltonian $\hat{H}$ in Eq. 1. In particular, we
find that in the weak-power limit the scattering matrix element $S_{ij}$,
defined by the ratio of the outgoing amplitude at port $i$ to the incoming
amplitude at port $j$, is given by [23]

$S_{ij}=\delta_{ij}-\sum_{k=1,2,\dots}\frac{\Gamma\langle
k|\hat{n}_{j}|0\rangle\langle
0|\hat{n}_{i}|k\rangle}{i\Delta\omega_{k}+\Gamma_{k}/2},$ (2)

where $\ket{0}$ and $\ket{k}$ with $k=1,2,\dots$ are the ground state and the
excited states of the Hamiltonian in Eq. 1, $\Delta\omega_{k}$ is the detuning
of the excited eigenenergy associated with the excited state $\ket{k}$ from
the input driving frequency, $\Gamma$ is the coupling strength between the
circulator ring and the external ports, and $\Gamma_{k}$ is the decay rate of
the excited state $\ket{k}$ to the ground state $\ket{0}$ due to external
couplings.

##  III Spectroscopy and device characterisation

Figure 2: Reflection measurements from port 1 and fit of the device a) to an
external flux ($\phi$) sweep, b) to a charge bias ($n_{g1}$) sweep at a coil
voltage shown by the black dashed line in a). The reflection is measured using
a vector network analyzer (VNA) with a bandwidth of 15 kHz, allowing the VNA
to capture the drop in reflection before a quasiparticle-tunneling event. This
measurement is repeated 100 times, and the minimum for each frequency is
plotted as the grayscale density plot after background subtraction. The lines
in the inset of a) and in b) are the transition frequencies fitted to the
extracted minima for each frequency. There are four transition frequencies
(solid, dashed, dash-dotted, dotted) for each quasiparticle configuration
(black, purple, green, magenta) and for two charge configurations (two
different line thicknesses). Both fits have the same values of
$\\{E_{C},E_{J1},E_{J2},E_{J3}\\}/\hbar=\\{3.98,7.85,8.28,8.55\\}\times\text{GHz}$.
We show the full fit for a) in Fig. 7, .

We carried out spectroscopic measurements on the device using two ports of a
vector network analyzer (VNA). The source and receiver ports of the VNA were
respectively connected to one of the input and output microwave cable ports of
the device to measure reflection from or transmission through it.

Spectroscopic measurements are shown as a function of the reduced flux $\phi$
in Fig. 2a and the charge bias $n_{g1}$ in Fig. 2b. The density plots are the
minimum of 100 reflection measurements at a bandwidth of $15$ kHz. This
ensures that the transition frequency for each quasiparticle sector was
captured in at least one of the 100 measurements. We fit the transition
frequencies with the eigenenergies found by solving the eigensystem of the
Hamiltonian $\hat{H}$ in Eq. 1. This allows us to extract the device
parameters; in particular, the fit found the charging energy
$E_{C}/\hbar=3.98$ GHz and the Josephson energies
$E_{Ji}/\hbar=\\{7.85,8.28,8.55\\}\times\text{GHz}$, which yields
$\bar{E}_{J}/E_{C}\sim 2$ and matches the values from our room-temperature
resistance measurements. In Ref. [23], we showed that the tolerable asymmetry
of the junctions was 1% or less, which is determined by the requirement that
the junction asymmetry is lower than the external coupling bandwidth (which we
estimate to be $\Delta_{\mathrm{BW}}\sim 70$ MHz). In our device, the
asymmetry of the Josephson energies is much larger than the bandwidth ($\delta
E_{J}\approx 300$ MHz $\gg\Delta_{\mathrm{BW}}$), so the nonreciprocity
displayed by the device is limited by the junction asymmetry.

In Fig. 2a, the transition frequencies are grouped into clusters, each of
which contains eight closely located lines. These correspond to two different
charge-bias configurations, each consisting of four charge-parity sectors of
the superconducting loop [23] that exchange to each other due to intermittent
quasiparticle-tunneling events across the islands. The sectors are represented
by the charge parities of two superconducting islands, including
$\textsf{e-e},\textsf{e-o},\textsf{o-e}$ and o-o, where e (o) means an even
(odd) number of electrons on a particular island. We note that the
superconducting loop is galvanically isolated, so the total charge of the
islands is conserved. This makes the charge-parity configuration of the device
depend on two out of three islands only, thus justifying the four charge-
parity sectors. As shown later, we fit a hidden Markov model to stochastic,
experimental time-series data to characterise the populations and lifetimes of
the quasiparticle sectors.

