Chiral cavity quantum electrodynamics
In the format provided by the
authors and unedited
Supplementary information
https://doi.org/10.1038/s41567-022-01671-3
Chiral Cavity Quantum Electrodynamics
John Clai Owens,
1, 2
Margaret G. Panetta,
1
Brendan Saxberg,
1
Gabrielle Roberts,
1
Srivatsan
Chakram,
3
Ruichao Ma,
4
Andrei Vrajitoarea,
1
Jonathan Simon,
1
and David Schuster
1
1
James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA
2
Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute,
California Institute of Technology, Pasadena, California 91125, USA
3
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA
4
Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA
(Dated: April 24, 2022)
SUPPLEMENTARY INFORMATION
A. Cryogenic Setup
1
B. Transmon Fabrication
1
C. Lattice Fabrication
3
D. Cryogenic Chiral Lattice Site Characterization
4
E. Lattice Disorder Characterization and
Compensation
5
F. Probing Lattice Dynamics and Spectra
5
G. Transmon Characterization
6
H. Spectroscopy of the Lattice Modes Coupled to the
Qubit
8
I. Structure of Pulsed Measurements with Qubit
9
J. Mode Dependence of Fig. 3 Swap Rate
9
K. Mode Dependence of Dispersive Shift
11
L. Numerical Modeling of the
|
g
0
⟩↔|
f
1
⟩
Oscillations in Fig. 3b
12
M. Estimating Uncertainties for Fig. 3
12
References
12
Supplement A: Cryogenic Setup
The cryogenic setup employed for measuring the lat-
tice coupled to the qubit is shown in Fig. S1 and is similar
to the setup used to measure the lattice prior to the in-
troduction of the qubit. The lattice is mounted on the
mixing chamber (MXC) plate of a Bluefors LD-250 dilu-
tion refrigerator at
∼
31 mK. The lattice sites depicted
in blue are connected to a 10-way cryogenic switch which
is in turn connected to a circulator so that these sites
can all connect to either an input line or an output line.
Sites in green are connected to only input lines and thus
can only be excited, not measured. The site (1
,
1) is
connected separately to a circulator so that it can be
measured and excited independently of the blue lattice
sites. The remaining six sites in the 5x5 lattice are not
connected to a either an input line or output line. The
qubit readout resonator is also connected to a circula-
tor so that reflection measurements can be performed to
measure the state of the qubit. The qubit is excited off-
resonantly through the readout resonator. Each output
line is filtered by an eccosorb filter to suppress accidental
high-frequency qubit excitation.
For the measurements performed without a qubit (see
Fig. 2 in the main text),
two
cryogenic switches were
used within the fridge, enabling direct probing of 20 sites,
while a 21st site was independently connected to a cir-
culator. The only sites not measured were the 4 chiral
cavities, whose modes are more localized at the bottom of
the cavity, making coupling to them via a dipole antenna
difficult without spoiling the quality factor.
A Keysight PNA-X N5242 is used to perform lattice
spectroscopy. For the qubit measurements, a Keysight
arbitrary waveform generator (M8195A, 64 GSa/s) is
used to synthesize a local oscillator signal near the
qubit frequency, while Berkeley Nucleonics 845-M mi-
crowave synthesizers provide separate local oscillator sig-
nals near lattice and dressing frequencies. The local oscil-
lators are then I/Q modulated by Keysight PXIe AWGs
(M3202A, multichannel, 1GS/s) to generate the indi-
vidual qubit drive, qubit readout, and dressing pulses.
The qubit drive and readout pulses are combined out-
side the fridge and sent to the readout resonator (Fig. S2
and Fig. S1). The reflected readout signal is routed
to the output line via circulators and amplified with a
HEMT amplifier at 4 K and additional room temperature
amplifiers (Miteq AFS3-00101200-22-10P-4, Minicircuits
ZX60-123LN-S+). The signal is then demodulated using
an IQ mixer and recorded using a fast digitizer (Keysight
M3102A, 500 MSa/s). A schematic layout of the room-
temperature components of the experimental setup can
be found in Fig. S2.
Supplement B: Transmon Fabrication
The transmon qubit is made of aluminum deposited
on a ø50
.
8mm, 430
μ
m thick C-place (0001) sapphire
wafer. The wafer was annealed at 1200
◦
C for 1.5 hours.
2
MXC plate ~31 mK
Still
4 K
50 K
300 K
...
...
