Superconducting metamaterials for waveguide quantum electrodynamics
Mohammad Mirhosseini,
1, 2
Eunjong Kim,
1, 2
Vinicius S. Ferreira,
1, 2
Mahmoud Kalaee,
1, 2
Alp Sipahigil,
1, 2
Andrew J. Keller,
1, 2
and Oskar Painter
1, 2,
∗
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA.
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA.
(Dated: February 7, 2018)
The embedding of tunable quantum emitters in a photonic bandgap structure enables the control
of dissipative and dispersive interactions between emitters and their photonic bath. Operation in
the transmission band, outside the gap, allows for studying waveguide quantum electrodynamics
in the slow-light regime. Alternatively, tuning the emitter into the bandgap results in finite range
emitter-emitter interactions via bound photonic states. Here we couple a transmon qubit to a su-
perconducting metamaterial with a deep sub-wavelength lattice constant (
λ/
60). The metamaterial
is formed by periodically loading a transmission line with compact, low loss, low disorder lumped
element microwave resonators. We probe the coherent and dissipative dynamics of the system by
measuring the Lamb shift and the change in the lifetime of the transmon qubit. Tuning the qubit
frequency in the vicinity of a band-edge with a group index of
n
g
= 450, we observe an anomalous
Lamb shift of 10 MHz accompanied by a 24-fold enhancement in the qubit lifetime. In addition,
we demonstrate selective enhancement and inhibition of spontaneous emission of different trans-
mon transitions, which provide simultaneous access to long-lived metastable qubit states and states
strongly coupled to propagating waveguide modes.
Cavity quantum electrodynamics (QED) studies the
interaction of an atom with a single electromagnetic
mode of a high-finesse cavity with a discrete spectrum
[1, 2]. In this canonical setting, a large photon-atom
coupling is achieved by repeated interaction of the atom
with a single photon bouncing many times between the
cavity mirrors. Recently, there has been much interest
in achieving strong light-matter interaction in a cavity-
free system such as a waveguide. Waveguide QED refers
to a system where a chain of atoms are coupled to a
common optical channel with a continuum of electromag-
netic modes over a large bandwidth. Slow-light photonic
crystal waveguides are of particular interest in waveguide
QED because the reduced group velocity near a bandgap
preferentially amplifies the desired radiation of the atoms
into the waveguide modes [3–5]. Moreover, in this con-
figuration an interesting paradigm can be achieved by
placing the resonance frequency of the atom inside the
bandgap of the waveguide [6–10]. In this case the atom
cannot radiate into the waveguide but the evanescent
field surrounding it gives rise to a photonic bound state
[8]. The interaction of such localized bound states has
been proposed for realizing tunable spin-exchange inter-
action between atoms in a chain [11, 12], and also for
realizing effective non-local interactions between photons
[13, 14].
While achieving efficient waveguide coupling in the op-
tical regime requires the challenging task of interfacing
atoms or atomic-like systems with nanoscale dielectric
structures [15–18], superconducting circuits provide an
entirely different platform for studying the physics of
∗
opainter@caltech.edu; http://copilot.caltech.edu
light-matter interaction in the microwave regime [19].
Development of the field of circuit QED has enabled
fabrication of fast and tunable qubits with long co-
herence times [20–22].
Moreover, strong coupling is
readily achieved in this platform due to the deep sub-
wavelength transverse confinement of photons attainable
in microwave waveguides and the large electrical dipole of
superconducting qubits [23]. Microwave waveguides with
strong dispersion, even “bandgaps” in frequency, can also
be simply realized by periodically modulating the geome-
try of a coplanar transmission line [24]. Such an approach
was recently demonstrated in a pioneering experiment by
Liu and Houck [25], whereby a qubit was coupled to the
localized photonic state within the bandgap of a mod-
ulated coplanar waveguide (CPW). Satisfying the Bragg
condition in a periodically modulated waveguide requires
a lattice constant on the order of the wavelength [26],
however, which translates to a device size of approxi-
mately a few centimeters for complete confinement of the
evanescent fields in the frequency range suitable for mi-
crowave qubits. Such a restriction significantly limits the
scaling in this approach, both in qubit number and qubit
connectivity.
