Quantum Electrodynamics in a Topological Waveguide
Eunjong Kim ,
1,2
,*
Xueyue Zhang,
1,2
,*
Vinicius S. Ferreira ,
1,2
Jash Banker,
1,2
Joseph K. Iverson,
2,3
Alp Sipahigil,
1,2
Miguel Bello,
4
Alejandro González-Tudela,
5
Mohammad Mirhosseini,
2,6
and Oskar Painter
1,2,3
,
†
1
Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute,
California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
3
AWS Center for Quantum Computing, Pasadena, California 91125, USA
4
Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid 28049, Spain
5
Instituto de Física Fundamental, IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
6
Gordon and Betty Moore Laboratory of Engineering, California Institute of Technology,
Pasadena, California 91125, USA
(Received 26 June 2020; revised 2 November 2020; accepted 13 November 2020; published 25 January 2021)
While designing the energy-momentum relation of photons is key to many linear, nonlinear, and quantum
optical phenomena, a new set of light-matter properties may be realized by employing the topology of the
photonic bath itself. In this work we experimentally investigate the properties of superconducting qubits
coupled to a metamaterial waveguide based on a photonic analog of the Su-Schrieffer-Heeger model.
We explore topologically induced properties of qubits coupled to such a waveguide, ranging from the
formation of directional qubit-photon bound states to topology-dependent cooperative radiation effects.
Addition of qubits to this waveguide system also enables direct quantum control over topological edge
states that form in finite waveguide systems, useful for instance in constructing a topologically protected
quantum communication channel. More broadly, our work demonstrates the opportunity that topological
waveguide-QED systems offer in the synthesis and study of many-body states with exotic long-range
quantum correlations.
DOI:
10.1103/PhysRevX.11.011015
Subject Areas: Atomic and Molecular Physics,
Condensed Matter Physics,
Quantum Information
I. INTRODUCTION
Harnessing the topological properties of photonic
bands
[1
–
3]
is a burgeoning paradigm in the study of
periodic electromagnetic structures. Topological concepts
discovered in electronic systems
[4,5]
have now been
translated and studied as photonic analogs in various micro-
wave and optical systems
[2,3]
. In particular, symmetry-
protected topological phases
[6]
, which do not require time-
reversal-symmetry breaking, have received significant
attention in experimental studies of photonic topological
phenomena, both in the linear and nonlinear regime
[7]
.One
of the simplest canonical models is the Su-Schrieffer-Heeger
(SSH) model
[8,9]
, which was initially used to describe
electrons hopping along a one-dimensional dimerized
chain with a staggered set of hopping amplitudes between
nearest-neighbor elements. The chiral symmetry of the SSH
model, corresponding to a symmetry of the electron ampli-
tudesfound on the two types ofsites in the dimer chain,gives
rise to two topologically distinct phases of electron propa-
gation. The SSH model, and its various extensions, have
been used in photonics to explore a variety of optical pheno-
mena, from robust lasing in arrays of microcavities
[10,11]
and photonic crystals
[12]
to disorder-insensitive third
harmonic generation in zigzag nanoparticle arrays
[13]
.
Utilization of quantum emitters brings new opportunities
in the study of topological physics with strongly interacting
photons
[14
–
16]
, where single-excitation dynamics
[17]
and topological protection of quantum many-body
states
[18]
in the SSH model have recently been inves-
tigated. In a similar vein, a topological photonic bath can
also be used as an effective substrate for endowing special
properties to quantum matter. For example, a photonic
waveguide which localizes and transports electromagnetic
waves over large distances can form a highly effective
*
These authors contributed equally to this work.
†
opainter@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
11,
011015 (2021)
Featured in Physics
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=
21
=
11(1)
=
011015(27)
011015-1
Published by the American Physical Society
quantum light-matter interface
[19
–
21]
for introducing
nontrivial interactions between quantum emitters. Several
systems utilizing highly dispersive electromagnetic wave-
guide structures have been proposed for realizing quantum
photonic matter exhibiting tailorable, long-range inter-
actions between quantum emitters
[22
–
26]
. With the
addition of nontrivial topology to such a photonic bath,
exotic classes of quantum entanglement can be generated
through photon-mediated interactions of a chiral
[27,28]
or
directional nature
[29,30]
.
