PRX QUANTUM
5,
030346 (2024)
Stabilizing Remote Entanglement via Waveguide Dissipation
Parth S. Shah
,
1,2,
†
Frank Yang
,
1,2,
†
Chaitali Joshi,
1,2,
†
and Mohammad Mirhosseini
1,2,
*
1
Moore Laboratory of Engineering,
California Institute of Technology
, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter,
California Institute of Technology
, Pasadena,
California 91125, USA
(Received 23 February 2024; revised 4 July 2024; accepted 9 August 2024; published 6 September 2024)
Distributing entanglement between remote sites is integral to quantum networks. Here, we demonstrate
the autonomous stabilization of remote entanglement between a pair of noninteracting superconducting
qubits connected by an open waveguide on a chip. In this setting, the interplay between a classical con-
tinuous drive—supplied through the waveguide—and dissipation into the waveguide stabilizes the qubit
pair in a dark state, which, asymptotically, takes the form of a Bell state. We use field-quadrature mea-
surements of the photons emitted to the waveguide to perform quantum state tomography on the stabilized
states, where we find a concurrence of 0.504
+
0.007
−
0.029
in the optimal setting with a stabilization time constant
of 56
±
4 ns. We examine the imperfections within our system and discuss avenues for enhancing fideli-
ties and achieving scalability in future work. The decoherence-protected steady-state remote entanglement
offered via dissipative stabilization may find applications in distributed quantum computing, sensing, and
communication.
DOI:
10.1103/PRXQuantum.5.030346
I. INTRODUCTION
Entanglement between distant physical systems is a cru-
cial resource for quantum information processing. Over
long distances, entanglement can make communication
secure against eavesdropping and resilient to loss [
1
–
3
].
On shorter length scales, entanglement between distant
noninteracting modules can help realize nonlocal gates in
a quantum computer [
4
–
7
]. Remote entanglement can be
created by the deterministic exchange of photons between
remote sites (see, e.g., Refs. [
8
,
9
]). Alternatively, mea-
surements can be used to “herald” entanglement in a
probabilistic fashion (see, e.g., Refs. [
10
,
11
]). Once cre-
ated, entangled states have to be protected until they can
be used, a task that can be achieved via storage in quan-
tum memories [
3
]. Current research is pursuing efficient
means of generation, distribution, and storage of remote
entanglement.
An entirely different approach to entanglement genera-
tion is
stabilization
using quantum reservoir engineering.
*
Contact
author:
mohmir@caltech.edu;
http://qubit.
caltech.edu
†
These authors contributed equally to this work.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license. Fur-
ther distribution of this work must maintain attribution to the
author(s) and the published article’s title, journal citation, and
DOI.
This method employs dissipation into a shared reser-
voir, in combination with continuous drives, to establish
entanglement between two or more parties. Intriguingly,
an engineered dissipation can not only create entangle-
ment but also protect it—indefinitely—from decoherence
[
12
–
22
]. Beyond entanglement generation, dissipative pro-
cesses have also been studied for a variety of other tasks
as an alternative to unitary gate operations [
13
,
23
–
25
].
Despite the wide interest in this area, however, stabiliz-
ing
remote
entanglement has remained elusive, primarily
due to the challenge of engineering shared dissipation for
remote sites.
Spontaneous emission into a photonic bath can provide
a shared dissipation channel for remote quantum emit-
ters. Such a system can be realized within the paradigm
of waveguide quantum electrodynamics (QED), where
two or multiple qubits—acting as quantum emitters—are
strongly coupled to a shared waveguide. In this set-
ting, the interference of photons emitted by the qubits
can give rise to the formation of collective states that
are protected from dissipation by their internal symme-
tries [
26
–
30
]. Theoretical work has proposed a variety
of methods for stabilizing these dark states [
31
–
42
].
However, experimental demonstrations of these propos-
als have remained out of reach, owing to the need for
components such as unidirectional or time-modulated
qubit-photon coupling and the injection of nonclassi-
cal states into the waveguide, which are challenging to
implement.
2691-3399/24/5(3)/030346(23)
030346-1
Published by the American Physical Society
SHAH, YANG, JOSHI, and MIRHOSSEINI
PRX QUANTUM
5,
030346 (2024)
Here, we stabilize remote entanglement by exploit-
ing the emission into a one-dimensional (1D) photonic
bath. In our experiment, a pair of superconducting qubits,
which are connected via an open microwave waveguide
on a chip, serve as remote quantum nodes. In following
a previous theoretical proposal [
43
], our approach offers
simplicity by relying on classical drives and conventional
bidirectional qubit-photon couplings. The dark state in
our experiment is formed via the precise tuning of the
qubit transition frequencies, which is done to match the
interqubit physical distance to the wavelength of the emit-
ted photons. Supplying a continuous drive through the
waveguide, we stabilize the dark state, which takes the
form of a Bell state under strong drives asymptotically.
