††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Fano-enhanced low-loss on-chip superconducting microwave circulator

N. Pradeep Kumar  Analog Quantum Circuits Pty. Ltd., Brisbane, QLD, Australia
School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia  Dat Thanh Le  Analog Quantum Circuits Pty. Ltd., Brisbane,
QLD, Australia  School of Mathematics and Physics, University of Queensland,
Brisbane, QLD 4072, Australia  Prasanna Pakkiam  Analog Quantum Circuits Pty.
Ltd., Brisbane, QLD, Australia  School of Mathematics and Physics, University
of Queensland, Brisbane, QLD 4072, Australia  Thomas M. Stace  Analog Quantum
Circuits Pty. Ltd., Brisbane, QLD, Australia  School of Mathematics and
Physics, University of Queensland, Brisbane, QLD 4072, Australia  Arkady
Fedorov  Analog Quantum Circuits Pty. Ltd., Brisbane, QLD, Australia  School
of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072,
Australia

###### Abstract

Ferrite-free circulators that are passive and readily integratable on a chip
are highly sought-after in quantum technologies based on superconducting
circuits. In our previous work, we implemented such a circulator using a
three-Josephson-junction loop that exhibited unambiguous nonreciprocity and
signal circulation, but required junction energies to be within $1\%$ of
design values. This tolerance is tighter than standard junction fabrication
methods provide, so we propose and demonstrate a design improvement that
relaxes the required junction fabrication precision, allowing for higher
device performance and fabrication yield. Specifically, we introduce large
direct capacitive couplings between the waveguides to create strong Fano
scattering interference. We measure enhanced ‘circulation fidelity’ above
$97\%$, with optimised on-resonance insertion loss of $0.2$ dB, isolation of
$18$ dB, and power reflectance of $-15$ dB, in good agreement with model
calculations.

##  I Introduction

Circulators are a paradigmatic example of non-reciprocal devices with a wide
use in telecommunication and microwave electronics [1, 2, 3]. They are also
indispensable for cryogenic microwave measurements where they are used to
route weak microwave signals while protecting the system of interest from
thermal noise caused by higher temperature stages [4, 5]. However,
conventional ferrite circulators are bulky and not compatible with
microfabrication nor with superconducting circuits, and thus unsuitable for
very-large-scale superconducting microwave networks. Given the drive to scale
up superconducting quantum computers, designs for integrated microwave
circulators on a chip are becoming critical [6, 7].

Many approaches to integrating circulators with other solid-state quantum
circuits involve the application of strong magnetic fields, either real or
synthesised with time-dependent control fields [8, 9, 10, 11, 12, 13, 14, 4,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. However, these approaches may also be
incompatible with microfabricated superconducting systems, or add AC-control
complexity.

In contrast, our recent results demonstrated the realisation of an on-chip
superconducting circulator with only passive (i.e., DC) control [25, 26],
based on a three-Josephson-junction loop [27, 28]. While nonreciprocity and
microwave circulation were evident [26], the device performance was limited by
asymmetry in Josephson junction energies. In particular, the device exhibited
a ‘circulation fidelity’ of $\sim 80\%$ (i.e., the fidelity of the measured
device scattering matrix relative to that of an ideal circulator)
corresponding to an insertion loss of 2 dB, when post-selected on a specific
quasiparticle configuration sector.

In this paper, we report the implementation of an improved design for a three-
junction circulator device, which has been analysed in Refs. [27, 28, 29, 25,
26]. As in earlier work, the core of the device comprises three
superconducting islands, indicated by different colours in Fig. 1a that are
connected to each other via Josephson junctions, and are capacitively coupled
to external waveguides. The key advance reported here is the inclusion of
shunt capacitors that directly couple the waveguides [23, 24, 30], indicated
as $C_{X}$ in Fig. 1a. This introduces an additional microwaves scattering
pathway, giving rise to a Fano-like interference effect [31, 32].

Our theoretical simulations of the proposed design predict that high
circulation fidelity, above $97\%$ (corresponding to an insertion loss of
$0.2$ dB), can be reached even when the spread in the Josephson energies is
$\sim 3\%$, which is achievable with standard electron beam lithography [33,
34]. We confirm our theoretical predictions with experimental measurements,
demonstrating significantly enhanced circulation performance.

As with previously reported devices, our experimental system still suffers
from significant quasiparticle hopping, so we use post-selection to
characterise and optimise the device performance within a single quasiparticle
sector. Relative to Ref. [26], we observe a ten-fold improvement in device
performance, with measured insertion loss of $\rm{IL}=0.2$ dB, isolation of
$\rm{IS}=18$ dB, and power reflectance of $\rm{R}=-15$ dB at resonance.

##  II Background

The electronic design of the capcitively-shunted circulator is shown in Fig.
1a. Figure 1b implements this device, with the capacitive shunts included in
the hexagonal structure linking the waveguides, which couple to off-chip
signal sources and analysers. The experimental setup is described in detail in
Sec. IV.

Figure 1: (a) Lumped-element circuit model of the capacitively-shunted
circulator. The device consists of three Josephson junctions arranged in a
loop that creates three superconducting islands (indicated by blue, green, and
red colours). The islands are capacitively coupled to external waveguides via
$C_{C}$, and the waveguides are also directly coupled to one-another by
capacitive shunts $C_{X}$. (b) Optical microscope image of a fabricated
device, where the centered triangle represents the loop formed by three
Josephson junctions and the outer ring represents the inter-waveguide
capacitances.

To measure the scattering response of the system, we drive it with input
voltage signals $V^{\mathrm{in}}_{i}$ and measure the output voltage signals
$V^{\mathrm{out}}_{j}$ scattered by the device to determine the scattering
matrix amplitudes $S_{ji}=V^{\mathrm{out}}_{j}/V^{\mathrm{in}}_{i}$ with
$i,j\in\\{1,2,3\\}$. We use an external DC flux bias threaded through the
junction loop and DC charge biases applied to the superconducting islands
($\Phi_{x}$ and $V_{g,(1,2,3)}$ in Fig. 1a) to control the operation of the
device.

Given the device’s scattering matrix $S$, we quantify its circulation
performance by defining the average clockwise and anticlockwise circulation
fidelities, and the average reflection respectively as

$\displaystyle{\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}$
$\displaystyle=$ $\displaystyle(|S_{13}|+|S_{32}|+|S_{21}|)/3,$ (1a)
$\displaystyle{\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}$
$\displaystyle=$ $\displaystyle(|S_{12}|+|S_{23}|+|S_{31}|)/3,$ (1b)
$\displaystyle{\mathcal{R}}$ $\displaystyle=$
$\displaystyle(|S_{11}|+|S_{22}|+|S_{33}|)/3.$ (1c)

An ideal clockwise circulator will have
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}=1$
and
$\mathcal{F}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}=\mathcal{R}=0$
[26].

Our earliest experimental implementation of the three-junction-loop circulator
(without shunt capacitors) was reported in Navarathna  _et al._ [25], where we
observed qualitative nonreciprocal behaviour, i.e. $S_{ij}\neq S_{ji}$ for
$i\neq j$, albeit with a low circulation fidelity $\sim 50\%$. Based on
detailed modelling, we attributed this low circulation performance to
electrical asymmetry in the Josephson junctions, which had a large relative
spread in Josephson energies $\delta E_{J}\sim 8.5\%$, where $\delta
E_{J}\equiv(\mathrm{max}_{j}[E_{J,j}]-\mathrm{min}_{j}[E_{J,j}])/\bar{E}_{J,j}$.

