PHYSICAL REVIEW APPLIED
22,
064047 (2024)
Microwave-optical entanglement from pulse-pumped electro-optomechanics
Changchun Zhong
,
1,
*
Fangxin Li
,
2
Srujan Meesala,
3
Steven Wood,
3,4
David Lake,
3,4
Oskar Painter
,
3,4,5
and Liang Jiang
6,
†
1
Department of Physics,
Xi’an Jiaotong University
, Xi’an, Shanxi 710049, China
2
Department of Physics,
University of Chicago
, Chicago, Illinois 60637, USA
3
Institute for Quantum Information and Matter,
California Institute of Technology
, Pasadena,
California 91125, USA
4
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology
, Pasadena, California 91125, USA
5
Center for Quantum Computing,
Amazon Web Services
, Pasadena, California 91125, USA
6
Pritzker School of Molecular Engineering,
University of Chicago
, Chicago, Illinois 60637, USA
(Received 2 August 2024; revised 2 October 2024; accepted 26 November 2024; published 13 December 2024)
Entangling microwave and optical photons is one of the promising ways to realize quantum transduction
through quantum teleportation. This paper investigates the entanglement of microwave-optical photon
pairs generated from an electro-optomechanical system driven by a blue-detuned pulsed Gaussian pump.
The photon pairs are obtained through weak parametric-down-conversion, and their temporal correlation
is revealed by the second-order correlation function. We then study the discrete variable entanglement
encoded in the time-bin degree of freedom, where entanglement is identified by Bell-inequality violation.
Furthermore, we estimate the laser-induced heating and show that the pulse-pumped system features lower
heating effects while maintaining a reasonable coincidence photon-counting rate.
DOI:
10.1103/PhysRevApplied.22.064047
I. INTRODUCTION
Efficiently converting quantum information between
different frequencies is crucial for scaling up distributed
quantum architecture and advancing modern quantum
networks [
1
,
2
]. Recently developing quantum computa-
tion and communication modules utilize diverse physi-
cal platforms operating at varying energy scales, such
as superconducting circuits in the microwave regime and
communicating photons in the optical frequency range. A
quantum transducer serves as a device designed to convert
quantum information between microwave and optical pho-
tons [
3
,
4
], enabling the coherent connection between these
two distinct modules.
The significant energy gap between microwave and
optical photons presents substantial technological chal-
lenges in terms of material nonlinearity and system noise
when converting quantum information between them.
In recent decades, various physical platforms, includ-
ing electro-optomechanics [
5
–
21
], electro-optics [
22
–
26
], magnons [
27
], Rydberg atoms [
28
–
30
], and oth-
ers, are under active investigation in order to achieve
the goal of quantum transduction [
3
,
4
]. Theoretically,
quantum transducers for different frequency modes can
*
Contact author: zhong.changchun@xjtu.edu.cn
†
Contact author: liang.jiang@uchicago.edu
be constructed using either beam-splitter coupling for
direct quantum conversion, known as direct quantum
transduction (DQT), or two-mode squeezing interaction
for generating microwave-optical entanglement, termed
entanglement-based quantum transduction (EQT). EQT is
expected to be more practical than DQT and has demon-
strated compatibility with current technology [
31
]. Indeed,
compared to DQT, EQT offers a larger quantum capac-
ity region for quantum information conversion within a
more extensive practical parameter space [
32
,
33
]. Further-
more, microwave-optical entanglement is compatible with
the renowned DLCZ protocol, which holds the potential to
directly link microwave quantum circuits [
34
,
35
].
An essential step of EQT is the efficient generation
of microwave-optical entanglement [
31
]. Prior analyses
of entanglement generation have primarily centered on
the model involving the steady-state generation through
two-mode squeezing interactions with a continuous laser
drive. This study, however, delves into a scenario where
a piezo-optomechanical system is driven by short pump
pulses, which generally does not have sufficient time to
reach a steady state. In comparison to continuous pump-
ing, the pulsed-drive model describes more accurately
the prevalent time-bin entanglement encoding scheme.
Additionally, the system with a shorter pump suffers less
from the laser-induced heating, e.g., in chip-scale quan-
tum transducers [
36
]. This potentially allows for higher
2331-7019/24/22(6)/064047(11)
064047-1
© 2024 American Physical Society
CHANGCHUN ZHONG
et al.
PHYS. REV. APPLIED
22,
064047 (2024)
instantaneous laser pump power, increasing the probabil-
ity of photon-pair generation while maintaining quantum
coherence.
