††thanks: Corresponding author<EMAIL_ADDRESS>
# Free-space quantum information platform on a chip

Volkan Gurses  Division of Engineering and Applied Science, California
Institute of Technology, Pasadena, CA, USA  Samantha I. Davis  Division of
Physics, Mathematics and Astronomy, Caltech, Pasadena, CA, USA  Alliance for
Quantum Technologies (AQT), California Institute of Technology, Pasadena, CA,
USA  Neil Sinclair  Alliance for Quantum Technologies (AQT), California
Institute of Technology, Pasadena, CA, USA  John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Maria Spiropulu  Division of Physics, Mathematics and Astronomy, Caltech,
Pasadena, CA, USA  Alliance for Quantum Technologies (AQT), California
Institute of Technology, Pasadena, CA, USA  Ali Hajimiri  Division of
Engineering and Applied Science, California Institute of Technology, Pasadena,
CA, USA

###### Abstract

Emerging technologies that employ quantum physics offer fundamental
enhancements in information processing tasks, including sensing,
communications, and computing [1, 2, 3]. Here, we introduce the quantum phased
array, which generalizes the operating principles of phased arrays [4] and
wavefront engineering [5] to quantum fields, and report the first quantum
phased array technology demonstration. An integrated photonic-electronic
system is used to manipulate free-space quantum information to establish
reconfigurable wireless quantum links in a standalone, compact form factor.
Such a robust, scalable, and integrated quantum platform can enable broad
deployment of quantum technologies with high connectivity, potentially
expanding their use cases to real-world applications. We report the first, to
our knowledge, free-space-to-chip interface for quantum links, enabled by 32
metamaterial antennas with more than 500,000 sub-wavelength engineered
nanophotonic elements over a 550 $\times$ 550 $\mathrm{\mu m^{2}}$ physical
aperture. We implement a 32-channel array of quantum coherent receivers with
30.3 dB shot noise clearance and 90.2 dB common-mode rejection ratio that
downconverts the quantum optical information via homodyne detection and
processes it coherently in the radio-frequency domain. With our platform, we
demonstrate 32-pixel imaging of squeezed light for quantum sensing,
reconfigurable free-space links for quantum communications, and proof-of-
concept entanglement generation for measurement-based quantum computing. This
approach offers targeted, real-time, dynamically-adjustable free-space
capabilities to integrated quantum systems that can enable wireless quantum
technologies.

## Introduction

Figure 1: Free-space quantum information platform. Conceptual breakdown of the
free-space quantum information platform (QIP) showing the key technology
demonstrations and potential applications.

The science and engineering of quantum systems have expanded in the last two
decades to enable technologies that can manipulate quantum information at
scale [6]. These technological developments have led to impressive
demonstrations across numerous architectures including superconducting qubits
[7], atom arrays [8], trapped ions [9], and integrated photonics [10].
Integrated photonics is robust against decoherence at room temperature and
varying operating conditions [11]. It can be integrated end-to-end with
electronics [12] and manufactured at high volume and high yield with CMOS
fabrication [13]. Therefore, it is a compelling candidate for real-world
quantum systems including mobile quantum devices. Establishing practical
quantum links with high fidelity [14] and a high degree of parallelization
[15], offered by free-space interconnects, is important for distributing
quantum information between these systems. To our knowledge, there has been no
free-space-to-chip interconnect suitable for point-to-point links due to the
high coupling loss in conventional free-space-to-chip interfaces and beam
divergence challenges in free-space links.

In this work, we demonstrate a large-scale yet compact, room-temperature
quantum information platform (QIP), integrated on a silicon photonic chip,
which we call the quantum phased array (QPA), that can establish
reconfigurable wireless links for free-space quantum information processing. A
large-area (550 $\times$ 550 $\mathrm{\mu m^{2}}$) metamaterial aperture
enables a low-loss interface between free space and the chip, and a 32-channel
array of self-stabilizing coherent receivers downconverts the quantum optical
signals to radio-frequency (RF) by homodyne detection for coherent processing
in the RF domain. Coherent processing with the quantum coherent receivers on
our QPA chip overcomes the geometric loss limitation [2], despite each channel
individually exhibiting high loss. Therefore, we expand the concept of
wavefront engineering with classical electromagnetic fields to the non-
classical domain. Moreover, coherently interfacing our QPA chip with RF
circuits in our platform enables optoelectronic processing of continuous
variable quantum information. In this work, we provide a proof-of-concept
realization of our platform shown in Fig. 1 with the QPA chip. We envision
this platform, which allows for free-space connectivity with end-to-end system
realization, to enable future mobile and wireless quantum technologies.

In the following, we outline the QPA system design, demonstrate multipixel
squeezed light imaging with 32 coherent channels, introduce wavefunction
engineering with quantum phased arrays, and implement a proof-of-concept
demonstration of entanglement generation with optoelectronic processing for
measurement-based quantum computing.

## Quantum phased array chip

Figure 2: Quantum phased array chip. a) Diagram and die photo (bottom right)
of the quantum phased array chip showing the building blocks including i) the
metamaterial antenna and ii) the quantum coherent receiver. b) Far-field
radiation pattern of the metamaterial antenna. c) LO power sweep
characterizing the shot noise clearance and the LO power knee for the QRX in
high shot noise clearance configuration. d) Output noise spectra at different
LO powers characterizing the shot-noise-limited bandwidth for the QRX in the
high bandwidth configuration. e) Squeezed light detection with a single-
channel QRX in the high bandwidth configuration.

We implement a QPA receiver system using a commercial silicon photonics
process, as shown in Fig. 2.

The QPA chip has 32 channels comprising antennas, waveguides, phase shifters,
and quantum coherent receivers. Free-space-to-chip coupling is enabled by a
large-area (550 $\times$ 550 $\mathrm{\mu m^{2}}$) fully-filled aperture
comprised of 32 metamaterial antennas (MMAs) that couple a collimated beam in
free-space to 32 single-mode waveguides on chip. Each MMA is 597
$\mathrm{\times}$ 16.7 $\mathrm{\mu m^{2}}$ in footprint. It is designed to
have a sufficiently large effective aperture (see Methods) suitable for low-
loss coupling with commercial fiber collimators, which are available for beam
diameters greater than 200 $\mathrm{\mu m}$. This design minimizes the mode
mismatch between the free-space beam and the on-chip aperture to enable low-
loss free-space-to-chip coupling, overcoming a limitation in conventional
nanophotonic antennas.

The simulated 3D radiation pattern for the resulting antenna design is shown
in Fig. 2b. The free-space-to-chip interface is characterized by the geometric
loss, which is the loss due to imperfect modal overlap of the incident free-
space beam and the on-chip aperture, and the MMA insertion loss, which
includes the propagation loss in the MMA and loss due to downward scattering.
The MMA has a measured (simulated) insertion loss of 3.82 dB (3.78 dB). The
total measured (simulated) geometric losses of 8 and 32 antenna apertures with
a collimated beam of 200 $\mathrm{\mu m}$ diameter, are 2.18 dB (2.03 dB) and
4.85 dB (4.50 dB), respectively. The minimum geometric loss can be attained by
adding amplitude weights to the channels, yielding a total measured
(simulated) loss of 1.14 dB (1.35 dB). This is at least an order of magnitude
lower than previously reported on-chip aperture designs [16, 17] and is
sufficiently low to start interfacing free-space quantum optics with photonic
integrated circuits (PICs).

The waveguides after the antennas are path-length matched and are connected to
32 quantum coherent receivers (QRX). Each receiver consists of a tunable Mach-
Zehnder interferometer (MZI), a pair of balanced Ge photodiodes, and a
transimpedance amplifier (TIA). The MZI interferes a signal field with a
strong local oscillator (LO) for homodyne detection. The LO is coupled to the
chip with a grating coupler and is split into 32 channels with a 1-to-32
splitter tree. The LO input to each channel hosts a thermo-optic phase shifter
(TOPS) for phase tuning. Each output of the MZI is sent to a photodiode, and
the currents at the outputs of the photodiodes are subtracted and amplified by
the TIA.

