of 27
Resonance Fluorescence of a Chiral Artificial Atom
Chaitali Joshi,
*
Frank Yang ,
*
and Mohammad Mirhosseini
Moore Laboratory of Engineering, California Institute of Technology, Pasadena, California 91125, USA
and Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
(Received 21 December 2022; revised 12 April 2023; accepted 26 April 2023; published 26 June 2023)
We demonstrate a superconducting artificial atom with strong unidirectional coupling to a microwave
photonic waveguide. Our artificial atom is realized by coupling a transmon qubit to the waveguide at
two spatially separated points with time-modulated interactions. Direction-sensitive interference arising
from the parametric couplings in our scheme results in a nonreciprocal response, where we measure a
forward/backward ratio of spontaneous emission exceeding 100. We verify the quantum nonlinear behavior
of this artificial chiral atom by measuring the resonance fluorescence spectrum under a strong resonant
drive and observing well-resolved Mollow triplets. Further, we demonstrate chirality for the second
transition energy of the artificial atom and control it with a pulse sequence to realize a qubit-state-dependent
nonreciprocal phase on itinerant photons. Our demonstration puts forth a superconducting hardware
platform for the scalable realization of several key functionalities pursued within the paradigm of chiral
quantum optics, including quantum networks with all-to-all connectivity, driven-dissipative stabilization of
many-body entanglement, and the generation of complex nonclassical states of light.
DOI:
10.1103/PhysRevX.13.021039
Subject Areas: Nonlinear Dynamics, Optics,
Quantum Physics
I. INTRODUCTION
Chiral light-matter interfaces have been long studied in
quantum optics
[1,2]
and promise a myriad of potential
opportunities for developing quantum networks with long-
range connectivity
[3
5]
and generating novel many-body
entangled states of light
[6,7]
and matter
[8,9]
. Light-matter
interaction is said to be
chiral
when the scattering of a
photon from an atom depends strongly on the photon
s
propagation direction in a one-dimensional (1D) wave-
guide
[10]
. This breaking of the symmetry of atom-
waveguide coupling to the right and left propagating modes
gives rise to a range of unique phenomena. For example,
a resonant photon impinging on a chiral atom strongly
coupled to a 1D waveguide acquires a nonreciprocal
π
phase shift conditioned on the state of the atom. This
remarkable effect can be exploited to realize entangling
gates between distant stationary qubits mediated by itin-
erant photons
[11,12]
. In the paradigm of waveguide
quantum electrodynamics (QED), coupling several chiral
two-level systems to a common waveguide results in
novel collective spin dynamics and the formation of exotic
nonequilibrium phases of entangled spin clusters
[9]
.
Conversely, complex nonclassical states of light such as
multidimensional cluster states and Fock states can be
generated efficiently using protocols that rely on determin-
istic chiral atom-photon interactions
[7,13]
. In addition,
chiral atom arrays can be used for generating many-photon
bound states
[14]
and for quantum nondemolition detection
of propagating photons
[15]
.
Chiral atom-photon interfaces have been realized in the
optical domain by coupling atoms and solid-state quantum
emitters to nanophotonic structures, where the strong
confinement of light results in the locking of the local
polarization of a photon to its direction of propagation
[10,16
20]
. Despite remarkable progress in these systems,
achieving strong unidirectional coupling with a chain of
emitters remains challenging due to the relative weakness
of interactions in the case of single atoms and the
environment-induced frequency disorder of solid-state
emitters. More recently, artificial atoms based on super-
conducting qubits have emerged as a powerful platform for
waveguide QED in the microwave domain. These systems
offer control over individual emitters and their coupling
to the environment, as well as the ability to tailor the
dispersion of the electromagnetic modes in waveguides.
Additionally, the relatively large wavelength of radiation at
the gigahertz band allows for the precise placement of
*
These authors contributed equally to this work.
mohmir@caltech.edu
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
s title, journal citation,
and DOI.