In the subsequent sections, we begin by analyzing a
model system for quantum transduction based on piezo-
optomechanics, where a blue-detuned Gaussian pump
pulse is applied. The system’s Hamiltonian features a time-
dependent squeezing strength with a Gaussian time profile.
We then investigate the temporal correlation of the result-
ing microwave-optical emission. With an overall weak
laser pump, pair-photon generation is the most probable
outcome (excluding vacuum). The pair-photon state and its
temporal correlation can be described through the bipho-
ton wave packet. Employing the Schmidt decomposition,
we further unveil the temporal mode structure of the wave
packets, wherein the fundamental zeroth mode dominates
the probability distribution. This output temporal mode
can effectively encode discrete variable entanglement in
time-bin degrees of freedom. We numerically verify this
time-bin entanglement through Bell-inequality violation
using the state-of-the-art experimental parameters. Finally,
we study a model for analyzing the laser-induced heating
in the system, comparing both continuous-laser pumps and
pulsed drives. Compared to the previous heating model
in Ref. [
36
], the current method is predictive for varied
pump pulses, which would enable us to theoretically opti-
mize the pump in order to balance the laser heating and
entanglement generation. Our findings demonstrate that
pulsed pumping allows for a significantly higher pump
power while maintaining the same average heating noise
as the continuous pump scheme. In practice, careful opti-
mization of the duration and strength of the pump pulses
is essential to suppressing laser-induced heating, which
ensures the quantum coherence, and achieving a rea-
sonable pair-generation rate for practical applications in
quantum networking.
II. THE MODEL SYSTEM
We consider a piezo-optomechanical system with a
short pump pulse. This system involves two main inter-
actions: piezoelectricity induces a microwave-mechanical
beam-splitter-type coupling; the mechanical mode simulta-
neously interacts with optical photons through optical scat-
tering forces [
37
,
38
]. With the pump pulse blue detuned,
the scattering interaction usually reduces to a two-mode-
squeezing coupling. We denote
ˆ
a
,
ˆ
b
,and
ˆ
c
as the optical,
mechanical, and electrical mode operators,
ω
o
,
ω
m
,and
ω
e
as the corresponding mode frequencies, respectively. The
total Hamiltonian is expressed as follows:
ˆ
H
/
=−
o
ˆ
a
†
ˆ
a
+
ω
m
ˆ
b
†
ˆ
b
+
ω
e
ˆ
c
†
ˆ
c
−
g
em
(
ˆ
b
†
ˆ
c
+
ˆ
b
ˆ
c
†
)
−
g
0
̄
n
o
(
t
)(
ˆ
a
†
ˆ
b
†
+ˆ
a
ˆ
b
)
.(1)
Here
g
em
is the piezomechanical coupling and
g
0
repre-
sents the one-photon optomechanical coupling rate. The
latter is further modified by the intracavity photon num-
ber, denoted as
̄
n
o
(
t
)
.Wedefine
g
om
(
t
)
:
=
g
0
√
̄
n
o
(
t
)
as the
optical-mechanical squeezing strength. It is worthwhile to
note that the photon number changes with time due to the
short pump pulse, resulting in a time-dependent squeezing
strength. In this paper, we consider a pump pulse with a
Gaussian time profile. As the optical cavity has a relatively
large dissipation rate
κ
o
over the optomechanical coupling
rate, which essentially puts us in the adiabatic regime, the
intracavity photon
̄
n
o
(
t
)
closely follows the pump-pulse
time profile, exhibiting a Gaussian time dependence.
III. THE BIPHOTON OUTPUT AND THE
SCHMIDT MODES
A. General theory
The piezo-optomechanical system is capable of generat-
ing entangled optical-microwave photons. Intuitively, the
two-mode squeezing interaction first entangles the opti-
cal and the mechanical modes, meanwhile the mechanical
excitation is swapped to the electrical mode with a separate
beam-splitter-type coupling. The pair-photon generated
from this process can be approximately described by the
biphoton wave packet [
39
]
|
ψ
=
dt
1
dt
2
f
(
t
1
,
t
2
)
ˆ
a
†
out
(
t
1
)
ˆ
c
†
out
(
t
2
)
|
0
.(2)
The lower index “out” indicates the corresponding out-
put fields in terms of the localized modes. The coefficient
f
(
t
1
,
t
2
)
is determined by the two-time correlation function,
which is given by
|
f
(
t
1
,
t
2
)
|
2
∝
⎧
⎨
⎩
ˆ
a
†
(
t
1
)
ˆ
c
†
(
t
2
)
ˆ
c
(
t
2
)
ˆ
a
(
t
1
)
,
t
1
<
t
2
ˆ
c
†
(
t
2
)
ˆ
a
†
(
t
1
)
ˆ
a
(
t
1
)
ˆ
c
(
t
2
)
,
t
2
<
t
1
.(3)
The biphoton wave packet can be uniquely decomposed
into orthogonal temporal modes through the Schmidt
decomposition [
40
]
f
(
t
1
,
t
2
)
=
∞
k
=
0
√
λ
k
f
o
k
(
t
1
)
f
e
k
(
t
2
)
.
