of 15
Quantum Entanglement between Optical and Microwave Photonic Qubits
Srujan Meesala,
1,2,3
,*,§
David Lake ,
1,2,3
,*
Steven Wood,
1,2,3
,*
Piero Chiappina,
1,2,3
Changchun Zhong ,
4
,
Andrew D. Beyer,
5
Matthew D. Shaw,
5
Liang Jiang ,
4
and Oskar Painter
1,2,3,6
,
1
Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology
,
Pasadena, California 91125, USA
2
Kavli Nanoscience Institute,
California Institute of Technology
, Pasadena, California 91125, USA
3
Institute for Quantum Information and Matter,
California Institute of Technology
,
Pasadena, California 91125, USA
4
Pritzker School of Molecular Engineering,
The University of Chicago
, Chicago, Illinois 60637, USA
5
Jet Propulsion Laboratory
,
California Institute of Technology
,
4800 Oak Grove Drive, Pasadena, California 91109, USA
6
AWS Center for Quantum Computing
, Pasadena, California 91125, USA
(Received 28 February 2024; revised 24 May 2024; accepted 5 August 2024; published 30 September 2024)
Entanglement is an extraordinary feature of quantum mechanics. Sources of entangled optical photons
were essential to test the foundations of quantum physics through violations of Bell
s inequalities. More
recently, entangled many-body states have been realized via strong nonlinear interactions in microwave
circuits with superconducting qubits. Here, we demonstrate a chip-scale source of entangled optical and
microwave photonic qubits. Our device platform integrates a piezo-optomechanical transducer with a
superconducting resonator which is robust under optical illumination. We drive a photon-pair generation
process and employ a dual-rail encoding intrinsic to our system to prepare entangled states of microwave
and optical photons. We place a lower bound on the fidelity of the entangled state by measuring microwave
and optical photons in two orthogonal bases. This entanglement source can directly interface telecom
wavelength time-bin qubits and gigahertz frequency superconducting qubits, two well-established
platforms for quantum communication and computation, respectively.
DOI:
10.1103/PhysRevX.14.031055
Subject Areas: Quantum Physics, Quantum Information
I. INTRODUCTION
The growing size and complexity of superconducting
quantum processors
[1]
suggests that future, large-scale
quantum computers based on superconducting qubits will
likelybemodularsystemswithinterconnectedprocessors
[2]
.
Toward this long-term goal, a quantum network architecture
has been envisioned with optical channels as low-loss, room-
temperature communication links between superconducting
processors cooled in separate dilution refrigerator nodes.
Such a network would leverage the complementary strengths
ofoptical photonsand superconductingmicrowave circuits to
reliably communicate and process quantum information,
respectively, and may also be used to realize quantum
communication protocols relevant for long-distance secure
communication and distributed sensing. An integral building
blockinthisarchitectureisatransducerthatenablesquantum-
coherent interactions between microwave and optical pho-
tons. An ideal microwave-optical quantum transducer would
perform bidirectional frequency conversion of quantum
states without any added noise or loss. While such an ideal
converter is not fundamentally forbidden, recent experimen-
tal efforts
[3
9]
have revealed various challenges including
low efficiency due to relatively weak optical nonlinearities
and technical noise due to parasitic optical absorption.
Moreover, external losses in photon collection, transmission,
and routing are expected to limit the feasibility of operations
between remote qubits by means of direct quantum state
transfer along optical links. The demanding requirements for
near-ideal transducers and communication channels can be
relaxed through the use of heralded schemes such as the
Duan-Lukin-Cirac-Zoller (DLCZ) protocol
[10]
and its
variants,whichhavebeenrealizedpreviouslywithavarietyof
optical emitters
[11
15]
. In these schemes, quantum inter-
ference and measurement are used to generate entanglement
*
These authors contributed equally to this work.
Present address: Department of Physics, Xi
an Jiaotong
University, Xi
an, Shanxi 710049, China.
Contact author: opainter@caltech.edu;
http://copilot.caltech.edu
§
Present address: Department of Electrical and Computer
Engineering and Smalley-Curl Institute, Rice University,
Houston, Texas 77005, USA.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
s title, journal citation,
and DOI.
PHYSICAL REVIEW X
14,
031055 (2024)
2160-3308
=
24
=
14(3)
=
031055(15)
031055-1
Published by the American Physical Society
between remote qubits, heralded by specific measurement
outcomes. The remote entangled states can then be used as a
resource for high-fidelity operations between qubits in
separate nodes by means of quantum teleportation
[16]
.
A simple adaptation of the DLCZ protocol for micro-
wave-optical quantum transduction requires the use of a
transducer to generate quantum states of single microwave
and optical photons in pairs, with some finite
probability
[17,18]
. By interfering the optical emission
from transducers in two separate nodes and performing
optical single-photon detection measurements, one can
herald remote entangled microwave states with a fidelity
insensitive to both the internal efficiency of the transducer
and external optical losses. Toward this goal, we recently
integrated piezo-optomechanical transducers with light-
robust superconducting circuits, enabling measurement of
nonclassical correlations between microwave and optical
photons generated via spontaneous parametric down-con-
version (SPDC)
[19]
. However, an observation of non-
classical statistics is insufficient to prove entanglement or
inseparability of the joint optical-microwave state
[20]
.
