ARTICLE
Superconducting metamaterials for waveguide
quantum electrodynamics
Mohammad Mirhosseini
1,2,3
, Eunjong Kim
1,2,3
, Vinicius S. Ferreira
1,2,3
, Mahmoud Kalaee
1,2,3
, Alp Sipahigil
1,2,3
,
Andrew J. Keller
1,2,3
& Oskar Painter
1,2,3
Embedding tunable quantum emitters in a photonic bandgap structure enables 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 electro-
dynamics in the slow-light regime. Alternatively, tuning the emitter into the bandgap results
in
fi
nite-range emitter
–
emitter interactions via bound photonic states. Here, we couple a
transmon qubit to a superconducting 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. 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
−
28 MHz accompanied by a 24-fold enhancement in the qubit
lifetime. In addition, we demonstrate selective enhancement and inhibition of spontaneous
emission of different transmon transitions, which provide simultaneous access to short-lived
radiatively damped and long-lived metastable qubit states.
DOI: 10.1038/s41467-018-06142-z
OPEN
1
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA.
2
Thomas J. Watson, Sr., Laboratory of Applied Physics, California
Institute of Technology, Pasadena, CA 91125, USA.
3
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125,
USA. Correspondence and requests for materials should be addressed to O.P. (email:
opainter@caltech.edu
)
NATURE COMMUNICATIONS
| (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications
1
1234567890():,;
C
avity quantum electrodynamics (QED) studies the inter-
action of an atom with a single electromagnetic mode of a
high-
fi
nesse 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
3
,
4
. Waveguide QED refers
to a system where a chain of atoms are coupled to a common
optical channel with a continuum of electromagnetic 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 ampli
fi
es the desired
radiation of the atoms into the waveguide modes
5
–
7
. Moreover, in
this con
fi
guration an interesting paradigm can be achieved by
placing the resonance frequency of the atom inside the bandgap
of the waveguide
8
–
11
. In this case, the atom cannot radiate into
the waveguide but the evanescent
fi
eld surrounding it gives rise to
a photonic bound state
9
. The interaction of such localized bound
states has been proposed for realizing tunable spin
–
exchange
interaction between atoms in a chain
12
,
13
, and also for realizing
effective non-local interactions between photons
14
,
15
.
While achieving ef
fi
cient waveguide coupling in the optical
regime requires the challenging task of interfacing atoms or
atomic-like systems with nanoscale dielectric structures
16
–
20
,
superconducting circuits provide an entirely different platform
for studying the physics of light
–
matter interaction in the
microwave regime
4
,
21
. Development of the
fi
eld of circuit QED
has enabled fabrication of tunable qubits with long coherence
times and fast qubit gate times
22
,
23
. Moreover, strong coupling is
readily achieved in coplanar platforms due to the deep sub-
wavelength transverse con
fi
nement of photons attainable in
microwave waveguides and the large electric dipole of super-
conducting qubits
24
. Microwave waveguides with strong disper-
sion, even
“
bandgaps
”
in frequency, can also be simply realized by
periodically modulating the geometry of a coplanar transmission
line
25
. Such an approach was recently demonstrated in a pio-
neering experiment by Liu and Houck
26
, whereby a qubit was
coupled to the localized photonic state within the bandgap of a
modulated coplanar waveguide (CPW). Satisfying the Bragg
condition in a periodically modulated waveguide requires a lattice
constant on the order of the wavelength, however, which trans-
lates to a device size of approximately a few centimeters for
complete con
fi
nement of the evanescent
fi
elds in the frequency
range suitable for microwave qubits. Such a restriction sig-
ni
fi
cantly 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 metamaterial
concept. Metamaterials are composite structures with sub-
wavelength components, which are designed to provide an
effective electromagnetic response
27
,
28
. Since the early micro-
wave work, the electromagnetic metamaterial concept has been
expanded and extensively studied across a broad range of
classical optical sciences
29
–
31
; however, their role in quantum
optics has remained relatively unexplored, at least in part due to
the lossy nature of many sub-wavelength components.
Improvements in design and fabrication of low-loss super-
conducting circuit components in circuit QED offer a new
prospect for utilizing microwave metamaterials in quantum
applications
32
. Indeed, high quality-factor superconducting
components such as resonators can be readily fabricated on a
chip
33
, and such elements have been used as a tool for achieving
phase-matching in near quantum-limited traveling wave
ampli
fi
ers
34
,
35
and for tailoring qubit interactions in a multi-
mode cavity QED architecture
36
.
