# On-chip parallel processing of quantum frequency combs for high-dimensional
hyper-entanglement generation

Liang Zhang  James C. Wyant College of Optical Sciences, The University of
Arizona, Tucson, Arizona 85721, USA  Chaohan Cui  James C. Wyant College of
Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA
Jianchang Yan  Research and Development Center for Solid State Lighting,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China  Yanan Guo  Research and Development Center for Solid State Lighting,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China  Junxi Wang  Research and Development Center for Solid State Lighting,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China  Linran Fan<EMAIL_ADDRESS>James C. Wyant College of Optical
Sciences, The University of Arizona, Tucson, Arizona 85721, USA

High-dimensional encoding and hyper-entanglement are unique features that
distinguish optical photons from other quantum information carriers, leading
to improved system efficiency and novel quantum functions. However, the
disparate requirements to control different optical degrees of freedom have
prevent the development of complete integrated platforms that is capable of
leveraging the complementary benefits of high-dimensional encoding and hyper-
entanglement at the same time. Here we demonstrate the chip-scale solution to
the generation and manipulation of high-dimensional hyper-entanglement. This
is achieved by the parallel processing of multiple quantum frequency combs in
the path domain. Cavity-enhanced parametric down-conversion with Sagnac
configuration is implemented to ensure the spectral indistinguishability.
Simultaneous entanglement in path and frequency is realized with high
dimensions. On-chip reconfiguration of the entanglement structure is also
demonstrated. We further present quantum interference in both entanglement
degrees of freedom with high visibility. Our work provides the critical step
for the efficient and parallel processing of quantum information with
integrated photonics.

The generalization of two-level quantum systems to high dimensions provides
the capability to verify quantum theories with stronger criteria, perform
quantum computing with better error resilience, and conduct quantum
communications with higher capacity and noise robustness Collins et al.
(2002); Cerf et al. (2002); Lanyon et al. (2009). Optical photons are the
ideal candidate for high-dimensional encoding Erhard et al. (2020). Various
degrees of freedom including orbital angular momentum Leach et al. (2010);
Fickler et al. (2012), frequency Xie et al. (2015); Reimer et al. (2016);
Jaramillo-Villegas et al. (2017); Kues et al. (2017), spatial Walborn et al.
(2010); Dada et al. (2011); Edgar et al. (2012), and temporal modes De
Riedmatten et al. (2002); Vagniluca et al. (2020), have been used for high-
dimensional photonic entanglement generation. Besides high-dimensional
encoding, simultaneous entanglement in different degrees of freedom (hyper-
entanglement) is another promising route to enhance the quantum information
processing capability with complementary features Deng et al. (2017). In
particular, deterministic quantum logic Sheng and Deng (2010) and complete
Bell measurement Walborn et al. (2003); Schuck et al. (2006); Barreiro et al.
(2008) can be realized with the assistance of hyper-entanglement. By
coherently combining high-dimensional encoding and hyper-entanglement, novel
photonic technologies can be developed to realize critical quantum functions
such as cluster state generation for universal quantum information processing
Reimer et al. (2019); Imany et al. (2019); Ciampini et al. (2016).

Enhanced phase stability and high nonlinear efficiency make integrated
photonics the ideal platform to implement complex large-scale quantum photonic
systems Elshaari et al. (2020); Wang et al. (2020). Among different degrees of
freedom, path and frequency are two promising candidates for on-chip quantum
information encoding. Path-encoding features the complete state manipulation
with arbitrary unitary transformation at the expense of large-size photonic
circuits. Frequency-encoding can generate large-scale quantum states with
ultra-compact devices Reimer et al. (2016); Jaramillo-Villegas et al. (2017);
Kues et al. (2017). However, the difficulty in high-efficiency quantum
frequency conversion prevents the demonstration of complex quantum state
transformation on chip Kobayashi et al. (2016); Fan et al. (2016); Lu et al.
(2018); Joshi et al. (2020).

