# Integrated Quantum Optical Phase Sensor

Hubert S. Stokowski1 Timothy P. McKenna2 Taewon Park1,3 Alexander Y. Hwang1
Devin J. Dean1 Oguz Tolga Celik1,3 Vahid Ansari1 Martin M. Fejer1 Amir H.
Safavi-Naeini1,4 1Department of Applied Physics and Ginzton Laboratory,
Stanford University, Stanford, California 94305, USA 2Physics & Informatics
Laboratories, NTT Research, Inc., Sunnyvale, California 94085, USA 3Department
of Electrical Engineering, Stanford University, Stanford, California 94305,
USA 4<EMAIL_ADDRESS>
###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

The quantum noise of light fundamentally limits optical phase sensors. A
semiclassical picture attributes this noise to the random arrival time of
photons from a coherent light source such as a laser. An engineered source of
squeezed states suppresses this noise and allows sensitivity beyond the
standard quantum limit (SQL) for phase detection. Advanced gravitational wave
detectors like LIGO have already incorporated such sources, and nascent
efforts in realizing quantum biological measurements have provided glimpses
into new capabilities emerging in quantum measurement. We need ways to
engineer and use quantum light within deployable quantum sensors that operate
outside the confines of a lab environment. Here we present a photonic
integrated circuit fabricated in thin-film lithium niobate that provides a
path to meet these requirements. We use the second-order nonlinearity to
produce a squeezed state at the same frequency as the pump light and realize
circuit control and sensing with electro-optics. Using a 26.2 milliwatts of
optical power, we measure (2.7 $\pm$ 0.2 )$\%$ squeezing and apply it to
increase the signal-to-noise ratio of phase measurement. We anticipate that
on-chip photonic systems like this, which operate with low power and integrate
all of the needed functionality on a single die, will open new opportunities
for quantum optical sensing.

Figure 1: Squeezing and quantum-enhanced measurement PIC.  a, Schematic of a
photonic integrated circuit for squeezing and quantum-enhanced measurement.
Light from an external, integrated laser is polarization-controlled and
coupled through a cleaved facet. The PIC splits light into two paths at the
tunable beam splitter: pump the squeezer and serve as the local oscillator.
The two beams are recombined at the signal interferometer to perform a
quantum-enhanced measurement of a small RF signal. The signal interferometer
also serves as a 50/50 beam splitter for squeezing characterization in BHD.
Light is detected off-chip with a balanced photoreceiver connected to the chip
with two lensed multimode fibers. Sum and difference signals provide feedback
for locking output splitting ratio and LO power level. b, Microscope picture
of a chip with eight PICs, the red rectangle highlights a single device, blue
outlines a single signal interferometer. A large-scale gold cross-pattern
covers the entire chip to prevent stray light from interfering with guided
modes. c, Microscope picture of an MZI serving as a signal interferometer. The
asymmetric design provides phase modulation to one of the arms of the
interferometer through gold electrodes routed to an RF probe. FPC – fiber
polarization controller, SHG – second harmonic generation, OPA – optical
parametric amplification, BHD – balanced homodyne detection, AWG – arbitrary
waveform generator.

Squeezed states of light exhibit fluctuations in their quadrature amplitude
below that of the quantum vacuum. Andersen  _et al._ (2016) This property
makes them indispensable resources for increasing the sensitivity of optical
measurements. In interferometry with classical light, signal-to-noise ratio
scales as $\sqrt{N}$ with the total number of detected photons $N$. As a
result, the imprecision in determining the optical phase, or phase
sensitivity, follows $1/\sqrt{N}$, referred to as the standard quantum limit
(SQL). To increase the SNR within the confines of the SQL, we must increase
$N$ – by using more optical power or extending the measurement time. These
approaches are often undesirable or impossible due to technical or fundamental
limitations. Intriguingly, the SQL can be overcome by injecting squeezed
states of light into the dark port of the interferometer Caves  _et al._
(1980). Ultimately, this approach may lead to a truly quantum-limited
Heisenberg scaling $1/N$ of sensitivity Caves  _et al._ (1980); Giovannetti
_et al._ (2004, 2006). Over the past forty years, impressive efforts have been
put into designing squeezed light sources and integrating them into the
gravitational wave detectors. These detectors now routinely obtain quantum-
enhanced sensitivity exceeding 2 dB Goda  _et al._ (2008); Aasi  _et al._
(2013). Similar ideas were also extended into other laboratory demonstrations,
e.g., in distributed phase sensing Guo  _et al._ (2020) and enhanced signal-
to-noise-ratio in biological sensing Taylor  _et al._ (2013) and microscopy
Casacio  _et al._ (2021).

We aim to demonstrate an _integrated_ quantum optical sensor that generates
and uses squeezed light to perform measurements beyond the SQL. Our sensor
moves much of the functionality often implemented on the optical table – e.g.,
generating pump light, locking phases, and implementing an interferometer,
onto the chip itself. We use thin-film lithium niobate (LN) Zhang  _et al._
(2017) to achieve all of the functions in a monolithic circuit. In the last
few years, integrated thin-film LN devices have demonstrated efficient light
generation covering wavelengths from mid-infrared to blue Wang  _et al._
(2018a); Lu  _et al._ (2020); Mishra  _et al._ (2021); Park  _et al._ (2022),
optical parametric oscillators Lu  _et al._ (2021); McKenna  _et al._ (2022),
dispersion engineering Jankowski  _et al._ (2020, 2021); Mishra  _et al._
(2022), EO modulation, tuning Wang  _et al._ (2018b); Celik  _et al._ (2022),
and frequency combs Zhang  _et al._ (2019). This article combines nearly all
of the aforementioned capabilities in a single LN chip and demonstrates an
integrated quantum sensor. We leverage the strongest electro-optic coefficient
of LN ($r_{33}=31~{}\text{pm/V}$) Boyd (2008) to actively control the power
and phase of light within a photonic integrated circuit (PIC) and interface it
with periodically poled LN (PPLN) waveguides that support efficient nonlinear
processes. The strong second-order optical nonlinear coefficient $\chi^{(2)}$,
accompanied by quasi-phase-matching in PPLN, allows us to achieve efficient
second harmonic generation (SHG) and squeezed light generation through optical
parametric amplification (OPA).