An alternative approach for tailoring dispersion in the
microwave domain is to take advantage of the meta-
material concept. Metamaterials are composite struc-
tures with sub-wavelength components which are de-
signed to provide an effective electromagnetic response
[27, 28]. Since the early microwave work, the electro-
magnetic metamaterial concept has been expanded and
extensively studied across a broad range of classical op-
tical sciences [29–32]; however, their role in quantum op-
tics has remained relatively unexplored, at least in part
due to the lossy nature of many sub-wavelength compo-
arXiv:1802.01708v1 [quant-ph] 5 Feb 2018
2
4
4.5
5
5.5
6
6.5
7
7.5
8
0
0.2
0.4
0.6
0.8
1
Transmission
0
1
k
x
(
π/
d
)
1
2
3
4
5
6
7
8
9
10
Frequency (GHz)
bare waveguide
dispersion
bandgap region
(imaginary
k
-vector)
ω
+,k
ω
-,k
a
b
c
symmetry axis
x
20 μm
1/2
L
r
1/2
L
r
C
r
C
k
Frequency (GHz)
Band Gap
FIG. 1.
Microwave metamaterial waveguide. a
, Dis-
persion relation of a CPW loaded with a periodic array of
microwave resonators. The dashed line shows the dispersion
relation of the waveguide without the resonators. Inset: cir-
cuit diagram for a unit cell of the periodic structure.
b
, Scan-
ning electron microscope (SEM) image of the fabricated ca-
pacitively coupled microwave resonator with a wire width of
500 nm. The resonator region is false-colored in purple, the
waveguide central conductor and the ground plane are colored
green, and the coupling capacitor is shown in orange. We have
used pairs of identical resonators symmetrically placed on the
two sides of the transmission line to preserve the symmetry of
the structure.
c
, Transmission measurement for the realized
metamaterial waveguide made from 9 unit cells of resonator
pairs with a wire width of 1
μ
m, repeated with a lattice con-
stant of
d
= 350
μ
m. The blue curve depicts the experimental
data and the red curve shows the lumped-element model fit
to the data.
nents. Improvements in design and fabrication of low-
loss superconducting circuit components in circuit QED
offer a new prospect for utilizing microwave metamateri-
als in quantum applications. Indeed, high quality-factor
superconducting components such as resonators can be
readily fabricated on a chip [33, 34], and such elements
have been used as a tool for achieving phase-matching
in near quantum-limited traveling wave amplifiers [35–
37] and for tailoring qubit interactions in a multimode
cavity QED architecture [38].
In this paper, we utilize an array of coupled lumped-
element microwave resonators to form a compact
bandgap waveguide with a deep sub-wavelength lattice
constant (
λ/
60) based on the metameterial concept. In
addition to a compact footprint, these sort of structures
can exhibit highly nonlinear band dispersion surrounding
the bandgap, leading to exceptionally strong confinement
of localized intra-gap photon states. We present the de-
sign and fabrication of such a metamaterial waveguide,
and characterize the resulting waveguide dispersion and
bandgap properties via interaction with a tunable super-
conducting transmon qubit. We measure the Lamb shift
and lifetime of the qubit in the bandgap and its vicinity,
demonstrating the anomalous Lamb shift of the funda-
mental qubit transition as well as selective inhibition and
enhancement of spontaneous emission for the first two
excited states of the transmon qubit.
We begin by considering the circuit model of a CPW
that is periodically loaded with microwave resonators as
shown in the inset to Fig. 1a. The Lagrangian for this
system can be constructed as a function of the node fluxes
of the resonator and waveguide sections Φ
b
n
and Φ
a
n
[40].
