of 28
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 Gonz ́alez-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, 28049 Madrid, Spain.
5
Instituto de F ́ısica 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.
(Dated: May 11, 2020)
While designing the energy-momentum relation of photons is key to many linear, non-linear, 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 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.
Harnessing the topological properties of photonic
bands [1–3] is a burgeoning paradigm in the study of
periodic electromagnetic structures.
Topological con-
cepts discovered in electronic systems [4, 5] have now
been translated and studied as photonic analogs in vari-
ous microwave 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 nonlin-
ear 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 hop-
ping amplitudes between nearest-neighbor elements. The
chiral symmetry of the SSH model, corresponding to a
symmetry of the electron amplitudes found on the two
types of sites in the dimer chain, gives rise to two topo-
logically distinct phases of electron propagation. The
SSH model, and its various extensions, have been used in
photonics to explore a variety of optical phenomena, from
robust lasing in arrays of microcavities [10, 11] and pho-
tonic crystals [12], to disorder-insensitive 3rd harmonic
generation in zigzag nanoparticle arrays [13].
Utilization of quantum emitters brings new opportu-
nities in the study of topological physics with strongly
interacting photons [14], where single-excitation dynam-
ics [15] and topological protection of quantum many-
These authors contributed equally to this work.
opainter@caltech.edu; http://copilot.caltech.edu
body states [16] in the SSH model have recently been
investigated. In a similar vain, a topological photonic
bath can also be used as an effective substrate for en-
dowing special properties to quantum matter. For exam-
ple, a photonic waveguide which localizes and transports
electromagnetic waves over large distances, can form a
highly effective quantum light-matter interface [17–19]
for introducing non-trivial interactions between quantum
emitters. Several systems utilizing highly dispersive elec-
tromagnetic waveguide structures have been proposed for
realizing quantum photonic matter exhibiting tailorable,
long-range interactions between quantum emitters [20–
23]. With the addition of non-trivial topology to such a
photonic bath, exotic classes of quantum entanglement
can be generated through photon-mediated interactions
of a chiral [24, 25] or directional nature [26, 27].
With this motivation, here we investigate the proper-
ties of quantum emitters coupled to a topological waveg-
uide which is a photonic analog of the SSH model [26].
Our setup is realized by coupling superconducting trans-
mon qubits [28] to an engineered superconducting meta-
material waveguide [29, 30], consisting of an array of
sub-wavelength microwave resonators with SSH topol-
ogy. Combining the notions from waveguide quantum
electrodynamics (QED) [18, 19, 31, 32] and topological
photonics [2, 3], we observe qubit-photon bound states
with directional photonic envelopes inside a bandgap and
cooperative radiative emission from qubits inside a pass-
band dependent on the topological configuration of the
waveguide. Coupling of qubits to the waveguide also al-
lows for quantum control over topological edge states,
enabling quantum state transfer between distant qubits
arXiv:2005.03802v1 [quant-ph] 8 May 2020
2
via a topological channel.
The SSH model describing the topological waveguide
studied here is illustrated in Fig. 1a. Each unit cell of
the waveguide consists of two photonic sites, A and B,
each containing a resonator with resonant frequency
ω
0
.
The intra-cell coupling between A and B sites is
J
(1 +
δ
)
and the inter-cell coupling between unit cells is
J
(1
δ
).
The discrete translational symmetry (lattice constant
d
)
of this system allows us to write the Hamiltonian in terms
of momentum-space operators,
ˆ
H/
~
=
k
(
ˆ
v
k
)
h
(
k
)
ˆ
v
k
,
where
ˆ
v
k
= (ˆ
a
k
,
ˆ
b
k
)
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
(
k
) =
(
ω
0
f
(
k
)
f
(
k
)
ω
0
)
.
(1)
Here,
f
(
k
)
≡−
J
[(1 +
δ
) + (1
δ
)
e
ikd
] is the momentum-
space coupling between modes on different sublattice,
which carries information about the topology of the sys-
tem. The eigenstates of this Hamiltonian form two sym-
metric bands centered about the reference frequency
ω
0
with dispersion relation
ω
±
(
k
) =
ω
0
±
J
2(1 +
δ
2
) + 2(1
δ
2
) cos (
kd
)
,
where the + (
) branch corresponds to the upper (lower)
frequency passband. While the band structure is depen-
dent 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 mid-
dle bandgap (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
=
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 topological or trivial phase in
the SSH model depends on the choice of unit cell, result-
ing in an ambiguity in defining the bulk properties. De-
spite this, considering the open boundary 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. 1a. The topological phase of the circuit
model is determined by the relative size of intra- and
inter-cell coupling between neighboring resonators, in-
cluding both the capacitive and inductive contributions.