Thus, the wave packet can be rewritten as
|
ψ
=
∞
k
=
0
λ
k
|
ψ
o
k
|
ψ
e
k
,(4)
where the states
|
ψ
o
k
=
dt
1
f
o
k
(
t
1
)
ˆ
a
†
out
(
t
1
)
|
0
and
|
ψ
e
k
=
dt
2
f
e
k
(
t
2
)
ˆ
c
†
out
(
t
2
)
|
0
are in the Hilbert space associated
with the optical and microwave temporal modes, respec-
tively. Pairs of temporal modes are excited with probability
λ
k
. In temporal-mode degrees of freedom, the output is
entangled with entanglement entropy
S
=−
k
λ
k
ln
λ
k
.
Methods for efficiently generating and manipulating them
is a current topic of interest in quantum information pro-
cessing [
41
].
064047-2
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PHYS. REV. APPLIED
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B. Pair photon generated from Gaussian pump pulse
We assume a pump pulse that generates a time-
dependent optomechanical squeezing strength
g
om
(
t
)
=
G
e
−
(
t
−
t
0
)
2
2
σ
2
g
0
.(5)
For demonstration, we pick
G
={
1, 20, 30
}
,
t
0
=
130 ns
and
σ
=
30 ns for the pump, as shown in Fig.
1(a)
.The
frequency landscape of the system and the pump is given
in Fig.
1(c)
. Using the experimentally feasible parameters
from Table
I
, we calculate the system dynamics by numer-
ically solving the Lindblad master equation with QuTip
software package [
42
]
̇
ρ
=−
i
[
ˆ
H
,
ρ
]
+
6
i
=
1
1
2
2
ˆ
L
i
ρ
ˆ
L
†
i
−{
ˆ
L
†
i
ˆ
L
i
,
ρ
}
,(6)
where
ρ(
t
)
denotes the system density operator including
the optical, mechanical, and electrical modes. The jump
operators are
ˆ
L
1
=
√
κ
o
ˆ
a
,
ˆ
L
2
=
κ
m
(
1
+
n
th,
b
)
ˆ
b
,
ˆ
L
3
=
√
κ
m
n
th,
b
ˆ
b
†
,
ˆ
L
4
=
√
κ
e
,
c
ˆ
c
,
ˆ
L
5
=
κ
e
,
i
(
1
+
n
th,
c
)
ˆ
c
,and
ˆ
L
6
=
√
κ
e
,
i
n
th,
c
ˆ
c
†
, where
κ
o
=
κ
o
,
c
+
κ
o
,
i
,
κ
m
,and
κ
e
=
κ
e
,
c
+
κ
e
,
i
are the total dissipation rates for optical, mechanical, and
electrical mode, respectively. The subindexes “
c
”and“
i
”
of the dissipation rates denote the corresponding outcou-
pling and intrinsic loss channels of a mode.
n
th,
b
(
n
th,
c
)is
the average thermal phonon (photon) number of the bath,
which couples to the mechanical (electrical) mode intrin-
sically. Note in experiment, the system is usually placed in
a several mK fridge, yielding a negligible thermal noise
for the mechanical or electrical modes with the mode
frequency on the order of several GHz. Thus, without
specifying otherwise we first take
n
th,
b
=
n
th,
c
=
0 in the
following discussions. A more practical case with heating
noise will be included in the end.
We first investigate the optical-microwave output state
correlation, which is determined by its second order corre-
lation function, as given by the amplitude of the biphoton
wave packet in Eq.
(3)
. This correlation function with dif-
ferent time delay can be obtained with the help of the
renowned quantum regression theorem [
43
]. Figure
1(b)
shows the result, where the shining blob appearing in the
upper-left corner indicates the probability in generating
paired photons. Obviously, there is a tendency in detecting
a microwave photon approximately 150 ns after detecting
an optical photon, which matches the expectation that it
takes time for the mechanical excitation to be swapped to
the microwave mode. It is worth noting that the microwave
photon has an increased temporal-mode width, as indicated
by the shape of the blob in Fig.
1(b)
.