The performance of a QRX is quantified by its shot noise clearance (SNC), LO
power knee ($\mathrm{P_{knee}}$), common-mode rejection ratio (CMRR), 3-dB
bandwidth ($\mathrm{BW_{3dB}}$) and shot-noise limited bandwidth
($\mathrm{BW_{shot}}$) [18, 19, 20, 21]. Our QRXs are characterized in two
configurations with two TIA designs: one used in the high bandwidth
configuration, optimal for communications, and another used in the high shot
noise clearance configuration, optimal for sensing (see Methods). These two
TIAs, which trade off bandwidth with noise floor and vice versa, are designed
to be interchangeably used with the photonic chip to overcome the fundamental
trade-off between SNC and bandwidth in balanced homodyne detection [18, 20].

In the high SNC (high bandwidth) configuration, the QRX has an SNC of 30.3 dB
(14.0 dB) and a $\mathrm{P_{knee}}$ of 12.5 $\mathrm{\mu W}$ (521 $\mathrm{\mu
W}$). To showcase the high-speed measurements achievable with the QRX in the
high bandwidth configuration, squeezed light was measured with a single QRX
channel up to the shot-noise-limited bandwidth of 3.50 GHz as seen in Fig, 2d
along with the output noise spectra at different LO photocurrents. In both
high SNC and high bandwidth configurations, the QRX has a CMRR of 90.2 dB at
1.1 MHz.

In the subsequent experiments, the QPA chip is packaged with electronics on
custom printed circuit boards. The QPA chip was wirebonded to an interposer to
fan out 104 electronic read-out and control lines to the interfaced
electronics. The interposer is plugged into a radio-frequency (RF) motherboard
with 50 $\mathrm{\Omega}$ coplanar waveguide outputs. The motherboard hosts a
32-channel TIA array in high SNC configuration and an CMRR auto-correction
circuit that is fed back to the MZIs on the PIC (see Methods). The RF outputs
are combined with a 32-to-1 power combiner and sent to an RF signal analyzer
(ESA). Before power combining, the outputs are also probed with high-impedance
outputs for independent recording of the 32-channel signals.

Figure 3: Squeezed light imaging. a) Experimental setup for the squeezed light
measurements. Squeezed light is generated off-chip and transmitted over free
space to the quantum phased array chip (blue, Photonics), which is interfaced
with electronics (orange, Electronics) for quantum information processing. b)
Squeezed light imaging measurement. Upper left: Wigner function at the source.
The blue and black contours correspond to the half-maximum points of the
squeezed vacuum and vacuum states, respectively. Middle: The sample mean and
variances of the pixel quadratures as a function of time. For each channel,
the sample variances are normalized to the mean variance. Right: Wigner
functions of the 32 pixel modes.

## Squeezed light imaging

We first operate the QPA chip as a 32-pixel quantum coherent imager. Squeezed
light is generated off-chip using off-the-shelf fiber-optic components at a
central wavelength of 1550 nm as shown in Fig. 3a. The squeezed light is sent
to a fiber collimator with a 200 $\mathrm{\mu m}$ beam diameter and is
transmitted to the chip over free space. At the chip aperture, the collimated
squeezed light is spatially distributed across the 32 antennas with a Gaussian
amplitude profile. The antennas define a set of 32 pixel modes
$\\{\hat{a}_{j}\\}$, where $\hat{a}_{j}$ is the bosonic annihilation operator
for the field coupled into the $j$th channel. Each channel outputs a voltage
proportional to the phase-dependent quadrature of its pixel mode,

$\displaystyle\hat{X}_{j}(\theta_{j})=\frac{1}{\sqrt{2}}(\hat{a}_{j}e^{i\theta_{j}}+\hat{a}_{j}^{\dagger}e^{-i\theta_{j}}),$
(1)

where $\theta_{j}$ is the phase of the $j$th pixel mode relative to the LO.
For a squeezed vacuum field, the quadrature mean
$\langle\Delta\hat{X}_{j}(\theta_{j})\rangle$ is zero, and the quadrature
variance is,

$\displaystyle\text{Var}\left(\hat{X}_{j}(\theta_{j})\right)$
$\displaystyle=\frac{\eta_{j}}{4}(e^{-2r}\cos^{2}{\theta_{j}}+e^{2r}\sin^{2}{\theta_{j}})+\frac{1-\eta_{j}}{4},$
(2)

where $r$ is the squeezing parameter, and $\eta_{j}$ is the effective
efficiency of channel $j$, which includes the effects of source loss, free-
space loss, on-chip loss, and RF loss.

To image the squeezed light, the output voltages are read out to a 32-channel
digitizer at a sampling rate of 20 MSa/s. A 0.5 Hz phase ramp is applied to
the LO off-chip to acquire voltage samples over various phases, and sample
means and variances are calculated over sets of 260,000 voltage samples. The
time evolution of the sample means and variances for all 32 pixel modes are
shown in Fig. 3b. The Gaussian profile of the incident field is observed in
the amplitude distribution of variances, with the edge channels corresponding
to the shot noise power levels for the vacuum state. Without phase locking the
signal and LO, phase drifts between the signal and LO paths in fiber optics
give rise to additional phase fluctuations. The fluctuations are coherent
across all 32 channels due to the preservation of quantum coherence over free
space and throughout the chip. For practical applications, phase-locking could
be implemented with an optical phase-locked loop [22] or by co-propagating the
LO and signal over free space with the transmitter counterpart to the QPA
receiver as illustrated in Fig. 1.

In Fig. 3b (left), the Wigner function of the source is plotted for the
squeezing parameter ($r=1.95$) as a function of the quadrature observables
($X$, $P$). The Wigner functions of the 32 pixel modes are also plotted in
Fig. 3b (right) for the same squeezing parameter as well as the phase and
geometric efficiency for each channel. The geometric efficiency of pixel $j$
is the fraction of the incident field that couples onto the $j$th channel. The
fixed phase relationships and geometric efficiency (loss) of the channels
manifest as variations in the orientation and the squeezing levels of the
Wigner functions, respectively (see Methods).

Figure 4: Wavefunction engineering. a) Conceptual illustration of wavefunction
engineering with squeezed light. b) Squeezing (orange) and antisqueezing
(blue) levels as a function of the number of channels combined relative to the
vacuum level (grey) after the chip is beamformed at the squeezed light
transmitter. c) Source characterization showing squeezing and antisqueezing
levels as a function of squeezer pump power with 32 channels combined. d)
Demonstration of programmable and spatially selective free-space quantum
links, showing the lack of quantum signal when the receiver is beamformed at
empty space (blue) and the reception of a quantum signal when the receiver is
beamformed at the transmitter (orange). The grey trace is the vacuum signal.
e) Squeezing and antisqueezing levels characterizing the beamwidth of the
quantum link for 8 (top) and 32 (bottom) channels combined. f) Squeezing and
antisqueezing levels characterizing the field of view of the QPA receiver for
8 (top) and 32 (bottom) channels combined.  Figure 5: Quantum optoelectronic
processing. a) Quantum circuit architecture for entanglement generation. The
free space operation ($\mathrm{U}$) corresponds to the change-of-basis matrix
mapping the spatial modes of the input state to the pixel modes, where $\rho$
represents the spatial coordinates in the aperture plane of the chip. Each
colored line represents a matrix element corresponding to the overlap of an
antenna and pixel mode function. A phase shifter $\varphi_{j}$ is applied to
each pixel mode, and each half of the array is combined in a 16:1 RF power
combiner ($\mathrm{G}$). The output voltages of the power combiners are
digitized and followed by a beamsplitter transformation ($S$). b) Emulated
optical circuit for two-mode Gaussian cluster state generation. c) The cluster
state inseparability (I) measured over time for a linear phase ramp. The data
for the squeezed vacuum and vacuum states are plotted in blue and gray,
respectively. The solid lines are the analytical expectations with a
sinusoidal fit to the squeezed data.

