of 11
Letter
https://doi.org/10.1038/s41586-019-1196-1
Cavity quantum electrodynamics with atom-like
mirrors
Mohammad Mirhosseini
1,2,3,8
, eunjong Kim
1,2,3,8
, Xueyue Zhang
1,2,3
, Alp Sipahigil
1,2,3
, Paul B. Dieterle
1,2,3
, Andrew J. Keller
1,2,3
,
Ana Asenjo-Garcia
3,4,5,6
, Darrick e. Chang
5,7
& Oskar Painter
1,2,3
*
1
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA, USA.
2
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.
3
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA.
4
Norman Bridge Laboratory of Physics, California Institute of Technology, Pasadena, CA,
USA.
5
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain.
6
Physics Department, Columbia University, New York, NY, USA.
7
ICREA-Institució
Catalana de Recerca i Estudis Avançats, Barcelona, Spain.
8
These authors contributed equally: Mohammad Mirhosseini, Eunjong Kim.
*
e-mail:
opainter@caltech.edu
NATURE|
www.nature.com/nature
SUPPLEMENTARY INFORMATION
https://doi.org/10.1038/s41586-019-1196-1
In the format provided by the authors and unedited.
2
I. SUPPLEMENTARY NOTE 1: SPECTROSCOPIC MEASUREMENT OF INDIVIDUAL QUBITS
The master equation of a qubit in a thermal bath at temperature
T
, driven by a classical field is given by
̇
ˆ
ρ
=
i
[
ˆ
H/
̄
h,
ˆ
ρ
] +
L
[ ˆ
ρ
], where the Hamiltonian
ˆ
H
and the Liouvillian
L
is written as [1]
ˆ
H/
̄
h
=
ω
p
ω
q
2
ˆ
σ
z
+
p
2
ˆ
σ
x
,
(S-1)
L
[ ˆ
ρ
] = ( ̄
n
th
+ 1)Γ
1
D
σ
] ˆ
ρ
+ ̄
n
th
Γ
1
D
σ
+
] ˆ
ρ
+
Γ
φ
2
D
σ
z
] ˆ
ρ.
(S-2)
Here,
ω
p
(
ω
q
) is the frequency of the drive (qubit), Ω
p
is the Rabi frequency of the drive, ̄
n
th
= 1
/
(
e
̄
q
/k
B
T
1) is
the thermal occupation of photons in the bath, Γ
1
and Γ
φ
are relaxation rate and pure dephasing rates of the qubit,
respectively. The superoperator
D
[
ˆ
A
] ˆ
ρ
=
ˆ
A
ˆ
ρ
ˆ
A
1
2
{
ˆ
A
ˆ
A,
ˆ
ρ
}
(S-3)
denotes the Lindblad dissipator. The master equation can be rewritten in terms of density matrix elements
ρ
a,b
a
|
ˆ
ρ
|
b
as
̇
ρ
e,e
=
i
p
2
(
ρ
e,g
ρ
g,e
)
( ̄
n
th
+ 1)Γ
1
ρ
e,e
+ ̄
n
th
Γ
1
ρ
g,g
(S-4)
̇
ρ
e,g
=
[
i
(
ω
p
ω
q
)
(2 ̄
n
th
+ 1)Γ
1
+ 2Γ
φ
2
]
ρ
e,g
+
i
p
2
(
ρ
e,e
ρ
g,g
)
(S-5)
̇
ρ
g,e
= ̇
ρ
e,g
;
̇
ρ
g,g
=
̇
ρ
e,e
(S-6)
With
ρ
e,e
+
ρ
g,g
= 1, the steady-state solution (
̇
ˆ
ρ
= 0) to the master equation can be expressed as
ρ
ss
e,e
=
̄
n
th
2 ̄
n
th
+ 1
1 + (
δω/
Γ
th
2
)
2
1 + (
δω/
Γ
th
2
)
2
+ Ω
2
p
/
th
1
Γ
th
2
)
+
1
2
2
p
/
th
1
Γ
th
2
)
1 + (
δω/
Γ
th
2
)
2
+ Ω
2
p
/
th
1
Γ
th
2
)
,
(S-7)
ρ
ss
e,g
=
i
p
th
2
(2 ̄
n
th
+ 1)
1 +
i δω/
Γ
th
2
1 + (
δω/
Γ
th
2
)
2
+ Ω
2
p
/
th
1
Γ
th
2
)
,
(S-8)
where
δω
=
ω
p
ω
q
is the detuning of the drive from qubit frequency, Γ
th
1
= (2 ̄
n
th
+ 1)Γ
1
and Γ
th
2
= Γ
th
1
/
2 + Γ
φ
are
the thermally enhanced decay rate and dephasing rate of the qubit.