Figure
1(d)
schematically shows the first several
Schmidt decomposed temporal modes of the output bipho-
ton wave packet. It is noted that the output state mainly
(a)
(c)
(d)
(b)
FIG. 1. (a) The time-dependent squeezing strength
g
om
(
t
)
that
takes a Gaussian form with varied
G
. (b) The normalized ampli-
tude of the output biphoton wave packet for
G
=
1. (c) The
frequency landscape for the piezo-optomechanical system under
a blue detuned pump. (d) The decomposed temporal modes of
the output wave packet with
G
=
1.
occupies the temporal zero mode (with
k
=
0), while other
higher-order modes are negligibly small. This is consis-
tent with the pump, which is also Gaussian in time. In the
following, we will focus on the temporal zero mode, dis-
cuss the corresponding pair photon-generation rate and the
time-bin encoding of the output entanglement.
IV. THE M-O PHOTON-PAIR GENERATION AND
THE TIME-BIN BELL STATE
A. The photon-pair generation rate
The photon-pair generation rate is an key quantity in
entanglement-based quantum transduction [
31
]. We can
theoretically calculate it in the framework of a second-
order correlation function [
44
]. Consider at time
t
dur-
ing the pump, the coincidence click probability can be
evaluated according to the formula
P
(
t
,
τ)
=
G
2
(
t
,
τ)
t
2
w
,(7)
where
G
2
(
t
,
τ)
=ˆ
a
†
out
(
t
)
ˆ
c
†
out
(
t
+
τ)
ˆ
c
out
(
t
+
τ)
ˆ
a
out
(
t
)
is
the second-order correlation function defined on the out-
put modes. Note we have the input-output relation
ˆ
a
out
=
ˆ
a
in
+
√
κ
o
,
c
ˆ
a
and
ˆ
c
out
=ˆ
c
in
+
√
κ
e
,
c
ˆ
c
.
a
in
and
c
in
are the
input bath-mode operators, which are assumed to be vac-
uum unless specified otherwise.
t
w
is the detection time
window and
τ
is the time delay, which essentially means
the time difference for recording an optical photon and
subsequently a microwave photon. Denoting
r
D
as the
experiment repetition rate, then the coincidence counting
064047-3
CHANGCHUN ZHONG
et al.
PHYS. REV. APPLIED
22,
064047 (2024)
TABLE I.
The following experimentally feasible parameters are used in the numerical evaluations in the text (unless specified
otherwise).
g
em
/
MHz
κ
e
,
i
/
MHz
κ
e
,
c
/
MHz
κ
o
,
i
/(GHz)
κ
o
,
c
κ
m
/
kHz
g
0
/
kHz
ω
o
/
THz
ω
m
/
GHz
ω
e
/
GHz
η
o
η
e
T
o
T
e
D
o
/
Hz
D
e
/
Hz
2
π
×
1.2 2
π
×
0.55 2
π
×
1.25 2
π
×
0.65
κ
o
,
i
2
π
×
150 2
π
×
260 2
π
×
190 2
π
×
52
π
×
50.80.910
−
2
0.5
∼
10
10
3
rate within
t
w
is given as
R
cc
(
t
,
τ)
=
P
(
t
,
τ)
r
D
.(8)
Using the quantum version of moment factoring theorem
[
45
] and the fact
r
D
t
w
∼
1, we get
R
cc
(
t
,
τ)
R
a
+
R
c
,(9)
where
R
a
=
a
†
out
(
t
)
a
out
(
t
)
c
†
out
(
t
+
τ)
c
out
(
t
+
τ)
t
w
is
called
the
accidental
counting
rate
while
R
c
=
a
†
out
(
t
)
c
†
out
(
t
+
τ)
c
out
(
t
+
τ)
a
out
(
t
)
t
w
is the correlated
counting rate. Note Eq.
(9)
gives the coincidence photon-
counting rate at time
t
after the pump, and the total rate
can be obtained by taking the sum over the entire pump
duration
R
cc
(τ )
=
k
=
0
G
2
(
t
k
,
τ)
t
w
,
with
t
k
=
t
i
+
k
×
t
w
t
f
t
i
G
2
(
t
,
τ)
dt
,
(10)
where
t
i
(
t
f
) denotes the initial (final) time of the pump
duration. The second line of integration takes into account
that the time
t
w
is much smaller than the output photon
linewidth. Obviously, the coincidence rate is
τ
dependent.
As shown in Fig.
2(a)
by numerical calculation (details
given in the following sections), the rate is maximized with
time delay
τ
150 ns, which again matches the expecta-
tion that the time (approximately
κ
e
/
g
2
em
) is needed to swap
the mechanical mode excitation to the electrical mode.