Strictly speaking, this circuit model breaks chiral sym-
metry of the original SSH Hamiltonian [3, 9], which en-
sures the band spectrum to be symmetric with respect to
ω
=
ω
0
. Nevertheless, the topological protection of the
edge states under perturbation in the intra- and inter-
cell coupling strengths remains valid under certain condi-
tions, 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 [33].
For detailed analysis of the modeling, symmetry, and ro-
bustness of the circuit topological waveguide see Apps. A
and B.
The circuit model is realized using standard fabrica-
tion techniques for superconducting metamaterials dis-
cussed in Refs. [29, 30], where the coupling between sites
is controlled by the physical distance between neighbor-
ing resonators. Due to 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. 1b (the values of intra- and inter-cell distances
are interchanged in the trivial phase). We find a good
agreement 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 (Fig. 1c,d).
For the topological configuration, the observed peak in
the waveguide transmission spectrum at 6.636 GHz in-
side the MBG signifies the associated edge state physics
in our system.
The non-trivial properties of the topological waveguide
can be accessed by coupling quantum emitters to the en-
gineered 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 spe-
cially designed resonators before connection to external
ports (see Fig. 2a). The tapering sections at both ends
of the array are designed to reduce the impedance mis-
match to the external ports (
Z
0
= 50 Ω) at frequencies
in the upper passband (UPB). This is crucial for reduc-
ing ripples in the waveguide transmission spectrum in
the passbands [30]. The device contains 14 frequency-
tunable transmon qubits [28] coupled to every site on the
7 unit cells in the middle of the array (labeled Q
α
i
, where
i
=1-7 and
α
=A,B are the cell and sublattice indices,
respectively). Properties of Device I and the tapering
section are discussed in further detail in Apps. C and D,
respectively.
For qubits lying within the middle bandgap, the topol-
ogy of the waveguide manifests itself in the spatial pro-
file of the resulting qubit-photon bound states. When
the qubit transition frequency is inside the bandgap, the
emission of a propagating photon from the qubit is for-
bidden 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 ex-
cited state and a waveguide photon with exponentially
localized photonic envelope [34, 35]. Generally, bound
states with a symmetric photonic envelope emerge due
to the inversion symmetry of the photonic bath with re-
spect to the qubit location [22]. In the case of the SSH
photonic bath, however, a directional envelope can be
realized [26] for a qubit at the centre 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
3
a
C
v
C
w
C
0
C
0
L
0
L
0
J
(1+
δ
)
J
(1−
δ
)
...
...
100
μ
m
c
b
d
M
v
M
w
A
B
A
B
0
kd
6
6.5
7
7.5
Frequency (GHz)
π
UBG
MBG
LBG
...
...
-50
-25
0
-50
-25
0
|S
21
| (dB)
6
6.5
7
7.5
Frequency (GHz)
P1
P2
P1
P2
t
w
t
v
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 intra- and inter-cell coupling
J
(1
±
δ
) (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 intra- and inter-cell 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
= 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 zoomed-in view of the section between resonators where planar wires of thickness
(
t
v
,t
w
) = (10
,
2)
μ
m (indicated with black arrows) control the intra- and inter-cell distance between neighboring resonators,
respectively.
c
, Dispersion relation of the realized waveguide according to the circuit model in panel
a
. Upper bandgap (UBG)
and lower bandgap (LBG) are shaded in gray, and middle bandgap (MBG) is shaded in green.
d
, Waveguide transmission
spectrum
|
S
21
|
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 App. A
for details) with parameters
L
0
= 1
.
9 nH,
C
0
= 253 fF, coupling capacitance (
C
v
,C
w
) = (33
,
17) fF and mutual inductance
(
M
v
,M
w
) = (
38
,
32) pH in the trivial phase (the values are interchanged in the case of topological phase). The shaded
regions correspond to bandgaps in the dispersion relation of panel
c
.
to an edge state (refer to App. E for a detailed descrip-
tion). 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 consist-
ing of the remaining 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. 2b), 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 direction-
ality reduces away from the center of the MBG, and is
effectively absent inside the upper or lower bandgaps.
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 waveg-
uide of Device I (see Fig. 2c). The external coupling
rate
κ
e
,p
(
p
= 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 localiza-
tion [22, 29, 36]. To find the reference frequency
ω
0
where
the bound state becomes most directional, we measure
the external linewidth of the bound state seen from each
port as a function of qubit tuning. 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
= 6
.
