Squeezed light is optical beams with variance below the shot noise level. They are a key resource for quantum technologies based on photons, and they can be used to achieve better precision measurements and improve security in quantum key distribution channels and as a fundamental resource for quantum computation. Here, we demonstrate an integrated source of squeezing based on four-wave mixing that requires a single laser pump, measuring 0.45 dB of broadband quadrature squeezing at high frequencies. We identify and verify that the current results are limited by excess noise produced in the chip and propose ways to reduce it. Calculations suggest that an improvement in the optical properties of the chip achievable with existing technology can develop scalable quantum technologies based on light.

Squeezed states of light are a fundamental building block of quantum optics, as they are capable of generating entangled states in the continuous variable (CV) regime.1 For this reason, they are the basis to demonstrate fundamental physics principles and develop quantum technologies. For example, squeezed light has been used to generate entanglement,2 as a resource required for quantum teleportation,3 and produce Schrodinger cat states.4 In quantum key distribution, squeezed states can be employed to enhance security5 with high compatibility with conventional optical communication technology. In sensing, squeezing has been demonstrated as the optimal resource used in interferometers to achieve sub-Shot Noise Level (SNL) measurements,6 and its use for gravitational wave detection has demonstrated outstanding broadband sensitivity.7 Finally, the hybridization of CV with single photons has been recently proposed to achieve high fidelity logical operations8,9 for the development of quantum computers able to operate with error correction protocols.10 The use of squeezing, rather than single photons, allows for the unconditional generation of entanglement,3 allowing deterministic schemes to achieve quantum advantages over the classical approach.

Efforts have been focused on the development of integrated circuits for CV photonics, including entangled circuits,11 and homodyne detectors for quantum states.12 Numerous demonstrations of on-chip generation based on spontaneous parametric down-conversion have been achieved in periodically poled lithium niobate (PPLN) waveguides at a telecommunication wavelength.13 Even though recent results of on-chip generation and manipulation of squeezed states of ∼2 dB are remarkable,14,15 the diffusion fabrication process results in low modal confinement, long interaction lengths, and high bending losses, which strongly limit the potential application for complex on-chip experiments and prevent the direct integration of either photodetectors or superconductive single photon detectors (SSPD). Silicon nitride (SiN) is a promising material for quantum optics application, thanks to the wide transparency spectrum and the absence of two-photon absorption up to visible wavelengths. SiN photonics has demonstrated the generate single photons,16,17 complex optical circuits,18 and SSPD scalable to high photon numbers.19 Recently, highly enhanced light–matter interactions in silicon nitride (SiN) resonators based on four-wave mixing have been used for efficient creation of twin beam squeezed states20,21 and degenerate squeezed beams based on complex pumping and optical circuit structure,22,23 achieving the level of measured squeezing around 1.5 dB.

Here, we use a single ring resonator and single pump by taking advantage of self-phase modulation and generate squeezed states by the Kerr effect, as proposed by Hoff et al.24 Contrary to the original theoretical scheme, we produce two counter propagating bright squeezed states that reinterfere in an integrated Sagnac interferometer to generate a single quadrature squeezed state.25 This technique has been widely investigated in experiments based on optical fibers26 as a way to reject spurious noise and reduce the optical power to avoid detector saturation. We measure 0.45 dB and infer 1 dB of broadband quadrature squeezing at telecommunication wavelengths using an on-chip SiN microring structure that was fabricated with CMOS compatible material.

The optical device was designed to maximize the third order non-linearity in the ring resonators (see the supplementary material). The photonic device was fabricated from a plain silicon wafer on which 2 μm of low loss thermal SiO2 is grown by wet oxidation. Then, we deposit 500 nm plasma enhanced chemical vapor deposition (PECVD) SiN with a 5:2 ratio of NH3:SiH4,17 showing a refractive index of 1.96 at 1550 nm. Samples are diced and spun with 450 nm of positive resist CSAR. Next, the photonic circuits are exposed with an electron beam lithography system JEOL JBX9300FS, developed and etched with an inductively coupled plasma (ICP) RIE in fluorine chemistry. The remaining resist is removed and the sample is annealed in N2 for 3 h at 1200° to further decrease the material losses around 1550 nm. Finally, 1.2 μm of SiO2 is deposited as top cladding by using PECVD liquid tetraethoxysilane (TEOS). The SiN photonic circuit consists of a 2 × 2 multi-mode interference (MMI) coupler with the output ports connected in a loop to form an integrated Sagnac interferometer. Inside of this loop, four microring resonators were designed with different resonant frequencies in the highly overcoupled regime (with different escape efficiency >70%). This is achieved, thanks to a short single mode section that pushes the optical field out of the waveguide in order to achieve high coupling ideality.27 An optical image of the photonic chip is reported in Fig. 1(b). The device was characterized by performing transmission measurements with a tunable laser. Figure 1(a) displays the transmission spectrum of the overcoupled ring resonator with an escape efficiency of 77% used for the experiment. The ring has 30 μm radius and a loaded Q-factor of 238 000, from which propagation losses of 0.32 dB/cm can be extracted.