While nonclassical statistics can be verified through a
measurement of intensity correlations, entanglement veri-
fication additionally demands a measurement of phase
coherence on the correlated photon pairs. Here, we
upgrade our platform to operate with dual-rail encoded
photonic qubits and perform measurements in comple-
mentary bases to extract the requisite intensity correlations
and phase coherence, allowing an unambiguous observa-
tion of discrete-variable entanglement with a quantum
transducer. Recently, a bulk electro-optic transducer was
used to infer continuous variable entanglement though a
measurement of two-mode squeezing between microwave
and optical fields referred to the output ports of the
device
[21]
. However, the continuous-variable entangle-
ment in such states is susceptible to photon loss
[22]
and is
challenging to distribute in an optical quantum network
given current experimental capabilities. In contrast, we
prepare Bell states of dual-rail optical and microwave
photonic qubits, each containing exactly one photon. Such
Bell states intrinsically allow for detection of loss errors
and entanglement purification and are fundamental build-
ing blocks in well-established discrete variable quantum
communication protocols
[23,24]
. For the optical photonic
qubit, we use a time-bin encoding, the preferred choice in
optical quantum communication due to its robustness to
fluctuations in optical path length
[25]
. For the microwave
photonic qubit, we use a modified time-bin encoding
involving two orthogonal modes which naturally arise
from hybridized acoustic and electrical resonances in our
device. We verify entanglement by correlating microwave
quantum state tomography results with detection of a
single optical photon in a chosen time bin or of a coherent
superposition of time bins achieved by passing the optical
emission through a fiber optic time-delay interferometer.
II. MICROWAVE-OPTICAL ENTANGLED
PAIR SOURCE
Figure
1(a)
shows a simplified schematic of our micro-
wave-optical entanglement source, which we operate in a
dilution refrigerator setup at a temperature of approximately
20 mK. Details of the device geometry and fabrication
process have been provided in previous work
[19]
. Pump
laser pulses are used to excite a piezo-optomechanical
transducer containing a silicon optomechanical crystal
resonator which supports optical and acoustic resonances
(a)
(b)(c)
FIG. 1. Microwave-optical entanglement source. (a) Simplified
schematic of various components of the chip-scale microwave-
optical entanglement source, not shown to scale to aid presentation.
The terminals of the superconducting kinetic inductance resonator
are galvanically connected to the electrical terminals of the piezo-
optomechanical transducer, shown in detail in (b). The optical
cavity in the transducer is coupled to an optical waveguide, which
terminatesonthe leftedgeof thechip,wherealensedopticalfiberis
used to launch pump pulses into the device and to collect emitted
optical photons in the reverse direction. Microwave (MW) photons
emitted by the device are capacitively coupled from the super-
conducting resonator to an on-chip transmission line and even-
tually collected in a
50
Ω
coax cable. (b) Illustration of the SPDC
process in the transducer where a pump photon at frequency
ω
p
decays into optical and microwave excitations at frequencies
ω
o
and
ω
m
, respectively, due to the parametric optomechanical
interaction at a rate
G
om
. The microwave excitation is shared
between the transducer acoustic mode and the electrical mode of a
superconducting kinetic inductance (KI) resonator, which are
strongly hybridized with the piezoelectric interaction strength
g
pe
. Simulated profiles of the optical electric field (left) and
microwave acoustic displacement field (right) in the transducer
areshown.(c)SchematicforgenerationofBellstatesbetweendual-
rail optical and MW photonic qubits. Two consecutive Gaussian
pump pulses induce emission of single optical and MW photons
into early and late modes centered at their respective frequencies.
Theoretically calculated intensity envelopes of these modes are
shown when the device is excited with two Gaussian pump pulses
separated by a time delay
T
d
¼
279
ns, which ensures orthogon-
ality of early and late modes used for the dual-rail encoding.
SRUJAN MEESALA
et al.
PHYS. REV. X
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031055-2
at frequencies
ω
o
2
π
×
200
THz and
ω
m
2
π
×
5
GHz,
respectively. Colocalized telecom photons and microwave
phonons in this wavelength-scale resonator can interact via
radiation pressure and the photoelastic effect
[26]
. By tuning
the pump laser frequency to the blue-detuned mechanical
sideband of the optical cavity with
ω
p
¼
ω
o
þ
ω
m
, we drive
a two-mode squeezing interaction described by the
Hamiltonian
ˆ
H
¼
G
om
ð
ˆ
a
ˆ
b
þ
ˆ
a
ˆ
b
Þ
, where
ˆ
a
and
ˆ
b
are
the annihilation operators for the internal optical and acous-
tic modes, respectively, of the transducer
[27]
.Here,
G
om
¼
ffiffiffiffiffi
n
p
p
g
om
is the strength of the parametric optomechanical
interaction, where
n
p
is the number of intracavity pump
photons and
g
om
=
2
π
¼
270
kHz is the optomechanical
coupling rate at the single-photon and -phonon level in
the device under study. In this setting, we can induce SPDC
of a pump photon into a photon-phonon pair at frequencies
ω
o
and
ω
m
, as illustrated in Fig.