With this motivation, here we investigate the properties of
quantum emitters coupled to a topological waveguide which
is a photonic analog of the SSH model, following the
theoretical proposal in Ref.
[29]
. Our setup is realized by
coupling superconducting transmon qubits
[31]
to an engi-
neered superconducting metamaterial waveguide
[32,33]
,
consisting of an array of subwavelength microwave reso-
nators with SSH topology. Combining the notions from
waveguide quantum electrodynamics (QED)
[20,21,34,35]
and topological photonics
[2,3]
, we observe qubit-photon
bound states with directional photonic envelopes inside a
band gap and cooperative radiative emission from qubits
inside a passband dependent on the topological configura-
tion of the waveguide. Coupling of qubits to the waveguide
also allows for quantum control over topological edge states,
enabling quantum state transfer between distant qubits via a
topological channel.
II. DESCRIPTION OF THE TOPOLOGICAL
WAVEGUIDE
The SSH model describing the topological waveguide
studied here is illustrated in Fig.
1(a)
. Each unit cell of the
waveguide consists of two photonic sites,
A
and
B
, each
containing a resonator with resonant frequency
ω
0
. The
intracell coupling between
A
and
B
sites is
J
ð
1
þ
δ
Þ
and
the intercell coupling between unit cells is
J
ð
1
−
δ
Þ
.
The discrete translational symmetry (lattice constant
d
)
of this system allows us to write the Hamiltonian in terms
of momentum-space operators,
ˆ
H=
ℏ
¼
P
k
ð
ˆ
v
k
Þ
†
h
ð
k
Þ
ˆ
v
k
,
where
ˆ
v
k
¼ð
ˆ
a
k
;
ˆ
b
k
Þ
T
is a vector operator consisting
of a pair of
A
and
B
sublattice photonic mode operators,
and the
k
-dependent kernel of the Hamiltonian is
given by
h
ð
k
Þ¼
ω
0
f
ð
k
Þ
f
ð
k
Þ
ω
0
:
ð
1
Þ
Here,
f
ð
k
Þ
≡
−
J
½ð
1
þ
δ
Þþð
1
−
δ
Þ
e
−
ikd
is the momen-
tum-space coupling between modes on different sublatti-
ces, which carries information about the topology of the
system. The eigenstates of this Hamiltonian form two
symmetric bands centered about the reference frequency
ω
0
with dispersion relation
(a)
(c)
(b)(d)
FIG. 1. Topological waveguide. (a) Top: schematic of the SSH model. Each unit cell contains two sites
A
and
B
(red and blue circles)
with intracell and intercell coupling
J
ð
1
δ
Þ
(orange and brown arrows). Bottom: an analog of this model in electrical circuits, with
corresponding components color coded. The photonic sites are mapped to
LC
resonators with inductance
L
0
and capacitance
C
0
, with
intracell and intercell coupling capacitance
C
v
,
C
w
and mutual inductance
M
v
,
M
w
between neighboring resonators, respectively
(arrows). (b) Optical micrograph (false colored) of a unit cell (lattice constant
d
¼
592
μ
m) on a fabricated device in the topological
phase. The lumped-element resonator corresponding to sublattice
A
(
B
) is colored in red (blue). The insets show enlarged view of the
section between resonators where planar wires of thickness
ð
t
v
;t
w
Þ¼ð
10
;
2
Þ
μ
m (indicated with black arrows) control the intracell and
intercell distance between neighboring resonators, respectively. (c) Dispersion relation of the realized waveguide according to the circuit
model in (a). Upper band gap (UBG) and lower band gap (LBG) are shaded in gray, and middle band gap (MBG) is shaded in green.
(d) Waveguide transmission spectrum
j
S
21
j
across the test structure with 8 unit cells in the trivial (
δ
>
0
, top) and topological (
δ
<
0
,
bottom) phase. The cartoons illustrate the measurement configuration of systems with external ports 1 and 2 (denoted P1 and P2), where
distances between circles are used to specify relative coupling strengths between sites and blue (green) outlines enclosing two circles
indicate unit cells in the trivial (topological) phase. Black solid curves are fits to the measured data (see the Appendix
A
for details)
with parameters
L
0
¼
1
.