FIG. 1.

Experimental setup and optical characterization. (a) Transmission spectrum of the SiN chip, showing the overcoupled resonance with a loaded Q-factor of 238 000 used for squeezing generation. (b) Optical image of the photonic chip; scale bar is 200 µm. The waveguide adjacent to the MMI output is used to facilitate coupling into the chip. (c) Schematic setup for on-chip generation of broadband quadrature squeezed states. PMF—polarization maintaining fiber, EDFA—erbium doped fiber, COL—collimator amplifier, HWP—half waveplate, QWP—quarter waveplate, PBS—polarizing beam splitter, DUT—device under test, MMF—multimode fiber, and ESA—electronic spectrum analyzer.

FIG. 1.

Experimental setup and optical characterization. (a) Transmission spectrum of the SiN chip, showing the overcoupled resonance with a loaded Q-factor of 238 000 used for squeezing generation. (b) Optical image of the photonic chip; scale bar is 200 µm. The waveguide adjacent to the MMI output is used to facilitate coupling into the chip. (c) Schematic setup for on-chip generation of broadband quadrature squeezed states. PMF—polarization maintaining fiber, EDFA—erbium doped fiber, COL—collimator amplifier, HWP—half waveplate, QWP—quarter waveplate, PBS—polarizing beam splitter, DUT—device under test, MMF—multimode fiber, and ESA—electronic spectrum analyzer.

Close modal

Figure 1(c) shows a schematic of the experimental setup for generation of the squeezed state (see the supplementary material). An input beam in diagonal polarization was coupled to the photonic chip via high numerical aperture aspheric lens. In the chip, the horizontal polarization is equally split in the 50/50 MMI and coupled to the ring resonator producing two counter-propagating bright squeezed states, thanks to the self-phase modulation based on the third-order nonlinear Kerr effect.26 The beams reinterfere on the MMI and produce an attenuated quadrature squeezed state at the output port, while the majority of the pump is rejected in the input port. At the same time, vertical polarization is unequally split by the MMI (59/41 ratio); the beams counter-propagate back toward the beam splitter where they partially interfere with lower visibility so that a mW-level beam of light copropagates alongside the quadrature squeezed state and can be used as a local oscillator. This configuration is chosen to simplify the control of the experiment, since there is high phase stability between the local oscillator and squeezed state beams as they share the same optical path. Both beams were out-coupled via an additional lens. Using four waveplates, we were able to measure the noise power at different quadratures with a polarization homodyne detection scheme (see the supplementary material).

The noise spectrum of the light collected from the chip is presented in Fig. 2(a) for three different input powers in the input waveguide (26 mW, 39 mW, and 52 mW), while panel (b) presents the spectrum normalized to the shot noise. Squeezing is observed spanning a frequency range of 300 MHz, with a maximum reduction of noise of 0.45 dB. The inferred level of squeezing corrected for the measurement efficiency is estimated to be 1 dB. The squeezing level above Ω ∼ 800 MHz decreases due to the response of the detector and the spectral properties of the ring resonator. This bandwidth is comparable to the one observed in down conversion parametric oscillators by monolithic cavities, but an order of magnitude greater than bow tie configurations, offering high data rate for quantum communication and cryptography while requiring modest power requirements. No squeezing is observed at low frequencies, as excess noise above the SNL is present up to Ω ∼ 500 MHz. We assign the origin of this noise to the thermorefractive effect: statistical variations in the temperature of the chip drive refractive index fluctuations through the thermo-optic coefficient of the material, introducing phase noise in the propagating beam. The slow diffusion of these random temperature fluctuations results in a noise that decays with the square of the frequency28 Ω−2. In the supplementary material, we investigate the characteristics of the excess noise to experimentally confirm its thermorefractive origin and provide power, frequency, and temperature dependent measurements that corroborate our model.