1(b)
. Single phonons from
the SPDC process are converted into single microwave
photons in a niobium nitride superconducting kinetic induct-
ance resonator tuned into resonance with the transducer
acoustic mode via an external magnetic field perpendicular
to the device. This conversion process is mediated by a
compact aluminum nitride piezoelectric component in the
transducer and occurs at a coherent rate
g
pe
=
2
π
¼
1
.
2
MHz,
which exceeds the intrinsic damping rates of the acoustic and
electrical modes in the device under study (see Appendix
A
).
Finally, as shown in Fig.
1(a)
, both optical and microwave
photons emitted from the device decay into on-chip wave-
guides and are routed into a lensed optical fiber and a
50
Ω
microwave coaxial cable, respectively.
To prepare entangled microwave-optical states, we excite
the device with two consecutive pump pulses. Each pump
pulse can produce a microwave-optical photon pair in well-
defined temporal modes, separated in time and centered at
frequencies
ω
m
and
ω
o
in the microwave and optical outputs,
respectively. While the envelopes of the microwave and
optical emission are not identical, they are expected to be
correlated, since the SPDC process produces single phonons
and optical photons in pairs. Based on the theoretically
expected intensity envelopes of
early
and
late
modes
in the output ports of the device shown in Fig.
1(c)
, we adopta
modified time-bin encoding to define dual-rail photonic
qubits. The optical early and late mode envelopes adiabati-
cally follow the pump pulses, since the optomechanical
interaction strength
G
om
is much smaller than the decay rate
of the optical cavity,
κ
o
. We use two Gaussian pump pulses
with two sigma width
T
p
¼
96
ns to define time-bin modes
for the optical photonic qubit. On the other hand, the
microwave early and late mode envelopes are determined
by the time evolution of a single phonon scattered into the
hybridized microwave electroacoustic resonator system.
Given the piezoelectric coupling strength and the decay rates
of the microwave and acoustic modes in the device under
study, such a phonon preferentially decays into the micro-
wave output waveguide with a damped oscillatory envelope.
To satisfy orthogonality between the early and late modes
used to encode the microwave photonic qubit, we space the
pump pulses by
T
d
¼
279
ns, matching the electroacoustic
oscillation period which is independently measured by
microwave electrical spectroscopy (see Appendix
D
).
Under this condition, if a phonon were scattered by the early
pump pulse, it would be swapped into the electrical resonator
after the time
T
d
, leaving the acoustic mode in the vacuum
state for the action of the late pump pulse.
In this setting, the joint wave function of early and late
modes in the optical and microwave output ports of the
device can be written as
j
Ψ
i
j
00
i
o
j
00
i
m
þ
ffiffiffiffi
p
p
ðj
10
i
o
j
10
i
m
þ
e
i
φ
p
j
01
i
o
j
01
i
m
Þþ
O
ð
p
Þ
;
ð
1
Þ
where
j
kl
i
o
(
j
kl
i
m
) denotes the direct product of a
k
-photon
state in the early mode and an
l
-photon state in the late mode
ontheoptical (microwave)output port.The phase
φ
p
issetby
the relative phase between early and late pump pulses. When
the scattering probability
p
1
,the
O
ð
p
Þ
terms may be
neglectedanddetectionofa singleoptical photoncan beused
to postselect an entangled state between an optical photonic
qubit in the
fj
10
i
o
;
j
01
i
o
g
manifold and a microwave
photonic qubit in the
fj
10
i
m
;
j
01
i
m
g
manifold. The state
vectors within curly brackets define the native measurement
basis of the photonic qubits, which we call the
Z
basis in
referencetothenorthandsouthpolesoftheBlochsphere.We
refer to a rotated measurement basis on the equator of this
Bloch sphere as the
X
basis. To verify entanglement, we
characterize correlations between the photonic qubits in
these two orthogonal bases. In our experiments, we operate
with Gaussian optical pulses with a peak power of 83 nW
corresponding to an intracavity optical photon number
n
p
¼
0
.
8
and two sigma width
T
p
¼
96
ns, which leads to
p
¼
R
4
j
G
om
ð
t
Þj
2
dt=
κ
o
¼
1
.
0
×
10
4
. In comparison with
previous work on measuring nonclassical correlations
[19]
,
weoperatewithpumppulsesofshorterdurationbyafactorof
2 to define well-resolved optical time bins and reduce
transducer noise from pump-induced heating. This change
in transducer operation conditions is primarily enabled by an
increase in the bandwidth of the pump filters in our optical
measurement chain and the improved external optical col-
lection efficiency
η
opt
¼
5
.
5
×
10
2
(see Appendix
B
). With
a pulse repetition rate
R
¼
50
kHz, we detect heralding
events at a rate
R
click
¼
0
.
26
s
1
. Microwave phonon to
photon conversion is expected to occur with efficiency
η
mw
¼
0
.
59
based on the piezoelectric coupling rate and
microwave damping rates in our device.