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
metamaterial concept. In addition to a compact footprint, these
sort of structures can exhibit highly nonlinear band dispersion
surrounding the bandgap, leading to exceptionally strong con-
fi
nement of localized intra-gap photon states. We present the
design and fabrication of such a metamaterial waveguide, and
characterize the resulting waveguide dispersion and bandgap
properties via interaction with a tunable superconducting trans-
mon qubit. We measure the Lamb shift and lifetime of the qubit
in the bandgap and its vicinity, demonstrating the anomalous
Lamb shift of the fundamental qubit transition as well as selective
inhibition and enhancement of spontaneous emission for the
fi
rst
two excited states of the transmon qubit.
Results
Band-structure analysis and spectroscopy
. We begin by con-
sidering the circuit model of a CPW that is periodically loaded
with microwave resonators as shown in the inset to Fig.
1
a. The
Lagrangian for this system can be constructed as a function of the
node
fl
uxes of the resonator and waveguide sections
Φ
b
n
and
Φ
a
n
37
.
Assuming periodic boundary conditions and applying the rotat-
ing wave approximation, we derive the Hamiltonian for this
system and solve for the energies
h
ω
±
;
k
along with the corre-
sponding eigenstates ±
;
k
ji¼
^
α
±
;
k
0
ji
as (see Supplementary
Note 1)
ω
±
;
k
¼
1
2
Ω
k
þ
ω
0
ðÞ
±
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω
k
ω
0
ðÞ
2
þ
4
g
2
k
q
;
ð
1
Þ
^
α
±
;
k
¼
ω
±
;
k
ω
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω
±
;
k
ω
0
2
þ
g
2
k
r
^
a
k
þ
g
k
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω
±
;
k
ω
0
2
þ
g
2
k
r
^
b
k
:
ð
2
Þ
Here,
^
a
k
and
^
b
k
describe the momentum-space annihilation
operators for the bare waveguide and bare resonator 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, respec-
tively. The operators
^
α
±
;
k
represent quasi-particle solutions of the
composite waveguide, where far from the bandgap the quasi-
particle is primarily composed of the bare waveguide mode, while
in the vicinity of
ω
0
most of its energy is con
fi
ned in the
microwave resonators.
Figure
1
a 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
uncoupled 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 square of the wavenumber (
ω
−
ω
c
−
)
∝
1/
k
2
.
The analysis above has been presented for resonators which are
capacitively coupled to a waveguide in a parallel geometry; a
similar band structure can also be achieved using series inductive
coupling of resonators (see Supplementary Note 1 and Supple-
mentary Fig. 1).
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z
2
NATURE COMMUNICATIONS
| (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications
Physical realization using lumped-element resonators
.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 corre-
sponding to the fundamental resonance frequency
25
,
33
. However,
it is possible to signi
fi
cantly reduce the footprint of a resonator by
using components that mimic the behavior of lumped elements.
We have used the design presented in ref.
38
to realize resonators
in the frequency range of 6
–
10 GHz. This design provides com-
pact resonators by placing interdigital 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.
1
b). The
symmetry of this geometry results in the suppression of the
second harmonic frequency and thus the elimination of an
undesired bandgap at twice the fundamental resonance frequency
of the band-gap waveguide. A more subtle design criterion is that
the resonators be of high impedance. Use of high impedance
resonators allows for a larger photonic bandgap and greater
waveguide
–
qubit coupling. For the waveguide QED application of
interest this enables denser qubit circuits, both spatially and
spectrally.
The impedance of the resonators scales roughly as the inverse
square-root of the pitch of the wires in the spiral coils.
Complicating matters is that smaller wire widths have been
found to introduce larger resonator frequency disorder due to
kinetic inductance effects
39
. Here, we have selected an aggressive
resonator wire width of 1
μ
m and fabricated a periodic array of
N
=
9 resonator pairs coupled to a CPW with a lattice constant of
d
=
350. The resonators are arranged in identical pairs placed on
opposite sides of the central waveguide conductor to preserve the
symmetry of the waveguide. In addition, the center conductor of
each CPW section is meandered over a length of 210
μ
msoasto
increase the overall inductance of the waveguide section which
also increases the bandgap. Further details of the design criteria
and lumped element parameters are given in Supplementary
Note 2. The fabrication of the waveguide is performed using
electron-beam deposited Al
fi
lm (see Methods). Figure
1
c shows
the measured power transmission through such a
fi
nite-length
metamaterial waveguide. Here 50-
Ω
CPW segments, galvanically
coupled to the metamaterial waveguide, are used at the input and
output ports. We
fi
nd a midgap frequency of 5.83 GHz and a
bandgap extent of 1.82 GHz for the structure. 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.
