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: September 14, 2021)
Cavity quantum electrodynamics, which explores the granularity of light by coupling a resonator
to a nonlinear emitter [1], has played a foundational role in the development of modern quantum
information science and technology. In parallel, the field of condensed matter physics has been
revolutionized by the discovery of underlying topological robustness in the face of disorder [2–4],
often arising from the breaking of time-reversal symmetry, as in the case of the quantum Hall
effect. In this work, we explore for the first time cavity quantum electrodynamics of a transmon
qubit in the topological vacuum of a Harper-Hofstadter topological lattice [5]. To achieve this,
we assemble a square lattice of niobium superconducting resonators [6] and break time-reversal
symmetry by introducing ferrimagnets [7] before coupling the system to a single transmon qubit.
We spectroscopically resolve the individual bulk and edge modes of this lattice, detect vacuum-
stimulated Rabi oscillations between the excited transmon and each mode, and thereby measure
the synthetic-vacuum-induced Lamb shift of the transmon. Finally, we demonstrate the ability to
employ the transmon to count individual photons [8] within each mode of the topological band
structure. This work opens the field of chiral quantum optics experiment [9], suggesting new routes
to topological many-body physics [10, 11] and offering unique approaches to backscatter-resilient
quantum communication.
I. INTRODUCTION
Materials made of light are a new frontier in quantum
many-body physics [12]; relying upon non-linear emitters
to generate strong photon-photon interactions and ultra-
low-loss meta-materials to manipulate the properties of
the individual photons, this field explores the interface
of condensed matter physics and quantum optics whilst
simultaneously producing novel devices for manipulating
light [13, 14]. Recent progress in imbuing photons with
topological properties [15], wherein the photons undergo
circular time-reversal-breaking orbits, promises opportu-
nities to explore photonic analogs of such solid-state phe-
nomena as the (fractional) quantum Hall effect [2, 3],
Abrikosov lattices [16], and topological insulators [4].
In electronic materials, the circular electron orbits re-
sult from magnetic or spin-orbit couplings [4]. Unlike
electrons, photons are charge-neutral objects and so do
not directly couple to magnetic fields. There is thus an
effort to generate synthetic magnetic fields for photons
and more generally to explore ideas of topological quan-
tum matter in synthetic photonic platforms. Significant
progress in this arena has been made in both optical-
and microwave- topological photonics: in silicon photon-
ics [17, 18] and optics [19, 20], synthetic gauge fields have
been achieved while maintaining time-reversal symme-
try by encoding a pseudo-spin in either the polarization
or spatial mode. In RF and microwave meta-materials,
both time-reversal-symmetric [21, 22] and time-reversal-
symmetry-broken models have been explored, with the
T-breaking induced either by coupling the light to fer-
rimagnets in magnetic fields [7, 23] or by Floquet engi-
neering [24].
To mediate interactions between photons, a nonlinear
emitter or ensemble of nonlinear emitters must be in-
troduced into the system [25]. This has been realized
for optical photons by coupling them to Rydberg-dressed
atoms, providing the first assembly of two-photon Laugh-
lin states of light [26]. In quantum circuits, a 3-site lat-
tice of parametrically-coupled transmon qubits enabled
observation of chiral orbits of photons/holes [24], and a
1
×
8 lattice of transmons enabled exploration of Mott
physics [27]. In nanophotonics, a topological interface
enabled helical information transfer between a pair of
quantum dots [28].
In this work, we demonstrate a scalable architecture for
probing interacting topological physics with light. Build-
ing on prior room-temperature work [7], we demonstrate
a 5
×
5 array of superconducting resonators that acts as
a quarter flux (
α
=
1
4
) Hofstadter lattice [5], exhibiting
topological bulk and edge modes for the photons that
reside within it. We couple a single transmon qubit to
the edge of this system, and enter, for the first time,
the regime of strong-coupling cavity quantum electrody-
namics for a highly nonlinear emitter interacting with the
spectrally resolved modes of a topological band structure.