Assuming periodic boundary conditions and applying the
rotating wave approximation, we derive the Hamiltonian
for this system and find the eigenstates and energies to
be (see App. A),
ω
±
,k
=
1
2
[
(Ω
k
+
ω
0
)
±
√
(Ω
k
−
ω
0
)
2
+ 4
g
2
k
]
,
(1)
ˆ
α
±
,k
=
(
ω
±
,k
−
ω
0
)
√
(
ω
±
,k
−
ω
0
)
2
+
g
2
k
ˆ
a
k
+
g
k
√
(
ω
±
,k
−
ω
0
)
2
+
g
2
k
ˆ
b
k
,
(2)
where ˆ
a
k
, and
ˆ
b
k
describe the momentum-space anni-
hilation operators for the bare waveguide and bare res-
onator sections, the index
k
denotes the wavevector, and
the parameters Ω
k
,
ω
0
, and
g
k
quantify the frequency
of traveling modes of the bare waveguide, the resonance
frequency of the microwave resonators, and coupling rate
between resonator and waveguide modes, respectively.
The operators ˆ
α
±
,k
represent quasi-particle solutions of
the composite waveguide, where far from the bandgap the
quasi-particle is primarily composed of the bare waveg-
uide mode, while in the vicinity of
ω
0
most of its energy
is confined in the microwave resonators.
Figure 1a depicts the numerically calculated energy
bands
ω
±
,k
as a function of the wavevector
k
. It is evident
that the dispersion has the form of an avoided crossing
between the energy bands of the bare waveguide and the
uncoupled resonators. For small gap sizes, the midgap
frequency is close to the resonance frequency of uncou-
pled resonators
ω
0
, and unlike the case of a periodically
modulated waveguide, there is no fundamental relation
tying the midgap frequency to the lattice constant in this
case. The form of the band structure near the higher cut-
off frequency
ω
c
+
can be approximated as a quadratic
function (
ω
−
ω
c
+
)
∝
k
2
, whereas the band structure near
the lower band-edge
ω
c
−
is inversely proportional to the
3
metamaterial
waveguide
resonator
qubit
100
μ
m
CPW input
XY
Z
SQUID loop
25
μ
m
Z
b
100
μ
m
10
0
10
-5
10
-10
LDOS (GHz
-1
μm
-1
)
10
8
10
4
10
0
8
3
4
5
7
6
Frequency (GHz)
8
3
4
5
7
6
Band Gap
a
c
Localization length,
l=d
FIG. 2.
Disorder effects and qubit-waveguide coupling. a
, Calculated localization length for a metamaterial waveguide
with structural disorder and resonator loss are shown as blue dots. The waveguide parameters are determined from the fit to
a lumped element model with resonator loss to the transmission data in Fig. 1. Numerical simulation has been performed for
N
= 100 unit cells, averaged over 10
5
randomly realized values of the resonance frequency
ω
0
, with the standard deviation
δω
0
/ω
0
= 0
.
5%. The red curve outside the gap is an analytic model based on Ref. [39].
b
, SEM image of the fabricated
qubit-waveguide system. The metamaterial waveguide (gray) consists of 9 periods of the resonator unit cell. The waveguide is
capacitively coupled to an external CPW (red) for reflective read-out. Bottom left inset: The transmon qubit is capacitively
coupled to the resonator at the end of the array. The Z drive is used to tune the qubit resonance frequency by controlling
the external flux bias in the superconducting quantum interference device (SQUID) loop. The XY drive is used to coherently
excite the qubit. Top right inset: capacitively coupled microwave resonator.
c
, Calculated local density of states (LDOS) at
the qubit position for a metamaterial waveguide with a length of 9 unit cells and open boundary conditions. The band-edges
for an infinite structure are marked with dashed red lines.
square of the wavenumber (
ω
−
ω
c
−
)
∝
1
/k
2
. The anal-
ysis above has been presented for resonators which are
capacitively coupled to a waveguide in a parallel geome-
try; a similar band structure can also be achieved using
series inductive coupling of resonators (see App. A).
A coplanar microwave resonator is often realized by
terminating a short segment of a coplanar transmission
line with a length set to an integer multiple of
λ/
4, where
λ
is the wavelength corresponding to the fundamental
resonance frequency [24, 33, 41]. However, it is possi-
ble to significantly reduce the footprint of a resonator by
using components that mimic the behavior of lumped el-
ements. We have used the design presented in Ref. [42]
to realize resonators in the frequency range of 6-10 GHz.