This stems from the fact that a large value of $\delta E_{J}$ determines the
frequency splitting between the nearly degenerate first and second excited
states of the loop. The interference of scattering pathways mediated by these
states is responsible for signal circulation in the device [29], so that when
this splitting is much larger than the waveguide coupling strength, an
external drive cannot simultaneously couple strongly to both of the excited
states, thus limiting the nonreciprocal interference.

Furthermore, the device reported in Ref. [25] was sensitive to charge
fluctuations. In particular, quasiparticle tunneling between the
superconducting islands created four quasiparticle sectors, each with a
distinct scattering response [29]. The average quasiparticle lifetime of the
device in Ref. [25] was found to be $\tau^{\rm{(qp)}}\sim 200\,\mu\mathrm{s}$.

The second iteration of the three-junction-loop circulator reported in Fedorov
_et al._ [26] (also without shunt capacitors) featured a more geometrically
symmetric design with a smaller spread in Josephson energies $\delta E_{J}\sim
2.2\%$. Additional design optimisation ensured that the capacitance matrix of
the system was electrically symmetric. With these improvements, the device
showed significant nonreciprocity and reached circulation fidelity $\sim 80\%$
(post-selected over quasiparticle sectors), for both clockwise and counter-
clockwise circulation. The average quasiparticle lifetime was also improved to
$\tau^{\rm{(qp)}}\sim 4\,\mathrm{ms}$, due to enhanced infrared shielding and
a change in electronic parameters to reduce the charge-parity-switching rates.

To attain higher circulation performance in the three-junction system
investigated in Ref. [26], modelling predicted that we need $\delta
E_{J}\lesssim 1\%$ [29], which is more constrained than standard junction
fabrication precision allows for. Instead, this level of precision can only be
reliably assured with post-fabrication treatments such as laser annealing
[35].

In the following sections, we demonstrate an alternative approach to improving
the circulation performance, based on Fano interference arising from the
inclusion of shunt capacitors. We first develop a detailed theoretical model
of the system, and use this to show that including relatively large waveguide
shunt capacitors in the circulator design relaxes the required junction
fabrication precision, so that circulation becomes more robust against
variations in Josephson junction energies. We then implement the device
design, shown in Fig. 1b, to demonstrate the experimental performance of the
system, confirming good agreement between theory and experiment, as well as
high quality circulation, after accounting for quasiparticle noise.

##  III Modelling and Analysis

In this section, we develop a theoretical input-output model, which we use to
show that introducing direct capacitive couplings between the waveguides, as
shown in Fig. 1a, enhances the circulation fidelity close to the ideal, even
when asymmetry in Josephson energies is relatively large.

###  III.1 SLH master equation

We derive a master equation for a three-junction loop capacitively coupled to
input-output waveguides, including direct waveguide shunt capacitors mutually
coupling the waveguides. We begin by first describing the bare Hamiltonian of
the junction loop [27, 28, 29]

$\displaystyle\hat{H}_{\rm loop}$ $\displaystyle=$ $\displaystyle
E_{C_{\Sigma}}\big{(}(\hat{n}^{\prime}_{1}-\tfrac{1}{2}(n_{0}+n_{g,1}-n_{g,3}))^{2}$
(2)
$\displaystyle+(\hat{n}^{\prime}_{2}+\tfrac{1}{2}(n_{0}+n_{g,2}-n_{g,3}))^{2}-\hat{n}^{\prime}_{1}\hat{n}^{\prime}_{2}\big{)}$
$\displaystyle-
E_{J,1}\cos(\hat{\phi}^{\prime}_{1}-\tfrac{1}{3}\phi_{x})-E_{J,2}\cos(\hat{\phi}^{\prime}_{2}-\tfrac{1}{3}\phi_{x})$
$\displaystyle-
E_{J,3}\cos(\hat{\phi}^{\prime}_{1}+\hat{\phi}^{\prime}_{2}+\tfrac{1}{3}\phi_{x}),$

which depends on a single charging energy $E_{C_{\Sigma}}$ (under the
assumption that the system capacitances are symmetric), three Josephson
energies $E_{J,j}$, three charge biases $n_{g,j}$, and a flux bias $\phi_{x}$.
In Eq. (2), the charge operators $\hat{n}^{\prime}_{1}$ and
$\hat{n}^{\prime}_{2}$, the conversed total charge $n_{0}$, and the phase
(difference) operators $\hat{\phi}^{\prime}_{1}$ and $\hat{\phi}^{\prime}_{2}$
are related to the original charge and phase operators of the superconducting
islands as follows, $\hat{n}^{\prime}_{1}=\hat{n}_{1}$,
$\hat{n}^{\prime}_{2}=-\hat{n}_{2}$,
$n_{0}=\hat{n}_{1}+\hat{n}_{2}+\hat{n}_{3}$,
$\hat{\phi}^{\prime}_{1}=\hat{\phi}_{1}-\hat{\phi}_{3}$, and
$\hat{\phi}^{\prime}_{2}=\hat{\phi}_{3}-\hat{\phi}_{2}$. The form of the
kinetic (charging) energy in Eq. (2) implies that the loop Hamiltonian
$\hat{H}_{\rm loop}$ actually depends on the relative bias charges between the
islands, $n_{g,1}-n_{g,3}$ and $n_{g,2}-n_{g,3}$. Therefore, tuning two charge
biases only, e.g., $n_{g,1}$ and $n_{g,2}$, as well as the flux bias
$\phi_{x}$ suffices to control the total operation of the circulator device.
In terms of its eigenbasis $\\{|k\rangle;k=0,1,2,\dots\\}$, $\hat{H}_{\rm
loop}$ is expressed as $\hat{H}_{\rm loop}=\sum_{k\geq
1}\hbar\omega_{k}|k\rangle\langle k|$, where $\omega_{k}$ is the loop
transition frequency from the ground state $|0\rangle$ to the excited state
$|k\rangle$ ($k\geq 1$).

We consider injecting single-mode weak coherent fields with coherent
amplitudes $\langle\hat{\bm{a}}^{\rm
in}\rangle=(\alpha_{1},\alpha_{2},\alpha_{3})^{\intercal}$ at a drive
frequency $\omega_{\rm d}$ to the three waveguide ports. Following the
‘Scattering-Lindblad-Hamiltonian’ (SLH) formalism [36], we model the coherent
input fields, the coupled waveguides, and the junction loop as three cascaded
systems with their SLH triples respectively given by

$\displaystyle G_{\rm d}$ $\displaystyle=$ $\displaystyle(\mathds{1}_{3\times
3},\hat{\bm{L}}_{\rm d},0),$ (3) $\displaystyle G_{\rm wg}$ $\displaystyle=$
$\displaystyle({\bf A},0,0),$ (4) $\displaystyle G_{\rm loop}$
$\displaystyle=$ $\displaystyle(\mathds{1}_{3\times 3},\hat{\bm{L}}_{\rm
loop},\hat{H}_{\rm loop}),$ (5)

where $\hat{\bm{L}}_{\rm
d}=(\alpha_{1}\hat{\mathds{I}},\alpha_{2}\hat{\mathds{I}},\alpha_{3}\hat{\mathds{I}})^{\intercal}$
denotes the coupling operators associated with the input drive fields and
$\hat{\bm{L}}_{\rm
loop}=(\sqrt{\Gamma}\hat{q}_{1,-},\sqrt{\Gamma}\hat{q}_{2,-},\sqrt{\Gamma}\hat{q}_{3,-})^{\intercal}$
denotes the coupling operators associated with the junction loop. Here
$\Gamma$ is the coupling strength between the junction loop and the waveguides
and $\hat{q}_{j,-}=\sum_{k<\ell}\langle
k|\hat{q}_{j}|\ell\rangle|k\rangle\langle\ell|$ are the upper triangularised
parts (in the junction loop eigenbasis) of the operators $\hat{q}_{j}$, where
$\hat{q}_{1}=\hat{n}^{\prime}_{1},\hat{q}_{2}=-\hat{n}^{\prime}_{2},$ and
$\hat{q}_{3}=-\hat{n}^{\prime}_{1}+\hat{n}^{\prime}_{2}$.

