of 12
ARTICLE
High-dimensional time-frequency entanglement in
a singly-
fi
ltered biphoton frequency comb
Xiang Cheng
1,7
, Kai-Chi Chang
1,7
, Murat Can Sarihan
1
, Andrew Mueller
2,3
, Maria Spiropulu
4
,
Matthew D. Shaw
2
, Boris Korzh
2
, Andrei Faraon
5
, Franco N. C. Wong
6
, Jeffrey H. Shapiro
6
&
Chee Wei Wong
1
High-dimensional quantum entanglement is a cornerstone for advanced technology enabling
large-scale noise-tolerant quantum systems, fault-tolerant quantum computing, and dis-
tributed quantum networks. The recently developed biphoton frequency comb (BFC) pro-
vides a powerful platform for high-dimensional quantum information processing in its
spectral and temporal quantum modes. Here we propose and generate a singly-
fi
ltered high-
dimensional BFC via spontaneous parametric down-conversion by spectrally shaping only the
signal photons with a Fabry-Pérot cavity. High-dimensional energy-time entanglement is
veri
fi
ed through Franson-interference recurrences and temporal correlation with low-jitter
detectors. Frequency- and temporal- entanglement of our singly-
fi
ltered BFC is then quan-
ti
fi
ed by Schmidt mode decomposition. Subsequently, we distribute the high-dimensional
singly-
fi
ltered BFC state over a 10 km
fi
ber link with a post-distribution time-bin dimension
lower bounded to be at least 168. Our demonstrations of high-dimensional entanglement and
entanglement distribution show the singly-
fi
ltered quantum frequency comb
s capability for
high-ef
fi
ciency quantum information processing and high-capacity quantum networks.
https://doi.org/10.1038/s42005-023-01370-2
OPEN
1
Fang Lu Mesoscopic Optics and Quantum Electronics Laboratory, Department of Electrical and Computer Engineering, University of California, Los An
geles,
CA 90095, USA.
2
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA.
3
Applied Physics, California
Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA.
4
Division of Physics, Mathematics and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA.
5
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
6
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
7
These authors contributed equally: Xiang Cheng,
Kai-Chi Chang.
email:
chengxiang@ucla.edu
;
uclakcchang@ucla.edu
;
cheewei.wong@ucla.edu
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1
1234567890():,;
Q
uantum entanglement, referred to as
spooky action at a
distance
1
, has enabled tremendous advances in both
fundamental science and engineering technologies
2
,
3
.As
a technological resource, quantum entanglement has revolutio-
nized communications
4
,
5
, computation
6
,
7
, metrology
8
, and
sensing
9
. High-dimensional entanglement
10
, owing to its higher
information capacity and resilience to noise, has been proposed
for noise-resilient large-alphabet quantum key distribution
11
16
and universal photonic quantum computation
17
20
. High-
dimensional quantum states of entangled photons have already
been demonstrated in a variety of degrees-of-freedom, such as
spatial modes
21
, orbital angular momentum
22
, optical
frequency
23
, and time bins
24
. Recently demonstrated BFCs
25
31
,
which can carry information in their spectro-temporal quantum
modes, are ideal candidates for high-dimensional quantum sys-
tems. Such BFCs can be produced by spontaneous parametric
down-conversion (SPDC) with post-generation
fi
ltering
25
,
28
,
29
,
31
,
optical parametric oscillation (OPOs) operating far below
threshold
32
,
33
, or spontaneous four-wave mixing in integrated
microring resonators
23
,
26
,
34
. The
fi
rst approach is realized by
sending SPDC-generated biphotons through a cavity, while the
second relies on cavity-enhanced SPDC, in which the biphoton
generation is both enhanced and spectrally con
fi
ned by the cavity
modes. The third approach utilizes third-order nonlinearity to
generate photon entanglement over a broad range of frequency
modes in a chip-scalable platform. These BFCs approaches are
usually doubly resonant, viz., the signal and idler
s spectra are
simultaneously tightly con
fi
ned to the cavity modes. On the other
hand, the singly-resonant BFC has been studied, theoretically, for
a sub-threshold OPO in which only the signal photons are
resonated by the OPO cavity
35
. Owing to the entanglement
between signal and idler photons from biphoton generation
process, the idler photons will then exhibit the comb-like spec-
trum corresponding to the OPO cavity
s internal mode
structure
36
. Recently, such a singly-resonant BFC has been
demonstrated to generate frequency-multiplexed photon pairs
over 1,000 frequency modes
37
, and also highly multi-mode
polarization-entangled photon pairs by means of a Sagnac
interferometer
38
. However, a typical OPO con
fi
guration employs
a cavity that is a few cm long, to accommodate the nonlinear
crystal used for SPDC. Hence its BFC
s free spectral range (FSR)
is limited to a few GHz. That bandwidth is not compatible with
off-the-shelf dense wavelength-division multiplexer/demulti-
plexer devices, whose channel bandwidths are 50 or 100 GHz.