Figure 1: High-dimensional hyper-entanglement generation scheme. a, Schematic
to show the device layout for the generation of high-dimensional hyper-
entanglement. The continuous-wave pump (green) is coupled onto the chip,
equally splitted with a Y-junction, and coupled into the integrated ring
cavity in both CW and CCW directions. Intra-cavity parametric down-conversion
generates frequency-entangled photon pairs across multiple cavity resonances.
Output photon pairs are interfered with tunable relative phase $\varphi$ on a
50/50 beamsplitter. b, Output entanglement structure with frequency
correlation in symmetric spectral modes across CW and CCW paths.

Here, we propose and demonstrate the on-chip generation and control of high-
dimensional hyper-entanglement leveraging the complementary features of path
and frequency encoding. Multiple quantum frequency combs with near-unity
spectral overlap are simultaneously generated with cavity-enhanced parametric
down-conversion in Sagnac configuration. We realize the parallel processing of
quantum frequency combs with uniform response and high visibility, leading to
the precise reconfiguration of the hyper-entanglement structure. High-
dimensional quantum interference is further performed to verify the high
quality of the entanglement generation and manipulation.

Hyper-entanglement generation scheme

The schematic to generate high-dimensional hyper-entanglement is shown in Fig.
1a. The single-frequency pump in the visible regime is equally splitted, and
coupled into the clockwise (CW) and counter-clockwise (CCW) directions of the
integrated photonic ring cavity. Pockels nonlinearity enables efficient
parametric down-conversion inside the cavity. With the large phase matching
bandwidth, photon pairs in the telecom regime are generated in the
superposition of multiple cavity resonances, forming quantum frequency combs.
The simultaneous generation of multiple quantum frequency combs in the same
cavity using Sagnac configuration ensures the near-identical spectral
distribution. The relative phase $\varphi$ between frequency combs can be
controlled by an on-chip phase shifter. After interference at the balanced
multimode interferometer, the output state can be written as

$\displaystyle|\psi_{out}\rangle=\frac{1}{\sqrt{2N+2}}\sum_{k=0}^{N}\sin\varphi(\hat{b}_{k}^{\dagger}\hat{b}_{-k}^{\dagger}-\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger})$
$\displaystyle|0\rangle$
$\displaystyle-\frac{1}{2\sqrt{2N+1}}\sum_{k=-N}^{N}\cos\varphi(\hat{b}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}+\hat{b}_{-k}^{\dagger}\hat{c}_{k}^{\dagger})$
$\displaystyle|0\rangle$ (1)

where $\hat{b}^{\dagger}$ and $\hat{c}^{\dagger}$ are the creation operators
for the upper and lower paths, $k$ labels the frequency mode, and the total
number of frequency modes is $2N+1$ (Supplementary Section I). The output
state is entangled in the frequency domain due to the energy conservation. At
the same time, the output state can be configured as a two-photon NOON state
in the path domain if the relative phase $\varphi$ is set to zero Kok et al.
(2002); Pryde and White (2003). Therefore, the output state is entangled in
both path and frequency, forming the high-dimensional hyper-entanglement (Fig.
1b).

Figure 2:  Generation of quantum frequency combs. a, Classical transmission
spectrum and single-photon spectrum of CW and CCW quantum frequency combs. b,
Measured frequency correlation of the CCW quantum frequency comb. c, Measured
frequency correlation of the CCW quantum frequency comb. d, Self-correlation
of photons generated from the single degenerate resonance (green) and five
resonances (blue). e, Dimension of the CW quantum frequency comb with
different numbers of resonance pairs. Upper (blue crosses) and lower (green
circles) bounds are obtained from the temporal self-correlation measurement
and the Schmidt decomposition of the frequency correlation matrix
respectively.

