of 21
Supplementary Information for: “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: August 1, 2018)
opainter@caltech.edu; http://copilot.caltech.edu
2
···
Φ
a
n
1
L
0
Φ
a
n
L
0
Φ
a
n
+1
···
C
0
C
g
Φ
b
n
+1
C
r
L
r
C
g
Φ
b
n
C
r
L
r
C
0
···
···
···
L
0
L
r
C
r
L
g
C
0
L
0
L
r
C
r
L
g
···
C
0
···
···
Supplementary Figure. 1.
Circuit diagram of metamaterial waveguide.
The waveguide can be made from periodic arrays
of transmission line sections loaded with capacitively coupled resonators (top), or inductively loaded resonators (bottom).
3
5.88
5.89
5.9
5.91
5.92
-50
-40
-30
-20
-10
Q
e
=325
Q
i
= 485562
5.88
5.89
5.9
5.91
5.92
-1.5
-1
-0.5
0
0.5
1
1.5
Frequenc
y (
GH
z)
Frequenc
y (
GH
z)
6.098
6.1
6.102
6.104
6.106
-40
-30
-20
-10
0
Q
e
=1098
Q
i
= 256332
6.098
6.1
6.102
6.104
6.106
-1
-0.5
0
0.5
1
)
z
H
G
(
y
c
n
e
u
q
e
r
F
)
z
H
G
(
y
c
n
e
u
q
e
r
F
b
a
d
c
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
6.8
7
F
r
e
q
u
e
n
c
y
(
G
H
z
)
-0.01
-0.005
0
0.005
0.01
arg(
S
21
)
S
21
(dB)
S
21
(dB)
arg(
S
21
)
Supplementary Figure. 2.
Characterization of lumped element resonators a
, Optical and SEM images of microwave
resonator array chip. Middle: optical image of the chip with two arrays of coupled resonators on a 1
×
1 cm silicon chip. Left
and Right: SEM image (false-color) of the fabricated inductively (left) and capacitively (right) coupled microwave resonator
pairs. The resonator region is colored red and the waveguide central conductor is colored blue.
b-c
, Amplitude and phase
response of two capacitively-coupled microwave resonator pairs measured at the fridge temperature
T
f
7 mK. The legends
show the intrinsic (
Q
i
=
ω
0
i
) and extrinsic (
Q
e
=
ω
0
e
) quality factors extracted from a Fano line shape fit.
d
, Difference
between the measured and the expected design value of the resonance frequencies for 9 resonators with similar geometries and
wire widths of 500 nm. The dashed lines mark the standard deviation of the frequency difference, which is equivalent to a
normalized value of
σ
= 0
.
3%.
4
C
g
C
q
V
q
E
J
metamaterial waveguide
Z
line
R
L
Supplementary Figure. 3.
Circuit diagram for a transmon qubit coupled to a metamaterial waveguide
The resistive
termination is used to model radiation into the 50Ω coplanar waveguide.
5
45678
Bare qubit frequency (GHz)
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
T
1
lifetime (s)
6. 46
. 66
.
87
Bare qubit frequency (GHz)
0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
0. 9
1
T
1
lifetime (s)
× 10
-5
a
b
Supplementary Figure. 4.
Qubit lifetime as a function of resonance frequency. a
, Simulated qubit lifetime set by
radiation into the output CPW port (blue), and structural loss in the waveguide (red).
b
, Comparison of the experimental
results (open circles) with the simulated qubit lifetime (solid and dashed lines) near the first resonance dip in the upper
transmission band. The lifetime set by radiation into the output port and structural loss in the waveguide are shown as blue
and red solid lines, respectively. Both of these contributions have been adjusted to include a frequency independent intrinsic
qubit life time of 10
.
86
μ
s. The black dashed line shows the theoretical qubit excited state lifetime including all contributions.
6
band
frequency (GHz)
g/
2
π
(MHz)
Q
e
(
×
10
3
)
Q
i
(
×
10
3
)
lower
4.2131
15.3
49.47
74.99
lower
4.6012
19.74
35.09
76.25
lower
4.7395
18.14
43.58
75
lower
4.8044
16.53
94.17
75.59
lower
4.8373
14.03
152.06
73.77
lower
4.856
9.73
455.8
76.47
lower
4.8654
4.48
2100
72
upper
6.6768
39.44
15.74
68.06
upper
7.309
58.06
12.02
70.44
Supplementary Table. I.