$G_{\rm d}$ and $G_{\rm loop}$ represent three-port systems, while $G_{\rm
wg}$ represents a six-port scattering system, with three ‘exterior’ and three
‘interior’ ports; its $6\times 6$ scattering matrix $\bf A$ is derived in
Appendix A. As analysed in Appendix B, the ‘interior’ ports of $G_{\rm wg}$
are connected to the ports of $G_{\rm loop}$ in a feedback configuration. We
therefore apply both the SLH series and feedback rules [36] to cascade the
total drive-waveguide-loop SLH triple

$G_{\rm tot}=G_{\rm d}\triangleleft(G_{\rm wg}\hookleftarrow G_{\rm loop}),$
(6)

which collectively describes a three-port device. Here we have introduced a
new SLH composition notation, $A\hookleftarrow B$ to indicate that systems $A$
(outer) and $B$ (inner) are coupled in a feedback loop. Equation 6 is the
basis for the SLH modelling, and further details are provided in Appendix B,
and illustrated in Fig. 7c.

Given $G_{\rm tot}$, the Linblad master equation for the density operator
$\rho$ of the loop system in a frame rotating at the drive frequency
$\omega_{\rm d}$ is given by

$\dot{\rho}=-i[\hat{H}^{\prime}_{\rm
tot},\rho]+\sum_{j=1}^{3}\mathcal{D}[\hat{a}^{\rm out}_{j}]\rho,$ (7)

where
$\mathcal{D}[\hat{o}]\rho=\tfrac{1}{2}(2\hat{o}\rho\hat{o}^{\dagger}-\rho\hat{o}^{\dagger}\hat{o}-\hat{o}^{\dagger}{\hat{o}}\rho)$
and

$\displaystyle\hat{H}^{\prime}_{\rm tot}$ $\displaystyle=\hat{H}^{\prime}_{\rm
loop}+\hat{H}_{\rm s}+\hat{H}_{\rm d},$ (8) $\displaystyle\hat{\bm{a}}^{\rm
out}$ $\displaystyle=\hat{\bm{L}}_{\rm w\hookleftarrow l}+S_{\rm
w\hookleftarrow l}\hat{\bm{L}}_{\rm d}.$ (9)

Here $\hat{H}^{\prime}_{\rm loop}=\sum_{k\geq 1}(\omega_{k}-\omega_{\rm
d})|k\rangle\langle k|$, $\hat{H}_{\rm s}$ and $\hat{H}_{\rm d}$ represent the
frequency shifts and the driving fields to the junction loop system,
$\hat{\bm{a}}^{\rm out}=(\hat{a}^{\rm out}_{1},\hat{a}^{\rm
out}_{2},\hat{a}^{\rm out}_{3})^{\intercal}$ denotes the output fields, and
$\hat{\bm{L}}_{\rm w\hookleftarrow l}$ and $S_{\rm w\hookleftarrow l}$ are the
coupling operators and the scattering matrix of the feedback-reduced cascaded
system $G_{\rm wg}\hookleftarrow G_{\rm loop}$. Explicit expressions of the
operators in Eqs. (8) and (9) are given in Appendix B. Since
$V^{\mathrm{in}}_{j}=K\alpha_{j}$ and
$V^{\mathrm{out}}_{j}=K\langle\hat{a}^{\mathrm{out}}_{j}\rangle$, where $K$ is
a conversion factor and
$\langle\hat{\mathcal{O}}\rangle=\mathrm{Tr}(\hat{\mathcal{O}}\rho)$, the
scattering matrix $S$ with its elements
$S_{ji}=V^{\mathrm{out}}_{j}/V^{\mathrm{in}}_{i}=\langle\hat{a}^{\mathrm{out}}_{j}\rangle/\alpha_{i}$
can be computed numerically from Eq. 7 and Eq. 9, to obtain
$\langle\hat{a}^{\mathrm{out}}_{j}\rangle$.

Figure 2:  Optimised circulation fidelity,
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}$, as
a function of the Josephson-energy spread $\delta E_{J}$ for increasing values
of the inter-waveguide capacitances, $C_{X}=\\{0,75,150\\}\,\rm fF$. The
fidelity is computed from the scattering matrix $S$ computed with the master
equation in Eq. 7. The junction loop parameters are
$E_{C_{\Sigma}}/h=3.09\,\rm{GHz}$ and $E_{J,2}=15.03\,\mathrm{GHz}$, and we
allow variations in $E_{J,1/3}$ as $E_{J,1}=E_{J,2}(1-\delta E_{J}/2)$, and
$E_{J,3}=E_{J,2}(1+\delta E_{J}/2)$. The black circle marks the point where
$\delta E_{J}=3\%$, $C_{X}=75\,\rm fF$, yielding an optimised fidelity
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}\gtrsim
97\%$, which is consistent with the measured circulation fidelity in Fig. 5.

###  III.2 Reduced sensitivity to junction asymmetry

Figure 3: Circuit diagram of the experimental setup inside the dilution
refrigerator. The ports of the device are connected to radial microwave
switches to allow for bypassing the device and calibrating the input and
output lines. The inputs $V^{\rm in}_{j}$ are sent through attenuators to the
device. DC voltages $V_{g,j}$ are added to the RF input lines via bias tees at
room temperature. Each of the outputs $V^{\rm out}_{j}$ from the device goes
through an eccosorb filter, a circulator, two isolators, a band pass filter, a
HEMT amplifier, and room temperature amplifiers.

The inclusion of the waveguide shunt capacitors modifies the scattering
mechanism in the proposed circulator. In particular, in Appendix C we
adiabatically eliminate the loop degrees of freedom to approximate the system
scattering matrix, at low drive powers, as $S=({\mathds{1}}+R_{\rm
loop}).S_{\rm w\hookleftarrow l}$, where $R_{\rm loop}$ represents the
response of the junction loop to the external drives. This shows that
scattering of the whole capacitively-shunted-waveguide and junction-loop
system is mediated via two pathways, namely, a direct pathway represented by
the waveguide scattering $S_{\rm w\hookleftarrow l}$ and the loop-mediated
scattering represented by $R_{\rm loop}$. This gives rise to the Fano
interference effect with asymmetric line shapes in the transmission and
reflection spectra [31, 32].