III. MICROWAVE-OPTICAL INTENSITY
CORRELATIONS
We first measure the time-resolved microwave output
intensity from the device conditioned on the detection time
QUANTUM ENTANGLEMENT BETWEEN OPTICAL AND
...
PHYS. REV. X
14,
031055 (2024)
031055-3
of single optical photons. This allows us to characterize
microwave-optical intensity correlations in the
Z
basis. In
this measurement, depicted schematically in Fig.
2(a)
,
optical emission from the device is detected on a super-
conducting nanowire single-photon detector (SNSPD) after
transmission through a filter setup to reject pump photons.
Figure
2(b)
shows a histogram of single optical photon
detection times revealing two nearly Gaussian envelopes
associated with the SPDC signal. We use gating windows
of width
2
T
p
¼
192
ns centered around each pulse to
define the early and late optical modes
ˆ
A
e
and
ˆ
A
l
,
respectively. Simultaneously, as shown in Fig.
2(a)
, micro-
wave emission from the device is directed to an amplifi-
cation chain with a near-quantum-limited Josephson
traveling wave parametric amplifier (TWPA) as the first
stage. The amplified microwave signal is down-converted
in a room-temperature heterodyne receiver, and the result-
ing voltage quadratures are sent to an ADC card, allowing
us to record a digitized, complex-valued voltage trace for
each experimental trial. We capture emission at the micro-
wave resonance frequency
ω
m
using a digital filter matched
to the theoretically expected microwave emission envelope
f
ð
t
τ
Þ
, where
τ
is a variable readout delay (see
Appendix
D
). We then subtract independently calibrated
amplifier-added thermal noise of approximately 2.6 quanta
via a moment inversion procedure
[28]
. Upon postselecting
measurement records from trials which produce optical
clicks, we observe that the microwave intensity conditioned
on a late click is delayed with respect to that conditioned on
an early click as shown by the solid traces in Fig.
2(c)
.
These conditional signals contain a finite amount of pump-
induced thermal noise from the transducer. Since
p
1
,
such noise is simply given by the unconditional microwave
output intensity, shown with the dashed time trace in
Fig.
2(c)
. The ratio of the conditional and unconditional
microwave intensities yields the normalized microwave-
optical cross-correlation function
g
ð
2
Þ
AC
, which reaches a
maximum value of 6.8 for early optical clicks and 5.0 for
late optical clicks. Both values exceed the Cauchy-Schwarz
bound of 2 for thermal states and signify nonclassical
microwave-optical correlations
[29
31]
. By performing the
matched filter operation at optimal microwave readout
delays
T
e
and
T
l
¼
T
e
þ
T
d
, shown by the dotted vertical
lines in Fig.
2(c)
, which maximize the cross-correlation,
we define microwave early and late modes
ˆ
C
e
and
ˆ
C
l
,
respectively. Figure
2(d)
shows conditional occupations of
these modes with the symbol
n
ij
for the occupation of
microwave mode
j
conditioned on an optical click
detected in mode
i
. Using these four conditional microwave
mode occupations, we define the
Z
-basis visibility
V
z
¼ð
n
ee
n
el
n
le
þ
n
ll
Þ
=
ð
n
ee
þ
n
el
þ
n
le
þ
n
ll
Þ
. For a
Bell state without additional noise or microwave loss,
we expect
n
ee
¼
n
ll
¼
1
and
n
el
¼
n
le
¼
0
, resulting in
V
z
¼
1
. On the contrary, when the microwave and optical
intensities are fully uncorrelated, we expect
V
z
¼
0
. From
the data in Fig.
2(d)
, we find
V
z
¼
0
.
633

0
.
014
, indicat-
ing significant intensity correlations between early and late
modes in the microwave and optical outputs.
This observation of
Z
-basis correlations is also compat-
ible with a statistical mixture of early and late microwave-
optical photon pairs. To rule out this scenario, we character-
ize intensity correlations in the
X
basis, which are indica-
tive of the phase coherence of the entangled microwave-
optical state. On the optics side, the measurement basis
rotation is performed with a time-delay interferometer
inserted into the detection path as shown in Fig.
3(a)
.
The interferometer is built with a fiber delay line in one arm
to achieve the time delay
T
d
¼
279
ns required to interfere
early and late optical time bins. Additionally, the setup
imprints a relative phase between the time bins,
φ
o
, which is
(a)(b)
(c)(d)
FIG. 2.
Z
-basis intensity correlations. (a) Simplified schematic
of experimental setup used to detect correlations between MW
and optical emission in the
Z
basis of the dual-rail qubits. Shaded
green box on the MW detection path indicates postprocessing on
voltage traces from the heterodyne setup after analog-to-digital
conversion (ADC). (b) Histogram of single optical photon
detection times plotted as a time trace of optical count rate.
Shaded vertical regions indicate gating windows used to define
early and late optical time-bin modes
ˆ
A
e
and
ˆ
A
l
, respectively.