Measured resonance parameters for metamaterial waveguide.
The values are measured
for the waveguide of Figs. 2-4 in main text. The resonances are measured in reflection from the input 50-Ω CPW port. The
qubit-resonance coupling,
g
, is inferred from the anti-crossing observed as the qubit is tuned through each waveguide resonance.
7
SUPPLEMENTARY NOTE 1. BAND STRUCTURE ANALYSIS
A. Quantization of a periodic resonator-loaded waveguide
We consider the case of a waveguide that is periodically loaded with microwave resonators. Supplementary Figure
1 depicts a unit cell for this configuration. The Lagrangian for this system can be readily written as [1]
L
=
n
[
1
2
C
0
[
̇
Φ
a
n
]
2
a
n
Φ
a
n
1
]
2
2
L
0
+
1
2
C
r
[
̇
Φ
b
n
]
2
+
1
2
C
g
[
̇
Φ
a
n
̇
Φ
b
n
]
2
b
n
]
2
2
L
r
]
.
(1)
In order to find solutions in form of traveling waves, it is easier to work with the Fourier transform of node fluxes.
We use the following convention for defining the (discrete) Fourier transformation
Φ
a
,
b
κ
=
1
M
N
n
=
N
e
i
2
π
(
κ/M
)
n
Φ
a
,
b
n
,
(2)
where
M
= 2
N
+ 1 is the total number of periods in the waveguide. Using the Fourier relation we find the Lagrangian
in
k
-space as
L
=
κ
[
1
2
(
C
0
+
C
g
)
|
̇
Φ
a
κ
|
2
1
e
i
2
π
(
κ/M
)
2
|
Φ
a
κ
|
2
2
L
0
1
2
(
C
g
+
C
r
)
|
̇
Φ
b
κ
|
2
|
Φ
b
κ
|
2
2
L
r
C
g
̇
Φ
b
κ
̇
Φ
a
κ
+
̇
Φ
b
κ
̇
Φ
a
κ
2
]
.
(3)
To proceed further, we need to find the canonical node charges which are defined as
Q
a
,
b
κ
=
∂L
̇
Φ
a
,
b
κ
, and subsequently
derive the Hamiltonian of the system by using a Legendre transformation. Doing so we find
H
=
κ
[
Q
a
κ
Q
a
κ
2
C
0
+
1
e
i
2
π
(
κ/M
)
2
Φ
a
κ
Φ
a
κ
2
L
0
+
Q
b
κ
Q
b
κ
2
C
r
+
Φ
b
κ
Φ
b
κ
2
L
r
+
Q
a
κ
Q
b
κ
+
Q
a
κ
Q
b
κ
2
C
g
]
.
(4)
Here, we have defined the following quantities
C
0
=
C
g
C
r
+
C
g
C
0
+
C
0
C
r
C
g
+
C
r
,
(5)
C
r
=
C
g
C
r
+
C
g
C
0
+
C
0
C
r
C
g
+
C
0
,
(6)
C
g
=
C
g
C
r
+
C
g
C
0
+
C
0
C
r
C
g
.
(7)
The canonical commutation relation [Φ
i
κ
,Q
j
κ
] =
i
~
δ
i,j
δ
κ,κ
allows us to define the following annihilation operators
as a function of charge and flux operators
ˆ
a
κ
=
C
0
k
2
~
(
Φ
a
κ
+
i
C
0
k
Q
a
κ
)
,
(8)
ˆ
b
κ
=
C
r
ω
0
2
~
(
Φ
b
κ
+
i
C
r
ω
0
Q
b
κ
)
.