FIG. 2.

Measurements of the squeezing level. (a) Noise spectrum for measured squeezing and anti-squeezing and shot noise for an on-chip pump power of 52 mW. The video (VBW) and radio (RBW) bandwidths are set up at 100 kHz and 20 Hz, respectively. Each line is an average of five measurements where each measurement has a sweep time of 10 s. (b) Squeezing spectrum, for three different input powers before the 50/50 beam splitter, that has been corrected for the noise of the detector and normalized to the shot noise level.

FIG. 2.

Measurements of the squeezing level. (a) Noise spectrum for measured squeezing and anti-squeezing and shot noise for an on-chip pump power of 52 mW. The video (VBW) and radio (RBW) bandwidths are set up at 100 kHz and 20 Hz, respectively. Each line is an average of five measurements where each measurement has a sweep time of 10 s. (b) Squeezing spectrum, for three different input powers before the 50/50 beam splitter, that has been corrected for the noise of the detector and normalized to the shot noise level.

Close modal

We further analyze the prospects to suppress this effect and to achieve high level of squeezing. The unwanted noise depends on both the thermo-optic coefficient and the thermal fluctuations induced by the local environment of the ring. Since for many future experiments, the generation of non-Gaussian quantum states1 will require the integration of SSPDs operating at cryogenic temperature, it is expected that the production of the unwanted noise will decrease. Assuming that the ring resonator temperature is lowered to <3 K, we expect a reduction of this noise by 50 dB, since the thermorefractive effect scales as T2 and the thermo-optic coefficient of SiN decreases at low temperatures. In such a case, we would predict a measurable squeezing level of 1.4 dB at low frequencies under the same pumping conditions. This is limited by the trade-off between pump power, escape efficiency, and noise at room temperature in the current device. An alternative way to reduce the noise that does not require cryogenic operation relies on a more precise control of the photonic interferometer. It has been proposed that the Sagnac works as a purifier of quantum correlations from classically correlated noisy effects such as Brillouin and Raman scattering25 or even technical noise of the laser.29 Therefore, improving the contrast of our Sagnac interferometer from 23 dB to 60 dB using an interferometer30 could greatly reduce any classically correlated noise (see the supplementary material for the effect of the Sagnac interferometer).

In order to understand the dynamics of generated quantum correlated states, we use the theory developed by Hoff24 for the Kerr effect in SiN microring resonators. We verify this model in Fig. 3 by comparing the measured quadrature spectra with theoretical calculations. The assumption describes reasonably well the measured squeezing at high frequencies. This can be supported by the fact that the Ω−2 scaling of the thermorefractive noise makes it negligible at higher frequencies so that the noise begins to be dominated by the Kerr squeezing.

FIG. 3.

Theoretical and experimental squeezing. Comparison between the measured squeezed spectrum and theoretical prediction without thermorefractive noise where the calculations include the overall out-coupling losses.

FIG. 3.

Theoretical and experimental squeezing. Comparison between the measured squeezed spectrum and theoretical prediction without thermorefractive noise where the calculations include the overall out-coupling losses.

Close modal

Finally, we evaluate the possible on-chip generation of the quadrature squeezed state with the already fabricated waveguide using commercially available low pressure chemical vapor deposition and ultra-low loss SiN31 and the highest-Q SiN ring resonators32 with an intrinsic Q factor of 13 and 37 × 106, respectively. In Fig. 4, we calculate the amount of on-chip squeezing considering a ring escape efficiency of 95%. We use the waveguide and ring parameters reported in the respective demonstrations and assume that the increased coupling does not degrade the intrinsic Q factor. In both cases, the amount of squeezing converges to 13 dB, mainly limited by the escape efficiency. More interestingly, for the higher Q factor, the power required to achieve such a squeezing level is only 40 mW. Provided that there are no additional noise sources that limit the squeezing level, such results could introduce a wide variety of future applications for continuous variable encoding for quantum computing, as 10 dB is considered to be sufficient to achieve fault tolerant universal quantum computation.10 

FIG. 4.

Prediction of squeezing with current technology. Theoretically predicted levels of quadrature squeezing for commercially deposited SiN from Ref. 31 and research grade SiN from Ref. 32. The prediction is based on the waveguide and ring dimensions reported in the respective reference, assuming an escape efficiency of 95%.

FIG. 4.