In the degenerate parametric amplification process, photons are generated in
pairs with a fixed phase relationship through the interaction
$\hat{H}_{\text{I}}\propto(\hat{a}^{2}+\hat{a}^{\dagger 2})$. The resulting
state of the light is squeezed. Following the first demonstration of a
squeezed light source by parametric down-conversion (PDC) in an optical cavity
Wu  _et al._ (1986), numerous other approaches, including single-pass PDC
Kaiser  _et al._ (2016); Kashiwazaki  _et al._ (2020), semiconductor lasers,
optical fibers, atomic ensembles Andersen  _et al._ (2016), and ponderomotive
squeezing with mechanical oscillators Safavi-Naeini  _et al._ (2013) have
successfully demonstrated varied amounts of squeezed light. Optical materials
with intrinsic second- or third-order nonlinearities may be the most versatile
integrated sensors approach. Recent demonstrations of integrated photonic
circuits in silicon nitride Vaidya  _et al._ (2020); Zhang  _et al._ (2021)
and thin-film LN Dutt  _et al._ (2015); Chen  _et al._ (2022); Nehra  _et al._
(2022) have shown that low power and scalable sources of quantum light are
possible.

From a photonic system perspective, the squeezed source is only one part of a
larger optical sensor. Combining emerging squeezed light sources with more
complex PICs is a promising avenue for developing deployable optical sensors
with quantum-enhanced sensitivity. So far, achieving sub-shot-noise
sensitivity in optical metrology has required complex setups with multiple
modulators, lasers, and optical cavities. This has limited the domain of
quantum optical sensing to experiments with high complexity Goda  _et al._
(2008); Aasi  _et al._ (2013); Casacio  _et al._ (2021) that can address only
a few of the possible application spaces – even for experiments that use new
integrated quantum light sources. Our work combines emerging squeezed light
sources with a complex PIC to realize a nearly complete on-chip quantum
sensor, thus opening a promising avenue for deployable optical sensors with
quantum-enhanced sensitivity.

##  I Results

In this work, we use the X-cut thin-film of lithium niobate on insulator
(LNOI) platform to build a sensor with an integrated source of squeezed light
for quantum-enhanced measurement. Fig. 1a outlines a scheme of the PIC
implemented in our chip with three major components – input tunable beam
splitter, squeezer, and signal interferometer. We pattern eight such PICs in a
single chip, which we show on a microscope picture in Fig. 1b. We design the
squeezer to generate a sub-shot noise state of light at the same frequency as
the pump light, enabling interference with the local oscillator (LO) extracted
from the original beam. Active EO circuitry controls interferometers and the
LO phase, which enables using the system for both squeezing characterization
and quantum measurement. Using lensed SMF-28 fiber, we interface our device
with an off-the-shelf DBR laser and operate the system in CW mode at 1544 nm.
The output ports of the beamsplitter out-couple light into two lensed
multimode fibers, which we connect to a balanced photoreceiver off-chip. Both
the laser de Beeck  _et al._ (2021); Li  _et al._ (2022) and detectors Tasker
_et al._ (2021) can be further integrated with the PIC to increase the system
efficiency and improve the squeezing visibility.

###  I.1 Device Design

The squeezer subsystem comprises two PPLN waveguides and a series of
directional couplers for spectral filtering. FH light first enters a PPLN
waveguide designed to phasematch a second harmonic generation process using
periodic poling with a period of around 3.7 µm. Tight mode confinement and
strong second-order optical nonlinear coefficient of LNOI yield highly
efficient SHG. Next, we filter out the the residual FH by passing the light
through three directional couplers. By design, each directional coupler
transfers majority of FH while keeping the second harmonic (SH) light in the
original waveguide; hence the structure acts as a dichroic beamsplitter. The
SH light then loops back to another PPLN waveguide, realizing a phase-
sensitive OPA at the FH frequency, squeezing one quadrature and anti-squeezing
the other. The amount of squeezing generated in this structure is given by:

$\displaystyle\cfrac{\langle\delta\hat{\mathrm{X}}^{2}\rangle}{\langle\delta\hat{\mathrm{X}}_{\text{vac}}^{2}\rangle}=\text{exp}\Bigg{(}{-2L\,\sqrt{\eta
P_{\text{in}}}\,\text{tanh}\Big{(}L\,\sqrt{\eta\,P_{\text{in}}}\Big{)}}\Bigg{)},$
(1)

where $\langle\delta\hat{\mathrm{X}}^{2}\rangle$ is the variance of the
quadrature squeezed with respect to vacuum
$\langle\delta\hat{\mathrm{X}}_{\text{vac}}^{2}\rangle$, $L$ is the length of
the poled waveguides, $\eta$ is the normalized nonlinear conversion efficiency
Wang  _et al._ (2018a), and $P_{\text{in}}$ is the telecom pump power (see
Methods for details). Finally, the SH light is filtered out using the same
directional coupler design, and the squeezed state propagates to the signal
interferometer at the output of the PIC. We route the light filtered out after
the SHG and OPA sections to the chip facet and use them as monitor ports.

Active EO circuitry controls optical power splitting and the LO phase within
the PIC. The _tunable beam splitter_ at the input and the _signal
interferometer_ at the output are both intensity modulators. Both consist of a
Mach-Zehnder interferometer (MZI) with a phase modulator in one arm. The
former is typically DC-biased at phase $\phi_{1}$ either close to zero, to
direct most of the input light into the squeezer, or set to $\pi$, to send
most of the light into the LO and characterize shot noise. The latter MZI,
shown in Fig 1c, is always locked with a DC bias to make the power at the two
outputs equal for a balanced homodyne detection (BHD) measurement. We use the
same phase modulator design for quadrature selection in the squeezer path
through $\phi_{\text{LO}}$ control. In addition to DC bias, we apply RF tones
to the LO phase modulator and the signal interferometer. The RF phase
modulation on the LO serves as voltage-to-phase calibration and quantifies the
light leakage in the squeezer
$\sqrt{\varepsilon}~{}=~{}\alpha_{\text{LO}}/\alpha_{\text{Leakage}}$, while
the RF phase modulation imposed on the signal interferometer forms the signal
we measure.