(9)
Here, we have defined the resonance frequency for each mode as
k
=
4sin
2
(
kd/
2)
L
0
C
0
,
(10)
ω
0
=
1
L
r
C
r
,
(11)
8
where
k
= (2
πκ
)
/
(
Md
) is the wavenumber. It is evident that Ω
k
has the expected dispersion relation of a discrete
periodic transmission line and
ω
0
is the resonance frequency of the loaded microwave resonators. Using the above
definitions for ˆ
a
κ
,
ˆ
b
κ
ˆ
H
=
~
2
k
[
k
(
ˆ
a
k
ˆ
a
k
+ ˆ
a
k
ˆ
a
k
)
+
ω
0
(
ˆ
b
k
ˆ
b
k
+
ˆ
b
k
ˆ
b
k
)
g
k
(
ˆ
b
k
ˆ
b
k
)(
ˆ
a
k
ˆ
a
k
)
g
k
(
ˆ
a
k
ˆ
a
k
)(
ˆ
b
k
ˆ
b
k
)
]
,
(12)
along with the coupling coefficient
g
k
=
C
0
C
r
2
C
g
ω
0
k
=
C
g
ω
0
k
2
(
C
0
+
C
g
)(
C
r
+
C
g
)
.
(13)
An alternative structure for coupling microwave resonators is depicted in the bottom panel of Supplementary Figure
1. In this geometry, the coupling is controlled by the inductive element
L
g
. Repeating the analysis above for this
case, we find
k
=
4sin
2
(
kd/
2)
C
0
L
0
,
(14)
ω
0
=
1
C
r
L
r
,
(15)
g
k
=
L
0
L
r
2
L
g
ω
0
k
.
(16)
We have defined the modified inductance values as
L
0
=
L
g
L
r
+
LL
g
L
0
+
L
0
L
r
L
g
+
L
r
,
(17)
L
r
=
L
g
L
r
+
L
g
L
0
+
L
0
L
r
L
g
+
L
0
,
(18)
L
g
=
L
g
L
r
+
L
g
L
0
+
L
0
L
r
L
g
.
(19)
B. Band structure calculation with RWA
Using the rotating wave approximation, the Hamiltonian in Eq. (12) can be simplified to
ˆ
H
=
~
k
[
k
ˆ
a
k
ˆ
a
k
+
ω
0
ˆ
b
k
ˆ
b
k
+
g
k
(
ˆ
b
k
ˆ
a
k
+ ˆ
a
k
ˆ
b
k
)
]
.
(20)
Note that this approximation is applicable only when the coupling is sufficiently weak,
g
k

min(
ω
0
,
k
), and the
detuning is sufficiently small
|
ω
0
k
| 
(
ω
0
+ Ω
k
). Assuming Ω
k
and
ω
0
are of the same order, this condition is
satisfied when
C
g

2
(
C
0
C
r
).
The simplified Hamiltonian can be written in the compact form
ˆ
H
=
~
k
x
k
H
k
x
k
,
(21)
where
H
k
=
[
k
g
k
g
k
ω
0
]
,
x
k
=
[
ˆ
a
k
ˆ
b
k
]
.
(22)
9
We desire to transform the Hamiltonian to a diagonalized form
̃
H
k
=
[
ω
+
,k
0
0
ω
,k
]
.
(23)
It is straightforward to use the eigenvalue decomposition to find
ω
±
,k
as
ω
±
,k
=
1
2
[
(Ω
k
+
ω
0
)
±
(Ω
k
ω
0
)
2
+ 4
g
2
k
]
,
(24)
along with the corresponding eigenstates
,k
= ˆ
α
±
,k
|
0
, where
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
.
(25)
C. Band structure calculation beyond RWA
The exact Hamiltonian in Eq. (12) can be written in the compact form
ˆ
H
=
~
2
k
x
k
H
k
x
k
,
(26)
where
H
k
=
k
0
g
k
g
k
0
k
g
k
g
k
g
k
g
k
ω
0
0
g
k
g
k
0
ω
0
,
x
k
=
ˆ
a
k
ˆ
a
k
ˆ
b
k
ˆ
b
k
.
(27)
To find the eigenstates of the system, we can use a linear transform to map the state vector
̃x
k
=
S
k
x
k
such that
x
k
H
k
x
k
=
̃x
k
̃
H
k
̃x
k
with the transformed diagonal Hamiltonian matrix
̃
H
k
=
ω
+
,k
0
0
0
0
ω
+
,k
0
0
0
0
ω
,k
0
0
0
0
ω
,k
.