Now, let us consider the case where a qubit is coupled to the waveguide with decay rate of Γ
1D
. If we send in a
probe field ˆ
a
in
from left to right along the waveguide, the right-propagating output field ˆ
a
out
after interaction with
the qubit is written as [2]
ˆ
a
out
= ˆ
a
in
+
Γ
1D
2
ˆ
σ
.
The probe field creates a classical drive on the qubit with the rate of Ω
p
/
2 =
i
ˆ
a
in
Γ
1D
/
2. With the steady-state
solution of master equation (S-8) the transmission amplitude
t
=
ˆ
a
out
/
ˆ
a
in
can be written as
t
(
δω
) = 1
Γ
1D
th
2
(2 ̄
n
th
+ 1)
1 +
i δω/
Γ
th
2
1 + (
δω/
Γ
th
2
)
2
+ Ω
2
p
/
th
1
Γ
th
2
)
.
(S-9)
At zero temperature ( ̄
n
th
= 0) Eq. (S-9) reduces to [3, 4]
t
(
δω
) = 1
Γ
1D
2
1 +
i δω/
Γ
2
1 + (
δω/
Γ
2
)
2
+ Ω
2
p
/
1
Γ
2
)
.
(S-10)
Here, Γ
2
= Γ
φ
+ Γ
1
/
2 is the dephasing rate of the qubit in the absence of thermal occupancy. In the following, we
define the parasitic decoherence rate of the qubit as Γ
= 2Γ
2
Γ
1D
= Γ
loss
+ 2Γ
φ
, where Γ
loss
denotes the decay rate
of qubit induced by channels other than the waveguide. Examples of Γ
loss
in superconducting qubits include dielectric
loss, decay into slotline mode, and loss from coupling to two-level system (TLS) defects.
3
A. Effect of saturation
To discuss the effect of saturation on the extinction in transmission, we start with the zero temperature case of
Eq. (S-10). We introduce the saturation parameter
s
2
p
/
Γ
1
Γ
2
to rewrite the on-resonance transmittivity as
t
(0) = 1
Γ
1D
2
1
1 +
s
1
Γ
1D
2
(1
s
) =
(
1 +
s
Γ
1D
Γ
)(
Γ
Γ
+ Γ
1D
)
,
(S-11)
where the low-power assumption
s

1 has been made in the last step. For the extinction to get negligible effect from
saturation, the power-dependent part in Eq. (S-11) should be small compared to the power-independent part. This
is equivalent to
s <
Γ
/
Γ
1D
. Using the relation
p
=
1D
P
p
̄
q
between the driven Rabi frequency and the power
P
p
of the probe and assuming Γ

Γ
1D
, this reduces to
P
p
<
̄
q
Γ
4
.
(S-12)
In the experiment, the probe power used to resolve the extinction was -150 dBm (10
18
W), which gives a limit to
the observable Γ
due to our coherent drive of Γ
/
2
π
150 kHz.
B. Effect of thermal occupation
To take into account the effect of thermal occupancy, we take the limit where the saturation is very small (Ω
p
0).
On resonance, the transmission amplitude is expressed as
t
(0) = 1
Γ
1D
[(2 ̄
n
th
+ 1)Γ
1
+ 2Γ
φ
](2 ̄
n
th
+ 1)
1
Γ
1D
2
+
1
+ Γ
φ
1D
Γ
2
2
̄
n
th
,
(S-13)
where we have assumed the thermal occupation is very small, ̄
n
th

1. In the limit where Γ
1D
is dominating spurious
loss and pure dephasing rates (Γ
2
Γ
1D
/
2), this reduces to
t
(0)
t
(0)
|
T
=0
+ 4 ̄
n
th
(S-14)
and hence the thermal contribution dominates the transmission amplitude unless ̄
n
th
<
Γ
/
1D
.
Using this relation, we can estimate the upper bound on the temperature of the environment based on our measure-
ment of extinction. We have measured the transmittance of Q
1
at its maximum frequency (Figure S-1) before and
after installing a thin-film microwave attenuator, which is customized for proper thermalization of the input signals
sent into the waveguide with the mixing chamber plate of the dilution refrigerator [5]. The minimum transmittance
was measured to be
|
t
|
2
1
.
7
×
10
4
(2
.
1
×
10
5
) before (after) installation of the attenuator, corresponding to the
upper bound on thermal photon number of ̄
n
th
<
3
.