Figure 2:  PIC subsystems characterization.  a, Shot noise power spectral
density for different LO powers on the chip. We characterize our setup to find
LO power yielding signals about 10 dB above the electronic noise floor. b,
Integrated noise from Fig. 2a vs. LO power follows a perfect linear trend,
proving that the system is limited by shot noise at the target LO power of 7.8
mW. c, Example SHG phasematching curve, deviation from ideal $\text{sinc}^{2}$
shape results from small geometry variation along the waveguide length. d,
Measured on-chip SHG peak power vs. FH pump power. Linear fit yields
normalized efficiency of the nonlinear process of about
$1,000\%/(\mathrm{W\cdot cm}^{2})$. The inset shows SHG microscope image of
high-quality periodic poling of the thin-film lithium niobate. We pattern
waveguides in the area where poling extends through the entire thickness of
the LN film. Figure 3:  Squeezing and quantum-enhanced detection.  a, Power
spectral density for maximum squeezing (orange line) and maximum anti-
squeezing (blue line), collected while pumping the squeezer with 18.4 mW of
FH. The black line provides a shot noise measurement reference. We highlight
two RF peaks resulting from modulation of the LO phase shifter (blue
highlighted area) and signal interferometer (orange highlighted area). b,
Local oscillator calibration peak power vs. DC bias voltage of the phase
shifter. The solid line indicates the fit line, from which we find
$V_{\pi}=28.6\text{ V}$ and light leakage into squeezing path
$\varepsilon\approx 4.0\%$. c, Integrated noise power as a function of the LO
phase (calibrated in Fig. 3b). The grey-dashed area represents the
characteristic sub-shot noise behavior of a squeezed state. Misalignment of
the squeezed state and LO phases is caused by the path length difference
between the two paths. The green highlighted area represents the model,
including squeezer leakage, detection efficiency, and normalized efficiency.
The uncertainty of the normalized efficiency and phase sets the model bounds.
d, Quantum-enhanced measurement scheme – squeezed state feeds into a dark port
of the MZI along with the LO entering the bright port. Phase modulation
applied in one of the arms can be detected with enhanced SNR compared to a
classical measurement. e, Measurement of a small RF tone with noise floor
suppressed with squeezing. We align the LO phase with maximum squeezing by DC
tuning and measure the RF power spectrum through a BHD. For a $-83$ dBm
modulation, we observe about (4 $\pm$ 1)$\%$ improvement of the SNR compared
to a shot-noise limited measurement.

###  I.2 Subsystem Characterization

We calibrate the shot noise level by routing all the optical power to the
local oscillator path. For this purpose, we bias the tunable beam splitter at
the input with a DC voltage bias of 47 V. We achieve this by minimizing the
amount of light in the squeezer’s fundamental harmonic (FH) monitor port. In
this configuration, we use the local oscillator to probe the vacuum state sent
to the opposing port of the signal interferometer. For the BHD measurement, we
lock the signal interferometer to a splitting ratio of 50:50. We achieve this
by controlling the difference between the intensity of two outputs and locking
its DC part to zero. Next, we step the power of the input laser using an
external variable optical attenuator (VOA), such that the power in the local
oscillator waveguide varies between $0$ and $8.8~{}\text{mW}$ and record the
RF spectra of the balanced homodyne detector, as shown in Fig. 2a. Finally, we
integrate the noise power and plot it against the LO power in Fig. 2b,
observing a linear dependence as expected from a shot noise-limited
measurement. The mean squared error of the linear fit defines our uncertainty
of the shot noise measurement. Based on the shot noise calibration, we choose
LO power for the squeezing experiments at around $7.8~{}\text{mW}$, where the
optical shot noise dominates our detection. We highlight this operating point
in Fig. 2b.

The squeezer relies on the performance of the periodically poled waveguides in
the SHG and OPA sections. We characterize our periodic poling by measuring the
quasi-phase-matching transfer function of a diagnostic waveguide patterned
adjacent to our SHG and OPA segments such that it passes through the same
poling region (see Methods for measurement details). An example QPM transfer
function is shown in Fig. 2c. This measurement gives us the SHG response
decoupled from the rest of the PIC components. Note that the diagnostic
waveguide is 8.1 mm long, whereas the one in the primary device is 10 mm long.
Given the significant length of the fabricated waveguide, nonuniformities
distort the ideal $\mathrm{sinc}^{2}$ shape of the nonlinear transfer function
Kuo (2021) and reduce the peak nonlinear efficiency. We measure the intensity
of generated SH light as a function of pump power controlled by external VOA.
In Fig. 2d, we plot data and linear fit to extract a maximum normalized
efficiency of about $1,000~{}\%/(\mathrm{W\cdot cm}^{2})$, lower than the
simulated $4,000~{}\%/(\mathrm{W\cdot cm}^{2})$ for an ideal poled structure.
We use this number further to estimate the amount of parametric amplification
and the expected squeezing level. Because of the nonuniformities, the
quadratic scaling of the efficiency with length is not exact. However, we
neglect this because the length difference between the test and device
waveguides is small. The inset of Fig. 2d shows a second harmonic microscope
image of periodic poling used in the fabrication process. Black stripes on the
sides of the picture are metal finger electrodes, and light grey shapes
stretching between them correspond to inverted crystal domains.

###  I.3 Squeezing and Quantum-Enhanced Sensing

We characterize squeezing by directing most of the laser light into the
squeezer and holding the LO power constant at around $7.8~{}\text{mW}$. In
this case, we drive the device with $125~{}\text{mW}$ of optical power off-
chip, of which $18.4~{}\text{mW}$ drives the SH generation directly. Given the
calibrated normalized efficiency, we estimate SH power at around
$3.4~{}\text{mW}$ to generate about (1.5 $\pm$ 0.1) dB of squeezing on the
chip. We perform a BHD measurement to verify this by beating the squeezed
state with the local oscillator at the signal interferometer. We lock the
phase shift within the output interferometer such that the DC part of the
difference signal is zero. At the same time, we lock the input beam splitter
to keep the sum of photodiode readouts at a constant value. We follow this
locking scheme to work around the slow phase drift observed in EO devices Zhu
_et al._ (2021); Wang  _et al._ (2022). A simple procedure that periodically
updates the voltage at the input beam splitter and signal interferometer at a
low frequency (around 1 Hz) is sufficient. Finally, we measure the power
spectral density, as shown in Fig. 3a, and reference it to the shot noise
measurement (black line). We perform this measurement as a function of the LO
phase, controlled through a DC voltage applied to a phase shifter in the LO
path. Blue and orange traces in Fig. 3a correspond to the maximum anti-
squeezing and maximum squeezing, respectively. In addition to probing
broadband noise, we introduce two RF signals corresponding to peaks at 72 MHz
and 60 MHz. The former originates from an RF signal applied to the LO phase
shifter, which we highlight in blue, and the latter corresponds to a tone on
the signal interferometer, which we highlight in orange.