This design provides compact resonators by placing in-
terdigital capacitors at the anti-nodes of the charge waves
and double spiral coils near the peak of the current waves
at the fundamental frequency (see Fig. 1b). Further, the
symmetry of the geometry results in the suppression of
the second harmonic frequency and thus the elimination
of an undesired bandgap at twice the fundamental reso-
nance frequency of the band-gap waveguide.
We fabricate individual resonator pairs using an
electron-beam deposited 120 nm Al film, patterned via
lift-off, on a high resistivity silicon wafer substrate of
thickness 500
μ
m (see Ref. [43] for further details of fab-
rication techniques). In this work we have made a pe-
riodic array of 9 resonator pairs with a wire width of 1
μ
m and coupled them to a CPW in a periodic fashion
with a lattice constant of 350
μ
m to realize a metama-
terial waveguide. The resonators are arranged in identi-
cal pairs placed on the opposite sides across the central
waveguide conductor to preserve the symmetry of the
waveguide. Figure 1c shows the measured power trans-
mission through such a finite-length metamaterial waveg-
uide. Here 50-Ω CPW segments, galvanically coupled to
the metamaterial waveguide, are used at the input and
output ports. We find the midgap frequency of 5
.
83 GHz
for the structure, and a gap frequency span of 1
.
82 GHz.
Using the simulated value of effective refractive index of
2
.
54, the midgap frequency gives a lattice constant-to-
wavelength ratio of
d/λ
≈
1
/
60.
Propagation of electromagnetic fields in the frequency
range within the bandgap is exponentially attenuated
with a localization length set by the imaginary part of
the wavenumber. In addition, statistical variations in
the electromagnetic properties of the periodic structure
result in random scattering of the traveling waves in the
transmission band. Such random scatterings can lead to
complete trapping of propagating photons in the pres-
ence of strong disorder and an exponential extinction for
weak disorder; a phenomenon known as the Anderson
4
5
5.5
6
6.5
Frequency (GHz)
-40
-20
0
20
40
60
Lamb shift (MHz)
5
5.5
6
6.5
Frequency (GHz)
0
5
T
1
lifetime (
μ
s)
15
10
a
b
FIG. 3.
Measured dispersive and dissipative qubit dy-
namics. a
, Qubit frequency Lamb shift versus frequency.
b
, Qubit lifetime versus frequency. The open circles show
experimental data and the solid line presents a theory fit.
The dashed red lines mark the estimated position of the
band-edge corresponding to an infinite-length structure and
the shaded grey regions correspond to anti-crossing with the
first individual resonances near the band-edges of the fi-
nite structure.
For calculating the Lamb shift, the bare
qubit frequency is calculated as a function of flux bias Φ as
~
ω
ge
=
√
8
E
C
E
J
(Φ)
−
E
C
using the extracted values of
E
C
,
E
J
, and assuming the symmetrical SQUID flux bias relation
E
J
(Φ) =
E
J,
max
cos(2
π
Φ
/
Φ
0
). The lifetime characterization
is performed in the time domain where the qubit is initially ex-
cited with a
π
pulse through the XY drive. The excited state
population, determined from the state-dependent dispersive
shift of a close-by band-edge waveguide mode, is measured
subsequent to a delay time during which the qubit freely de-
cays.
localization of light [44]. We have measured a random
standard deviation of 0
.
3% in the resonance frequency
of the fabricated lumped-element resonators. Figure 2a
shows the calculated localization length as a function
of frequency from numerical simulation of the indepen-
dently measured disorder and loss of the resonators in
the metamaterial waveguide (see App. C and D for fur-
ther details). Near the edges of the bandgap the local-
ization length from disorder dominates that from loss,
rapidly approaching zero at the lower band-edge where
the group index is largest and maintaining a large value
(6
×
10
3
periods) at the higher band-edge where the group
index is smaller. Similarly, the localization length in-
side the gap is inversely proportional to the curvature
of the energy bands [12]. Owing to the divergence (in
the loss-less case) of the lower band curvature for the
waveguide studied here, the localization length inside the
gap approaches zero near the lower band-edge frequency
as well. These results indicate that, even with practi-
cal limitations on disorder and loss in such metamaterial
waveguides, a range of photon length scales of nearly four
orders of magnitude should be accessible for frequencies
within a few hundred MHz of the band-edges.