Although the singly-resonant BFC offers a promising platform for
quantum information processing, its high-dimensional time-fre-
quency entanglement has not been carefully investigated. More-
over, distributing its high-dimensional entanglement
23
,
39
,
40
,
which is a critical precursor to its enabling large-scale high-
dimensional quantum communication and distributed networks,
remains a challenge. Indeed, high-dimensional time-frequency
entanglement distribution has yet to be demonstrated for the
singly-resonant BFC.
Here we propose a
fl
exible approach to generate a singly-
fi
ltered BFC state with the same temporal and spectral properties
as the OPO-based singly-resonant BFC by spectrally shaping only
the signal photon of the SPDC-generated pair with a Fabry-Pérot
cavity. We observe that this singly-
fi
ltered BFC exhibits the same
temporal correlation as the singly-resonant BFC
35
,
37
,
38
. More-
over, high-dimensional energy-time entanglement of such singly-
fi
ltered BFC is veri
fi
ed via Franson-interference recurrences over
16 times-bins. Specially, we verify the spectral phase coherence by
resolving the periodic oscillations in the cross-correlation with
single-sided decay using state-of-the-art low timing jitter super-
conducting nanowire single-photon detectors (SNSPDs), sup-
porting the high-dimensional entanglement of our BFC state. The
joint spectral intensity of the singly-
fi
ltered BFC is measured and
analyzed through Schmidt mode decomposition with 4.17
Schmidt number over 5 frequency bins. Recurrences of the
Franson-interference further enable the mapping of the BFC
s
time-binned joint temporal intensity, shown to have a Schmidt
number of 13.11 over 16 time bins, concurring with our high-
dimensional time-bin entangled state. Furthermore, entanglement
distribution of the singly-
fi
ltered BFC
s high-dimensional state is
demonstrated over a 10 km optical
fi
ber link in an asymmetric
con
fi
guration. The post-distribution BFC state
s high-
dimensional time-frequency entanglement is examined via non-
local interferometry, with up to 98.81 ± 0.61% visibility achieved
recurrently over 16 time-bins and with an averaged 96.70 ± 1.93%
visibility across 5 frequency-bin pairs. We also estimate the
Schmidt number to be 12.99 from the distributed non-local
quantum interference revivals, which lower bounds the time-
binned Hilbert-space dimensionality to be at least 168. Further-
more, we demonstrate proof-of-principle high-dimensional
quantum key distribution with our singly-
fi
ltered BFC, exploit-
ing the lower
fi
ltering loss of the singly-
fi
ltered con
fi
guration.
This
fi
rst high-dimensional time-frequency entanglement dis-
tribution paves the pathway in constructing practical long-
distance quantum networks.
Results
Generation and spectro-temporal characterization of a singly-
fi
ltered BFC
. Figure
1
a illustrates the experimental setup to
generate and characterize the singly-
fi
ltered BFC. The entangled
photon pairs are generated by a 16-mm long type-II periodically-
poled KTiOPO
4
(ppKTP) waveguide (AdvR Inc.) that was inte-
grated in a
fi
ber package for high
fl
uence and ef
fi
ciency
25
,
28
.A
658 nm Fabry-Pérot laser diode, stabilized by self-injection
locking through double-pass
fi
rst-order diffraction feedback
using an external grating, is used to pump the ppKTP waveguide.
The generated biphotons are orthogonally polarized and fre-
quency degenerate at 1316 nm with
245 GHz full-width half-
maximum (FWHM) bandwidth. The residual pump photons are
removed by a long-pass
fi
lter (LPF). A 1.3 nm bandpass
fi
lter
(BPF), i.e., 225 GHz FWHM bandwidth, is used to further clean
the biphoton spectrum. Then the signal and idler photons are
separated ef
fi
ciently by a polarizing beam splitter (PBS) due to the
type-II phase matching. The singly-
fi
ltered BFC is generated by
passing only the signal photons through a
fi
ber Fabry-Pérot cavity
(FFPC) (Luna Inc.). The idler photons, although not con
fi
ned in a
cavity, still exhibit a comb-like spectrum when heralded by
signal-photon detections, due to frequency entanglement
35
. The
FFPC, which has a 45.32 GHz FSR and 1.56 GHz FWHM band-
width, is stabilized with a high-performance temperature con-
troller. We note that if the FFPC is used in a doubly-
fi
ltered
con
fi
guration
when both the signal and idler photons are
fi
l-
tered
the FFPC
s polarization birefringence would result in
different post-
fi
ltering spectra for the signal and idler photons.