PHYSICAL REVIEW X
13,
021039 (2023)
Featured in Physics
2160-3308
=
23
=
13(2)
=
021039(27)
021039-1
Published by the American Physical Society
atoms along a waveguide to control photon-mediated
interactions and collective dissipation
[21,22]
. These
advantages have enabled several demonstrations of wave-
guide QED phenomena with superconducting artificial
atoms, including resonance fluorescence
[23
26]
, Dicke
superradiance and subradiance
[21,22,27,28]
, formation of
qubit-photon bound states
[29,30]
, and the realization of
long-range waveguide-mediated coupling for many-body
quantum simulations
[31]
. Despite this rapid progress,
studying chiral quantum optics with superconducting
qubits remains challenging due to the lack of an efficient
unidirectional interface for microwave photons.
Nonreciprocal transport of microwave photons is pos-
sible using devices based on ferromagnetic or ferrimagnetic
materials, which break Lorentz reciprocity. Recently, three-
dimensional qubit-cavity systems have successfully real-
ized chiral interactions using this approach
[32]
. However,
ferromagnetic devices such as circulators are not suitable
for on-chip integration due to their size, large magnetic
fields, and typically lossy response. Alternatively, low-loss
nonreciprocal components have been realized using
synthetic gauge fields
[33
36]
. While these experiments
demonstrate the nonreciprocal propagation of microwave
photons, a simple and scalable approach for realizing
on-chip chiral interactions with superconducting qubits
remains desirable. More recently, unidirectional emission
and absorption of microwave photons have been proposed
[11,37,38]
and demonstrated
[39
41]
using a pair of
entangled qubits. However, relying on the interference of
the emission from two distinct physical qubits limits the
chiral behavior to weak drives where at most a single
photon is exchanged with a radiative bath. This diluted
quantum nonlinear response forbids a direct realization of
strongly driven-dissipative quantum systems.
Here, we experimentally demonstrate a chiral artificial
atom consisting of a transmon qubit coupled to a trans-
mission line at two spatially separated points, operating
in the so-called
giant-atom
regime
[42
45]
. The emitted
field components from the two coupling points are
imparted a relative phase using time-modulated parametric
couplings. In this setting, chirality arises from the inter-
ference between the two emission pathways resulting from
the phase difference from the parametric couplings and the
direction-dependent phase delay from propagation in the
waveguide. We show highly directional atom-waveguide
coupling, with the rate of spontaneous emission to the
forward-propagating modes exceeding that of backward-
propagating modes by more than 2 orders of magnitude.
Relying on a single physical qubit as the emission source,
our scheme is robust against decoherence and preserves
quantum nonlinear response under strong drives. We
demonstrate this quantum nonlinearity using resonance
fluorescence measurements and observe Mollow triplets
under a strong resonant drive. The chiral response is further
shown to be continuously tunable and extends to the
transmon qubit
s second transition (
j
e
i
j
f
i
). Finally,
we use time-domain control to realize a qubit-state-
dependent response to traveling photons in the waveguide.
The minimal hardware overhead in our experiment, com-
bined with near-perfect directionality,
in situ
control of the
coupling, and access to higher-order chiral transitions,
provide a scalable platform for future studies of driven-
dissipative entanglement generation, quantum networks
with all-to-all coupling, and cascaded quantum systems
with superconducting qubits.
II. DESIGN OF THE ARTIFICIAL ATOM
As we show in Fig.
1(a)
, our experiment is based on a
planar transmon qubit (hereafter called the
emitter
)
coupled at two locations to an on-chip coplanar waveguide.
We use a dissipation port at each coupling point, which is
designed to realize a complex coupling rate to the wave-
guide modes with a well-defined phase (
φ
l;r
for the left or
right ports). The Hamiltonian for this system is given
by
ˆ
H
¼
ˆ
H
atom
þ
ˆ
H
field
þ
ˆ
H
int
, where
ˆ
H
atom
=
¼
ω
ge
j
e
ih
e
j
and
ˆ
H
field
=
¼
R
dk
ω
k
ð
ˆ
a
k;f
ˆ
a
k;f
þ
ˆ
a
k;b
ˆ
a
k;b
Þ
are the free
Hamiltonians for the atom and the field modes, respec-
tively. The integral in
ˆ
H
field
is performed over positive
values of the photonic wave vectors
k
, and the subscripts
denote forward (
f
) and backward (
b
) propagating modes
in the waveguide. The atom-waveguide interaction
Hamiltonian can be written as
ˆ
H
int
=
¼
X
i
l;r
Z
dk
½
̃
g
k;i
ˆ
σ
ð
ˆ
a
k;f
e
ikx
i
þ
ˆ
a
k;b
e
ikx
i
Þþ
H
:
c
:

;
ð
1
Þ
where
̃
g
k;l
ð
r
Þ
denotes the complex atom-waveguide cou-
pling strength at the left (right) coupling points,
ˆ
σ
is the
emitter
s lowering operator, and
x
l
ð
r
Þ
is the position of
the left (right) coupling point along the waveguide [see
Fig.