3
×
10
3
(1
.
1
×
10
3
). With the attenuator, this corresponds to
temperature of 43 mK, close to the temperature values reported in Ref. [5].
-10
0
10
10
-1
Before
After
10
-2
10
-3
10
-4
10
-5
5
-5
Detuning (MHz)
Transmittance, |
t
|
2
FIG. S-1.
Effect of thermal occupancy on extinction.
The transmittance of Q
1
is measured at the flux-insensitive point
before and after installation of customized microwave attenuator. We observe an order-of-magnitude enhancement in extinction
after the installation, indicating a better thermalization of input signals to the chip.
4
II. SUPPLEMENTARY NOTE 2: DETAILED MODELING OF THE ATOMIC CAVITY
In this section, we analyze the atomic cavity discussed in the main text in more detail, taking into account its higher
excitation levels. The atomic cavity is formed by two identical
mirror
qubits [frequency
ω
q
, decay rate Γ
1D
) to
waveguide (spurious loss) channel placed at
λ/
2 distance along the waveguide (Figure 1a). From the
λ/
2 spacing,
the correlated decay of the two qubits is maximized to
Γ
1D
, while the exchange interaction is zero. This results in
formation of dark state
|
D
and bright state
|
B
|
D
=
|
eg
+
|
ge
2
,
|
B
=
|
eg
〉−|
ge
2
,
(S-15)
which are single-excitation states of two qubits with suppressed and enhanced waveguide decay rates Γ
1D,D
= 0,
Γ
1D,B
= 2Γ
1D
to the waveguide. Here, g (e) denotes the ground (excited) state of each qubit. Other than the ground
state
|
G
〉≡|
gg
, there also exists a second excited state
|
E
〉≡|
ee
of two qubits, completing 2
2
= 4 eigenstates in the
Hilbert space of two qubits. We can alternatively define
|
D
and
|
B
in terms of collective annihilation operators
ˆ
S
D
=
1
2
(
ˆ
σ
(1)
+ ˆ
σ
(2)
)
,
ˆ
S
B
=
1
2
(
ˆ
σ
(1)
ˆ
σ
(2)
)
(S-16)
as
|
D
=
ˆ
S
D
|
G
and
|
B
=
ˆ
S
B
|
G
. Here, ˆ
σ
(
i
)
de-excites the state of
i
-th mirror qubit. Note that the doubly-excited
state
|
E
can be obtained by successive application of either
ˆ
S
D
or
ˆ
S
B
twice on the ground state
|
G
.
The interaction of qubits with the field in the waveguide is written in the form of
ˆ
H
WG
(
ˆ
S
B
+
ˆ
S
B
)
,
and hence
the state transfer via classical drive on the waveguide can be achieved only between states of non-vanishing transition
dipole
f
|
ˆ
S
B
|
i
. In the present case, only
|
G
〉 ↔ |
B
and
|
B
〉 ↔ |
E
transitions are available via the waveguide with
the same transition dipole. This implies that the waveguide decay rate of
|
E
is equal to that of
|
B
, Γ
1D,E
= 2Γ
1D
.
To investigate the level structure of the dark state, which is not accessible via the waveguide channel, we introduce
an ancilla
probe
qubit [frequency
ω
q
, decay rate Γ
1D,p
p
) to waveguide (loss) channel] at the center of mirror qubits.
The probe qubit is separated by
λ/
4 from mirror qubits, maximizing the exchange interaction to
Γ
1D,p
Γ
1D
/
2 with
zero correlated decay. This creates an interaction of excited state of probe qubit to the dark state of mirror qubits
|
e
p
|
G
〉↔|
g
p
|
D
, while the bright state remains decoupled from this dynamics.
The master equation of the three-qubit system reads
̇
ˆ
ρ
=
i
[
ˆ
H/
̄
h,
ˆ
ρ
] +
L
[ ˆ
ρ
], where the Hamiltonian
ˆ
H
and the
Liouvillian
L
are given by
ˆ
H
= ̄
hJ
[
ˆ
σ
(p)
ˆ
S
D
+ ˆ
σ
(p)
+
ˆ
S
D
]
(S-17)
L
[ ˆ
ρ
] = (Γ
1D,p
+ Γ
p
)
D
[
ˆ
σ
(p)
]
ˆ
ρ
+ (2Γ
1D
+ Γ
)
D
[
ˆ
S
B
]
ˆ
ρ
+ Γ
D
[
ˆ
S
D
]
ˆ
ρ
(S-18)
|
e
p
|
G
|
g
p
|
G
|
g
p
|
D
|
g
p
|
B
|
g
p
|
E
|
e
p
|
D
|
e
p
|
E
J
J
2
Γ
1D
Ω
WG
Γ
1D,p
Ω
XY
Γ
1D,p
Ω
XY
Γ
1D,p
Ω
XY
2
Γ
1D
Ω
WG
Interaction
Decay
Drive
|
e
a
|
0
|
g
a
|
0
|
g
a
|
1
|
g
a
|
2
|
e
a
|
1
|
e
a
|
2
2
g
g
γ
γ
γ
|
g
a
|
3
3
g
2
κ
κ
3
κ
...