However, in the present singly-
fi
ltered con
fi
guration, only the
signal photons pass through the FFPC and we further solve the
requirement for polarization birefringence elimination.
We
fi
rst characterize the temporal signature of the singly-
fi
ltered BFC. Our singly-
fi
ltered con
fi
guration generates the
(unnormalized) BFC state whose frequency-domain representa-
tion can be usefully approximated as:
j
ψ
N
m
¼
N
Z
d
Ω
f
ð
Ω

m
ΔΩ
Þ
sinc
ð
A
Ω
Þ
^
a
y
H
ω
p
2
þ
Ω

^
a
y
V
ω
p
2

Ω

j
0
i
;
ð
1
Þ
see Supplementary Note 1 for the exact result. Here:
^
a
y
H
and
^
a
y
V
are the creation operators for horizontally (signal) and vertically
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(idler) polarized photons;
ω
p
is the pump frequency; the sinc
function is the SPDC
s phase-matching function with
A
¼
1
:
39
=
π
B
PM
for
B
PM
=
245 GHz being the FWHM phase-
matching bandwidth;
ΔΩ
is the FFPC
s FSR in rad s
1
;
Ω
is the
detuning of the SPDC biphotons from their center frequency;
2
N
+
1
=
5 is the number of the cavity lines passed by the
bandwidth-limiting
fi
lter; and
f
ð
Ω

m
ΔΩ
Þ
is the spectral
amplitude of the FFPC
s
m
th cavity resonance, with
f
Ω
ðÞ¼
1
=
Δω
þ
i
Ω
½
;
i.e., a Lorentzian transmission whose FWHM
linewidth is 2
Δω
. The temporal representation of the singly-
fi
ltered BFC that is the dual of Eq. (
1
) is then (see Supplementary
Note 1 for the exact result):
j
ψ
Z
0
1
d
τ
exp
ð
Δω
τ
Þ
N
m
¼
N
sinc
Am
ΔΩ
ðÞ
cos
ð
m
ΔΩ
τ
Þ
^
a
y
H
ð
t
þ
τ
Þ
^
a
y
V
ð
t
Þj
0
i
ð
2
Þ
where we have used
ΔΩ
=
2
π

B
PM
:
The state
s temporal
behavior then has recurrences with repetition period
2
1
0
-1
-2
0
1000
2000
3000
4000
2
1
0
-1
-2
5000
S
2
&I
-2
S
1
&I
-1
S
0
&I
0
S
-1
&I
1
S
-2
&I
2
1.0
Pump power (mW)
1.5
2.0
2.5
0.0
0.1
0.2
0.3
Heralded g
2
(0)
b
c
d
a
Experiment
Theory t
j
= 74 ps
Theory t
j
= 7.8 ps
Signal-idler delay (ns)
-0.3
0.0
0.3
0.6
0.2
0.4
0.6
s
s
o
r
c
d
e
z
i
l
a
m
r
o
N-
n
o
i
t
a
l
e
r
r
o
c
function
0.0
0.8
1.0
Fig. 1 Generation and spectro-temporal characterization of singly-
fi
ltered biphoton frequency combs. a
Experimental schematic for generation and
characterization of a singly-
fi
ltered BFC. ppKTP: periodically-poled KTiOPO
4
waveguide; PBS polarizing beam splitter, FFPC
fi
ber Fabry
Pérot cavity, SPD
superconducting nanowire single-photon detector.
b
Experimental (black dots) and theoretical (red solid line and black shade) temporal second-order
cross-correlation function between signal and idler photons. Based on the 1.56 GHz cavity bandwidth and 45.32 GHz free spectral range (FSR) of our FFP
C,
we theoretically
fi
t our experimental results with different effective detector timing jitters. The periodic temporal oscillations of cross-correlation function
for the 45.32 GHz cavity
s singly-
fi
ltered BFC can be resolved when the effective timing jitter is set equal to or less than the cavity
s round-trip time
(detailed calculations are in Supplementary Note 1).