Prediction of squeezing with current technology. Theoretically predicted levels of quadrature squeezing for commercially deposited SiN from Ref. 31 and research grade SiN from Ref. 32. The prediction is based on the waveguide and ring dimensions reported in the respective reference, assuming an escape efficiency of 95%.

Close modal

In summary, we have studied the generation of broadband quadrature squeezing via self-phase modulation, from a CMOS-compatible SiN microring resonator in an integrated Sagnac interferometer, measured in the low CW pump regime. Contrary to the alternative PPLN sources that are several millimeter long and fabricated via proton beam lithography, our process takes full advantage of the standard commercial nanofabrication techniques, providing a truly scalable process that can provide hundreds of optical components. We provide demonstrations that the amount of squeezing is limited by the thermorefractive noise that is, in principle, present in all experiments based on self-phase modulation. This noise can be heavily reduced by suppressing the temperature fluctuations operating at cryogenic temperatures and improving the optical circuit. This would increase the measured amount of squeezing of the current sample under the same pumping conditions and provide sub-SNL light at low frequencies. The use of ultra-low loss SiN would result in much higher amount of squeezing for even lower pumping powers. Furthermore, on-chip entanglement can be achieved using only two ring resonators and simple integrated optic circuits,11 providing a basis for fundamental capabilities such as quantum teleportation, cryptography, and sensing. Finally, using time encoded sources of quadrature squeezed states of light, photon resolving SSPDs, delay lines, and integrated germanium photodetectors, could have the potential to achieve a fully integrated fault tolerant universal quantum computer33 and remove the limitation of out-coupling losses. The integration of all components in a single chip would make the experiment phase-stable without the need of additional locking electronics or polarization encoding. This will simplify both the design of the photonic device and the operation of the experiment while removing the need for many of the elements currently required in CV experiments. These prospects make SiN resonators excellent candidates to expand the applications of integrated sources of squeezed states of light to a broad range of future photonic quantum technology applications and go beyond the limits imposed by bulk optics.

See the supplementary material for details of the chip, polarization homodyne detection, noise characterization, and simulations of the Sagnac noise rejection.

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

The authors acknowledge the help of Z. Vernon and L. G. Helt for their extremely useful discussions of the origin of the low frequency technical noise and J. C. F. Matthews and R. Slavik for helpful advice. We also acknowledge support from the Southampton Nanofabrication Center. This work was supported by the H2020-FETPROACT2014 Grant QUCHIP (Quantum Simulation on a Photonic Chip; Grant Agreement No. 641039) and EPSRC (Grant No. EP/P003710/1).