9
nH,
C
0
¼
253
fF, coupling capacitance
ð
C
v
;C
w
Þ¼ð
33
;
17
Þ
fF and mutual inductance
ð
M
v
;M
w
Þ¼
ð
−
38
;
−
32
Þ
pH in the trivial phase (the values are interchanged in the case of topological phase). The shaded regions correspond
to band gaps in the dispersion relation of (c).
EUNJONG KIM
et al.
PHYS. REV. X
11,
011015 (2021)
011015-2
ω
ð
k
Þ¼
ω
0
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
1
þ
δ
2
Þþ
2
ð
1
−
δ
2
Þ
cos
ð
kd
Þ
q
;
where the
þ
(
−
) branch corresponds to the upper (lower)
frequency passband. While the band structure is dependent
only on the magnitude of
δ
, and not on whether
δ
>
0
or
δ
<
0
, deformation from one case to the other must be
accompanied by the closing of the middle band gap
(MBG), defining two topologically distinct phases. For a
finite system, it is well known that edge states localized on
the boundary of the waveguide at a
ω
¼
ω
0
only appear in
the case of
δ
<
0
, the so-called
topological
phase
[3,9]
. The
case for which
δ
>
0
is the
trivial
phase with no edge states.
It should be noted that for an infinite system, the topo-
logical or trivial phase in the SSH model depends on the
choice of unit cell, resulting in an ambiguity in defining the
bulk properties. Despite this, considering the open boun-
dary of a finite-sized array or a particular section of the
bulk, the topological character of the bands can be uniquely
defined and can give rise to observable effects.
We construct a circuit analog of this canonical model
using an array of inductor-capacitor (
LC
) resonators with
alternating coupling capacitance and mutual inductance as
shown in Fig.
1(a)
. The topological phase of the circuit
model is determined by the relative size of intracell and
intercell coupling between neighboring resonators, includ-
ing both the capacitive and inductive contributions. Strictly
speaking, this circuit model breaks chiral symmetry of the
original SSH Hamiltonian
[3,9]
, which ensures the band
spectrum to be symmetric with respect to
ω
¼
ω
0
.
Nevertheless, the topological protection of the edge states
under perturbation in the intracell and intercell coupling
strengths remains valid as long as the bare resonant
frequencies of resonators (diagonal elements of the
Hamiltonian) are not perturbed, and the existence of edge
states still persists due to the presence of inversion
symmetry within the unit cell of the circuit analog, leading
to a quantized Zak phase
[36]
. For detailed analysis of the
modeling, symmetry, and robustness of the circuit topo-
logical waveguide, see Appendixes
A
and
B
.
The circuit model is realized using fabrication tech-
niques for superconducting metamaterials discussed in
Refs.
[32,33]
, where the coupling between sites is con-
trolled by the physical distance between neighboring
resonators. Because of the near-field nature, the coupling
strength is larger (smaller) for smaller (larger) distance
between resonators on a device. An example unit cell
of a fabricated device in the topological phase is shown in
Fig.
1(b)
(the values of intracell and intercell distances are
interchanged in the trivial phase). We find a good agree-
ment between the measured transmission spectrum and a
theoretical curve calculated from a
LC
lumped-element
model of the test structures with 8 unit cells in both trivial
and topological configurations [Figs.
1(c)
and
1(d)
]. For
the topological configuration, the observed peak in the
waveguide transmission spectrum at 6.636 GHz inside
the MBG signifies the associated edge state physics in
our system.
III. PROPERTIES OF QUANTUM EMITTERS
COUPLED TO THE TOPOLOGICAL WAVEGUIDE
The nontrivial properties of the topological waveguide
can be accessed by coupling quantum emitters to the
engineered structure. To this end, we prepare device I
consisting of a topological waveguide in the trivial phase
with 9 unit cells, whose boundary is tapered with specially
designed resonators before connection to external ports [see
Fig.
2(a)
]. The tapering sections at both ends of the array
are designed to reduce the impedance mismatch to the
external ports (
Z
0
¼
50
Ω
) at frequencies in the upper
passband (UPB). This is crucial for reducing ripples in the
waveguide transmission spectrum in the passbands
[33]
.
Every site of the 7 unit cells in the middle of the array is
occupied by a single frequency-tunable transmon qubit
[31]
(the device contains in total 14 qubits labeled
Q
α
i
, where
i
¼
1
–
7
and
α
¼
A
,
B
are the cell and sublattice indices,
respectively). Properties of device I and the tapering section
are discussed in further detail in Appendixes
C
and
D
,
respectively.