## Wavefunction engineering

Geometric loss is a longstanding challenge in quantum communications [2]. In a
point-to-point free-space quantum link, a transmitter encodes a quantum signal
in a beam of light that is sent to a receiver. The spot size of the beam
spreads with distance due to diffraction, resulting in geometric loss from the
overlap of the diverging spot size and the receiver aperture area.
Diffraction-induced geometric loss can result in severe signal loss that
ultimately limits the range and rate of quantum communication protocols [23,
24, 25, 26]. The effect of geometric loss is apparent in Wigner functions of
Fig. 3b, where the degree of squeezing per channel is reduced compared to that
of the transmitted state. In classical wireless communications and sensing,
beam divergence is controlled by wavefront engineering with transmitter or
receiver phased arrays. Wavefront engineering allows for active manipulation
of an electromagnetic field in a dynamic real-time fashion [27, 28].
Beamforming, or angular focusing, of an electromagnetic field is performed by
coherently combining elements in a phased array such that the signal field
constructively interferes at a selected angle [29]. Here, we extend
beamforming to quantum fields and demonstrate how wavefunction engineering
with quantum phased arrays can overcome geometric loss, enable spatial
selectivity, and establish reconfigurable and programmable quantum links.

Beamforming with a QPA receiver is illustrated in Fig. 4a. A quantized
electromagnetic field with annihilation operator $\hat{a}_{\text{in}}(\rho)$
is transmitted to a phased array of quantum coherent elements. The field
$\hat{a}_{\text{in}}(\rho)=\sum_{n}u_{n}(\rho)\hat{a}_{n}$ is expanded over a
set of independent modes with annihilation operators $\\{\hat{a}_{n}\\}$,
where $\rho$ represents the transverse coordinates in the plane orthogonal to
propagation. The modes have an associated set of orthonormal mode functions
$\\{u_{n}(\rho)\\}$ that correspond to photon-wavefunctions in second
quantization [30, 31, 32]. Due to diffraction, a portion of the field is
coupled onto each antenna, resulting in high geometric loss per channel. The
RF outputs of the coherent receivers are combined after applying a gain and
phase to each channel. In the large array limit, the combined RF output is
proportional to the quadrature of an output field,

$\displaystyle\hat{a}_{\text{out}}(\rho)=\sum_{n}\hat{a}_{n}\int
g(\rho)e^{i\phi(\rho)}u_{n}(\rho)d\rho,$ (3)

where $g(\rho)$ and $\phi(\rho)$ are the applied gain and phase profiles (see
Methods). The gain and phase profiles give rise to a programmable array mode
function $\mathcal{A}(\rho)=g(\rho)e^{i\phi(\rho)}$, which is used to engineer
the wavefunction of the input field. For a signal encoded in a mode
$\hat{a}_{n}$, perfect modal overlap, i.e. geometric efficiency, is achieved
by setting $\mathcal{A}(\rho)=u_{n}(\rho)$. Beamforming refers to the
optimization of the gain and phase profiles to maximize the geometric
efficiency. For the QPA chip, the minimum geometric loss of 1.14 dB for a 200
$\mu$m beam diameter sets an upper bound of 0.77 on the geometric efficiency,
which can approach unity in future chip iterations with specialized antenna
design and by scaling up the array.

By beamforming with the QPA chip, we experimentally overcome the geometric
losses of Fig. 3. After optimizing the LO phases for all 32 channels (see
Methods), squeezed light is transmitted to the chip through the fiber
collimator. A 1-Hz phase ramp is applied to the LO before coupling to the
chip, and the outputs of the channels are coherently combined with a 32:1 RF
power combiner. The combined output signal is sent to an RF signal analyzer,
which measures the noise power proportional to the quadrature variance in Eq.
2.

The improvement in geometric efficiency with beamforming is demonstrated in
Fig. 4b. The noise powers are measured for various number of combined
channels. The number of channels is increased symmetrically about the center
of the array, starting with only channel 16 connected to the power combiner.
The solid lines are a fit of the data to a model constructed from the
classically characterized signal-to-noise ratio for each channel combination.
The squeezing (antisqueezing) level improves from $-0.017(+0.077)\pm 0.012$ dB
for a single channel to $-0.064(+0.312)\pm 0.012$ dB for eight channels
combined, corresponding to an increase in the geometric efficiency by a factor
of $4.5$ (see Methods).

We perform a source characterization to confirm that the combined RF signals
correspond to squeezed light with increased efficiency. For all 32 channels
combined, the squeezing and antisqueezing levels for various squeezer pump
powers ($P$) are shown in Fig 4c. The solid lines correspond to a least-
squares fit of the data to a model, where the effective efficiency of the
system, $\eta$, and the spontaneous parametric down conversion (SPDC)
efficiency, $\mu$, are taken as floating parameters. We obtain $\eta=0.016$
and $\mu=0.038$ $[\text{mW}]^{-1/2}$, which is consistent with the SPDC
efficiency of the squeezer (see Methods).

Establishing spatially-selective free-space quantum links with our QPA is
illustrated in Fig. 4d. A quantum link is established by beamforming the QPA
chip at an angle at which no light is detected (blue). The phase settings are
reconfigured by applying a linear phase mask to the LO phase shifters, which
steers the link towards a squeezed light source (orange). In the first five
seconds, no quantum signal is observed, demonstrating successful spatial
filtering. In the next five seconds, after the quantum link is electronically
steered toward the source, the noise power modulations of the squeezed light
are observed.

The spatial selectivity, namely beamwidth, of a quantum link is characterized
in Fig. 4e. A classical phase calibration is performed for a beam of squeezed
light at a normal angle of incidence to the chip. The angle of incidence
($\theta$) of the beam is swept while the chip is kept beamformed to normal
incidence ($\theta=0$), for 8 and 32 channels combined. The squeezing and
antisqueezing levels as a function of $\theta$ are shown in Fig. 4e. The solid
lines are fits of the data to a model obtained from classical beamwidth
measurements. The beamwidths ($\mathrm{BW_{n}}$) for 3-dB loss are 0.41
degrees and 0.20 degrees with 8 channels and 32 channels combined,
respectively, demonstrating the increased spatial selectivity as the array is
scaled (see Methods).

The programmability of free-space quantum links over the field-of-view (FoV)
of the QPA chip is demonstrated in Fig. 4f. At each of the nine different
angles ($\mathrm{\theta}$), classical phase calibration is performed and the
optimal phase settings are recorded. The squeezed light is then transmitted to
the chip at each of the nine angles. For each angle, a quantum link is
programmed by applying the corresponding phase settings to the LO phase
shifters. The squeezing and antisqueezing levels for each angle are shown in
Fig. 4f for 8 and 32 channels combined. The solid lines are a fit to a model
obtained from classical FoV measurements. The 3-dB loss FoV for squeezed light
is 2.3 degrees and 2.7 degrees with 8 channels and 32 channels combined,
respectively (see Methods).

## Quantum optoelectronic processing

We perform a proof-of-concept demonstration of cluster state generation in a
measurement-based approach [33, 34] to illustrate the potential of quantum
optoelectronic systems. Cluster states are a class of entangled graph states
that form a resource for universal measurement-based quantum computation [35].
CV Gaussian cluster states can be generated by interfering squeezed states in
linear optical networks [36, 37]. Here we generate two-mode cluster state
correlations by implementing the equivalent linear optical network after
optoelectronic downconversion with an RF circuit.

The quantum circuit architecture is shown in Fig. 5a. A squeezed state is
transmitted over free space to the QPA chip and a phase ramp at a modulation
frequency of $f=0.5$ Hz is applied to the LO before coupling to the chip. The
RF outputs of the QRXs in each half of the array are sent to a 16:1 power
combiner. Beamforming is performed on all 32 channels such that the two
outputs of the power combiners are in phase. To improve the geometric
efficiency, the outermost 12 channels are disconnected from each 16:1 power
combiner, for a total of 8 pixel modes in Fig. 5a. The outputs of the power
combiners are digitized at a sampling rate of 100 MSa/s and an RF beamsplitter
transformation is emulated on the digitized quadratures (see Methods).