In Fig. 2b, we observed two different spectra that belong to the two different
charge-bias configurations. We attribute this to the presence of a charge
fluctuator located near the superconducting loop. Using our fit routine, we
found that the differences in the charge biases of the two configurations are
$\\{0.11,-0.03,0.04\\}$, indicating that the charge fluctuator is close to the
first superconducting island. Moreover, the transition frequencies in Fig. 2b
are periodic with $n_{g1}$, as expected for superconducting devices tuned with
offset charge biases [27, 28]. The periodicity is found to be three Cooper
pairs, which is consistent with the fact that there are three superconducting
islands in the device. In contrast to the assumption in Ref. [21], we do not
observe discontinuities in the spectrum due to changes in the total charge of
the three islands.

Besides the mentioned-above fast charge events, we furthermore observed slow
charge drift that occurred roughly every $10$ minutes. This set new values for
the charge offsets $\mathbf{n}_{g}$ and resulted in random change in the
transition frequencies. An example of this is shown in Fig. 8, where we
recorded the minimum amplitude of reflection for 100 frequency sweeps.

##  IV $S$-matrix characterisation

Using the four-port VNA and three output readout chains we performed three-
port characterisation of our device. Specifically, one port of the VNA was
used as the source and connected to a mechanical microwave switch to send
signals to one of the three input ports. The three readout chains were
connected to the other three ports of the VNA, which allows measuring three
S-parameters, $S_{1j}$, $S_{2j}$, and $S_{3j}$, at the same time, where $j$ is
the input port. By switching the input port $j$, we obtained the full
$S$-matrix.

We note that the scattering matrix measured by the VNA, M, is a composite of
the scattering matrix from the device, S, and the transmission matrices of the
input and output lines, which include attenuators and amplifiers, denoted
$\textbf{A}={\rm diag}\\{a_{1},a_{2},a_{3}\\}$ and $\textbf{B}={\rm
diag}\\{b_{1},b_{2},b_{3}\\}$ respectively. It follows that ${\bf M=B.S.A}$
where we assumed the external circuitry has perfect impedance matching.

To calibrate the line responses we first changed the flux through the coil
such that the frequency of the circulator is off-resonant with the signals
from the VNA, and then measured M. In this case, we assume that the true
$S$-matrix S is lossless and unitary. This gives us the conditions
$\textbf{S}\textbf{S}^{\dagger}=\mathbb{I}$ and
$\textbf{S}^{\dagger}\textbf{S}=\mathbb{I}$. We measure M, and then solve for
A, B, and S, which will be close to the identity matrix $\mathbb{I}$ with a
small fraction of the signal being transmitted. We next tuned the circulator
frequency to be resonant with that of the VNA signals and used the values of A
and B found in the non-resonant case to determine the on-resonance matrix S.

Figure 3: Experimentally extracted partial $S$-matrix elements for two
different charge-bias configurations. In (a) $|S_{12}|>|S_{21}|$, showing that
signals are transmitted more from port 2 to port 1 than from 1 to 2. In (b) a
different charge-bias configuration yields $|S_{12}|<|S_{21}|$, reversing the
non-reciprocity. The full 3-port $S$-matrix is shown in Fig. 9. (c) and (d)
show theoretical simulations corresponding to the experimental configurations
above, using the fitting results from Sec. III and averaging over the four
quasiparticle sectors. We see a good qualitative agreement between experiment
and theory.

Following this procedure, in Fig. 3 we show a part of the experimentally
extracted $S$-matrix, which reveals nonreciprocity in our device. In Fig. 3a,
we see that when the drive frequency is around 6.5 GHz, $|S_{12}|>|S_{21}|$.
At this frequency, the reflections $|S_{11}|$ and $|S_{22}|$ are also at
minima. Figure 3b shows that by tuning the charge-bias voltages $V_{1}$,
$V_{2}$ and $V_{3}$ (at constant flux bias) we can tune the device so that
$|S_{12}|<|S_{21}|$; that is, we are able to electronically change the
direction of signal transfer. We provide the full $S$-matrix for both
circulation directions in Appendix D (see Fig. 9).