A. Directional qubit-photon bound states
For qubits lying within the middle band gap, the
topology of the waveguide manifests itself in the spatial
profile of the resulting qubit-photon bound states. When
the qubit transition frequency is inside the band gap, the
emission of a propagating photon from the qubit is
forbidden due to the absence of photonic modes at the
qubit resonant frequency. In this scenario, a stable bound
state excitation forms, consisting of a qubit in its excited
state and a waveguide photon with exponentially local-
ized photonic envelope
[37,38]
. Generally, bound states
with a symmetric photonic envelope emerge due to the
inversion symmetry of the photonic bath with respect to
the qubit location
[24]
. In the case of the SSH photonic
bath, however, a directional envelope can be realized
[29]
for a qubit at the center of the MBG (
ω
0
), where the
presence of a qubit creates a domain wall in the SSH
chain and the induced photonic bound state is akin to an
edge state (see Appendix
E
for a detailed description). For
example, in the trivial phase, a qubit coupled to site
A
(
B
)
acts as the last site of a topological array extended to the
right (left) while the subsystem consisting of the remain-
ing sites extended to the left (right) is interpreted as a
trivial array. Mimicking the topological edge state, the
induced photonic envelope of the bound state faces
right (left) with photon occupation only on
B
(
A
)sites
[Fig.
2(b)
], while across the trivial boundary on the
left (right) there is no photon occupation. The opposite
directional character is expected in the case of the
topological phase of the waveguide. The directionality
QUANTUM ELECTRODYNAMICS IN A TOPOLOGICAL
...
PHYS. REV. X
11,
011015 (2021)
011015-3
reduces away from the center of the MBG, and is
effectively absent inside the upper or lower band gaps.
We experimentally probe the directionality of qubit-
photon bound states by utilizing the coupling of bound
states to the external ports in the finite-length waveguide of
device I [see Fig.
2(c)
]. The external coupling rate
κ
e;p
(
p
¼
1
, 2) is governed by the overlap of modes in the
external port
p
with the tail of the exponentially attenuated
envelope of the bound state, and therefore serves as a useful
measure to characterize the localization
[24,32,39]
. To find
the reference frequency
ω
0
where the bound state becomes
most directional, we measure the reflection spectrum
S
11
(
S
22
) of the bound state seen from port 1 (2) as a function of
qubit tuning. We extract the external coupling rate
κ
e;p
by
fitting the measured reflection spectrum with a Fano line
shape
[40]
.For
Q
B
4
, which is located near the center of
the array, we find
κ
e;
1
to be much larger than
κ
e;
2
at all
frequencies inside MBG. At
ω
0
=
2
π
¼
6
.
621
GHz,
κ
e;
2
completely vanishes, indicating a directionality of the
Q
B
4
bound state to the left. Plotting the external coupling at this
frequency to both ports against qubit index, we observe a
decaying envelope on every other site, signifying the
directionality of photonic bound states is correlated with
the type of sublattice site a qubit is coupled to. Similar
measurements when qubits are tuned to other frequencies
near the edge of the MBG, or inside the upper band gap
(UBG), show the loss of directionality away from
ω
¼
ω
0
(Appendix
F
).
(a)(d)
(c)
(e)
(b)
FIG. 2. Directionality of qubit-photon bound states. (a) Schematic of device I, consisting of 9 unit cells in the trivial phase with qubits
(black lines terminated with a square) coupled to every site on the 7 central unit cells. The ends of the array are tapered with additional
resonators (purple) with engineered couplings designed to minimize impedance mismatch at upper passband frequencies. (b) Theoretical
photonic envelope of the directional qubit-photon bound states. At the reference frequency
ω
0
, the qubit coupled to site
A
(
B
) induces a
photonic envelope to the right (left), colored in green (blue). The bars along the envelope indicate photon occupation on the
corresponding resonator sites. (c) Measured coupling rate
κ
e;p
to external port numbers,
p
¼
1
, 2, of qubit-photon bound states.
Left: external coupling rate of qubit
Q
B
4
to each port as a function of frequency inside the MBG. Solid black curve is a model fit to the
measured external coupling curves. The frequency point of highest directionality is extracted from the fit curve, and is found to be
ω
0
=
2
π
¼
6
.