30 dB
20 dB
30 dB
20 dB
Eccosorb
fi
lter
Quinstar
4-12 GHZ
isolator
Radiall
R574F32005
RF switch
LNF 8-12 GHz
single junction
circulator
Quinstar
double junction
circulator
XMA SS cryo
attenuator
XMA gold-plated
copper threaded
cryo attenuator
4-12 GHz
HEMT
SMA
feedthrough
Lines to 8
lattice sites
Lines to 10
lattice sites
Lattice
Transmon
Qubit drive
and readout
Lattice site drive
Readout
Miteq ampli
fi
er
Minicircuits
ZX60-123LN-S+
30 dB
Resonator + qubit drive
Readout
30 dB20 dB
Single lattice site drive
Readout
20 dB
20 dB
20 dB
20 dB
FIG. S1.
Partial diagram of cryogenic setup.
Ex-
periments involving the qubit were performed in a Bluefors
LD-250 dilution refrigerator operating at approximately 31
mK. The antennas coupled to some lattice sites (highlighted
in blue) are accessible via an input and readout line connected
by a circulator, allowing reflection measurements on individ-
ual sites. Other antennas coupled to lattice sites (highlighted
in green) are accessible via input line only.
The wafer was then cleaned with toluene, acetone, iso-
propanol, and DI water in an ultrasonic bath immedi-
ately prior to junction deposition. The transmon was
deposited in a single layer. The mask for deposition
was defined by electron-beam lithograthy using a Raith
EBPG5000 Plus E-Beam Writer. The mask was a bi-
layer of resist made of a stack of MMA and PMMA. The
capacitor pads were rectangles with dimensions 50 mi-
crons by 1100 microns and were written with the 40 nA
beam. The junctions were patterned Manhattan-style
using a finer 1 nA beam. In the intermediate region that
connects the junction to the capacitor a beam of 4 nA was
used. Before writing, a 50 nm layer of Au was thermally
evaporated onto the wafer in order to provide adequate
grounding the the electron beam. After writing the re-
sist, the resist stack was developed for 1.5 minutes in a
solution of 3 parts IPA and 1 part DI water chilled on a
Q
RF
LO
I
BNC 645 AWG
Trigger
Trigger
Trigger
Keysight M3202A
1Gs/s AWG
Keysight M3201A
500 Ms/s AWG
Keysight M3102A
500 Ms/s digitizer
SR 445A
preampli
fi
er
LO
LO
Keysight PXIe Chassis
Minicircuits
ZFSC-2-10G+
splitter
Marki MLIQ218L
1851 IQ mixer
HP 33320H
manual atten.
Minicircuits
ZX60-832-N-S+
ampli
fi
er
3 dB SMA atten.
20 dB SMA atten.
Digital atten.
0-30 dB
+ insertion loss
Minicircuits
VLFX-80+
lowpass
fi
lter
Micro Lambda
MLFI-1017 tunable
YIG bandpass
fi
lter
LO
Lattice site drive
Readout
Resonator +
qubit drive
Single lattice site drive
Minicircuits
ZVE-3W-83+ high
power ampli
fi
er
Q
RF
LO
I
Q
RF
LO
I
FIG. S2.
Schematic of room-temperature experimen-
tal setup.
In any given measurement, one of the three read-
out lines pictured in Fig. S1 is selected for use. Drives on
lattice sites, important for the pulsed measurements with se-
quences detailed in Fig. S11, are either passed directly into
the refrigerator after I/Q modulation or, in the case of strong
dressing tones, are filtered by a YIG (Yttrium Iron Garnet)
tunable bandpass filter and amplified before use.
6
◦
C cold plate. The junction was then deposited using
electron-beam evaporation in three steps. First, 80 nm
of aluminum was deposited at an angle of 40
◦
relative
to the plane of the wafer and parallel to a finger of the
junction. Next, the junction was oxidized in 20 mbar of
high purity oxygen for 10 minutes. Last, a 45 nm layer
of aluminum was deposited at the same evaporation an-
gle of 40
◦
but orthogonal to the first evaporation. After
evaporation, the rest of resist as well as the aluminum
attached to the resist were removed via lift-off in an 80
◦
C solution of PG remover for 3 hours. The wafer was
thereafter diced into the dimensions that fit the lattice
setup.
3
Supplement C: Lattice Fabrication
The lattice is machined from a solid block of niobium,
which exhibits a low
H
c
1
to allow magnetic fields to pen-
etrate the material, and a high
H
c
2
to ensure that the
field does not quench the superconductivity [1]. High
niobium purity is not required since the magnetic field
used to bias the YIG spheres already limits the quality
factor of the cavities. The resonators are arranged in a
5
×
5 square lattice, which is the minimum lattice size
that supports a clear distinction between bulk and edge.