We verify the phase-sensitive nature of the generated squeezed light by
stepping the LO phase shifter bias. Due to a small power leakage of the FH in
the squeezer path, $\varepsilon$, we can observe the RF peak introduced at the
LO phase shifter at 72 MHz. This signal is proportional to
$\varepsilon\,\text{sin}^{2}(V_{DC}/V_{\pi}\cdot\pi)$ (see Methods for
details). By fitting the peak power as a function of LO bias voltage in Fig.
3b, we establish the power leakage ratio $\varepsilon$ at around (4.0 $\pm$
1.0)$\%$, which we use to estimate the filter extinction ratio at around 18
dB. This amount of leakage into the squeezer enables our LO phase calibration
procedure, but negatively impacts the squeezing visibility. Using the fit, we
also find the half-wave voltage of our LO to be 28.6 $\pm$ 0.3 V and convert
the DC bias to phase. The half-wave voltage agrees with our design (see
Methods for details). In the squeezing measurement, in Fig. 3c, we plot the
integrated noise floor over the measurement bandwidth vs. calibrated LO phase,
excluding the two RF peaks and a small, broad feature between 67 and 72 MHz.
Plotted noise is normalized to shot noise, in which uncertainty is smaller
than the width of the line. We see expected periodic behavior, where the
minimum and maximum correspond to probing squeezed and anti-squeezed field
quadratures, respectively. Instead of performing fast phase sweeps, we
maintain the LO phase at each tuning point for roughly 13 seconds and obtain
the full RF spectrum. This highlights the stability of our system, a necessary
ingredient for a stable and metrologically useful source of squeezed light. We
measure (2.7 $\pm$ 0.2) $\%$ of squeezing and (12.3 $\pm$ 0.2) $\%$ of anti-
squeezing. Limited squeezing visibility and squeezing/anti-squeezing asymmetry
result from the detection chain efficiency of 20$\%$ (see Methods for more
details) and squeezer leakage $\varepsilon\approx 4\%$. Accounting for these
factors, we expect the level of squeezing on-chip to be around (1.5 $\pm$ 0.1)
dB, which agrees with our estimation based on the normalized efficiency. Total
detection efficiency can improve up to 83$\%$ by reducing optical loss.
Implementing straightforward PIC layout improvements to reduce the propagation
loss, improved fiber-to-chip coupling He  _et al._ (2019); Hu  _et al._
(2021); Ying  _et al._ (2021), and higher quantum efficiency photodiodes can
result in an achievable 4.5 dB squeezing or SNR improvement by 280$\%$ using
the same off-the-shelf integrated laser. See the Methods section for more
details.

Finally, we use the signal interferometer to perform a quantum-enhanced
measurement of an RF signal. In this case, the chip configuration is the same
as in the squeezing measurement; most light pumps the squeezer, and the LO
power is fixed at $7.8~{}\text{mW}$. We use the signal interferometer to
perform a quantum-enhanced measurement by injecting squeezed state into its
dark port, as shown in Fig. 3d. Based on our squeezing calibration, we set the
LO phase to align with the maximum observed squeezing and perform the
measurement. For that purpose, we apply a small RF tone with an amplitude of
-83 dBm, corresponding to the root-mean-square voltage of 15.8 µV at 60 MHz.
This signal can be detected in the BHD measurement, as shown in Fig. 3e, and
it is not sensitive to the LO phase. As a result, the measurement noise floor
is reduced with respect to the shot noise and results in about (4 $\pm$ 1)$\%$
SNR improvement. This data was taken at a different time than the LO phase
sweep dataset shown in Fig 3c, and so we attribute the slightly higher level
of observed squeezing to slightly different experimental conditions.

##  II Discussion

Complex quantum photonic integrated circuits, like the one presented in this
work, extend the reach of quantum technologies into new domains. The
integration allows us to generate quantum light and leverage its properties
within the same circuit, enabling more stable and lower power operation. These
critical advantages point toward new classes of mobile and deployable sensors
that are natively “quantum-compatible”. In the work we present here, we
benchmark a sensor by detecting voltage signals imparted as phase upon an
optical field. Many other interferometric measurements are today limited by
shot noise. These measurements, including refractive index sensing,
optomechanical force, acceleration Krause  _et al._ (2012), and mass sensing
Hanay  _et al._ (2012), will directly benefit from integrating the quantum
apparatus, as demonstrated here. Moreover, squeezing and squeezed light are
crucial elements of numerous quantum sensing, communicationsGehring  _et al._
(2015), and computing protocols Zhong  _et al._ (2020, 2021). Quantum PICs,
such as those shown here, are promising and scalable approaches for building
these systems.

##  III Data Availability

The data sets generated during and/or analyzed during this study are available
from the corresponding author on request.

## Author contributions

H.S.S. designed the device. H.S.S., T.P., and A.Y.H. fabricated the device.
H.S.S., V.A., T.P.M., and O.T.C. developed the fabrication process. M.M.F. and
A.H.S.-N. provided experimental and theoretical support. H.S.S., V.A., T.P.,
and D.J.D. performed the experiments and analyzed the data. H.S.S., V.A., and
A.H.S.-N. wrote the manuscript. H.S.S., V.A., T.P.M, and A.H.S.-N. conceived
the experiment, and A.H.S.-N. supervised all efforts.

## Competing interests

The authors declare no competing interests.

## Acknowledgements

H.S. and V.A. thank Kevin Multani, Christopher Sarabalis, and Michael Stefszky
for discussions and technical support. The authors wish to thank NTT Research
for financial and technical support. This work was supported by the DARPA
Young Faculty Award (YFA). Part of this work was performed at the Stanford
Nano Shared Facilities (SNSF) and Stanford Nanofabrication Facility (SNF),
supported by the National Science Foundation under award ECCS-2026822. H.S.
acknowledges support from the Urbanek Family Fellowship. V.A. acknowledges
support from Stanford Q-FARM Bloch Fellowship. D.D. acknowledges support from
the NSF GRFP (No. DGE-1656518).

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## Methods

## Fabrication

We fabricate our device with X-cut thin-film LNOI (lithium niobate on
insulator, NanoLN), following the procedure outlined in figure 4. The material
stack consists of 500 nm of lithium niobate bonded to a 2 µm-thick silica
layer on top of a silicon handle wafer, as shown in Fig. 4a. Fabrication
starts with poling the thin-film. In the first step (Fig. 4b), we pattern Cr
electrodes on an insulating layer of 100 nm SiO2 using electron beam
lithography (JEOL 6300-FS, 100-kV) and liftoff process. Poling period is
around 3.7 µm, and we design it for phasematching between our waveguide’s 1550
nm and 775 nm modes. Next, we apply high voltage pulses to flip crystal
domains Nagy and Reano (2019); McKenna  _et al._ (2022). After poling, we
remove the electrodes with chromium etchant and buffered oxide etchant (Fig.
4b).