In what follows, we provide numerical evidence that the waveguide shunt
capacitors reduce the sensitivity of the circulation fidelity to variations in
the Josephson junctions fabricated in the loop. To this end, we numerically
solve the master equation in Eq. 7, compute the output fields, and determine
the scattering matrix $S$.

For simulation purposes, we take $E_{C_{\Sigma}}/h=3.09\,\rm GHz$,
$E_{J,2}/h\equiv E_{J}/h=15.03\,\mathrm{GHz}$ to match experimental parameters
discussed in Sec. IV, following, and we allow $E_{J,1/3}$ to vary as
$E_{J,1}=E_{J}(1-\delta E_{J}/2)$, and $E_{J,3}=E_{J}(1+\delta E_{J}/2)$,
where $\delta E_{J}$ is the fractional spread in junction energies. For
$0\leq\delta E_{J}\leq 5\%$, we numerically optimise the flux and charge
biases, and the drive frequency to find the optimal clockwise circulation
fidelity at each $\delta E_{J}$. This ‘maximally asymmetric’ choice of
junction energies for $E_{J,1/3}$ relative to the mean value $E_{J}$
represents the worst-case-scenario circulation fidelity [29], so provides an
estimate of the worst-case effect of asymmetry on the circulation performance.

The simulation results are shown in Fig. 2, where we plot the optimised
circulation fidelity,
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}$, as
a function of the Josephson-energy spread $\delta E_{J}$ for three values of
the inter-waveguide capacitances $C_{X}\in\\{0,75,150\\}\,\mathrm{fF}$. We
observe that for small junction spread, the fidelity is high, i.e.,
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}\approx
1$ for $\delta E_{J}\lesssim 1\%$, regardless of the value of $C_{X}$. For
larger asymmetry, the fidelity improves with increasing $C_{X}$. In
particular, the curve for $C_{X}=75\,\mathrm{fF}$, which corresponds to our
experimental value, shows high circulation performance up to $\delta E_{J}\sim
3\%$. These simulations indicate that large inter-waveguide capacitances
substantially enhance the robustness of the circulation fidelity against
Josephson-junction asymmetry.

##  IV Experimental results

###  IV.1 Device fabrication and measurement

The device shown in Fig. 1b was composed of four layers of aluminium deposited
on a high resistivity silicon wafer with different thicknesses to reduce
quasiparticle tunneling by gap engineering [37]. The first layer of 100 nm
formed the capacitors and the basic structure of the junction loop. Standard
double-angle evaporation was then used to deposit two layers of aluminium, of
20 nm and 60 nm respectively, with a single oxidation step between the two
aluminium deposition stages to form three Josephson junctions. After
evaporation, the chip was diced and bonded on a holder suitable for cryogenic
measurements in a dilution refrigerator operating at a base temperature of 10
mK.

The device was characterised using a fast, room-temperature microwave switch
to sequentially direct the external drive to each of the three input ports for
100 $\mu$s, and measuring the response at the three output ports with a vector
network analyser (VNA), as shown in Fig. 3.

###  IV.2 Spectral response

We first measured the spectral response of the system as a function of the
magnetic flux in the junction loop provided by a small external coil mounted
on the bottom of the sample holder. In Fig. 4, the measured spectrum, which
features a characteristic Y-shape [28, 29, 25, 26], shows good agreement with
the theoretical model, and the fitting provides an estimate for the electronic
parameters of the device.

Specifically, the model fitting returns an on-site charging energy
$E_{C_{\Sigma}}/h=(2e)^{2}/(hC_{\Sigma})=3.09$ GHz, which corresponds to a
total island capacitance of $C_{\Sigma}=C_{g}+C_{C}+3C_{J}=37.5$ fF, where
$C_{g}$, $C_{C}$, and $C_{J}$ are the ground, waveguide-loop coupling, and
junction capacitances respectively. This is close to the design value of
$C_{\Sigma}=40$ fF, from designed capacitance values $C_{g}=2.3$ fF,
$C_{C}=27.5$ fF, and $C_{J}=3.5$ fF. The fitted values for the Josephson
energies are $E_{J,(1,2,3)}/h=\\{14.73,15.15,15.22\\}\,\rm GHz$ with a spread
$\delta E_{J}=3.2\%$. These values are reasonably consistent with the measured
room-temperature junction resistances
$R_{J,(1,2,3)}=\\{6.86,7,7.02\\}\,\mathrm{k}\Omega$ [38], noting that junction
resistances may drift slightly, relative to one-another, over several days.

###  IV.3 Scattering analysis

We next measured the full $3\times 3$ complex-valued scattering matrix $S$.

Figure 4: (a) Spectral response for voltage scattering from port 1 to port 2
while sweeping over the flux coil voltage. The raw voltage output at each
frequency is scaled so that the far-off-resonant background data (i.e. where
the flux coil voltage is $<0.4\textrm{ V}$) has zero mean and unit variance;
the gray-scale thus represents the output signal-to-noise ratio of $|S_{21}|$.
(b) Fitted model spectral lines superimposed on the measured spectrum,
including four distinct quasiparticle sectors; the model spectrum obscure the
experimental features across the flux and frequency scan, indicating high
quality agreement between theory and experiment. The horizontal features in
the data are attributed to weak resonances from reflections in the cabling.
The fitting process also converts the dimensional flux coil voltage into a
dimensionless flux bias $\phi_{x}=2\pi\Phi_{x}/\Phi_{0}$. Different colours
correspond to transition frequencies from the ground state to the first
excited state (blue), second (orange), third (green), and fourth (red). The
fourfold multiplet within the fitted spectrum is due to four quasiparticle
sectors.

The raw data obtained at the VNA includes the response of the scattering
matrix $S$ of the capacitively-shunted on-chip circulator, along with the
attenuation, amplification, and loss inside the cabling in the dilution
refrigerator. To isolate the device scattering matrix, $S$, we use a two-step
calibration procedure. In the first step, we introduced three radial microwave
switches at the mixing chamber stage inside the dilution refrigerator (see
Fig. 3). These switches allowed us to bypass the device and measure
transmission through all the possible combinations of the input and output
lines. Assuming that the lines were matched to $50\,\Omega$ and there were no
reflections at the switches, we calibrated out the transfer functions of the
cables and amplifiers up to the magnetic shield of the device [26]. In the
second step, we fitted the measured data to a model which accounts for
additional losses within the magnetic shield induced predominantly by the
eccosorb filters. This process yielded a unitary matrix that represents the
device response.

The sampling time for each scattering matrix measurement was
$\tau_{s}=300\,\mu\mathrm{s}$. As in our previous work [25, 26], we observed
the characteristic jumps between discrete output voltage states which are
classified and attributed to four different quasiparticle sectors with a
$K$-means classifier. The results of the quasiparticle analysis and
classification reported here exactly replicates the process we developed and
described in Ref. [26]. The typical characteristic dwell times for the four
quasiparticle sectors were measured as
$\tau^{\rm(qp)}_{1,2,3,4}=\\{3.48,3.61,4.23,3.14\\}\,\mathrm{ms}$, which is
comparable to those in our previous device [26]. This classification then
allowed us to compute the circulation fidelity in each of the four
quasiparticle sectors. We then tuned the charge and flux biases to maximise
the measured fidelity for one of the quasiparticle sectors. Fixing these
biases at a working point with a high circulation fidelity, we measured the
scattering matrix $S$ while scanning over the drive frequency.