Pockels quantum frequency comb

The complete photonic circuit is fabricated with aluminum nitride (AlN) on
sapphire wafer. The photonic ring cavity has a radius of 60 $\mu$m with free-
spectral range of 362 GHz. The phase matching for parametric down-conversion
is realized with modal dispersion engineering (Supplementary Section II) Guo
et al. (2017). We first performed experiments to characterize quantum
frequency combs. Single-photon spectrum is measured with narrow bandpass
filters and superconducting nanowire single photon detectors (SNSPDs). Similar
frequency mode locations and single-photon rate profiles are observed in both
CW and CCW directions (Fig. 2a). Different frequency modes in the same quantum
frequency comb are then separated by tunable filters to measure the
correlation (Fig. 2b and c). Strong correlation is only observed between
frequency modes located symmetrically to the degenerate wavelength at 1555.68
nm. The coincidence-to-accidental ratio above 25 dB has been achieved for all
resonance pairs. The lower bound of the quantum state dimension can be
obtained by the Schmidt decomposition of the frequency correlation matrix Law
et al. (2000); Fedorov and Miklin (2014). The Schmidt number scales linearly
with the number of resonance pairs (Fig. 2e). With ten resonance pairs, the
lower bound of the system dimension is estimated above 9.3 for each quantum
frequency comb.

We further estimate the upper bound of the quantum state dimension through the
self-correlation measurement in the time domain with Hanbury-Brown-Twiss setup
(Supplementary Section III). The peak value of the self-correlation function
at zero time delay $g^{(2)}(0)$ directly reflects the effective mode number
$N_{\mathrm{eff}}$ $(g^{(2)}(0)=1+\frac{1}{N_{\mathrm{eff}}})$ (Fig. 2d)
Christ et al. (2011). The effective mode number for all resonances is
approximately unity, showing the high purity of the generated quantum
frequency comb (one effective mode per resonance) Helt et al. (2010); Cui et
al. (2021). The upper bound of the total effective mode number can be
estimated by the summation of individual separable modes (Fig. 2e). A linear
relation between the upper bound of the system dimension and the resonance
pairs is clearly observed. With ten resonance pairs, an upper bound of
$10.39\pm 0.52$ is achieved. As the lower and upper bounds are both rounded to
ten, we show that the system dimension of each quantum frequency comb is ten.

Figure 3: Path-domain entanglement and quantum interference. a, Transmission
of the coherent light through a Mach-Zehnder interferometer. b, Self-
correlation (red) and cross-correlation (blue) of the degenerate photon pair
with visibility $93.8\pm 0.51\%$ and $93.5\pm 0.36\%$ respectively. c,
Correlation between photon pairs from non-degenerate resonance pairs (Mode 4)
with the same path (red) and different paths (blue) with visibility $92.3\pm
0.65\%$ and $96.3\pm 3\%$ respectively. d, Correlation between photon pairs
from five resonance pairs with the same path (red) and different paths (blue)
with visibility $86.67\pm 0.32\%$ and $89.38\pm 0.47\%$. e, Visibility of
quantum interference with different resonance pairs. f, Thermal-optic phase
shifter condition for different phases and resonance pairs.

Parallel processing in path domain

We verify the path-domain performance through tunable on-chip quantum
interference between CW and CCW quantum frequency combs. In contrast to
classical interference where only phase $\varphi$ is obtained, a $2\varphi$
phase shift is applied to the quantum frequency comb. The phase $\varphi$ is
controlled with therm-optic effect through a local heater (Supplementary
Section IV). The interference pattern of the coherent light through a Mach-
Zehnder interferometer is used as the classical reference (Fig. 3a). We first
isolate the degenerate resonance and measure the self-correlation of the upper
path $S_{0,0}$ and the cross-correlation between the two paths $C_{0,0}$ (Fig.
3b). Both self- and cross-correlation show period half of the classical
interference with complementary count rates. This agrees with Eq. (1) which
suggests $S_{0,0}\propto\cos^{2}\varphi$ and $C_{0,0}\propto\sin^{2}\varphi$.
The quantum interference shows high visibility of $93.8\pm 0.51\%$ without
background noise subtraction.