(c) Quanta in the transducer microwave output mode defined by a
filter function matched to the theoretically expected emission
envelope,
f
ð
t
τ
Þ
centered at the MW resonance frequency
ω
m
(see Appendix
D
). The variable readout delay
τ
is shown on the
x
axis, and the occupation of the mode for a given
τ
is shown on the
y
axis. Blue and yellow traces show MW output quanta
conditioned on early and late optical clicks, respectively, and
the dashed gray trace shows unconditional MW output quanta,
which correspond to transducer-added noise. Dotted vertical lines
indicate readout delays
T
e
and
T
l
used to define early and late
MW modes
ˆ
C
e
and
ˆ
C
l
, respectively. The conditional traces are an
average over approximately
3
×
10
5
heralding events. Shaded
regions around traces span a confidence interval of two standard
deviations about the mean. (d) Output quanta
n
ij
in MW mode
j
conditioned on an optical click in mode
i
, where
i
and
j
run over
the early and late modes denoted by e and l, respectively. Data in
this panel correspond to approximately
3
×
10
5
heralding events.
Error bars indicate

one standard deviation.
SRUJAN MEESALA
et al.
PHYS. REV. X
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031055-4
actively stabilized through feedback on a piezoelectric fiber
stretcher in one of the arms. The relative phase between the
pump pulses used to excite the transducer,
φ
p
, is controlled
using an electro-optic phase modulator
[32]
. With one
output port of the interferometer connected to the single-
photon detection path, clicks registered on the SNSPD
correspond to measurement of a single photon in the mode
ð
ˆ
A
e
þ
e
i
ð
φ
p
þ
φ
o
Þ
ˆ
A
l
Þ
=
ffiffiffi
2
p
. The measurement phase
φ
p
þ
φ
o
can be independently calibrated by transmitting coherent
optical pulses through the interferometer (see Appendix
C
).
To perform a basis rotation on the microwave side, we add
the early and late complex voltage quadratures with a
relative phase
φ
m
in postprocessing as shown in Fig.
3(a)
.
After subtracting amplifier-added noise in a manner similar
to the
Z
-basis measurement, we measure the moments of
the microwave mode
ð
ˆ
C
e
þ
e
i
φ
m
ˆ
C
l
Þ
=
ffiffiffi
2
p
, averaged over
experimental trials. In Fig.
3(b)
, we show conditional
microwave output quanta measured in the
X
basis by
fixing
φ
m
¼
0
;
π
. As the optical phase is swept by tuning
φ
p
between the pump pulses, we observe correlation fringes
in the conditional microwave intensity, a clear signature of
coherence of the entangled microwave-optical state.
Because of the need for data at multiple optical phase
settings for this measurement, we use 3 times higher pump
power (
n
p
¼
2
.
4
) compared to the main dataset to speed up
acquisition. We then lower the pump power back to
n
p
¼
0
.
8
, the setting used in the
Z
-basis measurements,
and repeat the
X
-basis measurement for two optical
phase settings
φ
p
þ
φ
o
¼
0
.
56
π
;
1
.
56
π
at which we define
the optical modes
ˆ
A

, respectively. We measure condi-
tional microwave output quanta in the modes
ˆ
C

for
φ
m
¼
0
.
56
π
;
1
.
56
π
, respectively, and obtain the results
shown in Fig.
3(c)
for the four combinations of the
X
-basis modes. In a manner similar to the
Z
-basis corre-
lation measurement, we define the
X
-basis visibility
V
x
,
expected to equal 1 for Bell states and 0 for an equal
statistical mixture of early and late microwave-optical
photon pairs. With the data in Fig.
3(c)
, we observe
V
x
¼
0
.
611

0
.
034
. As an additional consistency check,
we sweep the microwave readout phase
φ
m
in postprocess-
ing and find that the maximum in
V
x
occurs for the phase
settings
φ
m
¼
0
.
62
π
;
1
.
62
π
, offset by
0
.
06
π
from the
theoretically expected modes
ˆ
C

. This can be attributed
to a systematic offset in the calibrated optical phase arising
from small differences in optical frequency and polarization
between calibration and data acquisition (see Appendix
C
).
IV. QUANTUM STATE TOMOGRAPHY
The microwave and optical emission from the device
exhibit intensity correlations in both
Z
and
X
bases which
are characteristic of a Bell pair prepared from the pure state
in Eq.
(1)
via optical detection. However, since the
experimentally prepared states have finite transducer-added
noise and microwave loss, the conditional microwave
intensities have contributions from outside the computa-
tional subspace where the dual-rail photonic qubits are
defined. In order to characterize entanglement more pre-
cisely, we must measure both optical and microwave
outputs in the single-photon subspace. Since we operate
in an experimental regime where the scattering probability
p
1
, we fulfill the condition that nearly all optical
detection events arise from within the single optical photon
subspace. On the microwave side, we use statistical
moments of conditional heterodyne voltages to perform
maximum likelihood state tomography in the joint Fock
basis of the early and late modes
[28,33]
and project onto
(a)
(b)(c)
FIG. 3.
X
-basisintensitycorrelations.(a)Simplifiedschematicof
experimental setup used to detect correlations between MW and
optical emission in the
X
basis of the dual-rail qubits. A time-delay
interferometer in the optical path is used to interfere early and late
optical time bins with a relative phase
φ
o
. The interference
operation in MW detection with a relative phase
φ
m
is performed
in digital postprocessing as shown in the shaded green box.