In practice, the coincidence counting rate is also affected
by the photon transmission loss, detector dark count
(including pump leakage), and detector efficiency. The
pump leakage induced dark count is usually proportional
to the pump power. Denote
T
o
,
T
e
as the transmission coef-
ficients,
D
o
,
D
e
as the dark count rates, and
η
o
,
η
e
the
detector efficiencies for the optical and microwave pho-
tons, respectively. The coincidence rate Eq.
(9)
shall be
modified as
R
cc
(
t
,
τ)
R
a
+
R
c
,
(11)
where
R
a
=
(η
o
T
o
̄
n
o
out
(
t
)
+
D
o
)(η
e
T
e
̄
n
e
out
(
t
+
τ)
+
D
e
)
t
w
,
and
R
c
=
η
o
η
e
T
o
T
e
R
c
. we denote
̄
n
o
out
(
t
)
=
a
†
out
(
t
)
a
out
(
t
)
,
and
̄
n
e
out
(
t
+
τ)
=
c
†
out
(
t
+
τ)
c
out
(
t
+
τ)
. The typical
value of the parameters are given in Table
I
, and shall be
used in the following numerical estimations.
B. Time-bin Bell state
Using the system, it is convenient to generate Bell states
in time bins, which is a coding scheme resistant to pho-
ton loss. In each experiment, we pump the system with
two weak successive pulses separated in time by
t
,defin-
ing the two time bins. Due to the low pump power, one
and only one pair of photons can be generated with high
probability (excluding vacuum). We can further make the
(c)
(b)
(a)
FIG. 2. (a) The optical-microwave photon-generation rate in terms of the time delay. (b) The coincidence counts within 1 min with
varied angle
β
and fixed time delay
τ
=
150 ns. (c) The CHSH inequality violation curves. The dashed-green-horizontal lines give the
local and nonlocal boundary while the red dashed line marks the maximal violation value 2
√
2. The green, orange, and blue curves in
(a)–(c) correspond to the pumps with
G
={
30, 20, 1
}
.
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photons in the two time bins indistinguishable by control-
ling the time delay on both microwave and optical sides
(delaying the photon in the first time bin by time
t
or
accelerate the photon in the second time bin by time
t
)
[
46
], creating a Bell state
|
ψ
B
=
1
√
2
|
bin
1
e
|
bin
1
o
+|
bin
2
e
|
bin
2
o
,
(12)
where we denote the optical time-bin state
|
bin
i
o
=
ˆ
a
†
out,
i
|
0
and the microwave time-bin state
|
bin
i
e
=
ˆ
c
†
out,
i
|
0
with the time-bin index
i
={
1, 2
}
. The entangle-
ment in this state can be verified using different entan-
glement witnesses, e.g., the Bell-state fidelity [
47
]. In the
following, we shall discuss its Bell-inequality violation
with different parameters and compare the corresponding
photon-generation rates.
C. Coincidence counts
In experiment, the entanglement can be tested by count-
ing the correlated statistics with proper control and photon
detection. For instance, on the optical side, an imbalanced
Mach-Zehnder interferometer can be used to rotate the
optical time bin in the Bloch sphere [
31
,
46
]. If we fix
the qubits in the equator, the detection of a single optical
photon at the interferometer output will project the state
onto
|
ψ
±
o
=
1
/
√
2
(
|
bin
1
o
±
e
i
θ
|
bin
2
o
)
.
(13)
Note half of the photon rate will be lost since only delayed
photon in time bin 1 can interfere with nondelayed pho-
ton in time bin 2. On the microwave side, the photon can
first be converted to a transmon qubit excitation, which
effectively selects the state by later qubit rotation [
48
,
49
]
|
ψ
±
e
=
1
/
√
2
(
|
bin
1
e
±
e
i
φ
|
bin
2
e
)
.
(14)
The statistics
ˆ
a
†
out
(
t
)
ˆ
c
†
out
(
t
+
τ)
ˆ
c
out
(
t
+
τ)
ˆ
a
out
(
t
)
with spe-
cific configurations, e.g., given
θ
,
φ
, can be obtained by
repeating the experiment many times [
36
]. To simplify
the notation, we will omit the lower index “out” unless
required for clarity in the following discussions.