The inseparability criterion required to show cluster state entanglement is,

$\displaystyle
I=\text{Var}(\hat{P}_{4}-\hat{X}_{3})+\text{Var}(\hat{P}_{3}-\hat{X}_{4})<1$
(4)

where $\hat{X}_{i},\hat{P}_{i}$ are the quadrature operators for each cluster
state mode denoted by $i=3,4$ in Fig. 5b, and the variances are relative to
those of the vacuum state. The quadrature correlations $I$ as a function of
time are shown in Fig. 5b. We observe the sinusoidal signature expected for a
rotation of the measurement basis due to the LO phase modulation. We obtain a
minimum inseparability of $I=0.994\pm 0.002$, which violates the classical
bound by three standard deviations. The resolution of entanglement is enabled
by the high precision and stability offered by the chip-scale optoelectronics
(see Methods).

## Discussion and outlook

We demonstrate a compact, scalable free-space quantum information platform
integrated on a silicon photonic chip with more than 1000 functional
components on a 3 $\mathrm{\times}$ 1.8 $\mathrm{mm^{2}}$ footprint. We design
and implement a sub-wavelength engineered large active area metamaterial
aperture with more than 500,000 scattering elements and a 32-channel array of
state-of-the-art quantum coherent receivers. To our knowledge, we report the
first free-space coupling and processing of non-classical states to a system-
on-chip, providing an interface between free-space quantum optics and
integrated photonic systems. We expand classical wavefront engineering to
quantized electromagnetic fields and demonstrate wavefunction engineering with
quantum phased arrays that enable dynamically programmable, wireless quantum
links with spatial selectivity. With our integrated platform, we mitigate the
long-standing challenge of diffraction-induced geometric loss by coherently
combining the signals of the receiver outputs in RF thus enabling wireless
quantum communications. We realize a measurement-based entanglement generation
scheme with quantum optoelectronic processing by implementing operations on
downconverted quantum optical information with RF circuits.

Demonstrated improvements in component losses [38, 39, 40] chart a path toward
deployment of our platform to real-world applications including microscopes,
sensors [41], and quantum communication transceivers, as well as potential
investigations of fundamental physics research [42]. Interfacing integrated
quantum photonics with electronics in the same package enables novel
engineering opportunities in realizing large-scale room-temperature quantum
systems. Coherent processing of downconverted quantum optical information with
RF or microwave integrated circuits could enable a low-loss optoelectronic
approach to quantum information processing as a quantum analog of microwave
photonics [43].

Our novel quantum platform bridges integrated photonics, electronics, and
free-space quantum optics with multiple envisioned applications in fundamental
physics and engineering.

### Acknowledgements

We are grateful to Raju Valivarthi for technical support on the squeezed light
sources and for technical discussions, Pablo Backer Peral for aiding the
development of digitizer script, Esme Knabe for aiding the initial single-
channel QRX measurements, Debjit Sarkar for aiding the setup for the high
bandwidth squeezed light measurement, and Andrew Mueller for aiding figure
design. Support for this work was provided in part by the Carver Mead New
Adventures Fund and Alliance for Quantum Technologies’ (AQT) Intelligent
Quantum Networks and Technologies (INQNET) program. S.I.D. is in part
supported by the Brinson Foundation. M.S. is in part supported by the
Department of Energy under Grant No. SC0019219.