Another key difference between the classical and quantum interference is the
dispersion response. The interference pattern shifts depending on wavelengths
in the classical case. In the quantum case, the collective phase of the photon
pair determines the interference pattern, which is the fundamental principle
behind nonlocal dispersion cancellation Franson (1992). With the fixed sum
frequency, the total accumulated phase for symmetric resonances
($\varphi_{k}+\varphi_{-k}$) stays constant, leading to the same interference
pattern for different resonance pairs. We verify this by measuring the cross-
correlation between symmetric frequency modes in the same path $S_{k,-k}$, as
well as in different paths $C_{k,-k}$ (Fig. 3c). The interference pattern
stays unchanged, and high visibility is maintained (Fig. 3e). In particular,
the control condition of the phase shifter remains consistent for different
resonance pairs (Fig. 3f). This enables the parallel processing of quantum
frequency combs. The interference pattern and quality of quantum frequency
combs are maintained the same as individual frequency modes (Fig. 3d).

Figure 4: Quantum state reconfiguration. Correlation matrix between different
path and frequency modes under (a) bunching with $\varphi=0$, (b)
superposition with $\varphi=\pi/4$, and (c) splitting with $\varphi=\pi/2$
conditions.

The parallel processing of quantum frequency combs can be further examined
through the simultaneous measurement of the correlation in both path and
frequency domains. Quantum frequency combs can be reconfigured to exhibit
bunching, coherent superposition, and splitting between the two paths by
setting the phase shifter to be $\varphi=0$, $\pi/4$, and $\pi/2$ respectively
(Fig. 4a-c). Under the splitting condition, the CW and CCW quantum frequency
combs undergo the reverse Hong-Ou-Mandel process, leading to the deterministic
separation of the two photons Chen et al. (2007). Therefore, no correlation is
observed between any resonance pairs in the same path (Fig. 4a). Under the
bunching condition, the two photons are grouped into the same path. Therefore,
strong correlation can be observed within either path, but only between
symmetric frequency modes. This indicates the generation of simultaneous
entanglement in path and frequency domains. Under all photonic circuit
settings, the coincidence-to-accidental ratio remains above 10 dB for all
resonance pairs.

High-dimensional quantum interference

We further configure the output state under the splitting condition with
$\varphi=\pi/2$ and perform the quantum interference between two quantum
frequency combs with different time delays Hong et al. (1987) (Fig. 5a). A
tunable optical delay line is placed in one path to control the relative time
delay, and a fiber polarization controller is used to match the polarization
between the two paths. A pair of programmable filters are used to shape the
frequency profile in each path. After interference at a balanced fiber
beamspliter, the coincidence is recorded by SNSPDs. We first select only the
degenerate frequency mode in both paths. The standard Hong-Ou-Mandel
interference is observed with a single dip at zero time delay (Fig. 5b).
Visibility around $89.6\pm 2.5\%$ is achieved without subtracting the
background noise. The dip width is estimated around $1.57\pm 0.04$ ns, which
matches the linewidth of the degenerate frequency mode (Supplementary Section
II). We further program the filters to incorporate the first five resonance
pairs. We observe the fast oscillation of the coincidence under different time
delays due to the beating between different resonance pairs (Fig. 5c). Upon
zoom-in of the coincidence pattern at zero time delay, narrow coincidence dips
with high visibility can be observed, proving the intra-pair phase coherence
between symmetric resonances (Fig. 5d) Lingaraju et al. (2019). The revival of
the coincidence dip is also observed with delay period of 1.4 ps, which agrees
with the free-spectral range of the photonic ring cavity (362 GHz). The
visibility of the coincidence dips decreases with non-zero time delay due to
the finite temporal coherence of quantum frequency combs (Fig. 5e). The
coherence length is determined by the cavity loss. Therefore, the envelop of
coincidence dips matches the standard Hong-Ou-Mandel interference with a
single frequency mode (Fig. 5c).