621 GHz,
κ
e
,
2
completely vanishes, indicating
a directionality of the Q
B
4
bound state to the left. Plot-
ting 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 bandgap (UBG), show the loss of di-
rectionality away from
ω
=
ω
0
(App. F).
A remarkable consequence of the distinctive shape of
bound states is direction-dependent photon-mediated in-
teractions between qubits (Fig. 2d,e). Due to the site-
dependent shapes of qubit-photon bound states, the in-
teraction 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
=
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
= 1
.
7 (in units of lat-
tice constant), close to the estimated value from the
circuit model of our system (see App. C). While the-
ory predicts the coupling between qubits in the remain-
4
6.55
6.6
6.65
6.7
Frequency (GHz)
|S
22
|
6.618
6.624
6.618
6.62
6.622
6.624
0.6
1
6.621
|S
22
|
6.55
6.6
6.65
6.7
Frequency (GHz)
6.55
6.6
6.65
6.7
Bare Q
4
Frequency (GHz)
B
0.8
0.9
1
|S
11
|
A
Q
4
B
Q
4
A
Q
5
...
B
Q
4
a
d
...
B
Q
1
A
Q
1
A
Q
7
B
Q
7
P1
P2
c
e
...
...
P1
...
...
P2
κ
e,1
B
Q
j
κ
e,2
A
Q
i
Ext. Coupling Rate,
κ
e,
p
/2
π
(MHz)
Q
4
Frequency (GHz)
B
6.5
6.6
6.7
0
0.1
0.2
0.3
0.4
A
B
A
B
A
B
A
B
B
A
B
A
B
0
1
2
3
1
2
3
4
5
6
7
A
0
1
2
3
Qubit Index
κ
e,1
/2
π
κ
e,2
/2
π
−5
−6
−4
−3
−2
−1
0
1
2
3
i
j
|
g
ij
|
/
2
π
(MHz)
AB
0
10
20
30
j
= 7
j
= 6
j
= 4
|
g
ij
|
/
2
π
(MHz)
αα
0.2
0
0.4
0.6
1
2
3
4
5
|
i
j
|
AA,
j
= 7
BB,
j
= 4
BB,
j
= 1
BB,
j
= 2
b
κ
e,1
/2
π
κ
e,2
/2
π
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
= 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
= 6
.
621 GHz (vertical dashed orange line).
Top (Bottom)-right: external coupling rate of all qubits tuned to
ω
=
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
=
ω
0
.
Left: reflection from P1 (
|
S
11
|
) while tuning Q
B
4
across Q
A
4
(fixed). An avoided crossing of 2
|
g
AB
44
|
/
2
π
= 65
.
7 MHz is observed.
Right: reflection from P2 (
|
S
22
|
) while tuning Q
B
4
across Q
A
5
(fixed), indicating the absence of appreciable coupling. Inset to the
right shows a zoomed-in region where a small avoided crossing of 2
|
g
AB
54
|
/
2
π
= 967 kHz is measured. The bare qubit frequencies
from the fit are shown with dashed lines.
e
, Coupling
|
g
αβ
ij
|
(
α,β
∈ {
A,B
}
) between various qubit pairs (Q
α
i
,Q
β
j
) at
ω
=
ω
0
,
extracted from the crossing experiments similar to panel
d
. Solid black curves are exponential fits to the measured qubit-qubit
coupling rate versus qubit index difference (spatial separation). Error bars in all figure panels indicate 95% confidence interval,
and are omitted on data points whose marker size is larger than the error itself.
ing combinations to be zero, we report that coupling of
|
g
AA,BB
ij
|
/
2
π
.
0
.
66 MHz and
|
g
AB
ij
|
/
2
π
.
0
.
48 MHz (for
i > j
) are observed, much smaller than the bound-state-
induced coupling, e.g.,
|
g
AB
45
|
/
2
π
= 32
.
9 MHz. We at-
tribute such spurious couplings to the unintended near-
field interaction between qubits. Note that we find con-
sistent 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 to-
tal, the avoided crossing and external linewidth experi-
ments at
ω
=
ω
0
provide strong evidence of the shape of
qubit-photon bound states, compatible with the theoret-
ical photon occupation illustrated in Fig. 2b.
In the passband regime, i.e., when the qubit frequen-
cies lie within the upper or lower passbands, the topol-
ogy of the waveguide is imprinted on cooperative inter-
action 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
f
(
k
)
for
k
values in the first Brillouin zone (Fig. 3a). In the
topological (trivial) phase, the contour of
f
(
k
) encloses
(excludes) the origin of the complex plane, resulting in
the winding number of 1 (0) and the corresponding Zak
phase of
π
(0) [33]. 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 ex-
change interaction
J
ij
and correlated decay Γ
ij
between
5
c
...