1(a)
]. Assuming a waveguide with linear dispersion,
the photonic mode resonant with the
j
g
i
j
e
i
transition of
the atom (frequency
ω
ge
) has a wave vector
k
¼
ω
ge
=v
,
where
v
is the speed of propagation of the modes in the
waveguide. Denoting the atom-waveguide coupling
strength for this mode as
̃
g
l
ð
r
Þ
at the left (right) coupling
points, the decay rate of the atom to the waveguide is given
by
κ
em
;l
ð
r
Þ
¼
4
π
j
̃
g
l
ð
r
Þ
j
2
D
ð
ω
ge
Þ
, where
D
ð
ω
Þ
is the density of
states in the waveguide
[42]
. The emission field of the atom
acquires a phase
φ
l
ð
r
Þ
¼
arg
½
̃
g
l
ð
r
Þ

at the coupling points.
In addition, the distance
d
¼
x
r
x
l
sets the propagation
phase
φ
WG
¼
ω
ge
d=v
between the two coupling points.
When setting
d
¼
λ
=
4
(where
λ
¼
2
π
v=
ω
ge
is the wave-
length of the photons), a photon emitted from one coupling
point accumulates a
π
=
2
phase shift when propagating to
the adjacent coupling point. In this situation, setting the
JOSHI, YANG, and MIRHOSSEINI
PHYS. REV. X
13,
021039 (2023)
021039-2
relative phase
φ
c
¼
φ
r
φ
l
¼
π
=
2
results in a chiral
response, with the emission from the two ports interfering
constructively (destructively) in the forward (backward)
direction. Formally, the spatially nonlocal emitter-
waveguide coupling can be described using the SLH
formalism (see Appendix
B
), leading to a pair of input-
output relations for the forward- and backward-propagating
modes:
ˆ
a
f
out
¼
ˆ
a
f
in
þð
1
þ
e
i
ð
φ
c
φ
WG
Þ
Þ
ffiffiffiffiffiffiffi
κ
em
2
r
ˆ
σ
;
ð
2
Þ
ˆ
a
b
out
¼
ˆ
a
b
in
þð
1
þ
e
i
ð
φ
c
þ
φ
WG
Þ
Þ
ffiffiffiffiffiffiffi
κ
em
2
r
ˆ
σ
:
ð
3
Þ
Here,
ˆ
a
f
ð
b
Þ
in
is the input field for the forward-propagating
(backward-propagating) mode,
ˆ
a
f
ð
b
Þ
out
is the corresponding
output field, and we assume that the magnitude of the decay
rate at the two coupling points is equal and given by
κ
em
.
Solving the input-output relations yields the transmission
t
¼h
ˆ
a
f
out
i
=
h
ˆ
a
f
in
i
and the emitter
s rate of spontaneous
emission into the forward (backward) direction:
Γ
f
ð
b
Þ
1
D
=
κ
em
¼
1
þ
cos
ð
φ
c
φ
WG
Þ
:
ð
4
Þ
Note that, in principle, any waveguide length permits full
suppression of coupling to one waveguide direction (with
the exception of
d
¼
n
λ
=
2
, for
n
Z
). This maximum
chirality condition only coincides with the maximum
emitter external decay rate when
d
¼ð
2
n
þ
1
Þ
λ
=
4
, moti-
vating our choice of the waveguide length (
φ
WG
¼
π
=
2
),
which results in
Γ
b
1
D
¼
0
(
Γ
f
1
D
¼
2
κ
em
)at
φ
c
¼
π
=
2
.