...
a
b
FIG. S-2.
Level structure of the atomic cavity and linear cavity.
a. Level structure of the three-qubit system of probe
qubit and atomic cavity. Γ
1D,p
and 2Γ
1D
denotes the decay rates into the waveguide channel, Ω
XY
is the local drive on the
probe qubit, and Ω
WG
is the drive from the waveguide. The coupling strength
J
is the same for the first excitation and second
excitation levels, b. Level structure of an atom coupled to a linear cavity.
|
e
a
(
|
g
a
) denotes the excited state (ground state)
of the atom, while
|
n
is the
n
-photon Fock state of the cavity field.
g
is the coupling,
γ
is the decay rate of the atom, and
κ
is the photon loss rate of the cavity.
5
Here, ˆ
σ
(p)
±
are the Pauli operators for the probe qubit, 2
J
=
1D,p
Γ
1D
is the interaction between probe qubit and
dark state, and
D
[
·
] is the Lindblad dissipator defined in Eq. (S-3). The full level structure of the 2
3
= 8 states
of three qubits and the rates in the system are summarized in Fig. S-2a. Note that the effective (non-Hermitian)
Hamiltonian
ˆ
H
eff
in the main text can be obtained from absorbing part of the Liouvillian in Eq. (S-18) excluding
terms associated with quantum jumps.
To reach the dark state of the atomic cavity, we first apply a local gate
|
g
p
|
G
〉→|
e
p
|
G
on the probe qubit (Ω
XY
in Fig. S-2a) to prepare the state in the first-excitation manifold. Then, the Rabi oscillation
|
e
p
|
G
〉↔|
g
p
|
D
takes
place with the rate of
J
. We can identify
g
=
J
,
γ
= Γ
1D,p
+ Γ
p
,
κ
= Γ
in analogy to cavity QED (Fig. 1a and
Fig. S-2b) and calculate cooperativity as
C
=
(2
J
)
2
Γ
1
,
p
Γ
1
,
D
=
1D,p
Γ
1D
1D,p
+ Γ
p
1D
Γ
,
when the spurious loss rate Γ
is small. A high cooperativity can be achieved in this case due to collective suppression
of radiation in atomic cavity and cooperative enhancement in the interaction, scaling linearly with the Purcell factor
P
1D
= Γ
1D
/
Γ
. Thus, we can successfully map the population from the excited state of probe qubit to dark state of
mirror qubits with the interaction time of (2
J/π
)
1
.
Going further, we attempt to reach the second-excited state
|
E
= (
ˆ
S
D
)
2
|
G
of atomic cavity. After the state
preparation of
|
g
p
|
D
mentioned above, we apply another local gate
|
g
p
|
D
〉 → |
e
p
|
D
on the probe qubit and
prepare the state in the second-excitation manifold. In this case, the second excited states
|
e
p
|
D
〉 ↔ |
g
p
|
E
have
interaction strength
J
, same as the first excitation, while the
|
E
state becomes highly radiative to waveguide channel.
The cooperativity
C
is calculated as
C
=
(2
J
)
2
Γ
1
,
p
Γ
1
,
E
=
1D,p
Γ
1D
1D,p
+ Γ
p
)(2Γ
1D
+ Γ
)
<
1
,
which is always smaller than unity. Therefore, the state
|
g
p
|
E
is only virtually populated and the interaction maps
the population in
|
e
p
|
D
to
|
g
p
|
B
with the rate of (2
J
)
2
/
(2Γ
1D
) = Γ
1D,p
. This process competes with radiative
decay (at a rate of Γ
1D,p
) of probe qubit
|
e
p
|
D
〉 → |
g
p
|
D
followed by the Rabi oscillation in the first-excitation
manifold, giving rise to damped Rabi oscillation in Fig. 3f.