...
...
...
Bare Qubit Frequency (GHz)
7
7.2
7.4
6.8
Frequency (GHz)
7
7.2
7.4
b
a
-2
J
-
J
Re[
f
(
k
)]
-
J
0
J
Im[
f
(
k
)]
Triv.
Topo.
0
J
φ
tr
φ
tp
-1
0
1
J
i j
/
Γ
e
ω
ω
min
ω
max
|S
21
|
1
0
7
7.2
7.4
7
7.2
7.4
Frequency (GHz)
6.8
|
i
j
| = 2
FIG. 3.
Probing band topology with qubits. a
,
f
(
k
)
in the complex plane for
k
values in the first Brillouin zone.
φ
tr
(
φ
tp
) is the phase angle of
f
(
k
) for a trivial (topological)
section of waveguide, which changes by 0 (
π
) radians 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 individual qubit decay rate Γ
e
(only
kd
[0
) branch is plotted). Here, one qubit is coupled to
the A sublattice of the
i
-th unit cell and the other qubit is cou-
pled to the B sublattice of the
j
-th unit cell, where
|
i
j
|
= 2.
Blue (green) curve corresponds to a trivial (topological) inter-
mediate section of waveguide between qubits. The intercepts
at
J
ij
= 0 (filled circles with arrows) correspond to points
where perfect super-radiance occurs.
c
, Waveguide transmis-
sion spectrum
|
S
21
|
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
|
S
21
|
from theory and ex-
periment, respectively. Swirl patterns (highlighted by arrows)
are observed in the vicinity of perfectly super-radiant points,
whose number of occurrences differ by one between trivial and
topological waveguide sections.
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
[37, 38], where
φ
ij
=
k
|
x
i
x
j
|
is the phase length. In the case of our
topological waveguide, considering a pair of qubits cou-
pled to A/B sublattice on
i
/
j
-th unit cell, this argument
additionally collects the phase
φ
(
k
)
arg
f
(
k
) [26]. This
is an important difference compared to the regular waveg-
uide case, because the zeros of equation
φ
ij
(
k
)
kd
|
i
j
|−
φ
(
k
) = 0 mod
π
(2)
determine wavevectors (and corresponding frequencies)
where perfect Dicke super-radiance [39] occurs. Due to
the properties of
f
(
k
) introduced above, for a fixed cell-
distance ∆
n
≡|
i
j
|≥
1 between qubits there exists ex-
actly ∆
n
1 (∆
n
) frequency points inside the passband
where perfect super-radiance occurs in the trivial (topo-
logical) phase. An example for the ∆
n
= 2 case is shown
in Fig. 3b. Note that although Eq. (2) is satisfied at the
band-edge frequencies
ω
min
and
ω
max
(
kd
=
{
0
}
), they
are excluded from the above counting due to breakdown
of the Born-Markov approximation that is assumed in
obtaining the particular form of cooperative interaction
in this picture.
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 reso-
nant with each other. We measure the waveguide trans-
mission spectrum
S
21
during this tuning, keeping track
of the lineshape 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 res-
onance passes through the perfectly super-radiant points,
resulting in a swirl pattern in
|
S
21
|
.
Such patterns
arise from the disappearance of the peak in transmis-
sion associated with interference between photons scat-
tered by imperfect super- and sub-radiant states, resem-
bling the electromagnetically-induced transparency in a
V-type atomic level structure [40]. 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. 3c. Each qubit pair configura-
tion 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 mea-
surement indicate that the qubit pair (Q
A
2
,Q
B
4
) has ex-
actly one perfectly super-radiant frequency point in the
UPB. For the other qubit pair (Q
B
2
,Q
A
5
), with waveguide
section in the topological phase, two such points occur
(corresponding to ∆
n
= 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 lineshapes are in good qualitative agreement
with the theoretical expectation in Ref. [26]. Detailed
description of the swirl pattern and similar measurement
results for other qubit combinations with varying ∆
n
are
reported in App. G.
Finally, to explore the physics associated with topo-
logical edge modes, we fabricated a second device, De-
vice II, which realizes a closed quantum system with 7
unit cells in the topological phase (Fig. 4a). We denote
the photonic sites in the array by (
i
,
α
), where
i
=1-7 is
the cell index and
α
=A,B is the sublattice index. Due
to reflection at the boundary, the passbands on this de-