We conduct quantum state tomography to quantify the
degree of entanglement in the steady state of the system.
Repeating the experiment with different settings, we find
a maximum concurrence of 0.504
+
0.007
−
0.029
(95% confidence
interval) with a stabilization time constant of 56
±
4nsin
our system. We study the trade-off between the stabiliza-
tion time and the entanglement quality in the steady state
and compare it with a numerical model that considers the
noise sources in our experiment. Finally, we discuss future
improvements in the experimental parameters, showing the
feasibility of achieving fidelities exceeding 90% with rea-
sonable improvements in thermalization and qubit coher-
ence. Our experiment elucidates the practical challenges of
reservoir engineering with an open waveguide and marks
an important step toward realizing a modular network
architecture in which multinode remote entanglement is
created and accessed on demand via an open radiation
channel.
II. THEORETICAL CONCEPT
Our system includes two qubits coupled to a shared
waveguide with equal dissipation rates of
1D
[see
Fig.
1(a)
]. The qubit frequencies are offset symmetrically
with respect to a center frequency (
ω
1,2
=
ω
±
δ
). Further,
we choose the center frequency
ω
such that
=
m
λ
, where
is the physical distance between the qubits,
λ
is the wave-
length of radiation at
ω
,and
m
is an integer. We assume
that the qubits are driven via a classical coherent field at the
frequency
ω
, supplied through the waveguide. In a frame
rotating at the driving frequency, and after applying the
rotating-wave approximation (RWA), the Hamiltonian for
this system can be written as
ˆ
H
/
=
∑
i
=
1,2
δ
i
2
ˆ
σ
z
,
i
+
1
2
(
ˆ
σ
†
i
+
∗
ˆ
σ
i
)
.(1)
Here,
denotes the Rabi frequency of the drive and
δ
1,2
=±
δ
. While this Hamiltonian is separable, the inter-
ference of photons emitted by the two qubits in this setting
gives rise to suppression and enhancement of spontaneous
emission [
44
]. These effects can be taken into account by
rewriting the Hamiltonian in the basis of triplet and singlet
states
|
(
T
,
S
=
(
|
eg
±|
ge
)/
√
2), finding
ˆ
H
/
=
√
2
(
|
T
gg
|+|
ee
T
|
)
−
δ
(
|
S
T
|
)
+
H.c. (2)
In Fig.
1(b)
(left), we show the corresponding energy-
level diagram, including the coupling and dissipation
terms (for the derivation, see Appendix
B1
). As is evi-
dent, the singlet state is subradiant and is protected from
(c)
(a)
(b)
1 mm
Q1
Q2
100 μm
10 μm
Q3
Input
Output
Waveguide
ee
ee
T
T
S
Ω
eff
,B
Ω
Ω
–
δ
–
δ
δ
2
2Γ
1D
2Γ
1D
Γ
1D
Γ
1D
2Γ
1D
γ
eff
,B
γ
eff
,D
mλ
FIG. 1. The experimental setup. (a) A pair of quantum emitters coupled to a shared waveguide. The qubit transition frequencies
are offset in opposite directions with respect to a central frequency and their separation is set equal to the wavelength at this center
frequency. A continuous Rabi drive is supplied through the channel. (b) Left: the energy-level diagram of the system. The triplet state
|
T
superradiantly decays into the waveguide. The singlet state
|
S
is coherently coupled to the triplet but has no direct decay path into
the waveguide. Right: the energy diagram in a basis including the dark state
|
D
. Here, the population is pumped from the triplet into
the dark state, which is protected from decay into the waveguide. (c) An optical image of the fabricated device, where three transmon
qubits are coupled to a shared coplanar waveguide on a chip.
030346-2
STABILIZING REMOTE ENTANGLEMENT...
PRX QUANTUM
5,
030346 (2024)
direct dissipation into the waveguide. However, the singlet
state exchanges population with the waveguide indirectly
through coherent interaction with the triplet state, which is
superradiant. In the steady state, the combination of this
interaction and continuous drive through the waveguide
creates a superposition of the singlet and the ground states
that is completely dark to the waveguide. This stationary
dark state is given by (see Appendix
B2
and [
35
,
43
])
|
D
=
|
gg
+
α
|
S
√
1
+|
α
|
2
.(3)
Here,
α
=
/
√
2
δ
is a drive-power-dependent parameter
that sets the singlet fraction (
|
α
|
2
/(
1
+|
α
|
2
)
). As is evi-
dent, the stationary state is pure and for strong drives
(
|
α
|
1) approaches the maximally entangled singlet
state. As a result, by simply supplying the drive into the
waveguide, one can stabilize entanglement between the
qubits starting from an arbitrary initial state.