Disorder and Anderson localization
. Fluctuations in the elec-
tromagnetic properties of the metamaterial waveguide along its
length, such as the aforementioned resonator disorder, results
in random scattering of traveling waves. Such random scat-
tering can lead to an exponential extinction of propagating
photons in the presence of weak disorder and complete trap-
ping of photons for strong disorder, a phenomenon known as
the Anderson localization of light
40
. Similarly, absorption loss
in the resonators results in attenuation of wave propagation
which adds a dissipative component to the effective localization
of
fi
elds in the metamaterial waveguide. Figure
2
ashows
numerical simulations of the effective localization length as a
function of frequency when considering separately the effects of
resonator frequency disorder and loss (see Supplementary
Note 3 for details of independent resonator measurements used
to determine frequency variation (0.5%) and loss parameters
(intrinsic
Q
-factor of 7.2 × 10
4
) for this model). In addition to
the desired localization of photons within the bandgap, we see
that the effects of disorder and loss also limit the localization
length outside the bandgap. In the lower transmission band
where the group index is largest, the localization length is seen
to rapidly approach zero near the band-edge, predominantly
due to disorder. In the upper transmission band where the
group index is smaller, the local
ization length maintains a large
value of 6 × 10
3
periods all the way to the band-edge. Within
the bandgap the simulations show that the localization length is
negligibly modi
fi
ed by the levels of loss and disorder expected
in the resonators of this work, and is well approximated by the
periodic loading of the waveguide alone which can be simply
related to the inverse of the curvature of the transmission bands
of a loss-less, disorder-free structure
13
. These results indicate
that, even with practical 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 fre-
quencies within a few hundred MHz of the band-edges of the
gap (see Supplementary Note 4).
4
4.5
5
5.5
6
6.5
7
7.5
8
0
0.2
0.4
0.6
0.8
1
Transmission
01
k
x
(
/
d
)
1
2
3
4
5
6
7
0
/2
8
9
10
Frequency (GHz)
Bare waveguide
dispersion
Bandgap region
(imaginary
k
-vector)
+
, k
–
, k
ab
c
Symmetry axis
x
20
μ
m
1/2
C
k
1/2
C
r
L
r
L
r
L
0
C
0
C
r
C
k
L
r
Frequency (GHz)
Band Gap
Φ
a
n
Φ
b
n
Fig. 1
Microwave metamaterial waveguide.
a
Dispersion relation of a CPW
loaded with a periodic array of microwave resonators (red curve). The
green line shows the dispersion relation of the waveguide without the
resonators. Inset: circuit diagram for a unit cell of the periodic structure.
b
Scanning electron microscope (SEM) image of a fabricated capacitively
coupled microwave resonator, here 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 constant of
d
=
350
μ
m. The blue curve depicts
the experimental data and the red curve shows the lumped-element model
fi
t to the data
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z
ARTICLE
NATURE COMMUNICATIONS
| (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications
3
Anomalous Lamb shift near the band-edge
. To further probe
the electromagnetic properties of the metamaterial waveguide we
couple it to a superconducting qubit. In this work, we use a
transmon qubit
22
with the fundamental resonance frequency
ω
ge
/
2
π
=
7.9 GHz and Josephson energy to single electron charging
energy ratio of
E
J
/
E
C
≈
100 at zero
fl
ux bias (details of our qubit
fabrication methods can also be found in ref.
41
). Figure
2
b shows
the geometry of the device where the qubit is capacitively coupled
to one end of the waveguide and the other end is capacitively
coupled to a 50-
Ω
CPW transmission line. This geometry allows
for forming narrow individual modes in the transmission band of
the metamaterial, which can be used for dispersive qubit state
read-out
42
from re
fl
ection measurements at the 50-
Ω
CPW input
port (see Supplementary Note 2 and Supplementary Table 1).
Figure
2
e, f shows the theoretical photonic LDOS and spatial
photon energy localization versus frequency for this
fi
nite length
qubit
–
waveguide system. Within the bandgap the qubit is self-
dressed by virtual photons which are emitted and re-absorbed
due to the lack of escape channels for the radiation. Near the
band-edges surrounding the bandgap, where the LDOS is rapidly
varying with frequency, this results in a large anomalous Lamb
shift of the dressed qubit frequency
10
,
43
. Figure
3
a shows the
measured qubit transition frequency shift from the expected bare
qubit tuning curve as a function of frequency. Shown for com-
parison is the circuit theory model frequency shift of a
fi
nite
structure with
N
=
9 periods (blue solid curve) alongside that of an
in
fi
nite length waveguide (red dashed curve). It is evident that the
qubit frequency is repelled from the band-edges on the two sides of
the bandgap, a result of the strongly asymmetric density of states in
these two regions. The measured frequency shift at the lower fre-
quency band-edge is 43 MHz, in good agreement with the circuit
theory model. Note that at the lower frequency band-edge where
the localization length approaches zero due to the anomalous
dispersion (see Fig.