The photonic circuit is patterned with a FOX-16 mask and electron beam
lithography and transferred to the LN layer with an argon ion mill (Fig. 4d).
The etch depth is 300 nm, and the waveguide width is 1.2 µm.

Metal electrodes for EO tuning are patterned with the same process as the
poling electrodes but made out of 100 nm gold (Fig. 4e). The bottom layer of
electrodes and waveguides are covered with SiO2 deposited with a high-density
plasma process (Fig. 4f) Shams-Ansari  _et al._ (2022). To provide electrical
contact to the buried electrodes, we pattern vias by standard SiO2 etch with
fluorine chemistry and photolithography (Fig. 4g). Finally, we pattern a top
metal layer with the electron beam. This layer provides access to the buried
electrodes with an external probe and consists of 200 nm of gold and a 10 nm
chromium adhesion layer (Fig. 4h).

In the final step, we prepare the chip facets for light coupling by stealth
dicing with a DISCO DFL7340 laser saw Jankowski  _et al._ (2020). High-energy
optical pulses are focused on the substrate to create an array of damage
sites. They act as nucleation sites for crack propagation, resulting in a
uniform and smooth cleave.

## Electro-Optic simulation

We design the geometry of our electro-optic devices to provide phase-tuning
functionality without introducing excess loss to the quantum state of light.
We model our system using a finite-element mode solver (COMSOL). We first
define the electric-field-dependent refractive index Boyd (2008) as:

$\displaystyle
n_{\text{o}}^{\prime}=n_{\text{o}}-\frac{1}{2}r_{13}n_{\text{o}}^{3}E_{\text{z}}$
(2) $\displaystyle
n_{\text{e}}^{\prime}=n_{\text{o}}-\frac{1}{2}r_{33}n_{\text{e}}^{3}E_{\text{z}}.$
(3)

The elements of the electro-optic tensor $r_{33}=31~{}\text{pm/V}$ and
$r_{13}=10~{}\text{pm/V}$ modify ordinary and extraordinary indices of the
crystal as a result of the static field. We solve for the static electric
field and follow with eigenmode analysis in the system with modified
refractive index, as shown in Fig. 5a.

We find the half-wave-voltage-length product from the relationship between
effective index and applied voltage $V_{\pi}\,L=\lambda/({\partial
n_{\text{eff}}/\partial{V}})$. We plot the expected values along with the
propagation loss from metal proximity in Fig. 5b. In the device described in
the main text, we use a waveguide-electrode gap of 1.6 µm and an electrode
length of 2.5 mm. We expect the half-wave voltage in this geometry to be
around 30 V, which agrees with the value measured in the experiment. This gap
size allows us to keep the propagation loss induced by the metal below 0.05
dB/cm, which is negligible compared to the measured loss resulting from
fabrication imperfections around 0.7 dB/cm.

## Experimental Setup

We characterize fabricated PICs in two different setup configurations, one for
BHD and one for SHG characterization. In both cases, the chip temperature is
controlled with a thermistor (Thorlabs TH10K) and thermo-electric cooler
(Thorlabs TECF2S, MTD415TE, MTDEVAL1). We operate our device at 45 ∘C to
maximize SHG intensity at the low noise laser operating wavelength.

The setup for BHD measurements is shown in Fig. 1a. For squeezing and quantum-
enhanced measurements, we use an ultra-low noise, integrated DFB laser with an
emission wavelength of around 1544 nm (Thorlabs ULN15PC). The laser is
followed by a high extinction ratio fiber isolator (Thorlabs IOT-G-1550A) and
a fiber polarization controller. We introduce an additional variable optical
attenuator (HP 8156A) for the shot noise measurement to control the laser
power. Then, we couple light to the chip with a lensed SMF-28 fiber (OZ
Optics). We collect light into a balanced photoreceiver (Thorlabs PDB425C with
RF output conversion gain $1.25\cdot 10^{5}$ V/W) using multimode fibers
lensed (in-house) and mounted on a v-groove chip with a pitch matching the PIC
output waveguide separation. We collect RF spectra with an electronic spectrum
analyzer (Rohde $\&$ Schwarz FSW26) with a resolution bandwidth set to 60 kHz.
Tap ports for FH and SH light filtered within the squeezer subsystem are
collected into another lensed multimode fiber, separated by a combination of a
50/50 splitter and optical edgepass filters, and monitored with photodiodes:
Newport 1623 Nanosecond Photodetector for FH and Thorlabs APD440A for SH.
Electro-optic components on the chip are controlled with DC voltage supplied
with three TTi PLH250-P sources. LO phase modulator and the signal
interferometer receive RF signal from an arbitrary waveform generator (Rigol
DG4102). We mix RF signals and DC bias with bias-tees (ZFBT-4R2G+) and connect
to the chip with a triple ground-signal-ground (GSG) probe. We estimate the
total insertion loss of RF input at $1.22$ dB by combining the loss of two SMA
cables (FLC-4FT-SMSM+), DC block (Inmet 8039), and the bias-tee.

For the second harmonic characterization, we reconfigure the setup to work
with a continuously tunable c-band laser (Santec TSL-550, 1480–1630 nm)
passing through a variable optical attenuator (HP 8156A). Next, we tap off
5$\%$ of the light for power calibration using a power meter (Newport 918D-IR-
OD3R) and pass the light through a polarization controller. Light couples to
the chip through an SMF-28 lensed fiber and is collected by the same kind of
fiber at the output facet. Then, we outcouple light into free space and split
with a 1000 nm short-pass dichroic mirror (Thorlabs DMSP1000) and filter to
avoid cross-talk. Finally, we measure light intensity with avalanche
photodiodes (Thorlabs APD410A and Thorlabs APD410). We use variable optical
attenuators before APDs to prevent saturation and increase the dynamic range
of our measurement (HP 8156A and Thorlabs FW102C).