In Figs. 3c and d, we performed simulations to reconstruct the $S$-matrix
elements in Figs. 3a and b. To this end, we use the device parameters measured
in Sec. III and for each working point (including the external biases and the
signal frequency) we compute the $S$-matrix for all four quasiparticle
sectors, and then average over the sectors (with equal weights) to achieve an
averaged $S$-matrix. Comparing Figs. 3a and b to Figs. 3c and d respectively,
we found qualitative agreement between theory and experiment.

Given the deleterious effect of quasiparticles and junction fabrication
asymmetry on circulation, we computed the potential performance of the device
in two scenarios that remove these two effects. First, we computed the
performance assuming that the system dwells in a single quasiparticle sector.
The simulated $S$-matrix for this case is in Fig. 4a, where we see a much
stronger effect on resonance, $S_{11}\ll 1$, but only modest changes in the
off-diagonal elements. Second, we compute the performance when reducing the
junction asymmetry, in addition to suppressing quasiparticle tunneling. In
Fig. 4b, we show the $S$-matrix elements assuming 1% junction asymmetry, i.e.
$E_{J1}/E_{J2}=0.99$ and $E_{J3}/E_{J2}=1.01$. From these simulations we
estimate the bandwidth within a single quasiparticle sector to be around $70$
MHz.

Figure 4:  (a) Simulated $S$-matrix elements when the device stays in a single
quasiparticle (QP) sector with the same junction asymmetry as in the current
sample. (b) Simulated $S$-matrix with junction asymmetry reduced to 1%, with
$E_{J1}/E_{J2}=0.99$ and $E_{J3}/E_{J2}=1.01$. (c) Nonreciprocity and fidelity
defined in Eqs. (3) and (4) for the experimentally measured $S$-matrix, and
for the simulated $S$-matrix with the experimentally determined values of
$E_{J}$ but no QP tunnelling. (d) Simulated $S$-matrix with no QP tunnelling,
and 1% junction dispersion.

To quantify the observed and simulated circulation, we introduce the
_nonreciprocity_ , $\mathcal{N}$ [29], and the fidelity, $\mathcal{F}$ [30]

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle||\mathbf{S}-\mathbf{S}^{\intercal}||/\sqrt{8},$ (3)
$\displaystyle\mathcal{F}$ $\displaystyle=$ $\displaystyle
1-{{\sum}_{i,j}}\big{|}|S_{ij}|-|S^{\mathrm{ideal}}_{ij}|\big{|}/8,$ (4)

where $\lVert X\rVert=\sqrt{\mathrm{Tr}(XX^{\dagger})}$ denotes the norm of a
matrix $X$, $\sqrt{8}$ is a normalisation factor, and $S^{\rm ideal}_{ij}$ are
the elements of an ideal circulator scattering matrix. We comment more on the
nonreciprocity measure in Appendix E. .

Figure 5: a) Repeated measurement of $S_{11}$ of the device at a frequency of
$6.709$ GHz, an external flux $\phi=1.9$. The out-of-phase quadrature Q versus
time shows gaussian noise that switches between one of four unobserved states.
We fit the the full $3\times 3$ scattering matrix data, $\bf{S}$, with a four-
state hidden Markov model (HMM), using approximately 32k time samples, and
attribute the data to one of the four hidden states accordingly, indicated by
color. We interpret these states as the quasiparticle states seen Fig. 2. b)
Histogram of the $S_{11}$ data in the IQ plane, showing clustering of
$S_{11}$. We also show the $1\sigma$ confidence ellipse extracted from the HMM
(circles). We note that the separation in the hidden state data is greater
than observed here, which is merely a projection of $\bf{S}$ onto a
2-dimensional subspace.