To realize complex coupling strengths, we rely on a
periodic modulation of the photon hopping rates from the
(b)
(a)
(c)
FIG. 1. Chiral atom coupled to a waveguide. (a) Schematic of chiral atom-waveguide system. The emitter atom couples to the
waveguide at two points separated by a distance
d
¼
λ
=
4
. Time-modulated coupling imparts a phase of
φ
l
ð
r
Þ
at the left (right) point; the
relative phase
φ
c
¼
φ
r
φ
l
tunes interference between the two radiation pathways.
φ
c
¼
π
=
2
results in maximum coupling to forward-
propagating modes (blue) and no coupling to backward-propagating modes (red). Left-hand inset: varying
φ
c
results in coupling to
waveguide modes of opposite directions. (b) Optical image of the fabricated device. The emitter transmon (
E
, yellow) couples to a
microwave coplanar waveguide (WG, orange) at two points separated by
d
¼
4
.
590
mm. At each point, radiation to the waveguide is
mediated by a frequency-selective dissipation port containing a tunable coupler (
C
l
ð
r
Þ
, purple) and a filter cavity (
R
l
ð
r
Þ
, blue) directly
coupled to the waveguide (orange). A pair of flux bias lines (
Z
C;l
ð
r
Þ
, green) are used to drive the couplers parametrically.
(c) Transmission spectrum
j
t
j¼jh
ˆ
a
f
out
i
=
h
ˆ
a
f
in
ij
of the device under the experimental settings used for achieving maximum chirality.
Parametric driving of the couplers leads to a visible sideband for the emitter transmon (
E
1
) at the center of the measurement band.
C
l
ð
r
Þ
(
R
l
ð
r
Þ
) mark the resonances corresponding to the couplers (filter cavities).
RESONANCE FLUORESCENCE OF A CHIRAL ARTIFICIAL
...
PHYS. REV. X
13,
021039 (2023)
021039-3
emitter to the waveguide
[46]
. We achieve this by frequency
modulating a coupler device
[47
49]
that is capacitively
coupled to the emitter. In this configuration, the coupler is
modulated with a sinusoidal flux drive at the frequency
Δ
,
with an amplitude
ε
l
ð
r
Þ
and phase
φ
l
ð
r
Þ
, resulting in an
effective emitter-waveguide coupling term that picks up
the driving phase arg
½
̃
g
l
ð
r
Þ
φ
l
ð
r
Þ
[50]
. Consequently, the
relative phase between the two coupling pathways
φ
c
¼
π
=
2
can be precisely set by controlling the relative
phase between the flux modulation drives of the two
couplers. We point out that, beyond shifting the emitter
frequency by
Δ
, the periodic flux modulation also creates
additional undesired frequency components in the emitter
s
spectrum (separated by integer multiples of
Δ
, hereafter
referred to as the
sidebands
; see Ref.
[51]
), which can act
as parasitic decay channels into the waveguide. To suppress
these decay channels, each dissipation port in our experi-
ment contains a compact microwave resonator, which
filters the emission into the waveguide spectrally.
Figure
1(b)
shows an image of the full device, with the
emitter transmon, two frequency-tunable couplers, and the
filter resonators. Figure
1(c)
shows the transmission spec-
trum through the waveguide for a coherent drive, where we
can identify the resonant features corresponding to the filter
resonators (
R
l;r
), the couplers (
C
l;r
), and the first-order
sideband of the emitter qubit (
E
1
at
ω
E
1
¼
ω
E
þ
Δ
). For
clarity of visualization, the transmission trace in Fig.
1(c)
was recorded with input powers that result in moderate
saturation of the chiral atom, resulting in a low transmission
amplitude on resonance. We deliberately design the filter
resonators to have different resonance frequencies, with
their detuning far exceeding their external decay rates to the
waveguide. This condition is required to avoid mode
hybridization between the resonators via the photon-
mediated exchange interaction through the waveguide
[43]
. We also note that a similar concept with an alternative
approach to sideband filtering has been theoretically
proposed based on photonic crystal waveguides
[52]
.At
optimal settings for chirality, we set the flux drive ampli-
tudes of the couplers to achieve equal emitter-waveguide
couplings via both dissipation ports [see Fig.