Theoretically, this coincidence rate can be straightfor-
wardly calculated. In the Heisenberg picture, the optical-
microwave output-mode operators actually undergo the
transforms
a
1
a
2
=
cos
α/
2sin
α/
2
e
i
θ
−
sin
α/
2
e
−
i
θ
cos
α/
2
a
1
a
2
(15)
and
c
1
c
2
=
cos
β/
2sin
β/
2
e
i
φ
−
sin
β/
2
e
−
i
φ
cos
β/
2
c
1
c
2
(16)
corresponding to those time-bin mode rotations in the
whole range of the Bloch sphere. For any fixed
α
,
β
,
θ
and
φ
, one can record four different outcomes, e.g., simultane-
ously detecting the mode
a
1
and
c
1
, which as we discussed
before has the coincidence rate (take
θ
=
0,
φ
=
0, for
instance)
t
f
t
i
a
†
1
(
t
)
c
†
1
(
t
+
τ)
c
1
(
t
+
τ)
a
1
(
t
)
dt
=
t
f
t
i
a
†
1
(
t
)
a
1
(
t
)
c
†
1
(
t
+
τ)
c
1
(
t
+
τ)
+
a
†
1
(
t
)
c
†
1
(
t
+
τ)
c
1
(
t
+
τ)
a
1
(
t
)
dt
=
t
f
t
i
a
†
1
(
t
)
a
1
(
t
)
c
†
1
(
t
+
τ)
c
1
(
t
+
τ)
+
cos
2
α
−
β
2
a
†
1
(
t
)
c
†
1
(
t
+
τ)
c
1
(
t
+
τ)
a
1
(
t
)
dt
,
(17)
where the first equal sign is arrived at by the renowned
moment factoring theorem. The last equal sign is obtained
using Eq.
(15)
and
(16)
, and we use the fact that the opti-
cal (microwave) photons in time bin 1 are not correlated
with photons in time bin 2, e.g.,
ˆ
a
†
1
(
t
)
ˆ
c
†
2
(
t
+
τ)
=
0. Fur-
ther, suppose we collect data for a certain amount of time,
e.g.,
t
c
, we can get the corresponding coincidence counts,
denoted as
C
(α
,
β)
=
R
cc
(
ˆ
a
1
,
ˆ
c
1
)
×
t
c
.
(18)
D. CHSH inequality
The well-known CHSH inequality is a type of Bell
inequality, whose violation excludes the possibility of local
hidden variable theory. It is written as
|
S
|≤
2 with
S
=
σ
A
σ
B
+
σ
A
σ
B
+
σ
A
σ
B
−
σ
A
σ
B
,
(19)
where
σ
A
is the Pauli-
Z
operator along
A
direction, e.g.,
σ
A
=|
α
,
θ
α
,
θ
|−|
α
+
π
,
θ
+
π
α
+
π
,
θ
+
π
|
for
A
=
(α
,
θ)
in the Bloch sphere of the optical time bin.
064047-5
CHANGCHUN ZHONG
et al.
PHYS. REV. APPLIED
22,
064047 (2024)
Similarly, we have
σ
B
=|
β
,
φ
β
,
φ
|−|
β
+
π
,
φ
+
π
β
+
π
,
φ
+
π
|
for
B
=
(β
,
φ)
of the microwave time
bin. The term, e.g., like
|
α
,
θ
α
,
θ
|
is called an “event”
in quantum information theory [
50
], which corresponds
to the photon click that projects the photon state onto,
e.g.,
|
ψ
=
cos
α/
2
|
bin
1
o
+
e
i
θ
sin
α/
2
|
bin
2
o
.Thecor-
responding number of clicks in a given amount of time
equals exactly the term
C
(α
,
β)
that we have in Eq.
(18)
(with phases included). The other terms in Eq.
(19)
are
similar. Typically, the measurement setting for violating
CHSH is fixed in a plane. Without losing generality, we
choose the plane with
θ
=
φ
=
0, and
α
=
α
+
π/
2,
β
=
β
+
π/
2. Thus,
S
becomes
S
(α
,
β)
=
σ
α
σ
β
+
σ
α
σ
β
+
σ
α
σ
β
−
σ
α
σ
β
, (20)
with each term given by, e.g.,
σ
α
σ
β
=
C
(α
,
β)
+
C
(α
+
π
,
β
+
π)
−
C
(α
,
β
+
π)
−
C
(α
+
π
,
β)
C
(α
,
β)
+
C
(α
+
π
,
β
+
π)
+
C
(α
,
β
+
π)
+
C
(α
+
π
,
β)
.
(21)
Combining the above equations with Eq.
(18)
, the quantity
S
can be readily evaluated.
E. Numerical results
We first numerically calculate the coincidence rate from
Eq.
(17)
in terms of the time delay by further fixing
α
=
β
=
0, as shown in Fig.