In Section II, we introduce our superconducting topo-
logical lattice architecture compatible with transmon
qubits. We then characterize its properties both spectro-
scopically and spatially in Section III. In Section IV, we
couple a single transmon qubit to the lattice, employing
it to detect and manipulate individual photons in bulk
arXiv:2109.06033v1 [cond-mat.mes-hall] 9 Sep 2021
2
ab
c
N
S
Readout
cavity
Qubit drive
+ readout
Lattice
drive
g
r
g
l
FIG. 1.
Elements of chiral cavity quantum electrody-
namics. a,
The apparatus consists of a 5
×
5,
α
=
1
4
Hof-
stadter lattice [5] of resonators in which microwave photons
propagate as charged particles in a magnetic field, coupled to
a single qubit on the edge that is sensitive to the precise num-
ber of photons and their energies. Each site, implemented as a
coaxial resonator milled into a block of niobium [6], exhibits a
resonance frequency
ω
0
determined by the length of a central
post, and a nearest neighbor tunneling rate
J
determined by
the size of a machined coupling hole. The synthetic magnetic
field manifests as an Aharonov-Bohm flux
π/
2 when photons
hop around minimal closed loops (green), generated by the
spatial structure of the resonator modes: each 2-site by 2-site
plaquette includes one lattice site (red) that exhibits a
p
x
+
ip
y
orbital, while the other three sites exhibit
s
orbitals [7, 10].
The additional site (blue) on the system edge serves as read-
out cavity into which transmons may be inserted.
b,
p
x
+
ip
y
sites instead contain
three
posts and thus support three mi-
crowave modes (
s
,
p
x
±
ip
y
). Because our Hofstadter lattice
employs only the
p
x
+
ip
y
mode, we must isolate it: the
s
mode
is tuned away by the electromagnetic coupling between posts,
while a Yttrium-Iron-Garnet (YIG) ferrimagnet (black) cou-
ples primarily to the
p
x
−
ip
y
mode (due to the orientation of
the B-field of the red/blue bar magnet), thereby detuning it in
energy and isolating the
p
x
+
ip
y
mode.
c,
A transmon qubit
is inserted into a gap between readout (left) and lattice (right)
cavities on a sapphire carrier (turquoise), and couples to the
two cavities with Rabi frequencies
g
r
and
g
l
respectively. An
SMA connector (gold) allows direct microwave probing of this
readout cavity and thus the transmon.
and edge modes of the lattice and to measure the Lamb
shift of this synthetic vacuum. In Section V we conclude,
exploring the opportunities opened by this platform.
II. A SUPERCONDUCTING HOFSTADTER
LATTICE FOR MICROWAVE PHOTONS
In vacuum, photons are neither (i) massive, nor (ii)
charged, nor (iii) confined to two dimensions, the crucial
ingredients for quantum Hall physics [2]. To realize these
essentials, we follow the road-map laid out in [10]: mi-
crowave photons are trapped in a 2D array of microwave
resonators, and thereby confined to two transverse di-
mensions and imbued with an effective mass due to the
finite tunneling rate between the resonators. Rather
than attempting to
actually
imbue photons with electric
charge, we note that when electrons are confined to a
lattice, the entire impact of a magnetic field on their dy-
namics is encompassed by the Aharonov-Bohm-like phase
that they acquire when tunneling around closed trajec-
tories. We engineer this “Peierls phase” via the spatial
structure of the on-site lattice orbitals.
Fig. 1a shows the square Hofstadter lattice that we
have developed for this work. Each square in the diagram
is a lattice site, implemented as a resonator of frequency
ω
0
≈
2
π
×
9 GHz, tunnel-coupled to its nearest-neighbors
with
J
= 2
π
×
18 MHz (see SI C). Sites with counter-
clockwise red arrows exhibit modes with spatial structure
p
x
+
ip
y
, while all other sites have
s
-like modes. The phase
winding in a
p
x
+
ip
y
site causes photons tunneling in/out
from different directions to acquire a phase
φ
=
δθ
, where
δθ
is the angle between input and output directions [7].