A. Effect of phase length mismatch
Deviation of phase length between mirror qubits from
λ/
2 along the waveguide can act as a non-ideal contribution
in the dynamics of atomic cavity. The waveguide decay rate of dark state can be written as Γ
1D,D
= Γ
1D
(1
−|
cos
φ
|
),
where
φ
=
k
1D
d
is the phase separation between mirror qubits [2]. Here,
k
1D
is the wavenumber and
d
is the distance
between mirror qubits.
We consider the case where the phase mismatch ∆
φ
=
φ
π
of mirror qubits is small. The decay rate of the dark
state scales as Γ
1D,D
Γ
1D
(∆
φ
)
2
/
2 only adding a small contribution to the decay rate of dark state. Based on the
decay rate of dark states from time-domain measurement in Table S-2, we estimate the upper bound on the phase
mismatch ∆
φ/π
to be 5% for type-I and 3.5% for type-II.
B. Effect of asymmetry in
Γ
1D
So far we have assumed that the waveguide decay rate Γ
1D
of mirror qubits are identical and neglected the asym-
metry. If the waveguide decay rates of mirror qubits are given by Γ
1D,1
6
= Γ
1D,2
, the dark state and bright state are
redefined as
|
D
=
Γ
1D,2
|
eg
+
Γ
1D,1
|
ge
Γ
1D,1
+ Γ
1D,2
,
|
B
=
Γ
1D,1
|
eg
〉−
Γ
1D,2
|
ge
Γ
1D,1
+ Γ
1D,2
,
(S-19)
with collectively suppressed and enhanced waveguide decay rates of Γ
1D,D
= 0, Γ
1D,B
= Γ
1D,1
+ Γ
1D,2
, remaining fully
dark and fully bright even in the presence of asymmetry. We also generalize Eq. (S-16) as
ˆ
S
D
=
Γ
1D,2
ˆ
σ
(1)
+
Γ
1D,1
ˆ
σ
(2)
Γ
1D,1
+ Γ
1D,2
,
ˆ
S
B
=
Γ
1D,1
ˆ
σ
(1)
Γ
1D,2
ˆ
σ
(2)
Γ
1D,1
+ Γ
1D,2
.
(S-20)
6
With this basis, the Hamiltonian can be written as
ˆ
H
= ̄
hJ
D
(
ˆ
σ
(p)
ˆ
S
D
+ ˆ
σ
(p)
+
ˆ
S
D
)
+ ̄
hJ
B
(
ˆ
σ
(p)
ˆ
S
B
+ ˆ
σ
(p)
+
ˆ
S
B
)
,
(S-21)
where
J
D
=
Γ
1D,p
Γ
1D,1
Γ
1D,2
Γ
1D,1
+ Γ
1D,2
, J
B
=
Γ
1D,p
1D,1
Γ
1D,2
)
2
Γ
1D,1
+ Γ
1D,2
.
Thus, the probe qubit interacts with both the dark state and bright state with the ratio of
J
D
:
J
B
= 2
Γ
1D,1
Γ
1D,2
:
1D,1
Γ
1D,2
), and thus for a small asymmetry in the waveguide decay rate, the coupling to the dark state dominates
the dynamics. In addition, we note that the bright state superradiantly decays to the waveguide, and it follows that
coupling of probe qubit to the bright state manifest only as contribution of
(2
J
B
)
2
Γ
1D,1
+ Γ
1D,2
= Γ
1D,p
(
Γ
1D,1
Γ
1D,2
Γ
1D,1
+ Γ
1D,2
)
2
to the probe qubit decay rate into spurious loss channel. In our experiment, the maximum asymmetry
d
=
|
Γ
1D,1
Γ
1D,2
|
Γ
1D,1
1D,2
in waveguide decay rate between qubits is 0.14 (0.03) for type-I (type-II) from Table S-1, and this affects the decay
rate of probe qubit by at most
2%.
C. Fitting of Rabi oscillation curves
The Rabi oscillation curves in Fig. 3a and Fig. 4d are modeled using a numerical master equation solver [6, 7]. The
qubit parameters used for fitting the Rabi oscillation curves are summarized in Table S-1. For all the qubits, Γ
1D
was found from spectroscopy. In addition, we have done a time-domain population decay measurement on the probe
qubit to find the total decay rate of Γ
1
/
2
π
= 1
.
1946 MHz (95% confidence interval [1
.
1644
,
1
.
2263] MHz, measured
at 6.55 GHz). Using the value of Γ
1D
/
2
π
= 1
.
1881 MHz (95% confidence interval [1
.
1550
,
1
.
2211] MHz, measured at
6.6 GHz) from spectroscopy, we find the spurious population decay rate Γ
loss
/
2
π
= Γ
1
/
2
π
Γ
1D
/
2
π
= 6
.