To further probe the electromagnetic properties of the
metamaterial waveguide we couple it to a superconduct-
ing qubit. In this work we use a transmon qubit [20, 21]
with the fundamental resonance frequency
ν
ge
= 7
.
9 GHz
and Josephson energy to single electron charging energy
ratio of
E
J
/E
C
≈
100 at zero flux bias (details of our
qubit fabrication methods can also be found in Ref. [43]).
Figure 2b shows the geometry of the device where the
qubit is capacitively coupled to one end of the waveg-
uide and the other end is capacitively coupled to a 50-Ω
CPW transmission line. This geometry allows for form-
ing narrow individual modes in the transmission band of
the metamaterial, which can be used for dispersive qubit
state read-out [45] from reflection measurements at the
50-Ω CPW input port (see Fig. 2b). Within the bandgap
the qubit is self-dressed by virtual photons which are
emitted and re-absorbed due to the lack of escape chan-
nels for the radiation. Near the band-edges surrounding
the bandgap, where the LDOS is rapidly varying with fre-
quency, this can result in a large anomalous Lamb shift
of the dressed qubit frequency [9, 46]. To observe this
effect, we tune the qubit frequency using a flux bias [21]
and find the frequency shift by subtracting the measured
frequency from the expected frequency of the qubit as a
function of flux bias. Figure 3a shows the measured fre-
quency shift as a function of tuning. It is evident that the
qubit frequency is repelled from the band-edges on the
two sides, as a result of the asymmetric density of states
near the cut-off frequencies. The measured frequency
shift is approximately 10 MHz at the band-edges (0
.
2%
of the qubit frequency), in excellent agreement with the
circuit theory model (see App. E).
Another signature of the qubit-waveguide interaction
is the change in the rate of spontaneous emission of the
qubit. Tuning the qubit into the bandgap changes the
localization length of the waveguide photonic state that
dresses the qubit. Since the finite waveguide is connected
to an external port which acts as a dissipative environ-
ment, the change in localization length
`
(
ω
) is accompa-
nied by a change in the radiative lifetime of the qubit
T
rad
(
ω
)
∝
e
2
x/`
(
ω
)
, where
x
is the total length of the
waveguide. Figure 3b shows the measured qubit lifetime
(
T
1
) as a function of its frequency in the bandgap. It is
evident that the qubit lifetime drastically increases in-
side the bandgap, where spontaneous emission into the
output port is greatly suppressed due to the reduced lo-
calization length of the photon bound state. Deep within
the bandgap one observes the appearance of multiple nar-
row spectral features in the measured frequency depen-
dence of the qubit lifetime. These features, attributable
to parasitic “box” modes of our chip packaging, high-
light the ability of the metamaterial waveguide to enable
effectively-dissipation-free probing of the qubit’s environ-
ment over a broad spectral range (
>
1 GHz). As the
qubit frequency approaches the band-edges, the lifetime
is sharply reduced because of the increase in the localiza-
tion length of the waveguide modes. The slope of the life-
time curve at the band-edge can be shown to be directly
proportional to the group delay,
|
∂T
rad
/∂ω
|
=
T
rad
τ
delay
(see App. E). We observe a 24-fold enhancement in the
lifetime of the qubit near the upper band-edge, corre-
sponding to a maximum group index of
n
g
= 450 right
5
Delay,
τ
(
μ
s)
Excited state population
LDOS
LDOS
τ
= 11
μ
s
τ
= 1.1
μ
s
τ
= 2.7
μ
s
τ
= 5.7
μ
s
10
0
10
-1
10
0
10
-1
0
10
20
30
0
2
4
6
8
10
12
a
b
FIG. 4.
State-selective enhancement and inhibition of
radiative decay. a
, Measurement with the
g
-
e
transition
tuned into the bandgap, with the
f
-
e
transition in the lower
transmission band.
b
, Measurement with the
g
-
e
transition
tuned into the upper transmission band, with the
f
-
e
tran-
sition in the bandgap. For measuring the
f
-
e
lifetime, we
initially excite the third energy level
|
f
〉
via a two-photon
π
pulse at the frequency of
ω
gf
/
2. Following the popula-
tion decay in a selected time interval, the population in
|
f
〉
is
mapped to the ground state using a second
π
pulse. Finally
the ground state population is read using the dispersive shift
of a close-by band-edge resonance of the waveguide.
g
-
e
(
f
-
e
)
transition data shown as red squares (blue circles)
at the band-edge.