(28)
In order to preserve the canonical commutation relations, the matrix
S
k
has to be symplectic, i.e.
J
=
S
k
JS
k
, with
the matrix
J
defined as
J
=
1 0 0
0
0
1 0
0
0 0 1
0
0 0 0
1
.
(29)
A linear transformation (such as
S
k
) that diagonalizes a set of quadratically coupled boson fields while preserving their
canonical commutation relations is often referred to as a Bogoliubov-Valatin transformation. While it is generally
difficult to find the transform matrix
S
k
, it is easy to find the eigenvalues of the diagonalized Hamiltonian by exploiting
some of the properties of
S
k
. Note that since
J
=
S
k
JS
k
, the matrices
J
̃
H
k
and
JH
k
share the same set of eigenvalues.
The eigenvalues of
J
̃
H
k
are the two frequencies
ω
±
,k
, and thus we have
ω
2
±
,k
=
1
2
[
(
2
k
+
ω
2
0
)
±
(Ω
2
k
ω
2
0
)
2
+ 16
ω
0
k
g
2
k
]
.
(30)
10
D. Circuit theory derivation of the band structure
Consider the pair of equations that describe the propagation of a monochromatic electromagnetic wave of the form
v
(
x,t
) =
V
(
x
)
e
ikx
e
iωt
(along with the corresponding current relation) inside a transmission line
d
d
x
V
(
x
) =
Z
(
ω
)
I
(
x
)
,
d
d
x
I
(
x
) =
Y
(
ω
)
V
(
x
)
.
(31)
Here,
Z
(
ω
) and
Y
(
ω
) are frequency dependent impedance and admittance functions that model the linear response of
the series and parallel portions of a transmission line with length
d
. It is straightforward to check that the solutions
to these equation satisfy
k
(
ω
) =
nω/c
=
Z
(
ω
)
Y
(
ω
)
/d
. For a loss-less waveguide and in the absence of dispersion
we have
Z
(
ω
) =
iωL
0
and
Y
(
ω
) =
iωC
0
, and thus we find the familiar dispersion relation
k
(
ω
) =
ω
L
0
C
0
/d
.
Nevertheless, the pair of equations above remain valid for arbitrary impedance and admittance functions
Z
(
ω
) and
Y
(
ω
), provided that the dimension of the model circuit remains much smaller than the wavelength under consideration.
In this model, a real and negative quantity for the product
ZY
results in an imaginary wavenumber and subsequently
creates a stop band in the dispersion relation. This situation can be achieved by periodically loading a transmission
line with an array of resonators [2, 3]. Assuming a unit length of
d
we find
k
2
=
(
ω
c
)
2
n
2
[
1 +
2
e
nd
1
ω
2
0
ω
2
]
.
(32)
Here,
ω
0
is the resonance frequency, and
γ
e
is the external coupling decay rate of an individual resonator in the array.
For moderate values of gap-midgap ratio (∆
m
), the frequency gap can be found as
∆ =
c
nd
(
γ
e
ω
0
)
,
(33)
and
ω
m
=
ω
0
+ ∆
/
2. We have defined the gap as the range of frequencies where the wavenumber is imaginary.
Although a microwave resonator can be realized by using a two-elements
LC
-circuit, the three-element circuits in
Supplementary Figure 1 provide an additional degree of freedom which enables setting the coupling
γ
e
independent
of the resonance frequency
ω
0
. Using circuit theory, it is straightforward to show
ω
0
=
1
L
r
(
C
r
+
C
g
)
,
(34)
γ
e
=
Z
0
2
L
r
(
C
g
C
r
+
C
g
)
2
.
(35)
Here,
Z
0
is the characteristic impedance of the unloaded waveguide. It is easy to check that for small values of
C
g
/C
r
,
the resonance frequency is only a weak function of
C
g
. As a result, it is possible to adjust the coupling rate
γ
e
by
setting the capacitor
C
g
while keeping the resonance frequency almost constant. Supplementary Figure 1 also depicts
an alternative strategy for coupling microwave resonators to the waveguide. In this design, the inductive element
L
g
is used to set the coupling in a “current divider” geometry. We provide experimental results for implementation of
bandgap waveguide based on both designs in the next section.