5 kHz (with
uncertainty of 45.3 kHz) for the probe qubit. The value of spurious population decay rate is assumed to be identical
for all the qubits in the experiment. Note that the decaying rate of the envelope in the Rabi oscillation curve is
primarily set by the waveguide decay rate of the probe qubit Γ
1D,p
, and the large relative uncertainty in Γ
loss
does
not substantially affect the fit curve.
The dephasing rate of the probe qubit is derived from time-domain population decay and Ramsey sequence mea-
surements Γ
φ
= Γ
2
Γ
1
/
2. In the case of the mirror qubits, the table shows effective single qubit parameters inferred
from measurements of the dark state lifetime. We calculate single mirror qubit dephasing rates that theoretically
yield the corresponding measured collective value. Assuming an uncorrelated Markovian dephasing for the mirror
qubits forming the cavity we find Γ
φ,
m
= Γ
φ,
D
(See Supplementary Note 3). Similarly, the waveguide decay rate of
the mirror qubits is found from the spectroscopy of the bright collective state as Γ
1D,m
= Γ
1D,B
/
2. The detuning
between probe qubit and the atomic cavity (∆) is treated as the only free parameter in our model. The value of ∆
sets the visibility and frequency of the Rabi oscillation, and is found from the the fitting algorithm.
Type
Qubits involved
Γ
1D,p
/
2
π
(MHz)
Γ
1D,m
/
2
π
(MHz)
Γ
φ,
p
/
2
π
(kHz)
Γ
φ,
m
/
2
π
(kHz)
/
2
π
(MHz)
I
Q
2
, Q
6
1.19
13.4
191
210
1.0
II
Q
1
, Q
7
0.87
96.7
332
581
5.9
Dark compound
Q
2
Q
3
, Q
5
Q
6
1.19
4.3
191
146
0.9
Bright compound
Q
2
Q
3
, Q
5
Q
6
1.19
20.2
191
253
1.4
TABLE S-1.
Parameters used for fitting Rabi oscillation curves.
The first and second row are the data for 2-qubit
dark states, the third and fourth row are the data for 4-qubit dark states made of compound mirrors. Here, Γ
1D,p
1D,m
) is
the waveguide decay rate and Γ
φ,
p
φ,
m
) is the pure dephasing rate of probe (mirror) qubit, ∆ is the detuning between probe
qubit and mirror qubits used for fitting the data.
7
III. SUPPLEMENTARY NOTE 3: LIFETIME (
T
1
) AND COHERENCE TIME (
T
2
) OF DARK STATE
The dark state of mirror qubits belongs to the decoherence-free subspace in the system due to its collectively
suppressed radiation to the waveguide channel. However, there exists non-ideal channels that each qubit is coupled
to, and such channels contribute to the finite lifetime (
T
1
) and coherence time (
T
2
) of the dark state (See Table S-2).
In the experiment, we have measured the decoherence rate Γ
2
,
D
of the dark state to be always larger than the decay
rate Γ
1
,
D
, which cannot be explained by simple Markovian model of two qubits subject to their own independent noise.
We discuss possible scenarios that can give rise to this situation of Γ
2
,
D
>
Γ
1
,
D
, with distinction of the Markovian
and non-Markovian noise contributions.
There are two major channels that can affect the coherence of the dark state. First, coupling of a qubit to dissipative
channels other than the waveguide can give rise to additional decay rate Γ
loss
= Γ
1
Γ
1D
(so-called non-radiative
decay rate). This type of decoherence is uncorrelated between qubits and is well understood in terms of the Lindblad
form of master equation, whose contribution to lifetime and coherence time of dark state is similar as in individual
qubit case. Another type of contribution that severely affects the dark state coherence arises from fluctuations in qubit
frequency, which manifest as pure dephasing rate Γ
φ
in the individual qubit case. This can affect the decoherence of
the dark state in two ways: (i) By accumulating a relative phase between different qubit states, this act as a channel
to map the dark state into the bright state with short lifetime, and hence contributes to loss of population in the dark
state; (ii) fluctuations in qubit frequency also induces the frequency jitter of the dark state and therefore contributes
to the dephasing of dark state.
In the following, we model the aforementioned contributions to the decoherence of dark state. Let us consider two
qubits separated by
λ/
2 along the waveguide on resonance, in the presence of fluctuations
̃
j
(
t
) in the qubit frequency.