While the edge channels predominantly reside in sites on
the lattice perimeter, they have some participation in the
sites one removed from the edge, exponentially decaying
into the bulk. Each lattice site consists of a
λ
4
coaxial
resonator described in detail in our earlier work [2]. The
frequency of the lattice site is inversely dependent on the
length of the post in the center of the cavity. By adding
indium foil to the end of a post the site frequency can
be tuned to a precision better than 1 MHz. In order to
maximize the quality factor of the resonator, the depth
of the cavity is set so that the evanescent decay from the
post mode is much less than the residual resistive loss
of the superconductor. The niobium lattice is mounted
in a copper box that is screwed into the niobium in or-
der to adequately thermalize the niobium to the mixing
chamber plate of the dilution refrigerator in which it is
placed. Antennas are mounted onto the lid of the copper
box so that a single antenna protrudes into each lattice
site from the top of the cavity. The length of the antenna
sets the coupling quality factor of each lattice site. For
these measurements, each lattice site is weakly coupled to
its antenna so that the total quality factor of the lattice
is maximized.
Special care is required to maintain cryo- and qubit
compatibility with the ferrimagnetic elements and their
associated B-fields. The YIG spheres are located physi-
cally inside of their lattice cavities in order to couple to
the microwave modes, so magnetic field must be routed
to the YIG spheres through bulk superconductor. We
make the cavities out of the type II superconductor nio-
bium so that magnetic fields can penetrate it while it
is in the vortex superconducting state. We place YIG
spheres in divots machined between the three posts of
the lattice cavities hosting chiral modes. These spheres
rest freely until secured in place by their attraction to
the permanent magnets used for field bias. To create
the bias field, we place a 1.6 mm diameter permanent
neodymium cylindrical magnet in a hole outside of the
cavity but directly underneath each YIG sphere, leaving
a 0.3 mm thick layer of niobium between the magnet and
the inside of the cavity and the YIG (Fig. 1b). The prox-
imity of the magnet to the YIG sphere achieves a bias
field of
∼
0
.
2 T on the YIG sphere, while the small size
of the magnet minimizes the amount of field that passes
through the posts of the cavity and the amount of normal
or vortex state niobium in areas with large current flow.
This bias field achieves splittings between the two chiral
rotating modes of the YIG cavity of 2
π
×
200 MHz while
retaining cavity quality factors of 2
×
10
5
. Transmission
through the two chiral modes of the YIG cavity is shown
in Fig. S3.
An advantage of this use of YIG spheres is the
method’s robustness to orientation issues of the bias field
and YIG resonance. Disorder in coupling between YIG
sphere and ‘bright’ (
p
x
−
ip
y
) chiral mode does not map
to disorder in the Hofstadter Hamiltonian we have real-
ized. Because the Chern insulator is realized using the
chiral three-post cavity modes
p
x
+
ip
y
which are ‘dark’
to the YIG spheres, the exact form of the coupling of
the YIG resonances to the ‘bright’ chiral modes is less
important than the fact this coupling splits these modes
away from resonance and isolates the ‘dark’ modes for
use in the lattice Hamiltonian.
Cavity quality factors were measured after machining
the lattice, inserting the ferrites (YIG spheres) to relevant
cavities, and applying magnetic fields. After machining
but before applying any surface treatment, lattice sites
had quality factors of
∼
2
×
10
6
. Introducing the YIG
sphere to the cavity does not degrade the Q, so long
as no additional materials are added to hold the YIG
sphere in place. No such additional materials were used
for the lattice described in this paper. After applying the
magnetic field in the final configuration, cavity quality
factors dropped to
∼
2
×
10
5
, suggesting that the limiting
loss factor of the cavity modes is the resistive losses in
the normal regions created by the magnetic field piercing
the superconductor.
The cavities are coupled together via holes milled into
the back side of the niobium block. These holes open
paths between lattice sites that allow the Wannier func-
tions of the lattice sites to overlap with their neighbors,
creating coupling between sites. These coupler holes cre-
ate additional coupling via a virtual coupling mechanism,
in which the couplers act as higher frequency resonators.
In some (non-cryogenic) lattices we added a screw that
could tune the frequency of the coupler lower, allowing
us to achieve greater couplings between lattice sites (up
to 150 MHz). For the cryogenic lattice design we reduced
the tunneling between sites to
J
= 2
π
×
18 MHz. This was
done because we wanted to preserve the quality factor of
the lattice modes by using less magnetic field, which in
turn reduced the amount by which we broke time-reversal
symmetry. In order to increase both the quality factor of
the modes and the tunneling ratio, more effective meth-
ods of funneling magnetic field are required.