The extracted scattering matrix $S$ of the device in the optimised sector is
shown in Fig. 5a (lighter-coloured), where we see strong nonreciprocity with
$|S_{12}|\ll|S_{21}|$, $|S_{23}|\ll|S_{32}|$, and $|S_{31}|\ll|S_{13}|$ around
7.25 GHz. In addition, we observe asymmetric line shapes, which are indicative
of Fano interference, as well as noticeable background oscillations in some of
the scattering matrix elements, which we attribute to weak resonances arising
from back-reflection in the cabling (see Appendix E for more details).

We also show in Fig. 5a the results of model simulations (darker-coloured),
where we used the same device parameters obtained from the spectral fit in
Fig. 4 to compute the theoretical scattering matrix. The scattering matrix fit
yields a value of the inter-waveguide capacitance $C_{X}=76\,\mathrm{fF}$,
which is reasonably close to the design value from finite-element
electrostatic simulations (84 fF). The fit also returns the waveguide-loop
coupling strength of $270\,\mathrm{MHz}$, which is roughly double the value in
Fedorov  _et al._ [26] due to a larger design ratio $C_{C}/C_{\Sigma}$ [29].

Figure 5: (a) Spectra of optimised clockwise-circulating S-matrix elements. We
note the small oscillations with period $\sim 200\textrm{ MHz}$ are likely due
to weak reflections in the cabling, described in Appendix E. (b) Clockwise and
counter-clockwise fidelities for this set of S-matrix data. (c) Clockwise and
counter-clockwise fidelities of the S-matrix data optimally biased for
counter-clockwise circulation, showing that circulation direction in the
device can be changed electronically.

In Fig. 5b, we show the clockwise and counter-clockwise circulation
fidelities,
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}$ and
${\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}$
defined in Eq. 1, of the measured scattering spectrum in Fig. 5a (lighter-
coloured), along with the theoretical fidelities (darker-coloured). The peak
clockwise circulation fidelity measured in Fig. 5b is
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}=0.97$
with a correspondingly small counter-clockwise circulation fidelity of
${\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}=0.12$,
showing significant clockwise signal circulation. This is consistent with the
theoretical simulations in Fig. 2, where we predict that at a Josephson-energy
spread $\delta E_{J}=3.2\%$ circulation fidelity ${\mathcal{F}}>0.96$ is
achievable for a waveguide shunt capacitance of $C_{X}\sim 75\,\mathrm{fF}$.

Figure 6: (a) Clockwise circulation performance for the scattering data in
Fig. 5a, including the average insertion loss ${\rm
IL}={\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}^{2}$,
the average isolation ${\rm
IS}={\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}^{2}$,
and the average power reflectance ${\rm R}={\mathcal{R}}^{2}$. The range of
each curve indicated by a shaded region is computed from the smoothed maxima
and minima over the terms in the corresponding definition in Eq. 1. (b)
Counter-clockwise fidelity
$\mathcal{F}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}$ at an
off-resonant drive frequency $7.46\,\mathrm{GHz}$ as a function of the input
power obtained from both theory and experiment, showing the saturation power
of about $P_{3{\rm dB}}\approx-126\textrm{ dBm}$.

.

The direction of signal circulation in our device can be dynamically switched
with the external voltage control biases. Figure 5c shows the fidelities
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}$ and
${\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}$ versus
the drive frequency at a voltage-bias tuned for counter-clockwise circulation;
the results look very similar to those in Fig. 5b, except their roles are
exchanged. The peak counter-clockwise circulation fidelity measured in Fig. 5c
is
${\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{$\circlearrowleft$}}}=0.98$,
and
${\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{$\circlearrowright$}}}=0.05$
is correspondingly small.

For completeness, we present the scattering matrices for the other
quasiparticle sectors in Appendix D. As in Ref. [26], the clockwise
circulation performance in the other, unoptimised quasiparticle sectors is
significantly worse than the sector reported in Fig. 5a.

###  IV.4 Circulation performance and power dependence

We analyse the device performance as a clockwise circulator by defining the
average insertion loss ${\rm IL}$, the average isolation ${\rm IS}$, and the
average power reflectance ${\rm R}$ respectively as

${\rm
IL}={\mathcal{F}}_{\,\operatorname{\rotatebox{-90.0}{{\hbox{\circlearrowright}}}}}^{2},\hskip
14.22636pt{\rm
IS}={\mathcal{F}}_{\,\operatorname{\rotatebox{90.0}{{\hbox{\circlearrowleft}}}}}^{2},\hskip
14.22636pt{\rm R}={\mathcal{R}}^{2}.$ (10)

These quantities are computed from the scattering data in Fig. 5a and are
shown in Fig. 6a, where we find that at the resonance frequency $7.25$ GHz,
${\rm IL}=0.2$ dB, ${\rm IS}=18$ dB, and ${\rm R}=-15$ dB. In addition, Fig.
6a shows ${\rm IL}<1\,\rm dB$ over a bandwidth of $90\,\rm MHz$, while ${\rm
IS}>14\,\rm dB$ over a bandwidth of $85\,\rm MHz$. The circulator device
studied here thus exhibits a ten-fold improvement in the insertion loss
relative to the earlier device reported in [26] (for which ${\rm IL}=2\,\rm
dB$, ${\rm IS}=14\,\rm dB$, and ${\rm R}=-11\,\rm dB$ at resonance). Based on
the model predictions shown in Fig. 2, we attribute this to the inclusion of
the shunt capacitors, which reduces the required junction fabrication
precision.

Finally, to complete the characterisation of the device performance we
measured the saturation power. In Fig. 6b, we show the dependence of the
measured counter-clockwise fidelity on the input signal power (at an off-
resonance drive frequency $7.46\,\mathrm{GHz}$), as well as including
numerical simulations. The 3 dB compression point is $P_{3\rm
dB}\approx-126\,\rm dBm$, which is the same as in our previous circulator
device [26].

Microwave circulation in the device arises from quantum interference between
the ground and excited states of the junction loop. The junction-loop energy
spectrum is strongly anharmonic, and so we expect the device saturation power
to correspond to the arrival of one drive photon per excited-state lifetime,
$\tau_{e}=(\Gamma\,|\langle e|\hat{n}_{a}|g\rangle|^{2})^{-1}$. Using our
fitted device parameters, we find $\Gamma\approx 270\,\mathrm{MHz}$ and
$|\langle e|\hat{n}_{a}|g\rangle|^{2}\approx 0.3$, which implies a saturation
power $P_{\rm sat}\approx hf/\tau_{e}=-124\,\rm dBm$, consistent with the
experimentally measured value for $P_{3\rm dB}$ above.

##  V Discussion and Conclusions

In this work, we analysed an improved design for an on-chip superconducting
circulator based on a three-Josephson-junction loop. The key advance over our
earlier results [27, 28, 25, 26] was the addition of shunt capacitors between
the input-output waveguides to induce Fano interference between the scattering
pathways. We showed theoretically and experimentally that circulation in the
capacitively-shunted, three-junction loop is substantially less sensitive to
Josephson junction asymmetry and exhibits notable improvement in the
circulation performance.

This simple design modification increased the tolerable junction asymmetry to
$\delta E_{J}\lesssim 3\%$, removing the need for additional fabrication post-
processing to fine-tune the Josephson junctions towards higher symmetry. The
approach may also be applicable to other multi-component interference devices
in reducing the sensitivity to typical imprecision in component fabrication.