In Fig. 4c, we show the nonreciprocity $\mathcal{N}$ for both the
experimentally measured $S$-matrix in Fig. 3a, $\mathcal{N}_{\mathrm{exp}}$,
and the ‘no-quasiparticle-tunneling’ $S$-matrix in Fig. 4a,
$\mathcal{N}_{\mathrm{1\,sector}}$. We see that the nonreciprocity for the
latter is considerably improved, while the fidelities of the two cases,
$\mathcal{F}_{\mathrm{exp}}$ and $\mathcal{F}_{\mathrm{1\,sector}}$, are
comparable, indicating that a higher nonreciprocity is necessary but not
sufficient for better circulation.

Reducing the junction asymmetry to $1\%$, in addition to suppressing
quasiparticle tunneling, greatly boosts the device performance. Figure 4d
shows theoretical predictions for the fidelity and nonreciprocity in the
absence of quasiparticle fluctuations, and assuming parameter variations are
reduced to 1%. We see both the nonreciprocity and the fidelity are
substantially improved relative to the experimental results.

##  V Quasiparticle tunnelling

Due to the significant impact of quasiparticle tunneling, the measurement
setup was modified to measure the full $S$-matrix faster than the
quasiparticle tunnelling rate. For this purpose, we used a combination of
microwave sources, an AWG, and a digitizer (shown in Appendix F, Fig. 10),
which allows us to measure the $S$-matrix in $30~{}\mu$s.

In order to distinguish the four distinct states corresponding to the four
quasiparticle sectors, we use a relatively high power, -100 dBm, at the device
to enhance the single-shot signal-to-noise ratio for the scatter matrix
measurements. We took a continuous run of 32,768 time-series measurements of
the full scattering matrix, and then employed a hidden Markov model (HMM) to
analyse the time-series data (see Appendix G for details) [24, 25, 26]. At
each time step, the HMM is able to make a statistical inference about the
hidden quasiparticle state from the full $3\times 3$ scattering matrix, and
the emission statistics from the data within each hidden state.

In Fig. 5a, we show the Q-quadrature of the $S_{11}$ component of the measured
$S$-matrix, over time. We observe switching of the scattering matrix,
corresponding to jumps between the four quasiparticle sectors that were
observed in Fig. 2. At each time, we classify the data as being associated to
one of the four hidden states, indicated by the colours. In Fig. 5b, we plot
the data in the IQ plane which reveals four distinct clusters, one per
quasiparticle sector, along with the standard-error ellipse for each sector
extracted from the HMM fitting.

In these measurements, we were also able to extract the average $S$-matrix for
each quasiparticle sector. However, because of the high-power signals, our
$S$-matrix elements showed only very minor nonreciprocity. With quantum
limited amplifiers, one could classify the quasiparticle sectors at a lower
power. In this case, we expect to see higher circulation for one of the
quasiparticle sectors as theoretically described in Ref. [23].

The HMM provides additional information about the the dynamics of the jump
processes. In particular, we find that the jumps between the quasiparticle
states are well-described by Poisson processes with decay times of around 200
$\mu$s in each quasiparticle sector (see Appendix G, Fig. 11).

##  VI Conclusions

In conclusion, we fabricated and characterised a passive superconducting
circulator on a chip. Our device is fully compatible with superconducting
qubit fabrication and can be tuned with only DC control. Our experimental
results showed clear evidence of nonreciprocity, but not sufficient for high-
fidelity circulation.

We provide strong evidence that quasiparticle fluctuations and device
asymmetry are significant and detrimental to the operation of the device. We
showed that the device nonreciprocity and circulation can be substantially
improved by suppressing quasiparticle tunnelling such that the device stays in
one single quasiparticle sector, and reducing the disparity in the three
Josephson junctions energies to $1\%$ or less.

The precision of our Josephson junction fabrication is comparable to the
standard shadow evaporation fabrication. However, post-fabrication laser
annealing has been shown to improve the variation of junction normal
resistances to be $\sim 0.3\%$ of design values across a device [31]. This
level of asymmetry is suitable for reaching the necessary junction precision
$\lesssim 1\%$ required for high-quality circulation. In addition, recent
progress has been made in using different materials, including normal metals
[32] or low-gap superconductors [33], to suppress quasiparticle tunneling.
Together with possible reduction of charge sensitivity, these provide a
pathway to enhance the performance of the device we have demonstrated towards
practically useful components for on-chip superconducting quantum processors.