(b) Output quanta in the MW mode
ð
ˆ
C
e
þ
e
i
φ
m
ˆ
C
l
Þ
=
ffiffiffi
2
p
, for two
phase settings
φ
m
¼
0
(open circles) and
φ
m
¼
π
(filled squares),
conditioned on an optical click in the mode
ð
ˆ
A
e
þ
e
i
ð
φ
p
þ
φ
o
Þ
ˆ
A
l
Þ
=
ffiffiffi
2
p
at the output of the time-delay interferometer. The relative phase
between the pump pulses,
φ
p
, is varied along the horizontal axis,
while
φ
o
is kept constant at
0
.
31
π
. The uncertainty in the calibrated
optical phase over the duration of the measurement is

0
.
03
π
.
Solid and dashed lines are cosine fits. Data in this plot are acquired
at 3 times the pump power used for the main dataset and represent
an average over approximately
1
×
10
4
heralding events per optical
phase setting. All error bars indicate

one standard deviation.
(c)Output quanta
n
ij
in MW mode
j
conditioned on an optical click
in mode
i
, where
i
and
j
run over the
X
-basis MW and optical
measurement modes denoted by
þ
and
and corresponding to
phase settings
φ
m
¼
0
.
56
π
;
1
.
56
π
and
φ
p
þ
φ
o
¼
0
.
56
π
;
1
.
56
π
,
respectively, where
φ
o
is kept constant at
0
.
28
π
. The uncertainty in
the calibrated optical phase over the duration of the measurement is

0
.
04
π
. Data in this panel correspond to an average over
approximately
7
×
10
4
heralding events for
þ
and
optical
outcomes. All error bars indicate

one standard deviation.
QUANTUM ENTANGLEMENT BETWEEN OPTICAL AND
...
PHYS. REV. X
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031055 (2024)
031055-5
the single-photon subspace. This postselection operation is
strictly local and cannot generate microwave-optical entan-
glement. It can be implemented in practice by performing a
parity check
[17]
, a well-established capability in circuit
quantum electrodynamics.
Figure
4(a)
shows conditional density matrices
ρ
ð
e
Þ
,
ρ
ð
l
Þ
,
ρ
ðþÞ
, and
ρ
ð
Þ
of the microwave output state corresponding
to an optical click in the early, late,
þ
, and
modes,
respectively. For a pure Bell state, the entries highlighted
with colored bars in
ρ
ð
e
Þ
and
ρ
ð
l
Þ
are expected to have the
values
ρ
ð
e
Þ
10
;
10
¼
ρ
ð
l
Þ
01
;
01
¼
1
and
ρ
ð
e
Þ
01
;
01
¼
ρ
ð
l
Þ
10
;
10
¼
0
; like-
wise, the entries highlighted with colored bars in
ρ
ðþÞ
and
ρ
ð
Þ
are expected to have a magnitude of 0.5. The main
deviation in the measured conditional microwave states is
due to the nonzero vacuum component, primarily from
finite conversion efficiency
η
mw
¼
0
.
59
of a single phonon
into the microwave waveguide. Using the entries high-
lighted with colored bars in Fig.
4(a)
, which denote the
computational subspace of the dual-rail photonic qubit, we
obtain the conditional probability
p
ij
of a single microwave
photon in mode
j
conditioned on receipt of an optical click
in mode
i
(see Appendix
F
). Here,
i
and
j
run over early and
late (
þ
and
) modes for
Z
-(
X
-) basis measurements, and
the results are shown in Fig.
4(b)
. Error bars on the
probabilities account for statistical error obtained by boot-
strapping with replacement over the microwave dataset.
These conditional probabilities allow us to establish a lower
bound on the Bell state fidelity given by
[17,34]
F
lb
¼
1
2
ð
p
ee
þ
p
ll
p
el
p
le
þ
p
þþ
þ
p
−−
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
þ
p
þ
p
Þ
:
ð
2
Þ
We find that
F
lb
¼
0
.
794
þ
0
.
048
0
.
071
,whichexceedsthe
classical limit of 0.5 by over four standard deviations,
indicating the preparation of entangled microwave-optical
states. A simplemodelaccountingfor pump-inducedthermal
noise in the transducer (see Appendix
H
) predicts Bell state
fidelity exceeding 0.83, which agrees with the measured
lower bound, and suggests pump-induced heating as the
primarysourceofinfidelity. Inadditiontothe reductioninthe
fidelity due to pump-induced noise, we expect smaller
contributions due to dark counts and imperfections in optical
time-bin interference and microwave mode matching. From
these conditional probabilities, we calculate a lower bound
for the logarithmic negativity of the microwave-optical state,
E
lb
¼
0
.
65
(see Appendix
F
). Together with the microwave
collection efficiency and optical heralding rate, we find a
lower bound on the ebit rate,
η
mw
R
click
E
lb
¼
0
.
10
ebit
=
s.