The device insertion loss, isolation, and return loss, which are postselected
on the optimised quasiparticle sector, are comparable to the performance of
commercial ferrite circulators. The measured bandwidth, though not as large as
in commercial devices, is already sufficient for some practical applications,
for example, single qubit readout [39]. However, the saturation power and the
nonequilibrium quasiparticles, which cause random switching into and out of
the high-performance sector, remain barriers to making the device practically
useful.

###### Acknowledgements.

This work was funded through a commercial research contract with Analog
Quantum Circuits (AQC) Pty. Ltd. TMS and AK each declare a financial interest
in AQC. The authors acknowledge assistance from the Centre for Microscopy and
Microanalysis at the University of Queensland, and the Australian National
Fabrication Facility, ANFF-Q.

##  Appendix A Scattering between capacitively coupled waveguides

We decompose the circuit model in Fig. 1a into a system of capacitively
coupled waveguides and the three-junction loop. The waveguide system shown
Fig. 7a has three ‘exterior’ input/output ports denoted by $\bm{a}^{\rm
in/out}=(a^{\rm in/out}_{1},a^{\rm in/out}_{2},a^{\rm
in/out}_{2})^{\intercal}$ that couple to external fields, and three ‘interior’
input/output ports denoted by $\bm{b}^{\rm in/out}=(b^{\rm in/out}_{1},b^{\rm
in/out}_{2},b^{\rm in/out}_{2})^{\intercal}$ that couple to the junction loop.
The exterior ports couple to each other via the shunt waveguide capacitors
$C_{X,j}\equiv C_{X}$ and couple to the interior ports via coupling capacitors
$\tilde{C}_{C,j}\equiv\tilde{C}_{C}$. Later calculations will take the limit
$\tilde{C}_{C}\to\infty$ to account for the actually galvanic connection
between the exterior and interior ports.

In the following, we derive the $6\times 6$ scattering matrix $\bf A$ for the
system of waveguides in Fig. 7a. We assume that they are capacitively
connected to each other at the end points $x=0$. Following the theory of
lossless semi-infinite waveguides in Ref. [40], the voltage and current at the
end point $x=0$ of the waveguide $a_{j}$ (or $b_{j}$), denoted respectively as
$V_{a_{j}/b_{j}}$ and $I_{a_{j}/b_{j}}$, are given by

$\displaystyle V_{a_{j}/b_{j}}$ $\displaystyle=$ $\displaystyle
V_{a_{j}/b_{j}}^{\rm out}+V_{a_{j}/b_{j}}^{\rm in},$ (11) $\displaystyle
I_{a_{j}/b_{j}}$ $\displaystyle=$ $\displaystyle\frac{V_{a_{j}/b_{j}}^{\rm
out}-V_{a_{j}/b_{j}}^{\rm in}}{Z_{\rm wg}},$ (12)

where $V_{a_{j}/b_{j}}^{\rm out}$ and $V_{a_{j}/b_{j}}^{\rm in}$ are the
output and input voltages, and $Z_{\rm wg}$ is the waveguide impedance.

Figure 7: We describe the (a) capacitively coupled waveguides. Formally, we
take $\tilde{C}_{C}\rightarrow\infty$, to compute the transfer matrix for
${\bm{a}}^{\rm{in}}\rightarrow{\bm{b}}^{\rm{out}}$. (b) The $6\times 6$
scattering matrix $\bf{A}$ from Eq. 20, relating the triplets of input and
output modes of the capacitive shunts, where
${\bm{a}}^{\rm{in/out}}=({a_{1}}^{\rm{in/out}},{a_{2}}^{\rm{in/out}},{a_{3}}^{\rm{in/out}})$
are the ‘exterior’ inputs and outputs to/from the waveguides, and
${\bm{b}}^{\rm{in/out}}$ are the ‘interior’ input and output modes to/from the
junction loop. The junction loop (c) is described by (d) a $3\times 3$
junction loop scattering matrix $\bf{B}$ relating the loop input modes,
${\bm{c}}^{\rm{in}}$, to the loop output modes, ${\bm{c}}^{\rm{out}}$, as well
as the internal Hamiltonian evolution of the loop degrees of freedom. The
scattering matrix for the (e) waveguide-loop system from Fig. 1 is described
by the SLH formalism in which $\bf{A}$ and $\bf{B}$ are coupled in a feedback
loop, shown in (f), with the internal modes constrained so that
${\bm{b}}^{\rm{out}}={\bm{c}}^{\rm{in}}$ and
${\bm{b}}^{\rm{in}}={\bm{c}}^{\rm{out}}$ [36]. The heavier lines indicate
triplets of modes. Not shown here is the scattering system of the external
coherent drives that couple to the exterior ports of the waveguides.

We apply Kirchhoff’s current law at the coupling points $x=0$ of the
waveguides $a_{j}$ and $b_{j}$ and find that

$\displaystyle
I_{a_{j}}+\frac{V_{b_{j}}-V_{a_{j}}}{Z_{\tilde{C}_{j}}}+\sum_{j^{\prime}\neq
j}\frac{V_{a_{j^{\prime}}}-V_{a_{j}}}{Z_{C_{j,j^{\prime}}}}$ $\displaystyle=$
$\displaystyle 0,$ (13) $\displaystyle
I_{b_{j}}+\frac{V_{a_{j}}-V_{b_{j}}}{Z_{\tilde{C}_{j}}}$ $\displaystyle=$
$\displaystyle 0,$ (14)

where $Z_{C_{j,j^{\prime}}}=1/i\omega_{\rm d}C_{X}$ and
$Z_{\tilde{C}_{j}}=1/i\omega_{\rm d}\tilde{C}_{C}$. Using Eqs. (11) and (12),
we rewrite Eqs. (13) and (14) in terms of $V^{\mathrm{out}}_{a_{j}/b_{j}}$ and
$V^{\mathrm{in}}_{a_{j}/b_{j}}$

$\displaystyle\frac{V^{\rm out}_{a_{j}}-V^{\rm in}_{a_{j}}}{Z_{\rm
wg}}+\frac{V^{\rm out}_{b_{j}}+V^{\rm in}_{b_{j}}-V^{\rm out}_{a_{j}}-V^{\rm
in}_{a_{j}}}{Z_{\tilde{C}_{j}}}$ $\displaystyle+\sum_{j^{\prime}\neq
j}\frac{V^{\rm out}_{a_{j^{\prime}}}+V^{\rm in}_{a_{j^{\prime}}}-V^{\rm
out}_{a_{j}}-V^{\rm in}_{a_{j}}}{Z_{C_{j,j^{\prime}}}}$ $\displaystyle=$
$\displaystyle 0,$ (15) $\displaystyle\frac{V^{\rm out}_{b_{j}}-V^{\rm
in}_{b_{j}}}{Z_{\rm wg}}+\frac{V^{\rm out}_{a_{j}}+V^{\rm in}_{a_{j}}-V^{\rm
out}_{b_{j}}-V^{\rm in}_{b_{j}}}{Z_{\tilde{C}_{j}}}$ $\displaystyle=$
$\displaystyle 0.$ (16)