c
Measured frequency-correlation matrix of our 45.32 GHz singly-
fi
ltered BFC within the 245 GHz
SPDC bandwidth. The joint spectral intensity (JSI) is measured by using a pair of tunable bandpass
fi
lters with 300 pm bandwidths to select frequency bins
for the signal and idler photons. Only the diagonal elements of the frequency-correlation matrix show high coincidence counts, revealing the freque
ncy-bin
entanglement.
d
Measured heralded single-photon second-order auto-correlation function
g
(2)
(0) versus the pump power for each frequency-bin pair
(different colors represent corresponding frequency-bin pair in the JSI measurements). The minimum heralded
g
(2)
(0)
0.035 is measured for S
0
&I
0
pair
(green star) at 1.3 mW pump power with a heralding rate of 16 coincidences/s. All measured heralded
g
(2)
(0)
s are below the classical threshold, verifying
the high purity of the frequency-binned single photon states from our singly-
fi
ltered BFC.
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3
Δ
T
¼
2
π
=
ΔΩ
22.1 ps, i.e., the cavity
s round-trip time. We note
that for the doubly-
fi
ltered BFC, the integral in Eq. (
2
) spans from
negative to positive, which results in the symmetric temporal
behaviors
25
,
28
.
Figure
1
b shows the normalized second-order cross-correlation
function as a function of relative delay between signal and idler
photons (de
fi
ned as
τ
¼
t
idler

t
signal
), measured with super-
conducting nanowire single-photon detectors (SNSPDs,
80%
detection ef
fi
ciency, PhotonSpot Inc.). A single-sided exponential
decay of the cross-correlation function is the temporal signature
of a singly-resonant BFC
35
37
. The measured second-order cross-
correlation function matches well with our theoretical calcula-
tions assuming the detectors
combined root-mean-square timing
jitter is
t
j
=
3.4
Δ
T
=
74 ps (see Supplementary Note 1 for details).
Here the temporal oscillation signature of the cross-correlation
function is not fully resolved because the timing jitter is more
than 3 times the repetition time period. Low time-jitter SNSPDs
have been recently demonstrated, which enable better temporal
resolution
41
. Figure
1
b shows that with
t
j
=
0.35
Δ
T
=
7.8 ps the
temporal oscillation pro
fi
le of the 45.32 GHz singly-
fi
ltered BFC
can be observed; with
t
j
¼
0
:
035
4
T
¼
0
:
78 ps, that signature is
fully resolved, showing an oscillation period of 22.1 ps (see
Supplementary Note 1 for details).
Next we characterize the frequency correlation of our
45.32 GHz singly-
fi
ltered BFC through joint spectral intensity
(JSI) measurements. We use a 2 mW pump for SPDC generation
to minimize multiphoton emission and reduce cross-talk between
frequency bins. The JSI of our singly-
fi
ltered BFC is measured
using a pair of tunable BPFs for signal and idler photons
respectively. The BPFs have 300 pm bandwidth, which are able to
select only one frequency bin of our BFC. Signal-idler coincidence
counts are recorded while the tunable BPFs are set to different
combinations of the signal-idler frequency-bin pairs. Within the
245 GHz SPDC bandwidth, 5 frequency bins can be examined for
the signal and idler of our 45.32 GHz singly-
fi
ltered BFC. We
sweep the BPFs from
2to
+
2 frequency bins, with the 0th
frequency bin indicating the SPDC
s center frequency, i.e., half
the pump frequency. High values of photon coincidences are
measured only for symmetric frequency-bin pairs, shown as the
diagonal elements of the frequency correlation matrix in Fig.
1
c.
This behavior reveals the frequency correlation of the singly-
fi
ltered BFC, and is a characteristic of frequency-bin entangle-
ment. We note that the coincidence counts fall off at frequency
bin pairs away from the central bin, which results from the sinc-
squared spectra of the SPDC biphotons prior to signal
fi
ltering.
For each symmetric frequency-bin pair, we then measure the
signal-heralded second-order auto-correlation
g
(2)
(0) of the idler
photons. The measurement is performed using a Hanbury-Brown
and Twiss (HBT) interferometer in which the idler light from the
singly-
fi
ltered BFC is divided into two paths by a 50:50 beam
splitter for auto-correlation measurement heralded by signal-
photon detections. A pair of tunable BPFs (300 pm bandwidth) is
placed before the HBT interferometer to select different frequency
bins. The second-order auto-correlation
g
(2)
(0) is measured by
recording the three-fold coincidence counts between the HBT
interferometer
s output ports and the signal photons within a 2 ns
duration coincidence window. Detecting a signal photon heralds
the appearance of the idler photon, which exhibits non-classical
anti-bunching behavior. Figure
1
d shows the signal-heralded
g
(2)
(0) versus pump power for
fi
ve symmetric frequency-bin pairs.