In addition to radiative decay into the output channel,
losses in the resonators in the waveguide also contribute
to the qubit’s excited state decay. Using a low power
probe in the single-photon regime we have measured in-
trinsic
Q
-factors of 7
.
2
±
0
.
4
×
10
4
for the individual waveg-
uide modes between 4
.
6-7
.
4 GHz. The solid line in Fig. 3b
shows a fitted theoretical curve which takes into account
the loss in the waveguide along with a phenomenological
intrinsic lifetime of the qubit. While the measured life-
time near the upper band is in excellent agreement with
the theoretical model, the data near the lower band shows
significant departure from the model. We attribute this
departure in the lower band to the presence of a spurious
resonance or resonances near the lower band-edge. Possi-
ble candidates for such spurious modes include the asym-
metric “slotline” modes of the metamaterial waveguide,
which are weakly coupled to our symmetrically grounded
CPW line but may couple to the qubit. Further study
of the spectrum of these modes and possible methods for
suppressing them using cross-over connections [47] will
be a topic of future studies.
The sharp variation in the photonic LDOS near the
metamaterial waveguide band-edges may also be used to
engineer the multi-level dynamics of the qubit. A trans-
mon qubit, by construct, is a nonlinear quantum oscilla-
tor and thus it has a multilevel energy spectrum. In par-
ticular, a third energy level (
|
f
〉
) exists at the frequency
ω
gf
= 2
ω
ge
−
E
C
/
~
. Although the transition
g
-
f
is not al-
lowed because of the selection rules, the
f
-
e
transition is
allowed and has a dipole moment that is
√
2 larger than
the fundamental transition [20]. This is reminiscent of
the scaling of transition amplitudes in a harmonic oscil-
lator and results in a second transition lifetime that is half
of the fundamental transition lifetime for a uniform den-
sity of states in the electromagnetic bath. Nonetheless,
the sharply varying density of states in the metamaterial
can lead to strong suppression or enhancement of the
spontaneous emission for each transition. Figure 4 shows
the measured lifetimes of the two transitions for two dif-
ferent spectral configurations. In the first scenario, we
enhance the ratio of the lifetimes
T
eg
/T
fe
by situating the
fundamental transition frequency inside in the bandgap
while having the second transition positioned inside the
lower transmission band. The situation is reversed in the
second configuration, where the fundamental frequency
is tuned to be within the upper energy band while the
second transition lies inside the gap. In our fabricated
qubit, the second transition is 290 MHz lower than the
fundamental transition frequency at zero flux bias, which
allows for achieving large lifetime contrast in both con-
figurations.
Compact, low loss, low disorder superconducting meta-
materials, as presented here, can help realize more scal-
able superconducting quantum circuits with higher levels
of complexity and functionality in several regards. They
offer a method for densely packing qubits – both in spa-
tial and frequency dimensions – with isolation from the
environment by operation in forbidden bandgaps, and yet
with controllable connectivity achieved via bound qubit-
waveguide polaritons [5, 12]. Moreover, the ability to se-
lectively modify the transition lifetimes provides simul-
taneous access to long-lived metastable qubit states as
well as short-lived states strongly coupled to waveguide
modes. This approach realizes an effective Λ-type level
structure for the transmon, and can be used to create
state-dependent bound state localization lengths, quan-
tum nonlinear media for propagating microwave pho-
tons [14, 48, 49], or as recently demonstrated, to realize
spin-photon entanglement and high-bandwidth itinerant
single microwave photon detection [50, 51]. Combined,
these attributes provide a unique platform for studying
the many-body physics of quantum photonic matter [52–
55].
ACKNOWLEDGMENTS
We would like to thank Paul Dieterle, Ana Asenjo
Garcia and Darrick Chang for fruitful discussions re-
garding waveguide QED. This work was supported by
the AFOSR MURI Quantum Photo4nic Matter (grant