## Squeezer Performance

The integrated squeezer proposed in the main text takes advantage of the
second-order nonlinearity in LN waveguides. In this system, both SHG and OPA
can be described with coupled-mode equations for the degenerate three-wave
mixing process: Boyd (2008)

$\displaystyle\cfrac{\partial}{\partial z}A(z)$ $\displaystyle=$
$\displaystyle-i\sqrt{\eta}\,B(z)A^{\ast}(z)\,e^{-i\Delta kz},$ (4)
$\displaystyle\cfrac{\partial}{\partial z}B(z)$ $\displaystyle=$
$\displaystyle-i\sqrt{\eta}\,A^{2}(z)\,e^{i\Delta kz}.$ (5)

where $A(z)$ and $B(z)$ are amplitudes of the FH and SH modes with frequency
relationship $2\omega_{A}=\omega_{B}$. $\eta$ corresponds to normalized
efficiency, $\Delta k=2k_{A}-k_{B}$ is the phase mismatch between FH and SH
modes, which we compensate for with periodic poling. We solve these equations
to obtain the magnitude of the power normalized amplitude of the generated
SHG, assuming perfect phase matching:

$\displaystyle|B(z)|$ $\displaystyle=$
$\displaystyle|A(0)|\text{tanh}\Big{(}\sqrt{\eta}\,|A(0)|z\Big{)}.$ (6)

In the squeezer subsystem, FH light is filtered out after pumping the SHG, and
the SH stays in the waveguide to generate squeezing through OPA. To find the
amount of squeezing generated, we will first define field quadratures for the
FH mode as follows:

$\displaystyle\mathrm{X}$ $\displaystyle=$
$\displaystyle\cfrac{{A}+{A}^{\ast}}{2},$ (7) $\displaystyle\mathrm{Y}$
$\displaystyle=$ $\displaystyle i\,\cfrac{{A}-{A}^{\ast}}{2}.$ (8)

Next, we can use the equation 4 to find the evolution of the field quadratures
$\mathrm{X}$ and $\mathrm{Y}$ along the waveguide length as follows:

$\displaystyle\partial_{z}\Big{(}A(z)+A^{\ast}(z)\Big{)}$ $\displaystyle=$
$\displaystyle-i\sqrt{\eta}\Big{(}B(z)A(z)^{\ast}-B(z)^{\ast}A(z)\Big{)}$ (9)
$\displaystyle=$
$\displaystyle-\sqrt{\eta}|B(z)|\Big{(}A(z)+A(z)^{\ast}\Big{)}.$

In the last line we have chosen to set the phase of $B(z)$ to $-\pi/2$.
Performing a similar calculation for the time evolution of the $\mathrm{Y}$
quadrature, we arrive at:

$\displaystyle\partial_{z}\mathrm{X}$ $\displaystyle=$
$\displaystyle-\sqrt{\eta}|B(z)|\mathrm{X},$ (10)
$\displaystyle\partial_{z}\mathrm{Y}$ $\displaystyle=$
$\displaystyle\sqrt{\eta}|B(z)|\mathrm{Y}.$ (11)

Solving these equations yields:

$\displaystyle\mathrm{X}$ $\displaystyle=$
$\displaystyle\mathrm{X}(0)\,\text{exp}({-\sqrt{\eta}|B(z)|z}),$ (12)
$\displaystyle\mathrm{Y}$ $\displaystyle=$
$\displaystyle\mathrm{Y}(0)\,\text{exp}({\sqrt{\eta}|B(z)|z}).$ (13)

It is clear that the $\mathrm{X}$ quadrature is deamplified while $\mathrm{Y}$
quadrature is amplified. This amplification and deamplification process, when
applied to fields that have vacuum fluctuations, leads to anti-squeezing and
squeezing of the quadrature fluctuations. In other words, the variance for
quadrature operators under the action of amplification/deamplification is
given by:

$\displaystyle\langle\delta\hat{\mathrm{X}}^{2}\rangle$ $\displaystyle=$
$\displaystyle\text{exp}({-2\,\sqrt{\eta}\,|B(z)|\,z})\,\langle\delta\hat{\mathrm{X}}_{0}^{2}\rangle,$
(14) $\displaystyle\langle\delta\hat{\mathrm{Y}}^{2}\rangle$ $\displaystyle=$
$\displaystyle\text{exp}({2\,\sqrt{\eta}\,|B(z)|\,z})\,\langle\delta\hat{\mathrm{Y}}_{0}^{2}\rangle,$
(15)

where $\langle\delta\hat{\mathrm{X}}_{0}^{2}\rangle$ and
$\langle\delta\hat{\mathrm{Y}}_{0}^{2}\rangle$ are variances of the quantum
vacuum state. In the proposed PPLN squeezer subsystem of length $L$, combining
equations 6 and 14 defines squeezing as:

$\displaystyle\cfrac{\langle\delta\hat{\mathrm{X}}^{2}\rangle}{\langle\delta\hat{\mathrm{X}}_{\text{vac}}^{2}\rangle}=\text{exp}\Bigg{(}{-2L\,\sqrt{\eta
P_{\text{in}}}\,\text{tanh}\Big{(}L\,\sqrt{\eta\,P_{\text{in}}}\Big{)}}\Bigg{)}.$
(16)

Loss source | Loss (dB) | Determination
---|---|---
Waveguide propagation | 0.80 |  $Q$-factor measurement in a ring resonator on the same chip
Lensed multimode fiber collection | 5.40 | Coupling test to a straight waveguide
Detector quantum efficiency | 0.80 | Photodetector documentation
Table 1:  Summary of the sources of loss in the detection chain.

## Coherent leakage into the Squeezer and LO phase calibration

In our system, we observe non-perfect filtering of the squeezer subsection
pump at the fundamental frequency. This results from the limited extinction
ratio of the filter used on the chip to separate FH from SH after the SHG
section. This section explains how we quantify this leakage and use it to
calibrate the local oscillator phase.

Our model consists of a tunable beam splitter with two inputs (LO and squeezer
output) and two outputs (BHD ports). We calculate a result of a phase
modulation imposed on the LO arm on the BHD measurement result, depending on
the leakage level. The input state of the system is:

$\displaystyle\begin{bmatrix}A_{in}^{(1)}\\\
A_{in}^{(2)}\end{bmatrix}=\begin{bmatrix}\sqrt{\varepsilon}\alpha_{\text{LO}}(t)+\hat{a}\\\
\alpha_{\text{LO}}(t)e^{i\phi_{\text{LO}}}e^{i\phi_{\text{M}}(t)}+\hat{v}\end{bmatrix}.$
(17)

Here, $\varepsilon$ is the power ratio of the coherent leakage to the LO,
$\phi_{\text{LO}}$ is the LO phase with respect to the signal in the squeezer
path, $\phi_{\text{M}}$ is the RF phase modulation, $\hat{a}$ and $\hat{v}$
represent quantum fluctuations in the squeezer and LO paths, respectively. We
solve the beam splitter matrix equation:

$\displaystyle\begin{bmatrix}A_{out}^{(1)}\\\
A_{out}^{(2)}\end{bmatrix}=\begin{bmatrix}\text{sin}(\phi_{2}/2)&\text{cos}(\phi_{2}/2)\\\
\text{cos}(\phi_{2}/2)&-\text{sin}(\phi_{2}/2)\end{bmatrix}\begin{bmatrix}A_{in}^{(1)}\\\
A_{in}^{(2)}\end{bmatrix}$ (18)

to find the BHD signal generated. This model is equivalent to the signal
interferometer described in the main text. For no power leakage, it is biased
such that the phase is around $\phi_{2}=\pi/2$ but $\varepsilon\neq 0$ results
in an imbalance that we have to compensate for.