621
GHz (vertical dashed orange line). Top (Bottom), right: external coupling rate of all qubits tuned to
ω
¼
ω
0
measured
from port P1 (P2). The solid black curves in these plots correspond to exponential fits to the measured external qubit coupling versus
qubit index. (d) Two-dimensional color intensity plot of the reflection spectrum under crossing between a pair of qubits with frequency
centered around
ω
¼
ω
0
. Left: reflection from P1 (
j
S
11
j
) while tuning
Q
B
4
across
Q
A
4
(fixed). An avoided crossing of
2
j
g
AB
44
j
=
2
π
¼
65
.
7
MHz is observed. Right: reflection from P2 (
j
S
22
j
) while tuning
Q
B
4
across
Q
A
5
(fixed), indicating the absence of appreciable
coupling. Inset to the right shows an enlarged region where a small avoided crossing of
2
j
g
AB
54
j
=
2
π
¼
967
kHz is measured. The bare
qubit frequencies from the fit are shown with dashed lines. (e) Coupling
j
g
αβ
ij
j
(
α
;
β
∈
f
A; B
g
) between various qubit pairs (
Q
α
i
;Q
β
j
)at
ω
¼
ω
0
, extracted from the crossing experiments similar to (d). Solid black curves are exponential fits to the measured qubit-qubit
coupling rate versus qubit index difference (spatial separation). Error bars in all panels indicate 95% confidence interval, and are omitted
on data points whose marker size is larger than the error itself.
EUNJONG KIM
et al.
PHYS. REV. X
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011015 (2021)
011015-4
A remarkable consequence of the distinctive shape of
bound states is direction-dependent photon-mediated inter-
actions between qubits [Figs.
2(d)
and
2(e)
]. Because of the
site-dependent shapes of qubit-photon bound states, the
interaction between qubits becomes substantial only when
a qubit on sublattice
A
is on the left of the other qubit
on sublattice
B
; i.e.,
j>i
for a qubit pair (
Q
A
i
;Q
B
j
). From
the avoided crossing experiments centered at
ω
¼
ω
0
,we
extract the qubit-qubit coupling as a function of cell
displacement
i
−
j
. An exponential fit of the data gives
the localization length of
ξ
¼
1
.
7
(in units of lattice
constant), close to the estimated value from the circuit
model of our system (see Appendix
C
). While theory
predicts the coupling between qubits in the remaining
combinations to be zero, we report that couplings of
j
g
AA;BB
ij
j
=
2
π
≲
0
.
66
MHz and
j
g
AB
ij
j
=
2
π
≲
0
.
48
MHz (for
i>j
) are observed, much smaller than the bound-state-
induced coupling, e.g.,
j
g
AB
45
j
=
2
π
¼
32
.
9
MHz. We attribute
such spurious couplings to the unintended near-field
interaction between qubits. Note that we find consistent
coupling strength of qubit pairs dependent only on their
relative displacement, not on the actual location in the
array, suggesting that physics inside MBG remains intact
with the introduced waveguide boundaries. In total, the
avoided crossing and external linewidth experiments at
ω
¼
ω
0
provide strong evidence of the shape of qubit-
photon bound states, compatible with the theoretical photon
occupation illustrated in Fig.
2(b)
.
B. Topology-dependent photon scattering
In the passband regime, i.e., when the qubit frequencies
lie within the upper or lower passbands, the topology of the
waveguide is imprinted on cooperative interaction between
qubits and the single-photon scattering response of the
system. The topology of the SSH model can be visualized
by plotting the complex-valued momentum-space coupling
f
ð
k
Þ
for
k
values in the first Brillouin zone [Fig.