Three readout cavities for qubits are machined on the
edge of the lattice, though for the results shared in this
letter only a single qubit was introduced to one such cav-
ity. These cavities are designed to be much higher in fre-
quency (
∼
10
.
5 GHz) than the main lattice so that they
do not interact directly with it. The qubit is mounted so
that its capacitor pads act as antennas that can directly
couple the qubit both to the readout cavity and to the
lattice modes (see Fig. 1c). The antenna that couples to
the readout cavity is inserted into the bottom of this cav-
4
ity so that it can attain a low coupling Q (20,000), while
being short enough that the modes introduced by the
antenna are much higher in frequency than the readout
cavity.
Supplement D: Cryogenic Chiral Lattice Site
Characterization
A major innovation of this work was its design of
a way to break time-reversal symmetry in the lattice
while maintaining low-loss modes and compatibility with
qubits that are sensitive to magnetic fields. In prior work
we designed interactions between 3D microwave photons
and a DC magnetic field in a room temperature alu-
minum lattice [2, 3]. This kind of interaction is medi-
ated via a ferrite sphere (made of Yttrium Iron Garnet,
or YIG) placed inside a lattice cavity. A DC magnetic
field is applied to the YIG sphere through the cavity, tun-
ing magnon modes of the sphere into resonance with the
cavity modes. The hybridization of the chiral magnon
modes with the 3D cavity modes results in time-reversal-
symmetry-broken cavity modes. Lattice cavities popu-
lated with YIG spheres are engineered to host two degen-
erate modes. One of the modes’ magnetic fields at the
center of the cavity precesses clockwise, and the other
precesses counter-clockwise. This opposite chirality of
the modes creates a difference in the coupling strength
between the two cavity modes and the chiral YIG mode.
The mode that precesses with the same chirality as the
YIG sphere couples more strongly to the chiral YIG mode
than the mode with the opposite chirality. In Fig. S3, we
show the mode frequencies of the cavity as a function
of an applied uniform magnetic field on a cavity with a
YIG sphere inside. At low fields, the two cavity modes
are nearly degenerate, but as the bias magnetic field is
increased, the YIG mode comes into resonance with the
cavity and splits the cavity modes by up to 400 MHz.
The color of plotted data indicates the phase acquired
during transmission through the cavity via ports that
are placed 45
◦
apart with respect to the cavity center.
Transmission through one of the chiral modes results in
a phase accumulation of
π
2
radians, while the other chi-
ral mode acquires a phase of
3
π
2
=
−
π
2
. The observed
phase is twice the physical angle between the two ports
because we are comparing
S
21
and
S
12
to remove global
reciprocal phases accrued in cables.
In practice, routing the magnetic field to the ferrite
poses a challenge when using superconducting lattices,
as magnetic fields either are repelled by superconductors
or induce normal regions that greatly increase the cav-
ity loss. In Fig. S4 we show the effects of an increasing
magnetic field applied to a niobium cavity. The field is
uniform, applied externally at 12 Kelvin (K) before the
cavity superconducts. For each data point the cavity is
warmed to 12 K before the field is changed. At the fields
used to bias the YIG sphere to the cavity frequency (350
mT as shown in Fig. S3), the quality factor of the cav-
Transmission
FIG. S3.
Splitting
p
x
±
ip
y
modes in a B-field.
Chiral
cavity mode frequencies are measured as a function of mag-
netic field applied to the YIG sphere. This plot, drawn from
our earlier work [2], shows both the magnitude and phase of
measured transmission; the color scale at top right illustrates
how hue represents its phase while saturation represents its
magnitude normalized to a maximum transmission value. For
each slice in the y-direction, the cavity is warmed above the
T
C
of niobium to 12 K, so that the magnetic field on the
YIG sphere can be changed. The cavity is then cooled to 2 K
and the magnetic flux is locked in place by the superconduct-
ing transition of the cavity. Transmission is then measured
between two antennas placed 45
◦
apart (with respect to the
center of the cavity) so that the phase accumulated in trans-
mission can be measured. In this plot, we take the difference
between transmission from antenna 1 to 2 and transmission
from antenna 2 to 1 to isolate the non-reciprocal phase shift
in the cavity. The two cavity modes split in frequency when
the magnetic field is applied and the phase shifts of the modes
are
−
90
◦
and +90
◦
due to the opposite chirality of the modes.