This ensures that when photons tunnel around a closed
loop enclosing
n
plaquettes, they pick up an Aharonov-
Bohm-like phase
φ
loop
=
n
π
2
. Such a tight-binding model
with a flux per plaquette of
π/
2 is called a “quarter flux
Hofstadter lattice” [5].
To avoid seam losses and thus achieve the highest qual-
ity factors, the lattice structure is machined from a sin-
gle block of high-purity (RRR=300) niobium and cooled
to 30 mK to reduce loss and eliminate blackbody pho-
tons (see SI A). Sites are realized as coaxial resonators,
while tunneling between adjacent sites is achieved via
hole-coupling through the back side: as in [29], the cou-
pling holes are sub-wavelength and thus do not lead to
leakage out of the structure.
s
-orbital sites are imple-
mented as single post resonators, while
p
x
+
ip
y
sites are
realized with three posts in the same resonator, coupled
to a Yttrium-Iron-Garnet (YIG) ferrimagnet that ener-
getically isolates the
p
x
+
ip
y
mode from the (vestigial)
p
x
−
ip
y
and
s
modes (see Fig. 1b, SI Fig. S3). The T-
symmetry of the ferrimagnet is broken with the
∼
0
.
2
Tesla magnetic field of
∅
1
.
6 mm N52 rare-earth magnets
placed outside of the cavity, as close to the YIG as pos-
sible to minimize quenching of superconductivity (see SI
Fig. S4).
A single fixed-frequency transmon qubit on a sapphire
wafer is inserted between the top-left resonator in the
5
×
5 square lattice and the adjacent readout resonator
(see Fig. 1a); a zoom-in of this setup is shown in Fig. 1c.
III. PROBING THE TOPOLOGICAL LATTICE
We first characterize the mode structure of the topolog-
ical lattice itself in the linear regime, prior to introducing
the transmon nonlinearity. Fig. 2a shows the anticipated
energy spectrum of a semi-infinite strip
α
= 1
/
4 Hofs-
tadter lattice with four bands and topologically protected
edge channels living below the top band and above the
bottom band. In a finite system, these continuum bands
and edge channels fragment into individual modes satis-
fying the boundary conditions. Fig. 2b shows the mea-
sured response of the lattice when probed
spatially
both
within the bulk (left) and on the edge (right), with the
3
FIG. 2.
A superconducting Chern circuit.
The central ingredient of a chiral cavity QED platform is a long-lived,
spectrally-isolated chiral (unidirectional) mode to couple to a real or synthetic atom. For our experiments this mode is a
quantized edge-excitation of a synthetic Hall system realized in a
α
= 1
/
4 Hofstadter square lattice [5]. The numerically-
computed band structure of our implementation of this model is depicted in
a,
for an infinite strip geometry. The top and
bottom bands each exhibit a Chern number
C
=
−
1, while the middle two bands, which touch at Dirac points, have a total
Chern number
C
= +2; chiral edge channels exist above the bottom band and below the top band, as anticipated from the bulk-
boundary correspondence [4].
b,
shows microwave transmission spectra measured through our actual 5
×
5 lattice, where both
the bulk bands and chiral edges manifest as well-resolved resonant modes due to the finite system size. “Bulk” measurements
are performed by exciting and measuring at two distinct sites on the interior of the lattice, while “edge” measurements employ
two sites on the lattice perimeter. In
c,
we measure the spatial structure of the modes identified with arrows in
b
and observe
that, as expected, the mode residing predominantly in the interior of the lattice is located energetically within a lattice band,
while the one localized to the edge resides within an energy gap (see SI F for measurement details). In
d,
we excite a single
edge site at energies within the upper (red) and lower (blue) bulk gaps, and observe the response of the resulting traveling
excitation as a function of time averaged over the full perimeter (
main panels
) and vs. site index around the system edge
(
inset panels
). The insets demonstrate that upper and lower edge channels have opposite chiralities and reflect the numerous
orbits of the pulse before it damps away. The ability of a photon to undergo numerous round trips prior to decay is equivalent
to spectroscopic resolution of the individual edge modes.