By redrawing the energy-level diagram to include the
dark state,
|
D
, and a state orthogonal to it in the
{|
gg
,
|
S
}
subspace, (
|
B
=
(α
|
gg
−|
S
)/
√
1
+|
α
|
2
), one can find
the rate of pumping population into
|
D
. As shown in
Fig.
1(b)
(right), the population is dissipatively transferred
from
|
T
to
|
D
, where it is trapped due to the decoupling
of the dark state from the remainder of the energy levels.
The effective rate of this process can be found as (for the
derivation, see Appendix
B2
)
γ
eff,
D
=
2
1D
1
+
2
/
2
δ
2
.(4)
The reciprocal of this “pumping” rate
t
D
=
1
/γ
eff,
D
sets
the approximate time scale for stabilization of the dark
state and therefore sets the rate of entanglement gen-
eration. We highlight the competition between the sin-
glet fraction
|
α
|
2
/(
1
+|
α
|
2
)
and the relaxation time
t
D
=
(
1
+|
α
|
2
)/(
2
1D
)
, where achieving larger singlet
fractions—corresponding to more entanglement—requires
higher drive powers and longer relaxation times. This
trade-off is illustrated by the
δ
=
0 case, which corre-
sponds to stabilizing a pure Bell state
|
D
=|
S
but takes
infinitely long to stabilize
γ
eff,
D
→
0 (for further discus-
sion, see Appendixes
B2
and
G2
).
To successfully implement the stabilization protocol
under consideration, a physical platform must fulfill sev-
eral requirements. Most importantly, it is necessary to have
precise control over the qubit transition frequency and
to establish efficient interfaces between qubit and propa-
gating photons. Precise control over the qubit frequency
is needed to ensure that the interference of emitted pho-
tons leads to destructive interference at both the outputs
of the waveguide simultaneously, culminating in a dark
state. Formally, this condition can be articulated by defin-
ing the null spaces for the collective jump operators (see
Appendix
B2
)[
45
]. As shown in Ref. [
43
], deviations of
the drive frequency from the center frequency
ω
result
in the dark state having an additional loss mechanism,
thereby limiting the achievable entanglement. Efficient
qubit-photon interfaces are vital to minimize photon loss
during the emission and reabsorption processes among the
qubits, which can result in a reduced fidelity for the sta-
bilized state. A key metric in evaluating this effect is the
Purcell factor, defined as the ratio of the decay rate of an
individual qubit to the waveguide to its intrinsic decoher-
ence rate,
P
1D
=
1D
/
, where
=
2
2
−
1D
=
int
+
2
φ
.(
2
is the total qubit decoherence,
int
is the loss
to nonradiative channels, and
φ
is the pure dephasing.)
In addition to the factors mentioned, another key ingre-
dient is the characterization of the stabilized joint qubit
state. This task is particularly challenging in the pres-
ence of dissipation from the waveguide. To overcome this
challenge, characterization measurements need to happen
either on very short time scales or, alternatively, temporary
elimination of waveguide dissipation is needed during the
characterization process. In Sec.
III
, we detail an exper-
imental realization based on transmon superconducting
qubits that satisfies these requirements.
III. EXPERIMENT
The fabricated superconducting circuit used to realize
our experiment is shown in Fig.
1(c)
. The circuit consists of
three transmon qubits (1, 2, and 3), which are side coupled
to the same coplanar waveguide (CPW). Each qubit has a
weakly coupled charge control line (shown in orange) and
an external flux bias port (shown in green) for tuning its
transition frequency. Qubit 3 does not participate in any
of our experiments and is decoupled from the rest of the
system by tuning its frequency well away (
>
1 GHz) from
the other two qubits. The device fabrication is described in
Appendix
A
.
We first verify the required interference conditions by
looking for signatures of the subradiant and superradi-
ant states. In Fig.
2(a)
, we show the transmission through
the waveguide for weak microwave drives, measured as a
function of the flux bias of qubit 1. Meanwhile, qubit 2 has
its transition frequency fixed at
ω
such that the interqubit
separation along the waveguide equates to the correspond-
ing wavelength (
=
λ
). As the two qubits cross, we
note a broader resonant line shape, indicating the forma-
tion of a superradiant state [
44
]. In Fig.
2(b)
, we show
the waveguide transmission for two qubits that are pre-
cisely on resonance at
ω
(red) and for qubit 2 at
ω
while
qubit 1 is tuned out of the measurement window (blue).