## References

  * Degen  _et al._ [2017] C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Reviews of Modern Physics 89, 035002 (2017).
  * Gisin  _et al._ [2002] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Reviews of Modern Physics 74, 145 (2002).
  * Ladd  _et al._ [2010] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature 464, 45 (2010).
  * Braun [1909] K. F. Braun, Nobel Lecture, Nobel Foundation (1909).
  * Engheta and Ziolkowski [2006] N. Engheta and R. W. Ziolkowski,  _Metamaterials: physics and engineering explorations_ (John Wiley & Sons, 2006).
  * MacFarlane  _et al._ [2003] A. G. J. MacFarlane, J. P. Dowling, and G. J. Milburn, Quantum technology: the second quantum revolution, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 361, 1655 (2003).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Bluvstein  _et al._ [2024] D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, _et al._ , Logical quantum processor based on reconfigurable atom arrays, Nature 626, 58 (2024).
  * Pino  _et al._ [2021] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyenhuis, Demonstration of the trapped-ion quantum CCD computer architecture, Nature 592, 209 (2021).
  * Arrazola  _et al._ [2021] J. M. Arrazola, V. Bergholm, K. Brádler, T. R. Bromley, M. J. Collins, I. Dhand, A. Fumagalli, T. Gerrits, A. Goussev, L. G. Helt, _et al._ , Quantum circuits with many photons on a programmable nanophotonic chip, Nature 591, 54 (2021).
  * O’Brien  _et al._ [2009] J. L. O’Brien, A. Furusawa, and J. Vučković, Photonic quantum technologies, Nature Photonics 3, 687 (2009).
  * Atabaki  _et al._ [2018] A. H. Atabaki, S. Moazeni, F. Pavanello, H. Gevorgyan, J. Notaros, L. Alloatti, M. T. Wade, C. Sun, S. A. Kruger, H. Meng, _et al._ , Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip, Nature 556, 349 (2018).
  * Margalit  _et al._ [2021] N. Margalit, C. Xiang, S. M. Bowers, A. Bjorlin, R. Blum, and J. E. Bowers, Perspective on the future of silicon photonics and electronics, Applied Physics Letters 118, 220501 (2021).
  * Yin  _et al._ [2012] J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, _et al._ , Quantum teleportation and entanglement distribution over 100-kilometre free-space channels, Nature 488, 185 (2012).
  * Miller [2023] D. A. Miller, Why optics needs thickness, Science 379, 41 (2023).
  * Milanizadeh  _et al._ [2022] M. Milanizadeh, S. SeyedinNavadeh, F. Zanetto, V. Grimaldi, C. De Vita, C. Klitis, M. Sorel, G. Ferrari, D. A. Miller, A. Melloni, _et al._ , Separating arbitrary free-space beams with an integrated photonic processor, Light: Science & Applications 11, 197 (2022).
  * Ashtiani  _et al._ [2022] F. Ashtiani, A. J. Geers, and F. Aflatouni, An on-chip photonic deep neural network for image classification, Nature 606, 501 (2022).
  * Gurses  _et al._ [2023] V. Gurses, S. I. Davis, E. Knabe, R. Valivarthi, M. Spiropulu, and A. Hajimiri, A compact silicon photonic quantum coherent receiver with deterministic phase control, in  _2023 Conference on Lasers and Electro-Optics (CLEO)_ (2023) pp. 1–2.
  * Gurses and Hajimiri [2022] B. V. Gurses and A. Hajimiri, Performance limits of sub-shot-noise-limited balanced detectors, in  _Frontiers in Optics $+$ Laser Science 2022 (FIO, LS)_ (Optica Publishing Group, 2022) p. JW4A.32.
  * Bruynsteen  _et al._ [2021] C. Bruynsteen, M. Vanhoecke, J. Bauwelinck, and X. Yin, Integrated balanced homodyne photonic–electronic detector for beyond 20 GHz shot-noise-limited measurements, Optica 8, 1146 (2021).
  * Tasker  _et al._ [2021] J. F. Tasker, J. Frazer, G. Ferranti, E. J. Allen, L. F. Brunel, S. Tanzilli, V. D’Auria, and J. C. F. Matthews, Silicon photonics interfaced with integrated electronics for 9 GHz measurement of squeezed light, Nature Photonics 15, 11 (2021).
  * Taguchi  _et al._ [2022] Y. Taguchi, K. Oguchi, Z. Xu, D. Cheon, S. Takahashi, Y. Sano, F. Harashima, and Y. Ozeki, Phase locking of squeezed vacuum generated by a single-pass optical parametric amplifier, Optics Express 30, 8002 (2022).
  * Pirandola [2021] S. Pirandola, Limits and security of free-space quantum communications, Physical Review Research 3, 013279 (2021).
  * Kržič  _et al._ [2023] A. Kržič, S. Sharma, C. Spiess, U. Chandrashekara, S. Töpfer, G. Sauer, L. J. González-Martín del Campo, T. Kopf, S. Petscharnig, T. Grafenauer, _et al._ , Towards metropolitan free-space quantum networks, npj Quantum Information 9, 95 (2023).
  * Liao  _et al._ [2017] S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, _et al._ , Satellite-to-ground quantum key distribution, Nature 549, 43 (2017).
  * Ren  _et al._ [2017] J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, _et al._ , Ground-to-satellite quantum teleportation, Nature 549, 70 (2017).
  * Hansen [2009] R. C. Hansen,  _Phased array antennas_ (John Wiley & Sons, 2009).
  * Li  _et al._ [2018] A. Li, S. Singh, and D. Sievenpiper, Metasurfaces and their applications, Nanophotonics 7, 989 (2018).
  * Van Veen and Buckley [1988] B. D. Van Veen and K. M. Buckley, Beamforming: A versatile approach to spatial filtering, IEEE ASSP Magazine 5, 4 (1988).
  * Cohen-Tannoudji  _et al._ [1989] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,  _Photons and Atoms: Introduction to Quantum Electrodynamics_ (Wiley VCH, 1989).
  * Mandel and Wolf [1995] L. Mandel and E. Wolf,  _Optical Coherence and Quantum Optics_ (Cambridge University Press, 1995).
  * Grynberg  _et al._ [2010] G. Grynberg, A. Aspect, and C. Fabre,  _Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light_ (Cambridge University Press, 2010).
  * Armstrong  _et al._ [2012] S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, Programmable multimode quantum networks, Nature Communications 3, 1026 (2012).
  * Ferrini  _et al._ [2013] G. Ferrini, J.-P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, Compact Gaussian quantum computation by multi-pixel homodyne detection, New Journal of Physics 15, 093015 (2013).
  * Raussendorf and Briegel [2001] R. Raussendorf and H. J. Briegel, A one-way quantum computer, Physical Review Letters 86, 5188 (2001).
  * van Loock  _et al._ [2007] P. van Loock, C. Weedbrook, and M. Gu, Building Gaussian cluster states by linear optics, Physical Review A 76, 032321 (2007).
  * Yukawa  _et al._ [2008] M. Yukawa, R. Ukai, P. Van Loock, and A. Furusawa, Experimental generation of four-mode continuous-variable cluster states, Physical Review A 78, 012301 (2008).
  * Vahlbruch  _et al._ [2016] H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency, Physical Review Letters 117, 110801 (2016).
  * Michel  _et al._ [2010] J. Michel, J. Liu, and L. C. Kimerling, High-performance Ge-on-Si photodetectors, Nature Photonics 4, 527 (2010).
  * Boes  _et al._ [2023] A. Boes, L. Chang, C. Langrock, M. Yu, M. Zhang, Q. Lin, M. Lončar, M. Fejer, J. Bowers, and A. Mitchell, Lithium niobate photonics: Unlocking the electromagnetic spectrum, Science 379, eabj4396 (2023).
  * Rogers  _et al._ [2021] C. Rogers, A. Y. Piggott, D. J. Thomson, R. F. Wiser, I. E. Opris, S. A. Fortune, A. J. Compston, A. Gondarenko, F. Meng, X. Chen, G. T. Reed, and R. Nicolaescu, A universal 3D imaging sensor on a silicon photonics platform, Nature 590, 256 (2021).
  * Tse  _et al._ [2019] M. e. Tse, H. Yu, N. Kijbunchoo, A. Fernandez-Galiana, P. Dupej, L. Barsotti, C. Blair, D. Brown, S. Dwyer, A. Effler, _et al._ , Quantum-enhanced advanced LIGO detectors in the era of gravitational-wave astronomy, Physical Review Letters 123, 231107 (2019).
  * Marpaung  _et al._ [2019] D. Marpaung, J. Yao, and J. Capmany, Integrated microwave photonics, Nature Photonics 13, 80 (2019).
  * Beck [2000] M. Beck, Quantum state tomography with array detectors, Physical Review Letters 84, 5748 (2000).
  * Dawes  _et al._ [2003] A. M. Dawes, M. Beck, and K. Banaszek, Mode optimization for quantum-state tomography with array detectors, Physical Review A 67, 032102 (2003).
  * Fabre and Treps [2020] C. Fabre and N. Treps, Modes and states in quantum optics, Reviews of Modern Physics 92, 035005 (2020).
  * Ashtiani and Aflatouni [2021] F. Ashtiani and F. Aflatouni, Coherent multilayer photonic nanoantenna array with off-aperture phase adjustment, ACS Photonics 8, 3433 (2021).
  * Monzingo  _et al._ [2011] R. Monzingo, R. Haupt, and T. Miller,  _Introduction to Adaptive Arrays_ , Electromagnetic Waves Series (Institution of Engineering and Technology, 2011).
  * Leonhardt and Paul [1993] U. Leonhardt and H. Paul, Realistic optical homodyne measurements and quasiprobability distributions, Physical Review A 48, 4598 (1993).

## Methods

### Theory

Consider a quantized electromagnetic field $\hat{E}$ propagating over free
space in the $z$ direction towards a quantum phased array receiver. The field
is normally incident to the aperture plane at $z=0$, and can be decomposed
into positive and negative frequency components,
$\hat{E}=\hat{E}^{+}+\hat{E}^{-}$, where $\hat{E}^{\pm}$ are complex
conjugates. At the aperture plane, the positive frequency component of the
field can be expressed as,

$\displaystyle\hat{E}^{+}(\rho,t)=\sqrt{\frac{2\pi\hbar\omega}{L}}\sum_{n}\hat{a}_{n}u_{n}(\rho)e^{i\omega
t},$ (5)

where $\rho$ represents the transverse coordinates $(x,y)$, $\omega$ is the
frequency, and $L$ is the quantization length [44]. The field is expanded over
a complete set of independent modes with bosonic annihilation operators
$\\{\hat{a}_{n}\\}$ satisfying
$[\hat{a}_{n},\hat{a}_{m}^{\dagger}]=\delta_{nm}$. Relevant to fields in free
space are the Hermite-Gaussian modes and associated mode functions. In Eq. 5,
we assume a monochromatic treatment of the field. We note that the squeezed
light generated by SPDC in the experiments is broadband and that the following
analysis can be extended to multiple spectral modes.

The incident field to the aperture can be represented with the annihilation
operator,

$\displaystyle\hat{a}_{\text{in}}(\rho)=\sum_{n}\hat{a}_{n}u_{n}(\rho).$ (6)

The field coupled onto the $j$th channel of the receiver corresponds to the
modal overlap of the incident field and the $j$th antenna,

$\displaystyle\hat{a}_{j}$
$\displaystyle=\int\mathcal{E}_{j}(\rho)\hat{a}_{\text{in}}(\rho)d\rho=\sum_{n}U_{jn}\hat{a}_{n}$
(7)

where $\hat{a}_{j}$ is the creation operator for the $j$th pixel mode and
$\mathcal{E}_{j}(x)$ is the mode function for the $j$th antenna. The coupling
corresponds to a change-of-basis transformation $U$ between the input and
pixel modes,

$\displaystyle U_{jn}=\int\mathcal{E}_{j}(\rho)u_{n}(\rho)d\rho.$ (8)

For a signal encoded in a particular mode of a field with all other modes in
the vacuum state, imperfect overlap of antenna and signal mode functions
causes spurious vacuum modes to couple onto an antenna, resulting in geometric
loss.