While the “continuum” model described above provides a heuristic explanation for formation of bandgap in a
waveguide loaded with resonators, its results remains valid as far as
k

2
π/d
. To avoid this approximation, we can
use the transfer matrix method to find the exact dispersion relation for a system with discrete periodic symmetry [4].
In this case Eq. (32) is modified to
cos (
kd
) = 1
(
ω
c
)
2
n
2
d
2
2
ndγ
e
c
ω
2
ω
2
0
ω
2
.
(36)
Note that this relation still requires
d
to be much smaller than the wavelength of the unloaded waveguide
λ
= 2
πc/
(
).
11
E. Dispersion and group index near the band-edges
Equation (30) can be reversed to find the wavenumber
k
as a function of frequency. Assuming, a linear dispersion
relation of the form
k
=
n
k
/c
for the bare waveguide we find
k
=
c
ω
2
ω
2
c
+
ω
2
ω
2
c
.
(37)
Here,
ω
c
+
=
ω
0
and
ω
c
=
ω
0
1
4
g
2
k
/
(Ω
k
ω
0
) are the upper and lower cut-off frequencies, respectively. The quantity
g
2
k
/
(Ω
k
ω
0
) is a unit-less parameter quantifying the size of the bandgap and is independent of the wavenumber
k
.
The dispersion relation can be written in simpler forms by expanding the wavenumber in the vicinity of the two
band-edges
k
=
c
c
δ
for
ω
ω
c
,
c
+
c
δ
+
for
ω
ω
c
+
.
(38)
Here, ∆ =
ω
c
+
ω
c
is the frequency span of the bandgap and
δ
±
=
ω
ω
c
±
are the detunings from the band-edges.
The form of the dispersion relation Eq. (30) suggests that the maxima of the group index happens near the band-
edges. Having the wavenumber, we can readily evaluate the group velocity
v
g
=
∂ω/∂k
and find the group index
n
g
=
c/v
g
as
n
g
=
c
4(
δ
i
)
3
for
ω
ω
c
,
c
+
4∆(
δ
+
i
)
for
ω
ω
c
+
.
(39)
Note that we have replaced
δ
±
with
δ
±
i
to account for finite internal quality factor of the resonators in the
structure.
F. Coupling a Josephson junction qubit to a metamaterial waveguide
We consider the coupling of a Josephson junction qubit to the metamaterial waveguide. Assuming rotating wave
approximation (valid for weak coupling
f
k

ω
k
q
), the Hamiltonian of this system can be written as
ˆ
H
=
~
k
[
ω
k
ˆ
a
k
ˆ
a
k
+
ω
q
2
ˆ
σ
z
+
f
k
(
ˆ
a
k
ˆ
σ
+ ˆ
a
k
ˆ
σ
+
)
.
]
(40)
Here
f
k
is the coupling factor of the qubit to the waveguide photons, and
ω
k
=
ω
±
,k
, where the plus or minus sign is
chosen such that the qubit frequency
ω
q
lies within the band. Without loss of generality, we assume
f
k
to be a real
number. The Heisenberg equations of motions for the qubit and the photon operators can be written as
∂t
ˆ
a
k
=
k
ˆ
a
k
if
k
ˆ
σ
(41)
∂t
ˆ
σ
=
q
ˆ
σ
i
k
f
k
ˆ
a
k
(42)
The equation for ˆ
a
k
can be formally integrated and substituted in the equation for ˆ
σ
to find
∂t
ˆ
σ
=
q
ˆ
σ
i
k
f
k
e
k
(
t
t
0
)
ˆ
a
k
(
t
0
)
(43)
k
f
2
k
t
t
0
e
i
(
ω
k
)(
t
τ
)
ˆ
σ
(
τ
)d
τ.
We now use the Markov approximation to write ˆ
σ
(
τ
)
ˆ
σ
(
t
)
e
i
(
ω
q
)(
τ
t
)
, and thus
∂t
ˆ
σ
=
q
ˆ
σ
i
k
f
k
e
k
(
t
t
0
)
ˆ
a
k
(
t
0
)
(44)
k
f
2
k
(
t
t
0
e
i
(
ω
k
ω
q
)(
t
τ
)
d
τ
)
ˆ
σ
(
t
)
.