When a B-field is applied to the YIG sphere, the
p
x
−
ip
y
cav-
ity mode is split away, isolating the
p
x
+
ip
y
mode for use in
the Hofstadter lattice.
ity would be as low as 10
4
. Additionally, the Josephson
junction in the qubit is made out of aluminum, a type I
superconductor with a low
H
c
1
[4]. Simply applying an
external magnetic field would adversely effect both qubit
and cavity lifetimes.
To minimize the magnetic fields permeating the sys-
tem, we generate a
local
magnetic field in the vicin-
ity of the YIG spheres with small neodymium magnets
(
∼
1
.
5mm diameter). In order to achieve the required
field at each ferrite, we insert a magnet into a small hole
milled into the back the cavity (shown in Fig. 1) directly
beneath the ferrite. This minimizes the amount of nio-
bium which sees its superconductivity quenched by the
B-field, as the strongest part of the magnetic field is lo-
5
a
b
FIG. S4.
Quality factor of chiral cavities. a,
Quality
factor of a three post cavity as a function of magnetic field.
For every measurement, the cavity is warmed above niobium’s
superconducting transition of 9 K to 12 K so that the mag-
netic field can be changed. The cavity then is cooled to 2 K
within the magnetic field and the quality factor is measured.
The cavity quality factor decreases to 30000 at the field that
tunes the YIG to the cavity resonance.
b,
Quality factor of
a three post cavity with a YIG sphere as a function of tem-
perature while the magnetic field is held at zero. The YIG
sphere lowers the quality factor of the cavity but the quality
factors still reach
∼
2 million.
calized to the area between the ferrite and the magnet:
indeed, the bulk of the modal surface currents flows in
the posts of the cavity, locations where the B-field has
decayed substantially. As shown in Fig. S5, we generate
sufficient field to break the degeneracy between the two
chiral cavity modes by 200 MHz, whilst maintaining a
quality factor of 200,000. The splitting between these two
modes is a measure of how strongly time-reversal sym-
metry is broken. This splitting also limits the maximum
tunneling rate in the lattice, as the Hofstadter model as-
sumes one orbital per lattice site, requiring the tunneling
energy to remain small enough to avoid coupling to the
counter-rotating orbitals. The ratio between the tunnel-
ing rate and the loss rate in our cavities is then a measure
of how fast the dynamics are compared to the loss rate,
which is an important benchmark for the system that de-
termines how far the photons move within their lifetimes.
In this work, the ratio between tunneling and loss rates
is
18MHz
50kHz
≈
400.
Because the lattice harnesses the mode which is ‘dark’
to the YIG, we are insensitive to the precise magnitude
of the B-field, its orientation relative to the crystal axis,
or coupling to spin-wave modes.
9.1
9.2
9.3
9.4
9.5
Frequency (GHz)
0.5
1.0
1.5
Reflection (arb.)
FIG. S5.
Chiral cavity spectrum with a permanent
magnet.
The magnet is inserted into a hole bored from the
bottom of the cavity so that the magnet sits below the YIG
sphere with
∼
0.5 mm of niobium between the magnet and the
YIG sphere. The ensemble is then cooled to 2 K. The magnet
generates enough B-field to break the degeneracy of the two
chiral modes (
p
x
±
ip
y
) by
∼
200 MHz while still maintaining
a quality factor of 2
×
10
5
.
Supplement E: Lattice Disorder Characterization
and Compensation
Disorder in the lattice site frequencies must be con-
trolled to a level below other energy scales of the sys-
tem Hamiltonian (tunneling, particle-particle interac-
tions, and magnetic field interactions). Lattice sites are
tuned to degeneracy at room temperature, though the
different types of lattice sites (single post, three post
YIG-coupled) change frequency differently as the lattice
sites cool to 20 mK. We first measure the change in fre-
quency from cooling for the different lattice sites (
∼
23
MHz for single post cavities and
∼
40 MHz for cavities
with YIG spheres) and then adjust for the differential
at room temperature by adding indium to the top of
the cavity posts, which effectively lengthens the cavity
post and decreases the cavity frequency. Because indium
is malleable and superconducting, it attaches easily to
the cavity posts and does not decrease the cavity quality
factors. After modifying post lengths with indium, we
adjust each cavity frequency at 1K until the disorder in
lattice site frequencies is less than
±
1 MHz for the single
post cavities and
±
3 MHz for the YIG cavities. We tune
the lattice to
ω
l
≈
2
π
×
8
.
9 GHz for the measurements
without a qubit.
Supplement F: Probing Lattice Dynamics and
Spectra
In order to measure the linear response of the lattice,
we insert dipole antennas into each site and connect them
to a cryogenic switching network which is routed through
circulators to enable performance of reflection measure-
ments on most sites of the lattice (see SI A for an illustra-