2
a), boundary-effects in the
fi
nite structure do
not signi
fi
cantly alter the Lamb shift. Near the upper-frequency
band-edge, where
fi
nite-structure effects are non-negligible due to
the weaker dispersion and corresponding
fi
nite localization length,
we measure a qubit frequency shift as large as
−
28 MHz. This
againisingoodcorrespondencewiththe
fi
nite structure model;
the upper band-edge of the in
fi
nite length waveguide occurs at a
slightly lower frequency with a slightly smaller Lamb shift.
Enhancement and suppression of spontaneous emission
.
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 (see Fig.
2
f). Since
the
fi
nite waveguide is connected to an external port which acts as
a dissipative environment, the change in localization length
‘
ð
ω
Þ
is accompanied by a change in the lifetime of the qubit
T
rad
ð
ω
Þ/
e
2
x
=‘
ð
ω
Þ
, where
x
is the total length of the waveguide
(see Supplementary Note 5). 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 intrinsic
Q
-factors of 7.2 ± 0.4 × 10
4
for the individual waveguide reso-
nances between 4.6 and 7.4 GHz. Figure
3
b shows the measured
qubit lifetime (
T
1
) as a function of its frequency in the bandgap.
The solid blue curve in Fig.
3
b shows a
fi
tted theoretical curve
which takes into account the loss in the waveguide along with a
phenomenological intrinsic lifetime of the qubit (
T
l,
i
=
10.8
μ
s).
The dashed red curve shows the expected qubit lifetime for an
in
fi
nite waveguide length. Qualitatively, the measured lifetime of
Metamaterial
waveguide
Resonator
Qubit
100
μ
m
CPW input
XY
Z
SQUID loop
25
μ
m
Z
b
100
μ
m
10
0
LDOS
(GHz
–1
μ
m
–1
)
Frequency (GHz)
8
345
7
6
a
e
c
l
/
d
10
–5
10
–10
567
x
/
d
0
9
10
0
10
5
f
d
0
1
Fig. 2
Disorder effects and qubit
–
waveguide coupling.
a
Calculated localization length for a loss-less metamaterial waveguide with structural disorder (blue
circles). The nominal waveguide parameters are determined from the
fi
t to a lumped element model (including 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 vertical green lines represent the extent of the bandgap region. The red curve outside the gap is an analytic model
based on ref.
53
. For comparison, the solid black curve shows the calculated effective localization length without resonator frequency disorder but including
resonator loss.
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 re
fl
ective read-out.
c
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
fl
ux bias in the superconducting quantum
interference device (SQUID) loop. The XY drive is used to coherently excite the qubit.
d
Capacitively coupled microwave resonator.
e
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 (experimental
measurements of LDOS tabulated in Supplementary Table 1). The band-edges for the corresponding in
fi
nite structure are marked with vertical green lines.
f
Normalized electromagnetic energy distribution along the waveguide vs. qubit frequency for the coupled qubit
–
waveguide system. The vertical axis marks
the distance from the qubit (
x
/
d
) in units of the lattice period
d
ARTICLE
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z
4
NATURE COMMUNICATIONS
| (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications
the qubit behaves as expected; the qubit lifetime drastically
increases inside the bandgap region and is reduced in the trans-
mission bands. More subtle features of the measured lifetime
include multiple, narrow Fano-like spectral features deep within
the bandgap. These features arise from what are believed to be
interference between parasitic on-chip modes and low-
Q
modes
of our external copper box chip packaging. In addition, while the
measured lifetime near the upper band-edge is in excellent
agreement with the
fi
nite waveguide theoretical model, the data
near the lower band-edge shows signi
fi
cant deviation. We attri-
bute this discrepancy to the presence of low-
Q
parasitic reso-
nances, observable in transmission measurements between the
qubit XY drive line and the 50-
Ω
CPW port. Possible candidates
for such spurious modes include the asymmetric
“
slotline
”
modes
of the 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 suppres-
sing them will be a topic of future studies.
Focusing on the upper band-edge, we plot as an inset to Fig.