These equations can be concisely represented in matrix form

$\left[\mathds{1}+i\omega_{\rm d}Z_{\rm wg}{\bf
C}\right]\left[\begin{array}[]{c}\bm{V}^{\rm out}_{a}\\\ \bm{V}^{\rm
out}_{b}\end{array}\right]=\left[\mathds{1}-i\omega_{\rm d}Z_{\rm wg}{\bf
C}\right]\left[\begin{array}[]{c}\bm{V}^{\rm in}_{a}\\\ \bm{V}^{\rm
in}_{b}\end{array}\right],$ (17)

where $\bm{V}_{a/b}^{\rm in/out}=(V^{\rm in/out}_{a_{1}/b_{1}},V^{\rm
in/out}_{a_{2}/b_{2}},V^{\rm in/out}_{a_{3}/b_{3}})$, and the full $6\times 6$
capacitance matrix,

${\bf
C}=\left[\begin{array}[]{cc}{\bm{C}}_{X}-{\bm{C}}_{\Sigma}&{\bm{C}}_{C}\\\
{\bm{C}}_{C}&-{\bm{C}}_{C}\end{array}\right],$ (18)

is defined in terms of the $3\times 3$ partial capacitance matrices
${\bm{C}}_{\Sigma}=(\tilde{C}_{C}+2C_{X})\mathds{1}$,
${\bm{C}}_{C}=\tilde{C}_{C}\mathds{1}$, and

${\bm{C}}_{X}=\left[\begin{array}[]{ccc}0&C_{X}&C_{X}\\\ C_{X}&0&C_{X}\\\
C_{X}&C_{X}&0\end{array}\right].$ (19)

We solve Eq. 17 to obtain the shunt capacitor scattering matrix,

$\displaystyle{\bf A}$ $\displaystyle=\left[\mathds{1}+i\omega_{\rm d}Z_{\rm
wg}{\bf C}\right]^{-1}.\left[\mathds{1}-i\omega_{\rm d}Z_{\rm wg}{\bf
C}\right],$ (20)
$\displaystyle\equiv\left[\begin{array}[]{cc}\bm{A}_{11}&\bm{A}_{12}\\\
\bm{A}_{21}&\bm{A}_{22}\end{array}\right],$ (23)

which implicitly defines the $3\times 3$ submatrices $\bm{A}_{ij}$. This
scattering matrix is represented graphically in Fig. 7b. We take the limit
$\tilde{C}_{C}\to\infty$ and define $z=\omega_{\rm d}Z_{\rm wg}C_{X}$, so that

$\displaystyle{\bm{A}}_{11}$ $\displaystyle=$
$\displaystyle{\bm{A}}_{22}=\frac{z}{2i+3z}\left[\begin{array}[]{ccc}-2&1&1\\\
1&-2&1\\\ 1&1&-2\\\ \end{array}\right],$ (27) $\displaystyle{\bm{A}}_{12}$
$\displaystyle=$
$\displaystyle{\bm{A}}_{21}=\frac{z}{2i+3z}\left[\begin{array}[]{ccc}1+\tfrac{2i}{z}&1&1\\\
1&1+\tfrac{2i}{z}&1\\\ 1&1&1+\tfrac{2i}{z}\\\ \end{array}\right].$ (31)

Figure 8: Scattering matrices of the other three quasiparticle sectors,
obtained from the same data analysis that produced the scattering matrix in
Fig. 5a with high clockwise circulation. The (clockwise) circulation
performance in each of these sectors is substantially worse than in the
optimised sector.

##  Appendix B SLH master equation

We use the SLH formalism [36] to model the input-output network in our device.
It consists of the external coherent drives, the coupled waveguides, and the
junction loop. The drives are modeled as a three-port component following the
source model; its SLH triple is given by

$G_{\rm d}=(\mathds{1}_{3\times 3},\hat{\bm{L}}_{\rm d},0),$ (32)

where $\hat{\bm{L}}_{\rm
d}=(\alpha_{1}\hat{\mathds{I}},\alpha_{2}\hat{\mathds{I}},\alpha_{3}\hat{\mathds{I}})^{\intercal}$
with $\alpha_{j}$ the drive amplitudes. The coupled waveguides are a six-port
component without any coupling operators nor system Hamiltonian with its SLH
triple of the form

$G_{\rm wg}=({\bf A},0,0),$ (33)

where $\bf A$ is given in Eq. 20. The junction loop depicted in Figs. 7c and
7d is a three-port component represented by an SLH triple

$G_{\rm loop}=(\mathds{1}_{3\times 3},\hat{\bm{L}}_{\rm loop},\hat{H}_{\rm
loop}),$ (34)

where $\hat{\bm{L}}_{\rm
loop}=(\sqrt{\Gamma}\hat{q}_{1,-},\sqrt{\Gamma}\hat{q}_{2,-},\sqrt{\Gamma}\hat{q}_{3,-})^{\intercal}$
and $\hat{H}_{\rm loop}$ is given in the main text.

The total network is cascaded as

$G_{\rm tot}=G_{\rm d}\triangleleft G_{\mathrm{w}\hookleftarrow\mathrm{l}},$
(35)

where $G_{\mathrm{w}\hookleftarrow\mathrm{l}}=G_{\rm wg}\hookleftarrow G_{\rm
loop}$ describes the feedback loop concatenation between the interior ports of
$G_{\rm wg}$ and the ports of $G_{\rm loop}$, as described in Figs. 7e and 7f.
We compute $G_{\mathrm{w}\hookleftarrow\mathrm{l}}$ by first cascading
$G_{\mathrm{w}\triangleleft\mathrm{l}}=G_{\rm wg}\triangleleft(G_{3}\boxplus
G_{\rm loop})$, where $G_{3}=(\mathds{1},0,0)$ and find that

$G_{\mathrm{w}\triangleleft\mathrm{l}}=({\bf A},\left[\begin{array}[]{c}0\\\
\hat{\bm{L}}_{\rm loop}\end{array}\right],\hat{H}_{\rm loop}).$ (36)

$G_{\mathrm{w}\triangleleft\mathrm{l}}$ has six ports: its ‘upper’ ports
denoted as 1 do not involve any coupling operators and its ‘lower’ ports
denoted as 2 correspond to $\hat{\bm{L}}_{\rm loop}$ and are looped back to
themselves. We then use the feedback rule to eliminate these internal degrees
of freedom. The reduced three-port waveguide-loop SLH triple $G_{\rm
w\hookleftarrow l}=[G_{\rm w\triangleleft l}]_{2\hookleftarrow 2}$ is given by

$\displaystyle S_{\rm{w\hookleftarrow l}}$ $\displaystyle=$
$\displaystyle\bm{A}_{11}+\bm{A}_{12}(\mathds{1}-\bm{A}_{22})^{-1}\bm{A}_{21},$
(37a) $\displaystyle\hat{\bm{L}}_{\rm w\hookleftarrow l}$ $\displaystyle=$
$\displaystyle\bm{A}_{12}(\mathds{1}-\bm{A}_{22})^{-1}\hat{\bm{L}_{\rm
loop}},$ (37b) $\displaystyle\hat{H}_{\rm w\hookleftarrow l}\\!$
$\displaystyle=$ $\displaystyle\\!\hat{H}_{\rm
loop}-\tfrac{i}{2}(\hat{\bm{L}}_{\rm
loop}^{\dagger}\bm{A}_{22}(\mathds{1}-\bm{A}_{22})^{-1}\hat{\bm{L}}_{\rm
loop}-\rm{h.c.}).$