3(a)
]. In
the topological (trivial) phase, the contour of
f
ð
k
Þ
encloses
(excludes) the origin of the complex plane, resulting in the
winding number of 1 (0) and the corresponding Zak phase
of
π
(0)
[36]
. This is consistent with the earlier definition
based on the sign of
δ
. It is known that for a regular
waveguide with linear dispersion, the coherent exchange
interaction
J
ij
and correlated decay
Γ
ij
between qubits at
positions
x
i
and
x
j
along the waveguide take the forms
J
ij
∝
sin
φ
ij
and
Γ
ij
∝
cos
φ
ij
[41,42]
, where
φ
ij
¼
k
j
x
i
−
x
j
j
is the phase length. In the case of our topological wave-
guide, considering a pair of qubits coupled to the
A
(
B
)
sublattice on
i
th (
j
th) unit cell, this argument additionally
collects the phase
φ
ð
k
Þ
≡
arg
f
ð
k
Þ
[29]
. This is an impor-
tant difference compared to the regular waveguide case,
because the zeros of equation,
φ
ij
ð
k
Þ
≡
kd
j
i
−
j
j
−
φ
ð
k
Þ¼
0
mod
π
;
ð
2
Þ
determine wave vectors (and corresponding frequencies)
where perfect Dicke superradiance
[43]
occurs. Because of
the properties of
f
ð
k
Þ
introduced above, for a fixed cell
distance
Δ
n
≡
j
i
−
j
j
≥
1
between qubits there exists
exactly
Δ
n
−
1
(
Δ
n
) frequency points inside the passband
where perfect superradiance occurs in the trivial (topologi-
cal) phase. An example for the
Δ
n
¼
2
case is shown in
(c)
(b)
(a)
FIG. 3. Probing band topology with qubits. (a)
f
ð
k
Þ
in the
complex plane for
k
values in the first Brillouin zone.
φ
tr
(
φ
tp
)is
the phase angle of
f
ð
k
Þ
for a trivial (topological) section of
waveguide, which changes by 0 (
π
) rad as
kd
transitions from 0 to
π
(arc in upper plane following black arrowheads). (b) Coherent
exchange interaction
J
ij
between a pair of coupled qubits as a
function of frequency inside the passband, normalized to indi-
vidual qubit decay rate
Γ
e
[only
kd
∈
½
0
;
π
Þ
branch is plotted].
Here, one qubit is coupled to the
A
sublattice of the
i
th unit cell
and the other qubit is coupled to the
B
sublattice of the
j
th unit
cell, where
j
i
−
j
j¼
2
. Blue (green) curve corresponds to a trivial
(topological) intermediate section of waveguide between qubits.
The intercepts at
J
ij
¼
0
(filled circles with arrows) correspond to
points where perfect superradiance occurs. (c) Waveguide trans-
mission spectrum
j
S
21
j
as a qubit pair are resonantly tuned across
the UPB of device I [left, (
Q
A
2
;Q
B
4
); right, (
Q
B
2
;Q
A
5
)]. Top:
schematic illustrating system configuration during the experi-
ment, with left (right) system corresponding to an interacting
qubit pair subtending a three-unit-cell section of waveguide in the
trivial (topological) phase. Middle and bottom: two-dimensional
color intensity plots of
j
S
21
j
from theory and experiment,
respectively. Swirl patterns (highlighted by arrows) are observed
in the vicinity of perfectly superradiant points, whose number of
occurrences differs by one between trivial and topological
waveguide sections.
QUANTUM ELECTRODYNAMICS IN A TOPOLOGICAL
...
PHYS. REV. X
11,
011015 (2021)
011015-5
Fig.
3(b)
. Note that although Eq.
(2)
is satisfied at the band-
edge frequencies
ω
min
and
ω
max
(
kd
¼f
0
;
π
g
), they are
excluded from the above counting due to breakdown of the
Born-Markov approximation (see Appendix
G
).
To experimentally probe signatures of perfect super-
radiance, we tune the frequency of a pair of qubits across
the UPB of device I while keeping the two qubits resonant
with each other. We measure the waveguide transmission
spectrum
S
21
during this tuning, keeping track of the line
shape of the two-qubit resonance as
J
ij
and
Γ
ij
varies over
the tuning. Drastic changes in the waveguide transmission
spectrum occur whenever the two-qubit resonance passes
through the perfectly superradiant points, resulting in a
swirl pattern in
j
S
21
j
. Such patterns arise from the dis-
appearance of the peak in transmission associated with
interference between photons scattered by imperfect super-
radiant and subradiant states, resembling the electromag-
netically induced transparency in a V-type atomic level
structure
[44]
. As an example, we discuss the cases with
qubit pairs (
Q
A
2
;Q
B
4
) and (
Q
B
2
;Q
A
5
), which are shown in
Fig.