3
b
a zoom-in of the measured qubit lifetime along with theoretical
estimates of the different components of qubit decay. Here, the
qubit decay results from two dominant effects: detuning-
dependent coupling to the lossy resonances in the transmission
band of the waveguide, and emission into the output port of the
fi
nite waveguide structure. The former effect is an incoherent
phenomenon arising from a multi-mode cavity-QED picture,
whereas the latter effect arises from the coherent interference of
band-edge resonances which can be related to the photon bound
state picture and resulting localization length. Owing to the
weaker dispersion at the upper band-edge, the extent of the
photon bound state has an appreciable impact on the qubit
lifetime in the
N
=
9
fi
nite length waveguide. This is most telling
in the strongly asymmetric qubit lifetime around the
fi
rst
waveguide resonance in the upper transmission band. Quantita-
tively, the slope of the radiative component of the lifetime curve
in the bandgap near the band-edge can be shown to be
proportional to the group delay (see Supplementary Note 6),
∂
T
rad
=
∂
ω
jj
¼
T
rad
τ
delay
. The corresponding group index,
n
g
≡
τ
delay
/
x
, is a property of the waveguide independent of its
length
x
. Here, we measure a slope corresponding to a group
index
n
g
≈
450, in good correspondence with the circuit model of
the lossy metamaterial waveguide.
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 transmon qubit, by
construct, is a nonlinear quantum oscillator and thus has a
multilevel energy spectrum. In particular, a third energy level (|f
〉
)
exists at the frequency
ω
gf
¼
2
ω
ge
E
C
=
h
. Although the transi-
tion g
–
f is forbidden by selection rules, the f
–
e transition has a
dipole moment that is
ffiffiffi
2
p
larger than the fundamental
transition
22
. This is consistent with the scaling of transition
amplitudes in a harmonic oscillator and results in a second
transition lifetime that is half of the fundamental transition
lifetime for a uniform density of states in the electromagnetic
environment of the oscillator. The sharply varying density of
states in the metamaterial, on the other hand, 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 different spectral con
fi
gurations. In the
fi
rst
scenario, we enhance the ratio of the lifetimes
T
eg
/
T
fe
by situating
the fundamental transition frequency deep inside in the bandgap
while having the second transition positioned near the lower
transmission band. The situation is reversed in the second
con
fi
guration, where the fundamental frequency is tuned to be
near the upper frequency band while the second transition lies
deep inside the gap. In our fabricated qubit, the second transition
is about 300 MHz lower than the fundamental transition
frequency at zero
fl
ux bias, which allows for achieving large
lifetime contrast in both con
fi
gurations.
Discussion
Looking forward, we anticipate that further re
fi
nement in the
engineering and fabrication of the devices presented here should
enable metamaterial waveguides approaching a lattice constant-
to-wavelength ratio of
λ
/1000, with limited disorder and a
bandgap-to-midgap ratio in ex
cess of 50% (see Supplementary
Note 7). Such compact, low loss, low disorder superconducting
metamaterials can help realize more scalable superconducting
quantum circuits with higher levels of complexity and func-
tionality in several regards. They offer a method for densely
packing qubits
—
both in spatial and frequency dimensions
—
with isolation from the environment and controllable con-
nectivity achieved via bound qubit
–
waveguide polaritons
7
,
13
,
44
.
Moreover, the ability to selectively modify the transition life-
times provides simultaneous access to long-lived metastable
–40
–20
0
20
40
60
5
5.5
6
6.5
Frequency (GHz)
0
5
10
15
Lamb shift (MHz)
T
1
lifetime (
μ
s)
a
b
6.4
6.8
0
10
Fig. 3
Measured dispersive and dissipative qubit dynamics.
a
Lamb shift
of the qubit transition vs. qubit frequency.
b
Lifetime of the excited qubit
state vs. qubit frequency. Open circles show measured data. The solid
blue line (dashed red line) is a theoretical curve from the circuit model of
a
fi
nite (in
fi
nite) waveguide structure. For determining the Lamb shift
from measurement, the bare qubit frequency is calculated as a function
of
fl
ux bias
Φ
as
h
ω
ge
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
E
C
E
J
Φ
ðÞ
q
E
C
using the extracted values of
E
C
,
E
J
, and assuming the symmetric SQUID
fl
ux bias relation
E
J
(
Φ
)
=
E
J,
max
cos(2
π
Φ
/
Φ
0
)
22
. The lifetime characterization is performed in the time
domain where the qubit is initially excited 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 decays. Inset to
(
b
) shows a zoomed in region of the qubit lifetime near the upper band-
edge. Solid blue (red) lines show the circuit model contributions to
output port radiation (structural waveguide loss), adjusted to include a
frequency independent intrinsic qubit lifetime of 10.86
μ
s. The black
dashed line shows the cumulative theoretical lifetime
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z
ARTICLE
NATURE COMMUNICATIONS
| (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications
5