To correct for geometric loss, the outputs of the receivers are combined in RF
after applying a gain and phase shift to the output of each receiver. The
output signal measured with the RF signal analyzer is proportional to the
quadrature,

$\displaystyle\hat{X}_{\text{out}}=\frac{1}{\sqrt{2}}(\hat{a}_{\text{out}}e^{i(\omega-\omega_{LO})t}+\hat{a}_{\text{out}}^{\dagger}e^{-i(\omega-\omega_{LO})t}),$
(9)

where $\omega_{\text{LO}}$ is the frequency of the local oscillator, and the
downconverted frequency $\omega-\omega_{LO}$ is RF. The quadrature
$\hat{X}_{\text{out}}$ corresponds to an output field,

$\displaystyle\hat{a}_{\text{out}}=\sum_{j}g_{j}e^{i\phi_{j}}\hat{a}_{j}\approx\sum_{n}\hat{a}_{n}\int\mathcal{A}(\rho)u_{n}(\rho)d\rho,$
(10)

where $g_{j}$ is the gain and $\phi_{j}$ is the net phase applied to output
$j$. The approximation is taken in the limit of an array with a large number
of narrow pixels, where the gains and phase shifts approach continuous
amplitude and phase distributions $g(\rho)$ and $\phi(\rho)$, respectively. In
Eq. (10), $\mathcal{A}(\rho)=g(\rho)e^{i\phi(\rho)}$ is a programmable array
mode function that enables wavefunction engineering. For a signal encoded in a
mode $\hat{a}_{n}$, perfect modal overlap can be achieved by setting
$\mathcal{A}(\rho)=u_{n}(\rho)$ through beamforming. Due to the orthonormality
of $\\{u_{n}(\rho)\\}$, the vacuum noise contributions across all the
receivers destructively interfere, resulting in unity geometric efficiency.
For multimode fields, the signal in a particular mode can be uniquely selected
by setting $\mathcal{A}(\rho)$ to the desired mode function [44, 45, 46].
Furthermore, the QPA acts a tunable spatial filter, rejecting quantum signals
from angles that result in destructive RF interference for a given phase
setting. This defines an angular “beamwidth” for a quantum link established by
beamforming.

The transformations on the input modes can be extended to matrix operations
after coherent detection [34]. The overall class of operations that can be
performed by the QPA chip are,

$\displaystyle\vec{a}_{\text{out}}=MDU\vec{a}_{\text{in}},$ (11)

where the input modes are grouped into the vector
$\vec{a}_{\text{in}}=(\hat{a}_{1},\hat{a}_{2},...)$, $U$ is the change-of-
basis unitary, $D=\text{diag}(g_{1}e^{i\phi_{1}},g_{2}e^{i\phi_{2}},...)$, and
$M$ is a matrix implemented after coherent detection.

### Chip design

Decoherence due to loss is one of the biggest challenges for integrated
quantum photonics [11]. The most significant source of loss for free-space-
coupled integrated systems is geometric loss due to the mode mismatch between
an impinging beam mode and the aperture mode. In the case of a collimated beam
such as a beam transmitted from a large-aperture transmitter, beam divergence
due to diffraction is the primary cause of mode mismatch between the impinging
beam and the receiver aperture. Reducing this mode mismatch requires the QPA
receiver to have a large enough effective aperture. The effective aperture can
be increased either by arraying a large number of small-area antennas or
employing a single large-area antenna. We demonstrate both approaches in the
design by arraying 32 large-area antennas. Due to the planar routing
constraints and the resulting loss from having feed waveguides inside the
partially filled aperture, we demonstrate a 1D array. Multi-layer apertures
can be used to expand this concept into 2D arrays [47].

To maximize the effective aperture of a single antenna, the coupling strength
per area needs to be minimized. To achieve this, various antenna topologies
were simulated, and a parallelized waveguide metamaterial antenna design was
determined to have the lowest scattering strength per area while abiding by
the foundry design rules. Sixteen waveguides were connected and parallelized
with sub-wavelength gratings in the regions between the waveguides. The 0.82
$\mathrm{\mu m}$ wide waveguides keep a single mode confined throughout the
length of the antenna so that the phasefront of the coupled light across the
cross-section of the antenna is flat. At the end of the antenna active area, a
mode converter comprising a taper couples the light from 0.82 $\mathrm{\mu m}$
waveguides to 0.5 $\mathrm{\mu m}$ waveguides. A Y-junction-based 16-to-1
combiner tree combines all the outputs from a single antenna into a single
mode propagating in the 0.5 $\mathrm{\mu m}$ wide waveguide that is used to
route the quantum signal on the PIC. Three grating regions with apodized
coupling strengths to mode match the amplitude profile of the impinging beam
were designed, as seen in Fig. 2. The antenna design was verified using an
FDTD simulation. The physical footprint of the antenna is 597 $\times$ 16.7
$\mathrm{\mu m^{2}}$. Across 597 $\mathrm{\mu m}$ length starting from the
splitter tree, the 0 $\mathrm{\mu m}$ to 47 $\mathrm{\mu m}$ is the splitter
tree, 47 $\mathrm{\mu m}$ to 347 $\mathrm{\mu m}$ is the apodized grating duty
cycle region, 347 $\mathrm{\mu m}$ to 547 $\mathrm{\mu m}$ is the apodized
grating width region, and 547 $\mathrm{\mu m}$ to 597 $\mathrm{\mu m}$ is the
full width region. The aperture of the chip comprises 32 of these antennas
with 17.5 $\mathrm{\mu m}$ pitch to ensure sufficiently low crosstalk between
antennas. To aid with the free-space alignment of an impinging field to the
chip and ensure uniform response across all 32 antennas, two antennas are
added on each side of the aperture, resulting in 36 total antennas. One
antenna on each side is connected to a standard grating coupler to aid
alignment with optical input/output, and the other antenna is connected to a
photodiode to aid alignment with optical input/electronic output.

The QRX design comprises a tunable Mach-Zehnder interferometer (MZI) made out
of two 50:50 directional couplers and two diode phase shifters. Each phase
shifter is 100 $\mathrm{\mu m}$ long, comprising a resistive heater made out
of doped silicon with 1 $\mathrm{\Omega}$ resistance and a diode in series
with 1 V forward voltage. Doped Si is placed 0.9 $\mathrm{\mu m}$ away from
the waveguides to minimize loss from free carriers. The MZI is configured in a
push-pull configuration to extend the tuning range of the coupling
coefficients and is designed to provide sufficient tuning with $\mathrm{\pm}$5
V drivers. One branch of the MZI includes an optical delay with 90∘ phase
shift to set the nominal coupling of the MZI to 50:50. Fabrication
imperfections such as changes in the gap in the coupling region of the
couplers and surface roughness in the waveguides between the couplers shift
this ideal 50:50 coupling randomly throughout the chip. The tunability of the
MZIs allows correcting for these imperfections to set 50:50 coupling. The MZIs
are also designed to be symmetric to ensure a high extinction ratio.

After the MZI, the waveguides are adiabatically tapered to connect to a
balanced Ge photodiode pair with >20 GHz bandwidth at 3 V reverse bias,
>70$\%$ quantum efficiency, and <100 nA dark current. The QRX is surrounded by
a Ge shield to absorb stray light propagating in the chip substrate and
prevent it from coupling to the photodiodes. Each QRX output is connected to a
separate on-chip pad to be interfaced with a transimpedance amplifier (TIA)
and subsequent electronics for RF processing.

The LO is coupled to chip with a standard grating coupler and is sent to each
QRX through a 1-to-32 splitter tree. Each Y-junction in the splitter tree has
0.28 dB loss, and the grating coupler has 3.30 dB loss. Before the splitter
tree, a directional coupler on the LO waveguide is present to couple 1% of LO
power to a monitor photodiode for LO power monitoring. After the splitter
tree, a TOPS is included in each branch to tune the quadrature phase of each
channel for the phase calibration of the system. Each TOPS for phase tuning is
315 $\mathrm{\mu m}$ long, comprising a resistive heater made out of titanium
nitride above the waveguide with 630 $\Omega$ resistance.