Finally, the total SLH triple $G_{\rm tot}$ in Eq. 35 is

$\displaystyle S_{\rm tot}$ $\displaystyle=S_{\rm w\hookleftarrow l},$ (38a)
$\displaystyle\hat{\bm{L}}_{\rm tot}$ $\displaystyle=\hat{\bm{L}}_{\rm
w\hookleftarrow l}+S_{\rm w\hookleftarrow l}\hat{\bm{L}}_{\rm d},$ (38b)
$\displaystyle\hat{H}_{\rm tot}$ $\displaystyle=\hat{H}_{\rm w\hookleftarrow
l}-\tfrac{i}{2}(\hat{\bm{L}}_{\rm w\hookleftarrow l}^{\dagger}S_{\rm
w\hookleftarrow l}\hat{\bm{L}}_{\rm d}-\rm{h.c.}).$ (38c)

We decompose $\hat{H}_{\rm tot}$ as $\hat{H}_{\rm tot}=\hat{H}_{\rm
loop}+\hat{H}_{\rm s}+\hat{H}_{\rm d},$ where

$\displaystyle\hat{H}_{\rm s}$ $\displaystyle=$
$\displaystyle-\tfrac{i}{2}(\hat{\bm{L}}_{\rm
loop}^{\dagger}\bm{A}_{22}(\mathds{1}-\bm{A}_{22})^{-1}\hat{\bm{L}}_{\rm
loop}-\rm{h.c.}),\quad\quad$ (39a) $\displaystyle\hat{H}_{\rm d}$
$\displaystyle=$ $\displaystyle-\tfrac{i}{2}(\hat{\bm{L}}_{\rm w\hookleftarrow
l}^{\dagger}S_{\rm w\hookleftarrow l}\hat{\bm{L}}_{\rm d}-\rm{h.c.}),$ (39b)

which respectively describe the frequency shifts and the driving to
$\hat{H}_{\rm loop}$. Given these, we obtain the master equation for the
junction loop’s density operator (in a rotating frame at the drive frequency
$\omega_{\rm d}$)

$\dot{\rho}=-i[\hat{H}^{\prime}_{\rm
tot},\rho]+\sum_{j=1}^{3}\mathcal{D}[\hat{L}_{\mathrm{tot},j}]\rho.$ (40)

Equation (38b) in fact is nothing but the input-output relation, where
$\hat{\bm{L}}_{\rm tot}\equiv\hat{\bm{a}}^{\rm out}$, $\hat{\bm{L}}_{\rm
w\hookleftarrow l}$, and $S_{\rm w\hookleftarrow l}\hat{\bm{L}}_{\rm d}$
represent the output fields, the system’s response, and the input fields,
respectively. This reproduces the master equation Eq. 7 in the main text. We
note that when $C_{X}=0$, one finds $\hat{H}_{\rm s}=0$ and $S_{\rm
w\hookleftarrow l}=\mathds{1}$, which reduces Eq. 40 or Eq. 7 to the SLH
master equation used in Refs. [28, 29, 25, 26].

Figure 9: Comparison of the on-resonant scattering matrix data (solid blue) in
Fig. 5a with the off-resonant background response (dashed brown), showing very
similar oscillations away from the resonant frequency $7.25\,\rm GHz$. Also
shown is the computed shunt-capacitor scattering amplitudes (dot-dashed green)
from Eq. 38a, assuming $C_{X}=75.7$ fF. We attribute the period $\sim
200\textrm{ MHz}$ to weak reflections in cabling, over a length of about 0.5
m.

##  Appendix C Adiabatic elimination

We consider a semi-analytical derivation for the scattering matrix $S$ via
adiabatic elimination of the SLH triple [36], which provides us useful
insights into the operation of the proposed circulator. In particular, we
assume that the drive fields are weak so that the junction-loop system is
mostly populated in its ground state [28]. This allows us to separate the
Hilbert space of the loop into a fast subspace $\mathcal{F}=\\{|k\rangle,k\geq
1\\}$ that contains its excited states and a slow subspace
$\mathcal{S}=\\{|0\rangle\\}$ that contains only its ground state. We then
eliminate the dynamics of the fast subspace while considering only that of the
slow subspace (for more details, see Appendix B in Ref. [29]). By doing so, we
find that the scattering matrix $S$ within the slow subspace is given by

$S=({\mathds{1}}+R_{\rm loop}).S_{\rm w\hookleftarrow l},$ (41)

where $R_{\rm loop}$ represents the response of the junction loop to the
external coherent drives. We note that a similar expression to Eq. 41 was
derived in Ref. [41] for a resonator coupled to multiple ports in the presence
of a direct scattering channel between the ports. The matrix elements of
$R_{\rm loop}$ are [29]

$(R_{\rm loop})_{ij}=-\sum_{k>0}\frac{\langle
0|\hat{L}_{\mathrm{w\hookleftarrow l},i}|k\rangle\langle
k|\hat{L}_{\mathrm{w\hookleftarrow
l},j}|0\rangle}{i\Delta\omega_{k}+\Gamma_{k}/2},$ (42)

where $\Delta\omega_{k}$$=\omega_{k}-\omega_{\rm d}$ and $\Gamma_{k}=\langle
k|\hat{\bm{L}}_{\rm loop}^{\dagger}\bm{A}_{\rm s}\hat{\bm{L}}_{\rm
loop}|k\rangle$, with $\bm{A}_{\rm
s}=\bm{A}_{22}(\mathds{1}-\bm{A}_{22})^{-1}$. Here $\Gamma_{k}$ represents the
waveguide-induced decay rate as well as the frequency shift of the excited
state $|k\rangle$. We numerically confirm that the S-matrices computed via the
full SLH master equation Eq. 7 and via adiabatic elimination Eq. 41 agree very
well with each other.

##  Appendix D Scattering matrices of other quasiparticle sectors

In Fig. 8 we show the measured scattering matrices for the other quasiparticle
sectors obtained from the same data analysis that yielded the scattering
matrix reported in Fig. 5a. The clockwise circulation performance is evidently
poor for these, compared with the best sector in Fig. 5a.

##  Appendix E Background oscillations

The S-matrix elements shown in Fig. 5a and Fig. 8 have noticeable oscillatory
responses with peak-to-trough amplitude of about 0.2, and a characteristic
period of about 200 MHz. We attribute these to weak, resonant reflections in
the cabling, over a length of around 0.5 m.

Figure 9 reproduces the resonant circulation data (solid blue) in Fig. 5a, as
well as similar scattering data (dashed brown) taken with the flux-bias
selected to fully detune the device from the frequency band shown. The data
sets were taken several days apart. In addition, the pure theoretical
scattering matrix amplitudes for a pure shunt capacitor array, computed with
Eq. 38a, are shown (dot-dashed green), which broadly match the off-resonance
scattering amplitudes, apart from the small oscillations attributed to the
cable back-reflections.

Both on-resonant and off-resonant data sets have similar characteristic
oscillations with period $\sim 200\textrm{ MHz}$, indicating that these are
systematic, and reasonably stable. In some panels, far from the resonant
feature around 7.25 GHz, the oscillations match closely. In others, the
oscillations are qualitatively similar, but do not match exactly, suggesting
some subtle drifts over time.

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