V. CONCLUSION
The microwave photonic qubit emitted by our entangle-
ment source can be directly received by a dual-rail super-
conducting qubit
[17]
or a superconducting qutrit
[35]
, both
of which can be realized with standard transmons.
Following this state mapping operation, local operations
on the transmons can then be used to detect microwave
photon loss events, as demonstrated with current
technology
[35]
, and herald Bell states between a super-
conducting qubit and a time-bin optical photonic qubit. Our
entanglement generation approach, thus, comes with intrin-
sic robustness against loss errors in contrast with recent
work on continuous variable entanglement between micro-
wave and optical fields
[21]
. Additionally, Bell pair
preparation is the first step in well-established protocols
for entanglement purification
[23,24]
, a key requirement
for high-fidelity operations in a practical quantum network.
From a device engineering standpoint, a transmon module
can be connected to our transducer chip with minimal
impact of optical pump light on qubit coherence
[4,36
38]
.
In the near term, piezo-optomechanical transducers with
improved thermalization with the substrate through the use
(a)
(b)
FIG. 4. Quantum state tomography of conditional microwave
states.(a)Conditionaldensitymatrices
ρ
ð
e
Þ
,
ρ
ð
l
Þ
,
ρ
ðþÞ
,and
ρ
ð
Þ
ofthe
microwaveoutputstatecorrespondingtoearly,late,
þ
,and
optical
clicks, respectively, plotted in the joint Fock basis of early and late
microwave modes. The matrices are obtained from a maximum
likelihood reconstruction procedure performed over a joint Fock
space ofup to six photons in eachmode butare plotted in a truncated
space of up to two photons in each mode for better visualization.
Entries which are expected to be nonzero for a pure microwave-
optical Bell state are highlighted in color. (b) Conditional proba-
bility
p
ij
of a single photon in microwave mode
j
conditioned on
receipt of an optical click in mode
i
, calculated from the density
matrices in (a) after postselecting the single-photon subspace. Error
bars denote uncertainties of

one standard deviation.
SRUJAN MEESALA
et al.
PHYS. REV. X
14,
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031055-6
of two-dimensional optomechanical crystals
[39]
and
greater acoustic participation in silicon
[40,41]
can improve
transducer noise performance, enabling microwave-optical
entanglement generation rates of the order of
10
3
=
s.
Transducer operation in this performance regime can
facilitate the integration of superconducting qubit nodes
into optical quantum networks for applications in secure
communication
[42
44]
and distributed sensing
[45
47]
.
ACKNOWLEDGMENTS
The authors thank A. Butler, G. Kim, M. Mirhosseini,
and A. Sipahigil for helpful discussions and B. Baker and
M. McCoy for experimental support. We appreciate MIT
Lincoln Laboratories for providing the traveling-wave
parametric amplifier used in the microwave readout chain
in our experimental setup. NbN deposition during the
fabrication process was performed at the Jet Propulsion
Laboratory. This work was supported by the U.S. Army
Research Office (ARO)/Laboratory for Physical Sciences
(LPS) Cross Quantum Technology Systems program
(Grant No. W911NF-18-1-0103), the ARO/LPS Modular
Quantum Gates (ModQ) program (Grant No. W911NF-23-
1-0254), the U.S. Department of Energy Office of Science
National Quantum Information Science Research Centers
(Q-NEXT, Grant No. DE-AC02-06CH11357), the Institute
for Quantum Information and Matter (IQIM), an NSF
Physics Frontiers Center (Grant No. PHY-1125565) with
support from the Gordon and Betty Moore Foundation, the
Kavli Nanoscience Institute at Caltech, and the AWS
Center for Quantum Computing. L. J. acknowledges sup-
port from the AFRL (FA8649-21-P-0781), NSF (ERC-
1941583 and OMA-2137642), and the Packard Foundation
(2020-71479). S. M. acknowledges support from the IQIM
Postdoctoral Fellowship.
APPENDIX A: DEVICE AND EXPERIMENT
PARAMETERS
APPENDIX B: EXPERIMENTAL SETUP
The experimental setup used in this work is an
upgraded version of the one used in our recent demon-
stration of nonclassical microwave-optical photon
pairs
[19]
. Figure
5(a)
shows relevant upgrades to the
optical setup that enabled us to generate and characterize
microwave-optical Bell states. The microwave measure-
ment chain shown in Fig.
5(b)
and the heralding setup are
identical to the one used in Ref.
[19]
. Pump pulses used to
excite the transducer are generated via amplitude and phase
modulation of a continuous-wave external cavity diode
laser. Our pulse generation setup uses two acousto-optic
modulators (AOMs) in series for amplitude modulation and
an electro-optic modulator (EOM) for phase modulation.
By applying a rectangular voltage pulse of amplitude
V
eom
to the phase modulator over the duration of the late pump
pulse, we control the relative phase between the pump
pulses,
φ
p
introduced in the main text. Polarization of pump
light sent to the device is controlled using an electronic
fiber polarization controller (FPC). During long data
acquisition runs, we mitigate the impact of long-term
TABLE I. Independently calibrated losses of components along
optical detection path for
Z
-basis measurements. In
X
-basis
measurements, the time-delay interferometer setup adds extra
transmission loss of 0.89.