energies aligned to Fig. 2a. It is clear that the bulk spec-
trum exhibits modes within the bands, while the edge
spectrum exhibits modes within the bandgaps. We fur-
ther validate that the modes we have identified as “bulk”
and “edge” modes reside in the correct spatial location
by exciting modes identified with arrows in Fig. 2b and
performing full microscopy of their spatial structure in
Fig. 2c.
To demonstrate that the edge channels are indeed long-
lived and chiral (handed), we abruptly excite the system
at an edge site within each of the two bulk energy gaps
in Fig. 2d (see SI F). By monitoring the edge-averaged
response as the excitation repeatedly orbits the lattice
perimeter, we determine that the excitation can circle
the full lattice perimeter
>
20 times prior to decay (see
Fig. 2d). In the insets to Fig. 2d, we probe in both space
and
time, and observe that the excitations move in op-
posite directions in the upper and lower band gaps, as
anticipated from the bulk-boundary correspondence [4].
IV. COUPLING A QUANTUM EMITTER TO
THE TOPOLOGICAL LATTICE
To explore quantum nonlinear dynamics in the topo-
logical lattice we couple it to a transmon qubit (see Fig. 1
and SI G) which acts as a quantized nonlinear emitter
whose properties change with each photon that it ab-
sorbs. Unlike traditional cavity and circuit QED exper-
iments in which a nonlinear emitter couples to a single
mode of an isolated resonator, here the transmon couples
to all modes of the topological lattice. In what follows
we will induce a controlled resonant interaction between
the transmon and individual lattice modes, investigating
4
the resulting strong-coupling physics.
The
|
g
〉↔|
e
〉
transition of the transmon (
ω
q
≈
2
π
×
7
.
8
GHz) is detuned from the lattice spectrum (
ω
lattice
≈
2
π
×
9 GHz) by ∆
≈
2
π
×
1
.
2 GHz. We bring the trans-
mon controllably into resonance with individual lattice
modes via the dressing scheme in Fig. 3a inset (see Meth-
ods and ref. [30] for details); this dressing also gives us
complete control over the magnitude of the qubit-lattice
site coupling.
In Fig. 3a we tune the excited transmon into reso-
nance with individual lattice modes and observe vacuum-
stimulated Rabi oscillations (see SI I) of a quantized exci-
tation between the transmon and the mode. Comparing
with the predicted band structure shown in Fig. 3b, we
see that the transmon couples efficiently to both bulk and
edge modes of the lattice, despite being physically located
on the edge. This is because the lattice is only 5 sites
across, comparable to the magnetic length
l
B
∼
1
/α
= 4
sites, so the lattice site coupled to the transmon has sub-
stantial participation in both bulk and edge modes; fur-
thermore, the system is sufficiently small that the number
of bulk sites is comparable to the number of edge sites,
so all modes have approximately the same “volume.”
To unequivocally demonstrate strong coupling between
the transmon and a single lattice mode, we examine a
single frequency slice of Fig. 3a versus evolution time.
Fig. 3c shows such a slice and demonstrates high-contrast
oscillations that take several Rabi cycles to damp out, as
is required for strong light-matter coupling. For sim-
plicity, we choose our dressed coupling strength to be
less than the lattice mode spacing; stronger dressing to
explore simultaneous coupling to multiple lattice modes
opens the realm of super-strong-coupling physics [31, 32],
where the qubit launches wavepackets localized to smaller
than the system size.