### Chip fabrication

The QPA PIC was fabricated in the AMF 193-nm silicon-on-insulator (SOI)
process. The process has two metal layers (2000-nm thick and 750-nm thick) for
electronic routing, a titanium nitride heater layer, a 220-nm thick silicon
layer, a 400-nm thick silicon nitride layer, germanium epitaxy, and various
implantations for active devices. A process design kit (PDK) from the foundry
was provided, and the final design was completed and verified using the
KLayout software.

### Squeezed light generation

To generate squeezed light, continuous wave light from a fiber-coupled 1550 nm
laser is split into a signal path and a local oscillator (LO) path. The light
in each path is amplified by an erbium-doped fiber amplifier. After
amplification in the signal path, the 1550 nm coherent light is upconverted to
775 nm by a periodically poled lithium niobate (PPLN) waveguide via second
harmonic generation. The upconverted light is used as a continuous-wave pump
for Type 0 spontaneous parametric down-conversion (SPDC) with another PPLN
waveguide, which generates broadband light in a squeezed vacuum state at a
central wavelength of 1550 nm. A total of four PPLN waveguides were used to
generate squeezed light in the experiments. The squeezed vacuum light is sent
to a fiber optic collimator, which transmits the light over free space with a
flat phase front to the chip aperture. After amplification in the LO path, the
1550 nm coherent light is sent to a bulk lithium niobate electro-optic
modulator for phase control. The phase-modulated local oscillator is sent to a
cleaved fiber which is grating-coupled to the LO input of the chip.
Polarization controllers before the collimator and on the LO fiber are used to
optimize coupling efficiency to the chip.

### System electronics

The QPA chip is first packaged with an interposer board for fanning the
electronic input/output (IO) to/from the chip. The interposer board is
designed with a laser-milled cavity in the middle to place the QPA chip
surrounded by pads with blind vias for high-density routing. The chip and the
interposer are assembled so that the on-chip pads are level and parallel with
the on-board pads to shorten the bond wire length. The traces from the
interposer pads to the TIA inputs on the motherboard are minimized and spaced
sufficiently apart to minimize electronic crosstalk with 50 $\Omega$ coplanar
waveguide (CPW) transmission lines. The discrete TIA circuit on the
motherboard utilizes a FET-input operational amplifier (op-amp) with resistive
feedback. The op-amp IC (LTC6269-10) has a 4 GHz gain-bandwidth product and is
used with a 50 $\mathrm{k\Omega}$ feedback resistor. The capacitance of the
feedback trace is used to ensure sufficient phase margin while keeping the
closed-loop gain greater than 10 since the op-amp is decompensated. A 50
$\mathrm{\Omega}$ resistor is placed in series with the output of the TIA for
impedance matching and to dampen any oscillations from capacitive loading at
the output. The TIA outputs are routed with 50 $\Omega$ CPW transmission lines
to a high-speed, high-density connector to route the signals to data
acquisition.

The DC voltage across the TIA feedback resistor is used as the error signal
for the CMRR correction and drives an integrator circuit with a chopper-
stabilized op-amp IC (OPA2187) for low voltage offset, flicker noise, and
offset drift. The integrator unity-gain bandwidth is set close to DC to dampen
any oscillations in the CMRR auto-correction feedback. The integrator’s output
is fed back to the MZI on the QPA chip to correct the CMRR continuously. The
polarity of the integrator is designed to match with the polarity of the push-
pull MZI so that the correction circuit always maximizes the CMRR whether the
imperfections lead to negative or positive DC current from the balanced
photodiodes. The correction is limited by the dark current of each QRX and the
offset voltage at the input of each integrator, but offset correction can be
applied to each integrator to further maximize the CMRR. The CMRR auto-
correction circuit extracts an error signal from the TIA output, probing the
imperfect CMRR of each QRX and feeding it back to each respective push-pull
MZI to continuously correct the CMRR of the QRX array. This ensures shot-noise
limited noise floor during chip measurements, maximizing the shot noise
clearance and effective efficiency. A high-speed coaxial cable assembly is
used to connect to the high-density connector on the motherboard. The cable
first connects to a power board powering the active electronics on the
motherboard. This board also routes the output from the two photodiodes
connected to the two edge antennas of the aperture and the output from the
monitor photodiode connected to the LO coupler to current meters for
continuous monitoring of signal and LO alignment on the chip. Another cable
then connects the remaining IO to a splitter board that splits the 32 QRX
outputs for simultaneous imaging and RF data acquisition. The remaining
control lines for tuning the on-chip TOPS are connected to 32 digital-to-
analog converters (DACs) for independent phase tuning of each QRX.

### Data acquisition and readout

The 32 QRX outputs after the splitter is connected to boards that host SMA
connectors to interface with data acquisition (DAQ) equipment. One board, used
for parallelized 32-channel readout, connects to 32 channels of digitizers
with 100 MHz bandwidth, 100 MSa/s adjustable sampling rate and 14-bit
resolution. The digitizers are used in high-impedance mode to read out the
voltage of each QRX output for squeezed light imaging and during RF
measurements. The other board, used for RF single channel readout, connects to
a 32-to-1 RF power combiner assembly with an operating frequency range of
0.1-200 MHz. The output from the power combiner is connected to the ESA. For
squeezed light measurements in Fig. 4b,c,e and f, the ESA is configured to be
used in the zero-span mode at a center frequency of 5.5 MHz, with a resolution
bandwidth of 2 MHz, and a video bandwidth of 5 Hz. For the measurements in
Fig. 4, the video bandwidth is 10 Hz. Center frequency and resolution
bandwidth are selected to maximize the shot noise clearance after doing a
parameter sweep.

### Phase calibration

For each angle of incidence, we optimize the settings for the 32 LO TOPS such
that the quadratures for all pixel modes are aligned to the same phase.
Precise phase calibration is crucial to prevent additional loss due to vacuum
noise leaking into the combined output. Phase calibration is performed with a
1550 nm coherent state transmitted by the collimator and a 5 MHz phase ramp is
applied to the LO before coupling to the chip. The 5 MHz downconverted RF
signal after the QRX outputs are combined is used as feedback to the computer
to tune the on-chip TOPS iteratively. Various signal processing schemes and
algorithms have been developed for beamforming in classical phased arrays such
as random search, gradient search, direct matrix inversion, and recursive
algorithms [48]. We employ a modified gradient search algorithm by sweeping
phase settings of on-chip TOPS with an orthogonal mask set. We sweep the TOPS
voltage starting with large voltage steps and continuing with progressively
smaller voltage steps with each optimization iteration. Due to the Gaussian
amplitude front, edge channels contribute less SNR to the combined output.
Therefore, we sweep channel settings starting from the edge channels and
continuing to the middle channels. As each channel is tuned and the total SNR
improves, the proportional increase in SNR from element to element gets
smaller, leading to higher errors in the optimal phase setting of the last
channels that are swept. Therefore, for each optimization iteration, we
reverse the order of channels to be swept.

### Data analysis

The squeezing and antisqueezing levels are obtained from a statistical
analysis of the quadrature sample variances or noise powers. For the
experiments in Fig. 3(4,5), quadrature sample variances (noise powers) are
acquired for squeezed vacuum and vacuum states over an approximately uniform
distribution of phases, and histograms are constructed for the acquired data.
The squeezing and antisqueezing levels are estimated from the inflection
points of the probability density functions (PDFs) of quadrature variances,
which are obtained from the Gaussian kernel density estimates (KDEs) of the
histograms. The squeezing and antisqueezing level estimates correspond to the
locations of the peak slopes at the left (right) edges of the PDF,
respectively. In particular, the quadrature variances for the squeezing and
antisqueezing levels are identified from the peaks in the derivative of the
KDEs, which provide a well-defined measure of the edges of the quadrature
variance distribution. The same estimation procedure applied to the vacuum
data yields the standard deviation in the vacuum sample variance (shot noise
level). Error bars are obtained from the propagation of the vacuum standard
deviation on the squeezing and antisqueezing level estimates.