1.
C.
Weedbrook
,
S.
Pirandola
,
R.
García-Patrón
,
N. J.
Cerf
,
T. C.
Ralph
,
J. H.
Shapiro
, and
S.
Lloyd
, “
Gaussian quantum information
,”
Rev. Mod. Phys.
84
,
621
669
(
2012
).
2.
Z. Y.
Ou
,
S. F.
Pereira
,
H. J.
Kimble
, and
K. C.
Peng
, “
Realization of the Einstein-Podolsky-Rosen paradox for continuous variables
,”
Phys. Rev. Lett.
68
,
3663
3666
(
1992
).
3.
A.
Furusawa
,
J. L.
Sorensen
,
S. L.
Braunstein
,
C. A.
Fuchs
,
H. J.
Kimble
, and
E. S.
Polzik
, “
Unconditional quantum teleportation
,”
Science
282
,
706
709
(
1998
).
4.
A.
Ourjoumtsev
,
H.
Jeong
,
R.
Tualle-Brouri
, and
P.
Grangier
, “
Generation of optical ‘Schrödinger cats’ from photon number states
,”
Nature
448
,
784
786
(
2007
).
5.
T.
Gehring
,
V.
Händchen
,
J.
Duhme
,
F.
Furrer
,
T.
Franz
,
C.
Pacher
,
R. F.
Werner
, and
R.
Schnabel
, “
Implementation of continuous-variable quantum key distribution with composable and one-sided-device-independent security against coherent attacks
,”
Nat. Commun.
6
,
8795
(
2015
).
6.
C. M.
Caves
, “
Quantum-mechanical noise in an interferometer
,”
Phys. Rev. D
23
,
1693
1708
(
1981
).
7.
The LIGO Scientific Collaboration
, “
Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light
,”
Nat. Photonics
7
,
613
619
(
2013
).
8.
N.
Lee
,
H.
Benichi
,
Y.
Takeno
,
S.
Takeda
,
J.
Webb
,
E.
Huntington
, and
A.
Furusawa
, “
Teleportation of nonclassical wave packets of light
,”
Science
332
,
330
333
(
2011
).
9.
S.
Takeda
,
T.
Mizuta
,
M.
Fuwa
,
P.
van Loock
, and
A.
Furusawa
, “
Deterministic quantum teleportation of photonic quantum bits by a hybrid technique
,”
Nature
500
,
315
318
(
2013
).
10.
K.
Fukui
,
A.
Tomita
,
A.
Okamoto
, and
K.
Fujii
, “
High-threshold fault-tolerant quantum computation with analog quantum error correction
,”
Phys. Rev. X
8
,
021054
(
2018
).
11.
G.
Masada
,
K.
Miyata
,
A.
Politi
,
T.
Hashimoto
,
J. L.
O’Brien
, and
A.
Furusawa
, “
Continuous-variable entanglement on a chip
,”
Nat. Photonics
9
,
316
319
(
2015
).
12.
F.
Raffaelli
,
G.
Ferranti
,
D. H.
Mahler
,
P.
Sibson
,
J. E.
Kennard
,
A.
Santamato
,
G.
Sinclair
,
D.
Bonneau
,
M. G.
Thompson
, and
J. C. F.
Matthews
, “
A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers
,”
Quantum Sci. Technol.
3
,
025003
(
2018
).
13.
T.
Suhara
, “
Generation of quantum-entangled twin photons by waveguide nonlinear-optic devices
,”
Laser Photonics Rev.
3
,
370
393
(
2009
).
14.
F.
Lenzini
,
J.
Janousek
,
O.
Thearle
,
M.
Villa
,
B.
Haylock
,
S.
Kasture
,
L.
Cui
,
H.-P.
Phan
,
D. V.
Dao
,
H.
Yonezawa
,
P. K.
Lam
,
E. H.
Huntington
, and
M.
Lobino
, “
Integrated photonic platform for quantum information with continuous variables
,”
Sci. Adv.
4
,
eaat9331
(
2018
).
15.
F.
Mondain
,
T.
Lunghi
,
A.
Zavatta
,
E.
Gouzien
,
F.
Doutre
,
M.
De Micheli
,
S.
Tanzilli
, and
V.
D’Auria
, “
Chip-based squeezing at a telecom wavelength
,”
Photonics Res.
7
,
A36
A39
(
2019
).
16.
C.
Reimer
,
L.
Caspani
,
M.
Clerici
,
M.
Ferrera
,
M.
Kues
,
M.
Peccianti
,
A.
Pasquazi
,
L.
Razzari
,
B. E.
Little
,
S. T.
Chu
,
D. J.
Moss
, and
R.
Morandotti
, “
Integrated frequency comb source of heralded single photons
,”
Opt. Express
22
,
6535
6546
(
2014
).
17.
R.
Cernansky
,
F.
Martini
, and
A.
Politi
, “
Complementary metal-oxide semiconductor compatible source of single photons at near-visible wavelengths
,”
Opt. Lett.
43
,
855
858
(
2018
).
18.
C.
Taballione
,
T. A. W.
Wolterink
,
J.
Lugani
,
A.
Eckstein
,
B. A.
Bell
,
R.
Grootjans
,
I.
Visscher
,
D.
Geskus
,
C. G. H.