1(c)
]. A full
analytical analysis of the parametric waveguide coupling
and the spurious sideband suppression is provided in
Appendix
H
.
III. DEVICE PARAMETERS
In our experiment, the emitter is a transmon qubit
with a maximum frequency of
ω
E
=
2
π
¼
5
.
636
GHz.
The tunable couplers are flux modulated with a frequency
of
Δ
=
2
π
¼
805
MHz, creating the emitter first (blue)
sideband at a frequency of
ω
E
1
=
2
π
¼
ω
ge
=
2
π
¼
6
.
441
GHz
[see Fig.
1(c)
]. Flux control of the tunable couplers is
enabled by SQUID loops with two symmetric Josephson
junctions (see Appendix
A4
). The tunable couplers are
designed to operate in the transmon regime and are flux
biased to
ω
C;l
ð
r
Þ
=
2
π
¼
6
.
482
(6.402) GHz. At the experi-
ment operation settings, the filter cavity frequencies are
ω
R;l
ð
r
Þ
=
2
π
¼
6
.
184
(6.712) GHz (shifted from their
bare
values due to interaction with couplers). We control the
external coupling rate
Γ
f
1
D
=
2
π
of the chiral artificial atom by
changing the couplers
configuration (see Appendix
A3
).
The distance between the two coupling points along the
waveguide is 4.590 mm, which corresponds to a
λ
=
4
separation at
ω
E
1
=
2
π
¼
6
.
441
GHz [
λ
¼
2
π
c=
ð
ω
E
1
ffiffiffiffiffiffiffi
ε
eff
p
Þ
,
ε
eff
¼
6
.
45
]. Our fabrication methods and device parameters
are summarized in Appendix
A
.
IV. OBSERVATION OF CHIRALITY
We characterize the response of the atom by performing
transmission spectroscopy by applying a weak microwave
drive of variable frequency to one end of the waveguide.
This measurement is performed at a sufficiently low power
such that the atom
s excited state
s population remains
negligible. In a system with a partial directional response
(the most general case), coherent scattering from the atom
results in a Lorentzian line shape with a transmission
coefficient given by
t
ð
δω
Þ¼
1
Γ
f
1
D
i
δω
þ
Γ
tot
=
2
;
Γ
tot
¼
Γ
f
1
D
þ
Γ
b
1
D
þ
Γ
0
;
ð
5
Þ
where
δω
¼
ω
ω
ge
is the atom-drive detuning. Here,
Γ
f;b
1
D
are the rates of the atom
s spontaneous emission in the
forward and backward directions, and
Γ
0
is its intrinsic
decoherence rate. For a chiral atom in the strong coupling
regime, when the atom couples dominantly to the forward-
propagating modes (
Γ
f
1
D
Γ
b
1
D
þ
Γ
0
), we expect to see a
2
π
change in the phase imparted to the transmitted probe as
the frequency is swept across the resonance. In contrast,
this phase change cannot exceed
π
when the atom-
waveguide coupling to the forward- and backward-
propagating modes are symmetric (see Refs.
[53
55]
and
Appendix
A2
).
Figures
2(a)
and
2(b)
show the measurement results for
two different phase settings. As evident, when
φ
c
¼
π
=
2
,
we observe the canonical signature of chirality as a
2
π
phase across the resonance, which is consistent with our
expectation for
Γ
b
1
D
=
Γ
f
1
D
0
. Conversely, when we set
φ
c
¼
3
π
=
2
(via digital control of the phases of the
couplers
flux drives), the atom
s interaction with the
forward-propagating drive disappears, consistent with
Γ
f
1
D
=
Γ
b
1
D
0
. This phase-sensitive directional behavior
can be controlled with a fine resolution by varying
φ
c
in small steps across a full
2
π
range. Figure
2(c)
shows
the
Γ
f
1
D
obtained from a fit to the measured complex
JOSHI, YANG, and MIRHOSSEINI
PHYS. REV. X
13,
021039 (2023)
021039-4