Parameter
Value
Optical cavity to on-chip waveguide
0.50
Waveguide to lensed fiber
0.27
Circulator between excitation and detection
0.90
Total filter bank loss
0.55
(individual components below)
2
×
2
switches
0.79
Filters (
2
× cascaded)
0.79
Circulator in filter setup
0.88
SNSPD setup
0.83
Optical collection efficiency (
η
opt
)
5
.
5
×
10
2
TABLE III. Microwave-optical photon pair generation
parameters.
Symbol
Description
Value
T
p
Two sigma duration of pump pulse
96 ns
T
r
Repetition period of experiment
20
μ
s
T
d
Delay between pump pulses
279 ns

Peak power of pump pulse
83 nW
n
a
Peak intracavity photon number
0.78
p
click
Optical heralding probability
(single pulse)
5
.
2
×
10
6
R
click
Optical heralding rate (
¼
p
click
=T
r
)
0
.
26
s
1
p
SPDC scattering probability
1
.
0
×
10
4
η
opt
Optical collection efficiency
5
.
5
×
10
2
η
mw
Microwave conversion efficiency
0.59
TABLE II. Frequencies and coupling rates of transducer
internal modes.
Symbol
Description
Value
ω
o
=
2
π
Optical mode frequency
192.02 THz
ω
m
=
2
π
Microwave mode frequency
5.004 GHz
g
om
=
2
π
Optomechanical coupling rate
270 kHz
g
pe
=
2
π
Piezoelectric coupling rate
1.2 MHz
κ
e
;
o
=
2
π
External optical coupling rate
650 MHz
κ
i
;
o
=
2
π
Intrinsic optical loss rate
650 MHz
κ
m
=
2
π
Acoustic loss rate
150 kHz
κ
e
;
mw
=
2
π
External coupling rate of
superconducting resonator
1.2 MHz
κ
i
;
mw
=
2
π
Intrinsic loss rate of
superconducting resonator
550 kHz
κ
mw
=
2
π
Total damping rate of
superconducting resonator
1.75 MHz
QUANTUM ENTANGLEMENT BETWEEN OPTICAL AND
...
PHYS. REV. X
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polarization drifts along the excitation path through active
polarization control.
Pump pulses are sent to the transducer in the dilution
fridge (DF) setup through a circulator, which allows us to
direct optical emission along with reflected pump pulses to
the detection setup. At the beginning of the detection path,
a
2
×
2
switch allows us to route the signal to a home-built
time-delay interferometer for
X
-basis measurements or to
bypass it for
Z
-basis measurements. Prior to optical
detection on a SNSPD, the signal is passed through a
pump filter bank comprising two tunable Fabry-Perot
cavities. These cavities are locked on resonance with the
transducer optical resonance frequency
ω
o
using a refer-
ence tone derived from the pump laser [
pump filter lock
path colored in purple in Fig.
5(a)
]. Details of the locking
procedure, which is performed once every 4 min during
data acquisition, have been described previously
[19]
.
Compared with previous work, we upgrade to higher-
bandwidth filters (individual cavity bandwidth of 6 MHz
compared with 3.6 MHz in the old setup). Including a
circulator and two
2
×
2
fiber-optic switches, the filter bank
has a total on-resonance insertion loss of 2.6 dB at the
signal frequency while providing 105 dB extinction at the
pump frequency. This upgrade allows us to work with
pump pulses of shorter duration by a factor of 2 and ensure
that the optical time bins at the signal frequency are well
resolved after the dispersive effect of the pump filters. We
also obtain a significant increase in filter transmission for
the signal pulses and an overall improvement in the external
optical collection efficiency
η
opt
by a factor of 3. A detailed
budget of all losses in the optical detection chain is given on
Table
I
.
Optical time-bin interference for
X
-basis measurements
is realized in an asymmetric Mach-Zehnder interferometer
placed inside a thermally insulated enclosure. A relative
path difference of 53 m is achieved via a fiber spool of fixed
length in one of the arms of the interferometer. This path
difference is precalculated and, upon insertion of the fiber
spool into the setup, is experimentally verified to achieve
the desired time delay
T
d
¼
279
ns between the optical
time bins. Optical emission from the transducer is directed
into one of the two input ports of the interferometer, while
the single-photon detection path is connected to one of
the two output ports. The other input and output of the
(a)
(b)
FIG. 5. Schematic of experimental setup. (a) Schematic of optical setup highlighting essential components for the microwave-optical
Bell state experiment. A more detailed schematic is provided in previous work
[19]
. Dotted rectangular blocks encompass the pump
pulse generation module, the dilution fridge (DF) setup, interferometer used for
X
-basis optical measurements, and the pump filter bank.
Individual components are labeled in the key below the schematic. (b) Schematic of DF setup and the microwave amplification chain
with Josephson TWPA mounted on the mixing chamber (MXC) plate and high electron mobility transistor (HEMT) amplifier mounted
on the 4 K plate.
SRUJAN MEESALA
et al.
PHYS. REV. X
14,
031055 (2024)
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