###### Acknowledgements.

We thank Clemens Müller and Prasanna Pakkiam for useful discussions. This
research was supported by the Australian Research Council Centres of
Excellence for Engineered Quantum Systems (Projects No. CE170100009 and No.
CE110001013).

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##  Appendix A Fabrication and experiment setup

Figure 6: Schematic of the setup inside the dilution refrigerator. Each of the
three ports of the device is connected to a ferrite circulator to both send
and receive signals from the device. The inputs $I_{i}$ are sent through
attenuators to the device. A DC voltage $V_{i}$ is added to the RF input via
bias tees at room temperature. Each output $O_{i}$ from the device passes
through a circulator, two isolators, a band pass filter, a HEMT amplifier and
room temperature amplifiers.

The device was fabricated on a $500~{}\mu$m thick high-resistivity silicon
substrate. $300~{}\mu$m of the substrate was pre-diced on the back side to
allow cleaving of the chip rather than dicing. This is because dicing requires
an extra layer of spin-coating, which would increase the spread in $E_{Ji}$ of
the Josephson junctions. A bi-layer PMGI-PMMA resist stack was spin-coated on
the high resistivity silicon substrate before patterning using electron beam
lithography with a Raith EBPG. The patterned chip was developed and the native
silicon oxide was removed with a HF dip prior to loading the chip into a
Plassys MEB550 electron beam evaporator. In order to make the Josephson
junctions, two layers of aluminium were evaporated on the substrate which was
tilted at a 45 degrees to the vertical. The first and second layer of
thicknesses 20 nm and 60 nm were evaporated by rotating the substrate holder
along the normal of the substrate by 0 degrees and 90 degrees (planetary
rotation). This is done to make a quasiparticle trap via engineering the
superconducting gap [33]. The thinner layer of aluminium has a superconducting
gap higher than the rest of the aluminium, which acts as a potential barrier
for quasiparticle tunnelling. We still observe quasiparticle tunnelling with
the quasiparticle trap. However, we could not measure the reduction in the
tunnelling rate as we do not have a reference tunnelling rate for a sample
without a trap. After evaporation, the chip was cleaved along the pre-diced
lines using a cleaver and carefully placed tweezers. The resist was then
removed using PG remover heated to $70^{\circ}$ C. The room temperature
resistance of the Josephson junctions were determined using a probe station.
The $E_{Ji}$ were approximated to be 7.90 GHz, 8.10 GHz and 8.36 GHz,
corresponding to a separation of $5.6\%$.

The schematic of our experiment is shown in Fig. 6. The sample was loaded in
the base plate of a Bluefors LD dilution refrigerator and cooled down to $8$
mK. The three ports of the circulator were connected to input and output
microwave cables through three commercial ferrite circulators to measure
transmission and reflection from all three ports The flux through the loop
$\phi$ and the charge configuration $\textbf{n}_{g}$ is controlled by a DC
voltage source connected to a coil and three bias tees at room temperature
respectively.

##  Appendix B Fitting and simulated scattering matrix elements

Figure 7:  Same as Fig. 2a but a) raw data, and b) including the full fits.
The fitted lines are four transition frequencies (solid, dashed, dash-dotted,
dotted) for each quasiparticle configuration (black, purple, green, magenta)
and for two charge-bias configurations (two different line thicknesses).

The theoretical fit in Fig. 2 consists of the transitions from the ground
state to the excited states of the Hamiltonian $\hat{H}$ in Eq. 1. We find
these transitions by solving the eigenenergies of $\hat{H}$. To do this, we
first perform a coordinate transformation

$\displaystyle\hat{n}^{\prime}_{1}=\hat{n}_{1},\hskip
7.11317pt\hat{n}^{\prime}_{2}=-\hat{n}_{2},\hskip
7.11317pt\hat{n}^{\prime}_{3}=\hat{n}_{1}+\hat{n}_{2}+\hat{n}_{3}=n_{0},$ (5)
$\displaystyle\hat{\phi}^{\prime}_{1}=\hat{\phi}_{1}-\hat{\phi}_{3},\hskip
7.11317pt\hat{\phi}^{\prime}_{2}=\hat{\phi}_{3}-\hat{\phi}_{2},\hskip
7.11317pt\hat{\phi}^{\prime}_{3}=\hat{\phi}_{3},$ (6)

where $n_{0}$ is the conserved total charge number, which is set to be zero in
our simulations. In the new coordinates, the Hamiltonian is [23]