Roeloffzen
,
J. J.
Renema
,
I. A.
Walmsley
,
P. W. H.
Pinkse
, and
K.-J.
Boller
, “
8 × 8 reconfigurable quantum photonic processor based on silicon nitride waveguides
,”
Opt. Express
27
,
26842
26857
(
2019
).
19.
A.
Gaggero
,
F.
Martini
,
F.
Mattioli
,
F.
Chiarello
,
R.
Cernansky
,
A.
Politi
, and
R.
Leoni
, “
Amplitude-multiplexed readout of single photon detectors based on superconducting nanowires
,”
Optica
6
(
6
) ,
823
828
(
2019
).
20.
A.
Dutt
,
K.
Luke
,
S.
Manipatruni
,
A. L.
Gaeta
,
P.
Nussenzveig
, and
M.
Lipson
, “
On-chip optical squeezing
,”
Phys. Rev. Appl.
3
,
044005
(
2015
).
21.
V. D.
Vaidya
,
B.
Morrison
,
L. G.
Helt
,
R.
Shahrokhshahi
,
D. H.
Mahler
,
M. J.
Collins
,
K.
Tan
,
J.
Lavoie
,
A.
Repingon
,
M.
Menotti
,
N.
Quesada
,
R. C.
Pooser
,
A. E.
Lita
,
T.
Gerrits
,
S. W.
Nam
, and
Z.
Vernon
, “
Broadband quadrature-squeezed vacuum and nonclassical photon number correlations from a nanophotonic device
,”
Sci. Adv.
6
(
39
) ,
eaba9186
(
2020
).
22.
Y.
Zhao
,
Y.
Okawachi
,
J. K.
Jang
,
X.
Ji
,
M.
Lipson
, and
A. L.
Gaeta
, “
Near-degenerate quadrature-squeezed vacuum generation on a silicon-nitride chip
,”
Phys. Rev. Lett.
124
,
193601
(
2020
).
23.
Y.
Zhang
,
M.
Menotti
,
K.
Tan
,
V.
Vaidya
,
D.
Mahler
,
L.
Zatti
,
M.
Liscidini
,
B.
Morrison
, and
Z.
Vernon
, “
Single-mode quadrature squeezing using dual-pump four-wave mixing in an integrated nanophotonic device
,” arXiv:2001.09474 [quant-ph] (
2020
).
24.
U. B.
Hoff
,
B. M.
Nielsen
, and
U. L.
Andersen
, “
Integrated source of broadband quadrature squeezed light
,”
Opt. Express
23
,
12013
12036
(
2015
).
25.
M.
Shirasaki
and
H. A.
Haus
, “
Squeezing of pulses in a nonlinear interferometer
,”
J. Opt. Soc. Am. B
7
,
30
34
(
1990
).
26.
K.
Bergman
and
H. A.
Haus
, “
Squeezing in fibers with optical pulses
,”
Opt. Lett.
16
,
663
665
(
1991
).
27.
M. H. P.
Pfeiffer
,
J.
Liu
,
M.
Geiselmann
, and
T. J.
Kippenberg
, “
Coupling ideality of integrated planar high-Q microresonators
,”
Phys. Rev. Appl.
7
,
024026
(
2017
).
28.
N. L.
Thomas
,
A.
Dhakal
,
A.
Raza
,
F.
Peyskens
, and
R.
Baets
, “
Impact of fundamental thermodynamic fluctuations on light propagating in photonic waveguides made of amorphous materials
,”
Optica
5
,
328
336
(
2018
).
29.
T. C.
Ralph
and
A. G.
White
, “
Retrieving squeezing from classically noisy light in second-harmonic generation
,”
J. Opt. Soc. Am. B
12
,
833
839
(
1995
).
30.
C. M.
Wilkes
,
X.
Qiang
,
J.
Wang
,
R.
Santagati
,
S.
Paesani
,
X.
Zhou
,
D. A. B.
Miller
,
G. D.
Marshall
,
M. G.
Thompson
, and
J. L.
O’Brien
, “
60 dB high-extinction auto-configured Mach–Zehnder interferometer
,”
Opt. Lett.
41
,
5318
5321
(
2016
).
31.
Y.
Xuan
,
Y.
Liu
,
L. T.
Varghese
,
A. J.
Metcalf
,
X.
Xue
,
P.-H.
Wang
,
K.
Han
,
J. A.
Jaramillo-Villegas
,
A.
Al Noman
,
C.
Wang
,
S.
Kim
,
M.
Teng
,
Y. J.
Lee
,
B.
Niu
,
L.
Fan
,
J.
Wang
,
D. E.
Leaird
,
A. M.
Weiner
, and
M.
Qi
, “
High-Q silicon nitride microresonators exhibiting low-power frequency comb initiation
,”
Optica
3
,
1171
1180
(
2016
).
32.
X.
Ji
,
F. A. S.
Barbosa
,
S. P.
Roberts
,
A.
Dutt
,
J.
Cardenas
,
Y.
Okawachi
,
A.
Bryant
,
A. L.
Gaeta
, and
M.
Lipson
, “
Ultra-low-loss on-chip resonators with sub-milliwatt parametric oscillation threshold
,”
Optica
4
,
619
624
(
2017
).
33.
R. N.
Alexander
,
S.
Yokoyama
,
A.
Furusawa
, and
N. C.
Menicucci
, “
Universal quantum computation with temporal-mode bilayer square lattices
,”
Phys. Rev. A
97
,
032302
(
2018
).

Supplementary Material