2(a)
. Given three different
model pumps, the coincidence rate increases with the laser
pump strength. Figure
2(b)
depicts the coincidence counts
in 1 min while varying the angle
β
. Although a larger
pump gives higher coincidence counts, it also increases
the accidental counts, which could potentially ruin the Bell
test. Intuitively, it is known that larger pump power could
raise the chance of higher-order photon excitation, which
degrades the Bell state fidelity. This is confirmed by cal-
culating the quantity
S
as a function of
β
. The result is
shown in Fig.
2(c)
. All three curves go outside the region
|
S
|≤
2, clearly violating the CHSH inequality. Also, the
larger violation (blue curve) is obtained from lower laser
pump power, signifying the trade-off between Bell-state
fidelity and the photon-pair generation rate. In practice,
one would optimize for higher fidelity or higher photon-
generation rate depending on the specific application or
requirements.
V. LASER-HEATING ESTIMATION
A. A simplified model
Laser heating is one of the major problems in coherently
controlling the quantum transduction process. In experi-
ment, one needs a dilution refrigerator to maintain a cryo-
genic environment free of thermal noise. If the control laser
power used to activate the photon-pair generation of the
transducer is too high, it can lead to excess heating within
the transducer due to parasitic optical absorption, leading
to loss of coherent transduction of signals. For a typical
quantum transducer with a cw pump, the pump power must
be severely limited to avoid heating effects, greatly lim-
iting the mode-squeezing strength and the instantaneous
pair-generation probability. However, given the transient
heating of the transducer, in the case of a pulsed pump,
the peak power can be much larger while still avoid-
ing the deleterious effects of heating. This indicates that
pulsed pumping, in addition to being part of our time-bin
entanglement approach, can also potentially yield a much
higher instantaneous squeezing strength in the presence of
parasitic heating effects than that of cw pumping.
Here, we study the heating dynamics of the Gaussian
pulse pump and compare it with that of the cw pump
as a quantitative guide to experiments. For steady-state
bath population,the heating dynamics in nanoscale piezo-
optomechanical transducers is very well described by the
bottleneck
modelinRef.[
51
], which yields a prediction for
both the transient behavior of the thermal-bath occupancy
of the microwave (mechanical) mode and its damping rate
to the hot bath. Here we employ a much simpler model that
captures the essential transient behavior for varied pump
pulses in these devices, which is essential in experimental
design since the laser heating and the photon-pair rate form
two competing parameters in the entanglement generation.
In the long time limit, our model matches the scaling pre-
diction of the steady-state occupancy given in Ref. [
51
].
Roughly, the dynamics is governed by
(
P
heat
−
P
cool
)
×
t
=
CM
×
T
,
(22)
where
P
heat
and
P
cool
denote the laser-heating and system-
cooling power, respectively. In this model, we assume
the system is described by an effective temperature
T
p
(a
temperature distribution across the device would be more
practical and the discussion is given in the Appendix).
T
gives the temperature change.
t
is the time inter-
val.
C
and
M
are the specific heat capacity and mass
of the piezomaterial. In practice, the cooling power and
the heat capacity can be temperature dependent. Since
we care about the regime where the temperature varia-
tion is small, it is reasonable to take the linear dependence
of the cooling power on temperature. Assuming a con-
stant heat capacity, we solve the time dependence of the
device temperature with two different heating powers: (1)
a constant
P
heat
=
h
0
+
h
1
; (2) a Gaussian heating power
P
heat
=
h
0
+
h
2
exp
(
−
(
t
−
t
0
)
2
/
2
σ
2
)
, where
h
0
represents
the environment heating, and the second term in each case
is induced by the laser pump. For a complete comparison
with cw pump, we also consider applying the Gaussian
pump repeatedly with the periodicity
t
per
.
The results are shown in Fig.
3
(the detailed calculations
are put in Appendix ??). For a cw pump and a Gaussian
064047-6
MICROWAVE-OPTICAL ENTANGLEMENT...
PHYS. REV. APPLIED
22,
064047 (2024)
pulse pump that induces the same average thermal noise,
the allowed instantaneous intracavity photon of the pulsed
pump can be much larger than that of the cw pump. The
data in Figs.
3(a)
and
3(b)
confirm this intuition for various
pulse periods
t
per
and the width
σ
of the Gaussian pump.
The larger intracavity photon indicates a bigger coupling
strength, which suggests a higher probability in generating
paired photons while the average added thermal noise stays
the same.