$\displaystyle\hat{H}^{\prime}(n_{g1},n_{g2},n_{g3})$ (7) $\displaystyle=$
$\displaystyle
E_{C}\big{(}(\hat{n}^{\prime}_{1}-\tfrac{1}{2}(n_{0}+n_{g_{1}}-n_{g_{3}}))^{2}$
$\displaystyle+(\hat{n}^{\prime}_{2}+\tfrac{1}{2}(n_{0}+n_{g_{2}}-n_{g_{3}}))^{2}-\hat{n}^{\prime}_{1}\hat{n}^{\prime}_{2}\big{)}$
$\displaystyle-
E_{J1}\cos(\hat{\phi}^{\prime}_{1}-\tfrac{1}{3}\phi)-E_{J2}\cos(\hat{\phi}^{\prime}_{2}-\tfrac{1}{3}\phi)$
$\displaystyle-
E_{J3}\cos(\hat{\phi}^{\prime}_{1}+\hat{\phi}^{\prime}_{2}+\tfrac{1}{3}\phi),$

where $E_{C}$ is the charging energy. We then numerically diagonalise
$\hat{H}^{\prime}$ in the truncated discrete charge basis.

The Hamiltonian $\hat{H}^{\prime}(n_{g1},n_{g2},n_{g3})$ in Eq. (7) describes
one of the four quasiparticle sectors $\textsf{ee},\textsf{eo},\textsf{oe}$
and oo. The Hamiltonians of the other sectors are given by
$\hat{H}^{\prime}(n_{g1}-\tfrac{1}{2},n_{g2}+\tfrac{1}{2},n_{g3})$,
$\hat{H}^{\prime}(n_{g1},n_{g2}-\tfrac{1}{2},n_{g3}+\tfrac{1}{2})$, and
$\hat{H}^{\prime}(n_{g1}+\tfrac{1}{2},n_{g2},n_{g3}-\tfrac{1}{2})$. Four more
Hamiltonians are used to account for the presence of a charge fluctuator in
Fig. 2b, for which small fluctuations $\delta n_{gi}$ were added to $n_{gi}$.
In Fig. 7, we show the full fit of Fig. 2a, where one can see an excellent
overlap between the experimental data and the fitted transitions for the whole
range of the sweep.

The simulated scattering matrix elements in Figs. 3c and d and Figs. 4a and b
are computed using $S_{ij}$ given in Eq. 2. Within this expression, the charge
matrix elements $\langle k|\hat{n}_{j}|0\rangle$ and $\langle
0|\hat{n}_{i}|k\rangle$, the detunings $\Delta\omega_{k}$, and the decay rates
$\Gamma_{k}$ are found from the eigensystem of the Hamiltonian
$\hat{H}^{\prime}$ for each quasiparticle sector [23]. The coupling strength
$\Gamma$ can be varied in our simulation to fit to the bandwidth from the
experimental data. We also note that $S_{ij}$ in Eq. 2 is an approximate
expression for the $S$-matrix elements that is obtained from adiabatic
elimination of the open waveguide-ring system [23]. However, we found that
using $S_{ij}$ in Eq. 2 or $S_{ij}$ computed from solving an open-system SLH
master equation [22] to simulate the $S$-matrix yields essentially the same
results.

##  Appendix C Charge drifts and jumps

Figure 8: a) and b) Two examples of reflection measurements where $n_{g1}$ is
swept with an external voltage using the same method as in Fig. 2. Both the
plots are measured from left to right in time. We can see charge
reconfiguration events shown by the black arrows, where the transition
frequencies witness a discontinuous change.

Figure 8 shows reflection measurements where $n_{g1}$ is swept with an
external voltage using the same method as in Fig. 2. The plots were recorded
in time from left to right. Charge jumps manifest via sudden changes in the
transition frequencies, which are marked by black arrows in Fig. 8. Charge
drift can be observed as a gradual change in the periodic pattern in Fig. 8a.