By fitting Lorentzian line shapes to these spectra, we
find the radiative decay rate of the superradiant (triplet)
state,
1D,
T
/
2
π
=
18.3 MHz, which is nearly twice the
single-qubit decay rate [
(
1D,1
,
1D,2
)/
2
π
=
(
10.3, 10.7
)
MHz], pointing to the correct phase length between the
030346-3
SHAH, YANG, JOSHI, and MIRHOSSEINI
PRX QUANTUM
5,
030346 (2024)
(b)
(d)
(c)
(a)
0
500
1000
1500
2000
2500
3000
Time (ns)
10
–2
10
–1
10
0
Excited-state population
Single qubit
Subradiant
04080
0.5
1.0
0
1000
2000
3000
Time (ns)
0.5
1.0
–25
–20
–15
–10
–5
0
|
t
| (dB)
Single qubit
Superradiant
0.0
0.5
1.0
|
t
|
–0.378
–0.376
–0.374
–0.372
–20
–10
0
10
20
Detuning (MHz)
/
0
P
= –131 dBm
P
= –113 dBm
–6
–4
–2
0
2
4
6
Detuning (MHz)
–20
–10
0
10
20
Detuning (MHz)
0
2
4
6
PSD (W/Hz)
0.0
0.5
1.0
1.5
10
–23
FIG. 2. Characterizing the super- and subradiant collective states. (a) The transmission spectrum measured through the waveguide
as qubit 1 is frequency-tuned across qubit 2. The dashed line denotes the
≈
λ
point. (b) The transmission spectra for the single
qubit (qubit 2, blue) and two-qubit (red) settings, where a broader spectrum indicates the superradiant (triplet) state. (c) The inelastic
scattering spectrum. At higher drive powers (upper inset), the line shape includes contributions from both the triplet and singlet
states. Reducing the drive power results in a line shape predominantly set by the singlet state, which is fitted to a master-equation
simulation. (d) The measured relaxation lifetimes for an individual qubit (qubit 2) and the singlet state, yielding
T
1
=
16
±
1.9 ns and
T
1
=
910
±
47 ns, respectively. The insets show the linear-scale plots of the same data.
qubits. We extract individual-qubit Purcell factors (at
ω
)
of [
(
P
1D,1
,
P
1D,2
)
=
(
11.4, 10.7
)
].
Although the singlet is (ideally) protected from emission
to the waveguide, emission can be caused by dephasing or
imbalanced waveguide decay rates between qubits, which
prevent ideal destructive interference. The elastic scatter-
ing in this case is dominated by the bright-state response
and does not provide a good measure of emission from the
singlet. To probe the singlet, we instead measure the inelas-
tic scattering from the qubits using a spectrum analyzer
(see Appendix
A
). While the resonance fluorescence spec-
trum at higher powers contains contributions from both the
super- and subradiant states, a measurement done at a suffi-
ciently low drive power is dominated by the response from
the subradiant (singlet) state, as shown in Fig.
2(c)
[
26
].
A master-equation simulation fit to the inelastic scattering
profile confirms the presence of the subradiant state (for
a detailed discussion, see Appendix
C
). Furthermore, we
measure the population decay lifetimes of the singlet and
an individual qubit (described in Appendix
D
[
28
]), shown
in Fig.
2(d)
. We find a large contrast (a factor of over 50) in
the measured lifetimes, indicating that the qubit-frequency
configuration and the Purcell factors are suitable for the
realization of the stabilization protocol.
Having established the operation frequency, we proceed
with the stabilization and characterization of entangled
states. In this step, the qubits are detuned with respect
to the target frequency [see Fig.
3(a)
and Appendix
G2
].
Starting with a system at rest in
|
gg
, we apply a narrow-
band drive at
ω
to initiate the stabilization process. While
the drive is being supplied, the population is coherently
driven from
|
gg
to
|
T
, where it decays to the dark state
|
D
(
|
B
) with rate
γ
eff,
D
(
γ
eff,
B
). The population in
|
B
is
then continuously driven to
|
T
, while
|
D
decouples from
the dynamics. After supplying the drive for a finite dura-
tion of time, we turn it off and allow the qubits to decay
freely into the waveguide. We note here that the short
radiative lifetime of the qubits (caused by waveguide dis-
sipation) precludes the use of dispersive readout. As an
alternative, we characterize the state of the qubits using
the photons emitted into the waveguide. The photons orig-
inating from each qubit in our experiment are spectrally
distinguishable because of the large detuning between the
qubits. Using quadrature amplitude detection and mode
030346-4