We solve for the output port amplitudes and use them to find the BHD
differential signal defined as
$P_{\text{BH}}=|A_{out}^{(1)}|^{2}-|A_{out}^{(2)}|^{2}$. The result is given
by:

$\displaystyle P_{\text{BH}}$ $\displaystyle=$
$\displaystyle\text{cos}(\phi_{2})|\alpha_{\text{LO}}|^{2}(1-\varepsilon)+2\text{sin}(\phi_{2})\sqrt{\varepsilon}|\alpha_{\text{LO}}|^{2}\text{cos}(\phi_{\text{LO}})$
(19) $\displaystyle+$
$\displaystyle\text{cos}(\phi_{2})[\alpha_{\text{LO}}e^{i\phi_{\text{LO}}}\hat{v}^{\dagger}-\sqrt{\varepsilon}\alpha_{\text{LO}}\hat{a}^{\dagger}+{h.c}]$
$\displaystyle+$
$\displaystyle\text{sin}(\phi_{2})[\alpha_{\text{LO}}e^{i\phi_{\text{LO}}}\hat{a}^{\dagger}+\sqrt{\varepsilon}\alpha_{\text{LO}}\hat{v}^{\dagger}+{h.c}]$

We separate equation 19 into DC and AC parts. The former defines the locking
condition:

$\displaystyle\text{cos}(\phi_{2})(1-\varepsilon)+2\text{sin}(\phi_{2})\sqrt{\varepsilon}\text{cos}(\phi_{\text{LO}})=0,$
(20)

which is solvable analytically:

$\displaystyle\phi_{2}=\pm\text{arccos}\left[\pm\cfrac{2\sqrt{\varepsilon}\text{cos}(\phi_{\text{LO}})}{\sqrt{4\varepsilon\text{cos}^{2}(\phi_{\text{LO}})+\varepsilon^{2}-2\varepsilon+1}}\right].$
(21)

In addition to locking, we use the DC term to calibrate the LO phase and
directly measure the leakage $\varepsilon$. For this purpose, we introduce a
periodic modulation to the LO phase shift, such that
$\phi_{\text{LO}}=\phi_{\text{LO}}+\phi_{\text{M}}(t)=\phi_{\text{LO}}+\pi\,V_{\text{p-p}}/(2V_{\pi})\,\text{sin}(\Omega
t)$. $\Omega$ is the modulation frequency (here set to 72 MHz), $V_{\pi}$ is
the half-wave voltage, and $V_{\text{p-p}}$ is the peak-to-peak modulation
voltage. We focus on the DC part of the equation 19, apply a standard Taylor
series expansion to the phase modulation term, and write the resulting signal
in the frequency domain at the peak frequency of the modulation:

$\displaystyle P_{\text{RF}}(\Omega)=\cfrac{2\varepsilon
R^{2}P_{\text{LO}}^{2}}{Z}\left(\cfrac{\pi}{2}\right)^{3}\cfrac{V_{\text{p-p}}}{V_{\pi}}\text{sin}^{2}\left(\cfrac{V_{\text{DC}}}{V_{\pi}}\pi\right),$
(22)

$V_{\text{DC}}$ is the DC voltage bias at the LO phase shifter. Signal
strength depends on the local oscillator power $P_{\text{LO}}$, detection
responsivity $R$, the impedance of the detector $Z$, and the leakage
$\varepsilon$. We fit equation 22 in the main text to find $\varepsilon$ and
$V_{\pi}$.

## Impact of leakage on the squeezing visibility

The AC part of the equation 19 probes both the noise of the prepared quantum
state and the noise of the LO. It results in the noise measured in the main
text:

$\displaystyle P_{\text{BH}}^{\text{(AC)}}$ $\displaystyle=$
$\displaystyle\text{cos}(\phi_{2})[\alpha_{\text{LO}}e^{i\phi_{\text{LO}}}\hat{v}^{\dagger}-\sqrt{\varepsilon}\alpha_{\text{LO}}\hat{a}^{\dagger}+{h.c}]$
$\displaystyle+$
$\displaystyle\text{sin}(\phi_{2})[\alpha_{\text{LO}}e^{i\phi_{\text{LO}}}\hat{a}^{\dagger}+\sqrt{\varepsilon}\alpha_{\text{LO}}\hat{v}^{\dagger}+{h.c}].$

Note that it reduces to the usual BHD expression for $\varepsilon=0$ and
$\phi_{2}=\pi/2$. Our experiment introduces two modifications to the classical
BHD picture. One is the leakage $\varepsilon\neq 0$, which results in probing
the local oscillator noise $\hat{v}$, in addition to measuring the squeezed
state $\hat{a}$. The other modification is varying the splitting ratio of the
output beam splitter $\phi_{2}\neq\pi/2$. This adds a term proportional to
$\text{cos}(\phi_{2})$, resulting in changing the measured noise
characteristics. As a result, measured squeezing can be reduced. This section
explains how the power leakage $\varepsilon$ impacts observable squeezing. We
first rewrite the equation LABEL:eq:BHD_sig_AC with the quadrature operators
for the squeezing and LO paths
$\hat{a}=\delta\hat{\mathrm{X}}+i\delta\hat{\mathrm{Y}}$ and
$\hat{v}=\delta\hat{\mathrm{X}}_{\text{LO}}+i\delta\hat{\mathrm{Y}}_{\text{LO}}$:

$\displaystyle P_{\text{DIFF}}^{\text{(AC)}}$ $\displaystyle=$ $\displaystyle
2\,\text{sin}(\phi_{2})\,|\alpha_{\text{LO}}|\times$
$\displaystyle\left[\text{cos}(\phi_{\text{LO}})\delta\hat{\mathrm{X}}\right.+\left.\text{sin}(\phi_{\text{LO}})\delta\hat{\mathrm{Y}}+2\sqrt{\varepsilon}\delta\hat{\mathrm{X}}_{\text{LO}}.\right]$
$\displaystyle+$ $\displaystyle
2\,\text{cos}(\phi_{2})\,|\alpha_{\text{LO}}|\times$
$\displaystyle\left[\text{cos}(\phi_{\text{LO}})\delta\hat{\mathrm{X}}_{\text{LO}}\right.+\left.\text{sin}(\phi_{\text{LO}})\delta\hat{\mathrm{Y}}_{\text{LO}}-2\,\sqrt{\varepsilon}\delta\hat{\mathrm{X}}\right]$

This is an expected result for the BHD applied toward phase-sensitive probing
of the quadratures of a squeezed state with an additional term that probes the
noise of the LO. We assume the LO noise to be phase-insensitive and shot-
noise-limited. The signal variance is proportional to the power measured by
the spectrum analyzer and given by equation 25. We use this result as a model
for our squeezing measurement in the main text.

$\displaystyle\langle\delta P_{\text{DIFF}}^{\text{(AC)}2}\rangle$
$\displaystyle=$ $\displaystyle
4\,|\alpha_{\text{LO}}|^{2}\left[\,\text{cos}^{2}(\phi_{2})\left(\,\text{cos}^{2}(\phi_{\text{LO}})\langle\delta\hat{\mathrm{X}}_{\text{LO1}}^{2}\rangle+\text{sin}^{2}(\phi_{\text{LO}})\langle\delta\hat{\mathrm{X}}_{\text{LO2}}^{2}\rangle+4\,\varepsilon\,\langle\delta\hat{\mathrm{X}}_{\text{1}}^{2}\rangle\,\right)\right.$
(25) $\displaystyle+$
$\displaystyle\text{sin}^{2}(\phi_{2})\left(\,\text{cos}^{2}(\phi_{\text{LO}})\langle\delta\hat{\mathrm{X}}_{\text{1}}^{2}\rangle+\text{sin}^{2}(\phi_{\text{LO}})\langle\delta\hat{\mathrm{X}}_{\text{2}}^{2}\rangle+4\varepsilon\langle\delta\hat{\mathrm{X}}_{\text{\text{LO}1}}^{2}\rangle\,\right)$
$\displaystyle+$
$\displaystyle\left.2\,\text{sin}(2\phi_{2})\,\sqrt{\varepsilon}\,\text{cos}(\phi_{\text{LO}})\left(\,\langle\delta\hat{\mathrm{X}}_{\text{\text{LO}1}}^{2}\rangle-\langle\delta\hat{\mathrm{X}}_{\text{1}}^{2}\rangle\,\right)\,\right]$

## Effects of loss in the system

The total detection chain efficiency is a major challenge in characterizing
squeezed light. This is especially consequential when working with complex
PICs like the one described in the main text. In our experiment the total
detection chain efficiency was around $\zeta\approx\text{20}\%$, this leads to
the reduction of squeezing visibility:

$\displaystyle\left[\cfrac{\langle\delta\hat{\mathrm{X}}^{2}\rangle}{\langle\delta\hat{\mathrm{X}}_{\text{vac}}^{2}\rangle}\right]_{\text{}}=\left(\zeta\left[\cfrac{\langle\delta\hat{\mathrm{X}}^{2}\rangle}{\langle\delta\hat{\mathrm{X}}_{\text{vac}}^{2}\rangle}\right]_{\text{On-
Chip}}+(1-\zeta)\right).$ (26)

We summarize all the sources of loss that impact reported total detection
efficiency in table 1. In our measurement, the main limitation is a result of
the loss at the interface between the chip facet and collection fibers.

Reducing the overall losses in the system is a straightforward way to improve
the device’s performance. Reducing the fiber-to-chip coupling loss down to 0.6
dB has been established by engineering the waveguide termination. He  _et al._
(2019); Hu  _et al._ (2021); Ying  _et al._ (2021) Moreover, the propagation
loss can be reduced to 0.2 dB by reducing the waveguide length where the
squeezed state can experience loss. The quantum efficiency of the detector
used in this study can be improved to 99.5$\%$ using commercially available
photodiodes. We use these values to estimate potentially achievable squeezing
in the proposed device. In this case, the total detection loss is 0.8 dB, and
the total insertion loss of the laser is 0.6 dB. Using these values, we plot
the expected achievable squeezing with respect to the laser power in Fig. 6.
Assuming no squeezer leakage, we compare the calculated values in a system
with reduced loss (blue) to the current system with a total detection
efficiency of around 20$\%$ (orange). We estimate that the same integrated DFB
laser, as we used in the main text, would facilitate access to the blue-shaded
region and result in the maximum squeezing factor of 2.8 at the highest
achievable on-chip power of 96 mW.

Figure 4:  Device fabrication procedure.  a, We start with a 500 nm thin-film
LN on insulator. b, For periodic poling of the LN film, we use chromium
electrodes deposited on a 100 nm silica insulation layer. Next, we apply a
high-voltage pulse to invert crystal domains. c, Chromium electrodes and
silica layer are removed after poling using chromium etchant and buffered
oxide etchant. d, Waveguides are patterned using electron-beam lithography and
argon ion milling. e, We pattern a bottom layer of gold directly on LN to
increase the electric field within the waveguide region. f, Wavegues are
covered with around 700 nm of silica. g, We open vias in the cladding to
provide electrical contact to the buried electrodes. h, We pattern the top
layer of gold to connect our devices to the external probe.  Figure 5:  Design
of the electro-optic modulator.  a, Eigenmode solver solution for the
fundamental TE mode in the presence of a static electric field. White arrows
indicate the direction of the applied bias field. b, Half-wave voltage-length
product and metal-induced propagation loss as a function of the waveguide-
electrode gap. The dashed line corresponds to the geometry described in the
main text with a waveguide-electrode gap of 1.6 µm.  Figure 6:  Calculated
squeezing with reduced loss.  We calculate the expected squeezing for the
proposed device with reduced detection and laser coupling loss. We assume a
total laser coupling loss of 0.6 dB and a total detection loss of 0.8 dB. The
blue line corresponds to the expected squeezing after reducing optical loss in
the system. The orange line corresponds to the squeezing achievable with the
current system, with 20$\%$ detection efficiency. The shaded areas are
accessible with the current integrated DFB laser for both cases.