##  Appendix D Full experimentally extracted scattering matrix

Figure 9: Full scattering matrices of a) the partial one in Fig. 3a and b) the
partial one in Fig. 3b.

Figure 9 displays the full experimentally extracted scattering matrix for both
circulation directions in Figs. 3a and b. For both cases, nonreciprocity
exhibits clearly when comparing $S_{12}$ and $S_{21}$, but not with the pair
$S_{13}$ and $S_{31}$ or $S_{23}$ and $S_{32}$. Signal transmissions as seen
from the figure are also low.

##  Appendix E Measures of nonreciprocity and circulation

The measure of nonreciprocity $\mathcal{N}$ defined in Eq. (3) is too broad
for capturing circulation, as seen by considering a 3-port gyrator which has
the following scattering matrix

$\mathbf{S}_{\rm gyr}=\left[\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\
0&0&1\end{array}\right].$ (8)

For this matrix, $\mathcal{N}(\mathbf{S}_{\rm gyr})=1$, however it is a poor-
quality circulator (we note here that $\lVert\mathbf{S}_{\rm
gyr}-\mathbf{S}_{\rm gyr}^{\intercal}\rVert=\sqrt{8}$, explaining the
normalisation factor in Eq. (3)). We therefore introduce the fidelity
$\mathcal{F}$ defined in Eq. (4). For an ideal circulator with

$\mathbf{S}^{\rm ideal}=\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right]\text{or}\left[\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\
0&1&0\end{array}\right],$ (9)

we find $\mathcal{N}(\mathbf{S}^{\rm ideal})=\sqrt{3/4}$ and
$\mathcal{F}(\mathbf{S}^{\rm ideal})=1$.

##  Appendix F Time-series measurement setup

Figure 10: Schematic of the measurement hardware setup for the time series
measurements. The marker channels of the AWG are used to send digital signals
that gate the output of three microwave sources. The same measurement
frequency $f$ is the output from the three sources, one after the other in
time. The $S$-matrix is imprinted in the three outputs. The colors of the
outputs refer to the ideal clockwise circulation (green), ideal counter-
clockwise circulation (purple), and reflection (magenta). A fourth microwave
sources is used as an LO with a frequency of $f-100$ MHz for all three outputs
using splitters. The third output is further separated in frequency using
another mixer and a continuous sine wave used as the LO from an AWG channel.
The second and third outputs are then combined. The first output, along with
the combined output is recorded on the digitizer.

Figure 10 is a schematic diagram of our time-series measurement setup, which
uses four microwave sources, an AWG, and a digitizer. This facilitates fast
measurements of the $S$-matrix at a rate of $30~{}\mu\rm s$ to capture the
dynamics of quasiparticle events.

##  Appendix G Hidden Markov model

A hidden Markov model (HMM) [24, 25, 26] is a statistical tool that can be
used to predict the most likely states of a system given a sequence of
observed data. A HMM consists of a transition matrix that contains the
transition probabilities between hidden state, and the emission process that
describes the emitted data within each hidden state. If the process model is
not known beforehand, the Baum-Welch algorithm can be used to fit a model to
the observed data.

In our case, the time series of the full $3\times 3$ $S$-matrices are the
observed data, and the four quasiparticle configurations are the hidden
states. We used the pomegranate Python package [34] to fit the observed data.
Pomegranate first finds clusters of data using the k-means algorithm and then
uses the Baum-Welch algorithm to extract the HMM. With the fitted HMM, we can
then predict the most likely sequence of quasiparticle configurations that led
to the observed data. This predicted sequence of quasiparticle configurations
were used to classify the data in Fig. 5a.

##  Appendix H Quasiparticle jump statistics

We further analyse the statistics of the quasiparticle tunneling events in
Fig. 5. In particular, based on the jump times predicted by the HMM, we
estimate the lifetimes of the quasiparticle states, which are shown in Fig.
11.

Figure 11: Histograms of the lifetimes of the four quasiparticle states
(dots). The lifetimes are inferred from the jump times from the sequence
predicted by the hidden Markov model. The fit is to an exponential (line), and
the decay time is shown as $\tau_{\mathrm{fit}}$. The decay rate predicted by
the HMM is also shown as $\tau_{\mathrm{model}}$.