The master equation can be written as
̇
ˆ
ρ
=
i
[
ˆ
H/
̄
h,
ˆ
ρ
] +
L
[ ˆ
ρ
], where the Hamiltonian
ˆ
H
and the Liouvillian
L
are
given by
ˆ
H
(
t
) = ̄
h
j
=1
,
2
̃
j
(
t
σ
(
j
)
+
ˆ
σ
(
j
)
,
(S-22)
L
[
ρ
] =
j,k
=1
,
2
[
(
1)
j
k
Γ
1D
+
δ
jk
Γ
loss
]
(
ˆ
σ
(
j
)
ˆ
ρ
ˆ
σ
(
k
)
+
1
2
{
ˆ
σ
(
k
)
+
ˆ
σ
(
j
)
,
ˆ
ρ
}
)
.
(S-23)
Here, Γ
1D
loss
) is the decay rate of qubits into waveguide (spurious loss) channel. Note that we have assumed
the magnitude of fluctuation
̃
j
(
t
) in qubit frequency is small and neglected its effect on exchange interaction and
correlated decay. We investigate two scenarios in the following subsections depending on the correlation of noise that
gives rise to qubit frequency fluctuations.
A. Markovian noise
If the frequency fluctuations of the individual qubits satisfy the conditions for Born and Markov approximations,
i.e. the noise is weakly coupled to the qubit and has short correlation time, the frequency jitter can be described in
terms of the standard Lindblad form of dephasing [1].
More generally, we also consider the correlation between frequency jitter of different qubits. Such contribution can
arise when different qubits are coupled to a single fluctuating noise source. For instance, if two qubits in a system couple
to a magnetic field
B
0
+
̃
B
(
t
) that is global to the chip with
D
j
̃
j
/∂
̃
B
, the correlation between detuning of different
qubits follows correlation of the fluctuations in magnetic field, giving
̃
1
(
t
)
̃
2
(
t
+
τ
)
=
D
1
D
2
̃
B
(
t
)
̃
B
(
t
+
τ
)
〉 6
= 0.
Type
Qubits involved
Γ
1
,
D
/
2
π
(kHz)
Γ
2
,
D
/
2
π
(kHz)
I
Q
2
, Q
6
210
366
II
Q
1
, Q
7
581
838
Dark compound
Q
2
Q
3
, Q
5
Q
6
146
215
Bright compound
Q
2
Q
3
, Q
5
Q
6
253
376
TABLE S-2.
Decay rate and decoherence rate of dark states.
The first and second row are the data for 2-qubit dark
states, the third and fourth row are the data for 4-qubit dark states made of compound mirrors. Here, Γ
1
,
D
2
,
D
) is the decay
(decoherence) rate of the dark state.
8
The Liouvillian associated with dephasing can be written as [8]
L
φ,jk
[ ˆ
ρ
] =
Γ
φ,jk
2
(
ˆ
σ
(
j
)
z
ˆ
ρ
ˆ
σ
(
k
)
z
1
2
{
ˆ
σ
(
k
)
z
ˆ
σ
(
j
)
z
,
ˆ
ρ
}
)
,
(S-24)
where the dephasing rate Γ
φ,jk
between qubit
j
and qubit
k
(
j
=
k
denotes individual qubit dephasing and
j
6
=
k
is
the correlated dephasing) is given by
Γ
φ,jk
1
2
ˆ
+
−∞
d
τ
̃
j
(0)
̃
k
(
τ
)
.
(S-25)
Here, the average
〈·〉
is taken over an ensemble of fluctuators in the environment. Note that the correlated dephasing
rate Γ
φ,jk
can be either positive or negative depending on the sign of noise correlation, while the individual pure
dephasing rate Γ
φ,jj
is always positive.
After we incorporate the frequency jitter as the dephasing contributions to the Liouvillian, the master equation
takes the form
̇
ˆ
ρ
=
j,k
=1
,
2
{
[
(
1)
j
k
Γ
1D
+
δ
jk
Γ
loss
]
(
ˆ
σ
(
j
)
ˆ
ρ
ˆ
σ
(
k
)
+
1
2
{
ˆ
σ
(
k
)
+
ˆ
σ
(
j
)
,
ˆ
ρ
}
)
+
Γ
φ,jk
2
(
ˆ
σ
(
j
)
z
ˆ
ρ
ˆ
σ
(
k
)
z
1
2
{
ˆ
σ
(
k
)
z
ˆ
σ
(
j
)
z
,
ˆ
ρ
}
)}
,
(S-26)
We diagonalize the correlated decay part of the Liouvillian describe the two-qubit system in terms of bright and
dark states defined in Eq. (S-15). From now on, we assume the pure dephasing rate and the correlated dephasing
rate are identical for the two qubits, and write Γ
φ
Γ
φ,
11
= Γ
φ,
22
, Γ
φ,
c
Γ
φ,
12
= Γ
φ,
21
. For qubits with a large
Purcell factor (Γ
1D

Γ
φ
,
|
Γ
φ,
c
|
,
Γ
loss
), we can assume that the superradiant states
|
B
and
|
E
are only virtually
populated [9] and neglect the density matrix elements associated with
|
B
and
|
E
. Rewriting Eq. (S-26) in the basis of
{|
G
,
|
B
,
|
D
,
|
E
〉}
, the dynamics related to dark state can be expressed as ̇
ρ
D
,
D
≈−
Γ
1
,
D
ρ
D
,
D
and ̇
ρ
D
,
G
≈−
Γ
2
,
D
ρ
D
,
G
,
where
Γ
1
,
D
= Γ
loss
+ Γ
φ
Γ
φ,
c
,
Γ
2
,
D
=
Γ
loss
2
+ Γ
φ
.