B. The Bell test with heating noise
The above model enables us to simulate the entangled
photon source with laser-heating noise. We consider laser
heating that generates an effective time-dependent thermal
noise
n
th
(
t
)
that couples to the mechanical mode. As shown
in the Appendix, we fit our model to the experimental data
[
36
], and obtain a time-dependent thermal noise
n
th
(
t
)
=
(
e
ω
m
/
k
b
T
p
(
t
)
−
1
)
−
1
.
(23)
The thermal temperature is given by
T
p
(
t
)
=
b
e
−
at
f
(
t
)
+
d
a
1
−
1
−
a
d
T
0
e
−
at
,
(24)
where the function
f
(
t
)
=
[0.305
−
0.306Erf
(
2.013
−
10.408
t
/(
1
μ
s
))
]
μ
s. All other parameters are given in the
Appendix. The pump laser also generates the intracavity
photon
̄
n
o
(
t
)
=
n
m
e
−
(
t
−
t
0
)
2
2
σ
2
,
(25)
where
n
m
=
0.8,
t
0
=
160 ns and
σ
=
68 ns, which further
determines the optomechanical squeezing strength. Using
the above parameters, we numerically solve the master Eq.
(6)
again and calculate the Bell-inequality violation curve.
The results are shown in Fig.
3(c)
. The blue curve clearly
shows that the system is producing entangled pairs beyond
any classical descriptions, which matches the claim in Ref.
[
47
] based on nonclassical values of entanglement fidelity.
As a prediction for the trend, we also perform the sim-
ulation with varied pump-pulse widths
σ
={
42, 21, 11
}
ns, while the other parameters are kept the same. It is
shown that a larger violation can be achieved by decreasing
the pump-pulse width. Figure
3(d)
compares the trade-off
between obtainable entanglement fidelity and rate for the
different pump-pulse widths.
VI. DISCUSSION
The quantum transducer is an essential device in the
development of distributed quantum architectures. Recent
experimental demonstrations of microwave-optical entan-
glement, encoded either in the continuous variable or
discrete variable degrees of freedom [
47
,
52
], have shown
great promise in the realization of entanglement-based
quantum transduction [
31
]. Here we present a systematic
study of a pulsed entanglement-based scheme, utilizing
time-bin degrees of freedom for encoding. Our model
serves as a useful framework to analyze various physical
platforms; however, in order to highlight practical limita-
tions, and trade-offs between photon-pair generation rate
and Bell-state fidelity, we consider a state-of-the-art piezo-
optomechanical transducer [
36
–
38
]. Our results indicate
that such entanglement-based quantum transduction pro-
tocols should be able to be realized. In the future, it
would be helpful to consider different protocols, such as
a well-controlled pump-pulse shape, to further optimize
the entangled resources and to make full use of them in
connecting different quantum modules. We leave that for
future research.
ACKNOWLEDGMENTS
C.Z. thanks Mankei Tsang for helpful discussions.
We acknowledge supports from the ARO (Grants No.
W911NF-18-1-0020, No. W911NF-23-1-0077), ARO
MURI (Grant No. W911NF-21-1-0325), AFOSR MURI
(Grants No. FA9550-19-1-0399, No. FA9550-21-1-0209,
No. FA9550-23-1-0338), DARPA (Grants No. HR0011-
24-9-0359, No. HR0011-24-9-0361), NSF (Grants No.
OMA-1936118, No. ERC-1941583, No. OMA-2137642,
No. OSI-2326767, No. CCF-2312755), NTT Research,
and the Packard Foundation (Grant No. 2020-71479). C.Z.
thanks the start up support from Xi’an Jiaotong University
(Grant No. 11301224010717).
APPENDIX: A PHENOMENOLOGICAL HEATING
MODEL
Laser-induced device heating creates an effective ther-
mal bath. The microscopic dynamics of the optically
induced bath is described in detail by the
bottleneck
model in Ref. [
51
]. According to the model, the geom-
etry of the nanobeam imposes an effective phonon bot-
tleneck, which prevents a rapid thermalization between
the higher-frequency phonon modes and low-lying modes.
The absorbed optical power from the laser populates
the higher-frequency phonons, resulting in a buildup in
the bath phonon population above the bottleneck. This
elevated-temperature bath then couples to the lower-
lying modes—in particular, the breathing mode at 5
GHz—through elastic three-phonon scattering. The model
provides a experimentally verified scaling behavior of the
bath occupancy and damping rate assuming a steady-state
bath population created by the laser. In this paper, we pro-
vide a simple prediction model of the transient behavior of
the thermal bath.
For the transient behavior of the thermal bath in the
pulse-pumped scheme we consider, as a first step, a heating
064047-7