3(c)
. Each qubit pair configuration encloses a three-
unit-cell section of the waveguide; however, for the
(
Q
A
2
;Q
B
4
) pair the waveguide section is in the trivial phase,
whereas for (
Q
A
2
;Q
B
4
) the waveguide section is in the
topological phase. Both theory and measurement indicate
that the qubit pair (
Q
A
2
;Q
B
4
) has exactly one perfectly
superradiant frequency point in the UPB. For the other
qubit pair (
Q
B
2
;Q
A
5
), with waveguide section in the topo-
logical phase, two such points occur (corresponding to
Δ
n
¼
2
). This observation highlights the fact that while the
topological phase of the bulk in the SSH model is
ambiguous, a finite section of the array can still be
interpreted to have a definite topological phase. Apart
from the unintended ripples near the band edges, the
observed line shapes are in good qualitative agreement
with the theoretical expectation in Ref.
[29]
. The frequency
misalignment of swirl patterns between the theory and the
experiment is due to the slight discrepancy between the
realized circuit model and the ideal SSH model (see
Appendix
A
for details). Detailed description of the swirl
pattern and similar measurement results for other qubit
combinations with varying
Δ
n
are reported in Appendix
G
.
IV. QUANTUM STATE TRANSFER VIA
TOPOLOGICAL EDGE STATES
Finally, to explore the physics associated with topologi-
cal edge modes, we fabricated a second device, device II,
which realizes a closed quantum system with 7 unit cells in
the topological phase [Fig.
4(a)
]. We denote the photonic
sites in the array by (
i
,
α
), where
i
¼
1
–
7
is the cell index
and
α
¼
A
,
B
is the sublattice index. Because of reflection
at the boundary, the passbands on this device appear as sets
of discrete resonances. The system supports topological
edge modes localized near the sites
ð
1
;A
Þ
and
ð
7
;B
Þ
at the
boundary, labeled
E
L
and
E
R
. The edge modes are spatially
distributed with exponentially attenuated tails directed
toward the bulk. In a finite system, the nonvanishing
overlap between the envelopes of edge states generates a
coupling which depends on the localization length
ξ
and the
system size
L
as
G
∼
e
−
L=
ξ
. In device II, two qubits denoted
Q
L
and
Q
R
are coupled to the topological waveguide at
sites
ð
2
;A
Þ
and
ð
6
;B
Þ
, respectively. Each qubit has a local
drive line and a flux-bias line, which are connected to
room-temperature electronics for control. The qubits are
(a)
(b)
(c)
(d)
(e)
FIG. 4. Qubit interaction with topological edge modes. (a) Sche-
matic of device II, consisting of 7 unit cells in the topological
phase with qubits
Q
L
¼
Q
α
i
and
Q
R
¼
Q
β
j
coupled at sites
ð
i;
α
Þ¼ð
2
;A
Þ
and
ð
j;
β
Þ¼ð
6
;B
Þ
, respectively.
E
L
and
E
R
are
the left-localized and right-localized edge modes which interact
with each other at rate
G
due to their overlap in the center of the
finite waveguide. (b) Chevron-shaped oscillation of
Q
L
popula-
tion arising from interaction with edge modes under variable
frequency and duration of modulation pulse. The oscillation is
nearly symmetric with respect to optimal modulation frequency
242.5 MHz, apart from additional features at (219, 275) MHz due
to spurious interaction of unused sidebands with modes inside the
passband. (c) Line cut of (b) (indicated with a dashed line) at the
optimal modulation frequency. A population oscillation involving
two harmonics is observed due to coupling of
E
L
to
E
R
.
(d) Vacuum Rabi oscillations between
Q
L
and
E
L
when
Q
R
is
parked at the resonant frequency of edge modes to shift the
frequency of
E
R
, during the process in (c). In panels (c) and (d) the
filled orange circles (black solid lines) are the data from experi-
ment (theory). (e) Population transfer from
Q
L
to
Q
R
composed of
three consecutive swap transfers
Q
L
→
E
L
→
E
R
→
Q
R
. The
population of
Q
L
(
Q
R
) during the process is colored dark red
(dark blue), with filled circles and solid lines showing the
measured data and fit from theory, respectively. The light red
(light blue) curve indicates the expected population in
E
L
(
E
R
)
mode, calculated from theory.
EUNJONG KIM
et al.
PHYS. REV. X
11,
011015 (2021)
011015-6