(S-27)
Note that if the correlated dephasing rate Γ
φ,
c
is zero, Γ
1
,
D
is always larger than Γ
2
,
D
, which is in contradiction to
our measurement result.
We estimate the decay rate into non-ideal channels to be Γ
loss
/
2
π
= 6
.
5 kHz from the difference in Γ
1
and Γ
1D
of
the probe qubit, and assume Γ
loss
to be similar for all the qubits. Applying Eq. (S-27) to measured values of Γ
2
,
D
listed in Table S-2, we expect that the pure dephasing of the individual qubit is the dominant decay and decoherence
source for the dark state. In addition, we compare the decay rate Γ
1
,
D
and decoherence rate Γ
2
,
D
of dark states in
the Markovian noise model and infer that the correlated dephasing rate Γ
φ,
c
is positive and is around a third of the
individual dephasing rate Γ
φ
for all types of mirror qubits.
B. Non-Markovian noise
In a realistic experimental setup, there also exists non-Markovian noise sources contributing to the dephasing of the
qubits, e.g. 1
/f
-noise or quasi-static noise [10–12]. In such cases, the frequency jitter cannot be simply put into the
Lindblad form as described above. In this subsection, we consider how the individual qubit dephasing induced by non-
Markovian noise influences the decoherence of dark state. As shown below, we find that a non-Markovian noise source
can lead to a shorter coherence time to lifetime ratio for the dark states, in a similar fashion to correlated dephasing.
However, we find that the functional form of the visibility of Ramsey fringes is not necessarily an exponential for a
non-Markovian noise source.
We start from the master equation introduced in Eqs. (S-22)-(S-23) can be written in terms of the basis of
{|
G
,
|
B
,
|
D
,
|
E
〉}
,
̇
ˆ
ρ
=
i
̄
h
[
ˆ
H,
ˆ
ρ
] + (2Γ
1D
+ Γ
loss
)
D
[
ˆ
S
B
] ˆ
ρ
+ Γ
loss
D
[
ˆ
S
D
] ˆ
ρ,
(S-28)
where the Hamiltonian is written using the common frequency jitter
̃
c
(
t
)
[
̃
1
(
t
) +
̃
2
(
t
)]
/
2 and differential
frequency jitter
̃
d
(
t
)
[
̃
1
(
t
)
̃
2
(
t
)]
/
2
ˆ
H
(
t
)
/
̄
h
=
̃
c
(
t
) (2
|
E
〉〈
E
|
+
|
D
〉〈
D
|
+
|
B
〉〈
B
|
) +
̃
d
(
t
) (
|
B
〉〈
D
|
+
|
D
〉〈
B
|
)
.
(S-29)
9
Here,
ˆ
S
B
and
ˆ
S
D
are defined in Eq. (S-16). From the Hamiltonian in Eq. (S-29), we see that the common part of
frequency fluctuation
̃
c
(
t
) results in the frequency jitter of the dark state while the differential part of frequency
fluctuation
̃
d
(
t
) drives the transition between
|
D
and
|
B
, which acts as a decay channel for the dark state.
For uncorrelated low-frequency noise on the two qubits, the decoherence rate is approximately the standard deviation
of the common frequency jitter
̃
c
(
t
)
2
. The decay rate in this model can be found by modeling the bright state as a
cavity in the Purcell regime, and calculate the damping rate of the dark state using the Purcell factor as
4
̃
d
(
t
)
2
/
Γ
B
.
As evident, in this model the dark state’s population decay rate is strongly suppressed by the large damping rate of
bright state Γ
B
, while the dark state’s coherence time can be sharply reduced due to dephasing.