### Measurement characterization

For each quantum measurement in the reported experiments, a classical
measurement is also taken to characterize the system. The classical
measurements are taken using the same photonic and electronic hardware chain
as the quantum measurements to ensure consistency. For the experiment in Fig.
3b, a classical multipixel image is taken by sending a coherent state as
signal while the LO phase is ramped at 5 MHz. The 5 MHz tone from each channel
is digitized by the imaging readout, and its corresponding amplitude is
measured. For the experiments in Figs. 4b,e and f, a coherent state is sent as
signal while the LO phase is ramped at 5 MHz. The 5 MHz tone at the output of
the power combiner is measured on the ESA for each measurement setting. For
each channel combination in Fig. 4b, an SNR is calculated by taking the ratio
of this signal power to the respective vacuum noise acquired from the squeezed
light measurement.

### Wigner function calculation

For the calculations of the Wigner functions in Fig. 3b, the experimental
squeezing parameter $r=1.95$ is plugged into the Wigner function
$W(X,P,r,\theta,\eta)$ of a squeezed vacuum state [49], setting $\theta=0$ and
$\eta=1$ to obtain the Wigner function at the source. The Wigner function for
each pixel mode is obtained by plugging the squeezing parameter, phase, and
geometric efficiency into $W(X,P,r,\theta,\eta)$. The phases are estimated
from a sinusoidal fit to the sample variances of each channel over a region of
the data where the phase modulation was approximately uniform. From the
squeezing parameter, the effective efficiency of each channel is estimated
using,

$\displaystyle\eta=\frac{(A-1)\exp(2r)}{(\exp(2r)-1)(A+\exp(2r))},$ (12)

where $A=\Delta X_{+}^{2}/\Delta X_{-}^{2}$ is the ratio of the antisqueezing
($\Delta X_{+}^{2}$) to squeezing ($\Delta X_{-}^{2}$) levels. The geometric
efficiencies of the pixels are calculated from the effective efficiencies of
the channels divided by their total sum.

### Theoretical modeling

The theoretical models in Fig. 4 are constructed from the classical data of
the measurement characterizations using,

$\displaystyle\Delta X_{\pm}^{2}=\eta e^{\pm 2r}+1-\eta,$ (13)

where $\Delta X_{\pm}^{2}$ are the squeezing ($-$) and antisqueezing ($+$)
levels and $r$ is the squeezing parameter, and $\eta$ is the effective
efficiency of the system.

For Fig. 4b, the model is obtained from Eq. 13 with $\eta\propto\text{SNR}$. A
least-squares fit is performed by taking the proportionality constant
($\eta_{c}$) to the classical SNR data as the only free parameter, with the
squeezing parameter bounded in the range $r=0.748\pm 0.019$. Using SNR data
normalized to its peak value, we obtain optimal parameters of $\eta_{c}=0.021$
and $r=0.761$.

For Fig. 4c, the model is obtained from Eq. 13 with $r=\mu\sqrt{P}$, where $P$
is the squeezer pump power and $\mu$ is the SPDC efficiency, and a least-
squares fit is performed taking the $\mu$ and $\eta$ as free parameters. The
optimal parameters are reported in the main text.

For Fig. 4e, the models are obtained from Eq. 13 and $\eta$ proportional to
classical beamwidth data for 8 and 32 channels combined. For each data set, a
least-squares fit is performed taking the proportionality constant
($\eta_{c}^{(n)}$) to the classical beamwidth data as the only free parameter,
with the squeezing parameter bounded in the range $r=0.607\pm 0.015$. Using
beamwidth data normalized to their peak powers, we obtain optimal parameters
of $\eta_{c}^{(8)}=0.019$, $\eta_{c}^{(32)}=0.014$, and $r=0.611$. The 8 and
32 channel beamwidths are characterized directly from the squeezed light data
by extracting the effective efficiencies using Eq. 12. With linear
interpolation, angles corresponding to 0.5 effective efficiency are found to
calculate the beamwidths.

For Fig. 4f, the models are obtained from Eq. 13 and $\eta$ proportional to
the classical radiation pattern of a single antenna. For each data set, a
least-squares fit is performed taking the proportionality constant
($\eta_{c}^{(n)}$) to the classical radiation pattern as the only free
parameter, with the squeezing parameter bounded in the range $r=0.865\pm
0.043$. Using the radiation pattern data normalized to its peak power, we find
optimal parameters of $\eta_{c}^{(8)}=0.017$, $\eta_{c}^{(32)}=0.015$, and
$r=0.908$. The 8 and 32 channel FoVs are characterized directly from the
squeezed light data in the same way as beamwidth characterization using Eq.
12.

The squeezing parameters for the models are obtained from independent
characterizations of the sources.

### Cluster state generation

Cluster states of up to eight modes have been demonstrated with bulk
multipixel homodyne detection systems by programming virtual optical networks
in digital post-processing [33]. The virtual networks mix different spatial
regions in a beam of light to match the detection basis to an entangled
spatial mode basis. This method of entanglement generation allows for highly
compact and versatile implementations of Gaussian quantum computation in the
measurement-based model [34], which can be scaled to a higher number of modes
by interfacing quantum PICs like the QPA chip with special-purpose RF or
microwave ICs.

With our architecture in Fig. 5a, the overall transformation on the input
field can be summarized as,

$\displaystyle\vec{a}_{\text{out}}=S(G\oplus G)DU\vec{a}_{\text{in}},$ (14)

where $U$ is the free-space change-of-basis unitary mapping the input modes to
pixel modes, $D=\text{diag}(e^{i\phi_{1}},e^{i\phi_{2}},...)$,

$\displaystyle G\oplus G$ $\displaystyle=\begin{pmatrix}1&1&1&1&0&0&0&0\\\
0&0&0&0&1&1&1&1\end{pmatrix}$ (15)

is the gain matrix of the RF power combiners, and

$\displaystyle S$ $\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i\\\
i&1\end{pmatrix}$ (16)

is the beamsplitter matrix. The transformation of $S$ is performed on the
digitized quadratures as an emulation of an RF directional coupler, where
complex matrix elements are implemented as a $\pi/2$ phase shift. For a two-
mode Gaussian cluster state generated with $S$, the cluster state correlations
are given by,

$\displaystyle\text{Var}(\hat{X}_{3}(\theta)-\hat{P}_{4}(\theta))$
$\displaystyle=\text{Var}(\hat{X}_{1}(\theta)),$ (17)
$\displaystyle\text{Var}(\hat{X}_{4}(\theta)-\hat{P}_{3}(\theta))$
$\displaystyle=\text{Var}(\hat{X}_{2}(\theta)),$ (18)

where $\text{Var}(\hat{X}_{i}(\theta))$ for $i=1,2$ is given by Eq. 2 for
squeezed modes, such that the right-hand side is zero at $\theta=0$ in the
limit of large squeezing parameter and low loss. We note that in our
experiment, the inseparability given by Eq. 4 has a lower bound of 0.5 since
the squeezed light was generated in a single mode. This lower bound can be
overcome by transmitting multiple squeezed modes to the chip, allowing for the
generation of large cluster states up to 32 modes.

### Chip losses

On-chip losses consist of 3.78 dB loss from simulated antenna insertion loss,
0.321 dB loss from waveguide propagation loss, and 1.52 dB loss from
photodiode quantum efficiency. This results in a total expected on-chip loss
of 5.62 dB. In addition to on-chip losses, there is also the geometric loss
due to the mode mismatch between the aperture and the collimated beam and the
insertion loss of the collimator. The on-chip losses are verified
experimentally by sending 200 $\mathrm{\mu m}$ collimated beam to the chip
aperture after setting all QRXs to the unbalanced (100:0) configuration and
summing all QRX currents. For 0.452 mW input power, the output current is
0.0615 $\mathrm{\mu A}$, resulting in an insertion loss of 8.66 dB. For a 200
$\mathrm{\mu m}$ collimated beam, the geometric loss is 1.14 dB, the insertion
loss of the collimator is 0.8 dB, and the insertion loss of the connectors is
expected to be <1 dB. De-embedding these losses from the measurement, the on-
chip losses are measured to be 5.72 dB, which agrees well with the 5.62 dB
expected loss.