Figure 5:  High-dimensional quantum interference. a, Experiment setup for
quantum interference. FPC, fiber polarization controler; ODL, optical delay
line; TCSPC, time-correlated single photon counter. b, Normalized coincidence
rate between the two SNSPDs with photon pairs in the degenerate resonance. c,
Normalized coincidence rate between the two SNSPDs with photon pairs from five
resonance pairs. Theoretical envelop of the coincidence rate is also plotted.
d, Measured coincidence rate (black dot) and theoretical calculation (red)
near zero time delay, corresponding to the red-shaded area in c. e, Measured
coincidence rate (black dot) and theoretical calculation (blue) at time delay
around 1.1 ns, corresponding to the blue-shaded area in c.

Discussion

In addition to parametric down-conversion for efficient quantum state
generation, Pockels nonlinearity can also provide strong electro-optic effect.
High-frequency optical modulation and efficient pulse shaping have been
demonstrated using integrated AlN photonic devices with performance exceeding
commercial fiber-based and free-space modulators Fan et al. (2019). The
coherent combination of different frequency modes can be realized with
efficient phase modulation, which can verify the inter-pair phase coherence of
quantum frequency combs Kues et al. (2017); Lingaraju et al. (2019). With on-
chip programmable filters, the complete coherent transformation of the high-
dimensional hyper-entanglement state in both path and frequency domains can be
realized on chip based on integrated Pockels nonlinearity Khan et al. (2010);
Lukens and Lougovski (2017).

Scalability and functionality are the ultimate criteria to evaluate any
integrated quantum photonic systems. We have provided a new prospective by
proposing and demonstrating the coherent combination of path- and frequency-
encoding on the same chip. Parametric down-conversion with nanophotonic cavity
enhancement leads to the efficient generation of quantum frequency combs.
High-dimensional hyper-entanglement has been generated by the parallel
processing of quantum frequency combs. We further present the on-chip high-
fidelity state control and high-visibility quantum interference. The unique
Pockels nonlinearity can enable the integration of complete path-frequency
control on a single integrated platform. This work can significantly expand
the dimension of quantum photonic systems while simultaneously enhancing the
quantum functionality, providing new opportunities for photonic quantum
information processing.

Acknowledgments LZ, CC, and LF acknowledge the support from U.S. Department of
Energy, Office of Advanced Scientific Computing Research, (Field Work Proposal
ERKJ355); Office of Naval Research (N00014-19-1-2190); National Science
Foundation (ECCS-1842559). LF acknowledges the support from II-VI foundation.
Device is fabricated in the OSC cleanroom at the University of Arizona, and
the cleanroom of Arizona State University. SNSPDs are supported by NSF MRI
INQUIRE.

Author contributions

The experiments were conceived by LF and CC. The device was designed and
fabricated by LZ and LF. Measurements were performed by LZ, LF, and CC.
Analysis of the results was conducted by LZ, LF, and CC. The AlN wafer was
provided by JY, YG, and JW. All authors participated in the manuscript
preparation.

Methods

AlN wafer growth. The AlN wafer is grown on sapphire substrate with c-plane
(0001) by MOCVD. Trimethylaluminum (TMA), ammonia (NH3) and hydrogen (H2) are
used as the aluminum source, nitrogen source, and carrier gas, respectively.
The growth started with a 25-nm low-temperature AlN buffer layer at 550 oC,
followed by the growth of 1-$\mu$m high quality AlN layer at 1200 oC.

Device fabrication. The device is fabricated with 1-$\mu$m AlN film on
sapphire grown by MOCVD. We use electron-beam lithography to exposure FOX-16
resist to define the photonic circuit. After development in
Tetramethylammonium hydroxide (TMAH), the pattern is transferred to AlN layer
with reactive ion etching (RIE) using Cl2/BCl3/Ar chemistry. Then 2-$\mu$m
SiO2 is deposited by PECVD as the cladding layer. Finally, electrodes for
phase shifter is defined with S1813 photoresist using photolithography,
followed by Ti/Pt (5/200nm) deposition and lift-off in Acetone.

Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request

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