When a qubit is tuned towards resonance with a sin-
gle cavity mode it experiences level repulsion [33] and
then an avoided crossing at degeneracy. The situation is
more complex for a qubit coupled to a full lattice, where
one must account for interactions with
all
lattice modes,
both resonant and non-resonant. In total these couplings
produce the resonant oscillations observed in Fig. 3c plus
a frequency-dependent shift due to level repulsion from
off-resonant lattice modes, which may be understood as a
Lamb shift from coupling to the structured vacuum [34].
We quantify this Lamb shift by comparing the frequen-
cies of the modes observed in linear lattice spectroscopy,
as in Fig. 2a but with the transmon present (see SI H), to
those observed in chevron spectroscopy in Fig. 3a. These
data are shown in the lower inset to Fig. 3a. When the
qubit is tuned near the low-frequency edge of the lat-
tice spectrum it experiences a downward shift from all of
the modes above it, and when it is tuned near the upper
edge of the lattice, it experiences a corresponding upward
shift. These two extremes smoothly interpolate into one
another as modes move from one side of the qubit to
the other. There is also a near-constant Stark shift of
∼
3.5 MHz arising from the classical dressing tone. To
our knowledge, this is the first measurement of the Lamb
shift of a qubit in a synthetic lattice vacuum.
Finally, we demonstrate the ability to count photons
within an individual lattice mode. If the transmon were
coupled to a single lattice site and not to the full lattice,
each photon in that site would shift the qubit
|
g
〉 ↔ |
e
〉
transition by 2
χ
, where
χ
≈
g
2
l
∆
×
α
q
∆+
α
q
≈
2
π
×
5 MHz, and
α
q
is the transmon anharmonicity. This photon-number-
dependent shift, and thus the intra-cavity photon num-
ber, can be measured by performing qubit spectroscopy
detected through the readout cavity (see SI J). When the
transmon is coupled to a lattice rather than an isolated
cavity, the
χ
shift is diluted by the increased volume of
the modes. In Fig. 3d, we inject a coherent state into
the highlighted mode in Fig. 3a and then perform qubit
spectroscopy to count the number of photons within the
mode. The observed spectrum corresponds to a coherent
state with ̄
n
≈
1
.
4, with the individual photon occu-
pancies clearly resolved. Indeed, when we perform this
experiment as a function of the amplitude of the coher-
ent excitation pulse (Fig. 3d, inset), we find a continuous
evolution from vacuum into a Poisson distribution over
the first six Fock states.
V. OUTLOOK
In this work we have demonstrated a photonic materi-
als platform that combines synthetic magnetic fields for
lattice-trapped photons with a single emitter. This has
enabled us to explore interactions between the individual
modes of a topological system and the non-linear excita-
tion spectrum of the emitter, entering for the first time
the realm of fully-granular chiral cavity QED and thus
demonstrating the ability to count and manipulate indi-
vidual photons in each mode of the lattice. We anticipate
that coupling a transmon to a longer edge would enable
qubit-mediated photon-induced deformation of the edge
channel (in the “super strong” coupling limit of the edge
channel [31, 35]), as well as universal quantum compu-
tation via time-bin-encoding [36] or blockade engineer-
ing [37]. Introduction of a qubit to the bulk of this sys-
tem would allow investigation of the shell-structure of a
Landau-photon polariton [11], a precursor to Laughlin
states. Addition of a second qubit on the edge would al-
low chiral, back-scattering-immune quantum communi-
cation between the qubits [9]. Scaling up to one qubit
per site will enable dissipative stabilization [38–41] of
fractional Chern states of light [10] and thereby provide
a clean platform for creating anyons and probing their
statistics [42].
VI. METHODS
The transmon qubit (see SI B) has a
|
g
〉 ↔ |
e
〉
tran-
sition frequency of
ω
q
= 2
π
×
7
.
8 GHz, compared with