The pump power is set to 1.3 mW, 1.8 mW, and 2.4 mW
respectively. At low pump power, the heralded
g
(2)
(0) values for
all
fi
ve frequency-bin pairs are below 0.1, showing high single-
photon purity of the frequency-
fi
ltered states from our singly-
fi
ltered BFC. We observe that the heralded
g
(2)
(0) is proportional
to the pump power, due to the Poisson statistics of the SPDC
emission. With increased pump power, we note that the heralded
g
(2)
(0) for each frequency-bin pair increases because of multi-pair
emissions. At higher pump power, the heralded
g
(2)
(0) for S
0
&I
0
(S
0
and I
0
denote the central frequency bin for signal and idler
photons, respectively) frequency-bin pair is still below 0.1. The
heralded
g
(2)
(0) values for S
+
2
&I
-2
and S
-2
&I
+
2
increase faster
than is the case for S
+
1
&I
-1
and S
-1
&I
+
1
when the pump power is
increased. This may be due to the coincidence counts
fall off in
Fig.
1
c, which implies that the frequency bin pairs away from the
degeneracy will have worse signal-to-noise-ratio with increasing
noise photons introduced by the stronger pump. In addition, we
measured the heralded
g
(2)
(0) for the singly-
fi
ltered BFC state
without selection of a frequency-bin pair by using only a
broadband BPF (225 GHz bandwidth) to clean up the biphoton
spectrum (detailed in Supplementary Note 2). A signal-heralded
g
(2)
(0)
0.154 is measured for our singly-
fi
ltered BFC with
0.2 mW pump power. We also obtain a
g
(2)
(0)
0.130 with
0.6 mW pump power by sending the signal photons to the HBT
interferometer while all the idler photons are sent to an SNSPD
whose detections provided heralding for the three-fold coin-
cidence measurement. Both heralded
g
(2)
(0) values are well below
the classical threshold, demonstrating a high-purity heralded
single-photon state preparation from our singly-
fi
ltered BFC.
High-dimensional energy-time entanglement witnessed via
intrinsic temporal oscillations and Franson-interference
recurrences
. We verify the coherence of our high-dimensional
singly-
fi
ltered BFC states via temporal second-order cross-cor-
relation. We extract the two-photon time-correlation from joint
temporal intensity (JTI), namely, the temporal cross-correlation
measurements. In order to resolve the temporal correlation of our
singly-
fi
ltered BFC, the effective timing jitter should be equal to
or less than the cavity round-trip time. We employ two state-of-
the-art impedance-matched differential SNSPDs with low timing
jitter
42
while preserving a moderate detection ef
fi
ciency, in
combination with a multi-channel low-jitter time tagger (Swabian
Tagger X) to perform the cross-correlation measurements
between the signal and idler photons. First, we characterize the
combined system timing jitter of the two differential SNSPDs
using our SPDC photon source
the jitter adds in quadrature
and obtain a full-width at half maximum of
21.6 ps from the
cross-correlation function (see Supplementary Note 5), which is
comparable to our 45.32 GHz cavity round-trip time. This allows
us to measure the temporal correlation oscillations of our singly-
fi
ltered BFC and observe distinct correlation peaks, as shown in
the datapoints of Fig.
2
a. This periodic oscillation arises from the
coherent interference of biphotons
different frequency modes,
with a temporal spacing of 22.0 ps (corresponds to 45.45 GHz
FSR), matching well with the cavity round-trip time of 22.1 ps.
Figure
2
b shows another cross-correlation measurement with a
15.15 GHz cavity. The temporal spacing in the correlation peaks
is 66.8 ps, implying a 14.97 GHz cavity FSR and matching our
selected cavity. Supported by our exact theory on the temporal
oscillations (Supplementary Note 1) with the phase-sensitive
cross-spectrum of the post-
fi
ltered baseband
fi
eld operator in the
presence of detector jitter [vis-à-vis Supplementary Eqs. (2) and
(4)], the measured intrinsic temporal oscillations demonstrate
that our singly-
fi
ltered BFC is coherently generated with a
fl
at
spectral phase
38
,
43
.
We next characterize the high-dimensional energy-time
entanglement of a singly-
fi
ltered BFC by means of its Franson-
interference recurrences. We use a 1.3 nm bandwidth BPF to
clean the SPDC spectrum before separating the signal and idler
photons with a PBS. The signal photons are passed through a
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