[1]Eran Lustig These authors contributed equally to this work. These authors
contributed equally to this work. 1]Edward L. Ginzton Laboratory, Stanford
University, Stanford, 94305, CA, USA

# Emerging Quadrature Lattices of Kerr Combs

<EMAIL_ADDRESS>Melissa A. Guidry  Daniil M. Lukin  Shanhui Fan  Jelena
Vučković  [

###### Abstract

A quadrature lattice is a coupled array of squeezed vacuum field quadratures
that offers new avenues in shaping the quantum properties of multimode light
[1, 2, 3]. Such lattices are described within the framework of non-Hermitian,
non-dissipative physics and exhibit intriguing lattice phenomena such as
lattice exceptional points, edge-states, entanglement and non-Hermitian skin
effect, offering fundamentally new methods for controlling quantum
fluctuations [1, 4]. Nonlinear resonators are suitable for studying multimode
pair-generation processes and squeezing which are non-dissipative in
$\chi^{(2)}$ and $\chi^{(3)}$ materials [5, 6, 7, 8, 9, 10, 11, 12], but
observing non-Hermitian lattice phenomena in photonic quadrature lattices was
not achieved. Remarkably, in dissipative Kerr microcombs [13], which have
revolutionized photonic technology, such lattices emerge and govern the
quantum noise that leads to comb formation. Thus, they offer a unique
opportunity to realize quadrature lattices, and to study and manipulate
multimode quantum noise which is essential for any quantum technology. Here,
we experimentally study non-Hermitian lattice effects in photonic quadrature
lattices for the first time. Our photonic quadrature lattices emerge at Kerr
microcomb transitions, allowing us to observe fundamental connections between
dispersion symmetry, frequency-dependent squeezed supermodes, and non-
Hermitian lattice physics in an integrated setup. Our work unifies two major
fields, quantum non-Hermitian physics and Kerr combs, and opens the door to
utilizing dissipative Kerr combs to experimentally explore rich non-Hermitian
physics in the quantum regime, engineer quantum light, and develop new tools
to study the quantum noise and formation of Kerr combs.

##  1 Introduction

In recent years, there has been growing interest in photonic quadrature
lattices and their quantum properties [1]. Unlike conventional photonic
lattices, where the lattice sites are coherent field amplitudes, in photonic
quadrature lattices, each discrete site consists of two optical quadratures
that are coupled to other quadratures. Since the quadrature representation
forms the natural basis for continuous variable (CV) [14] quantum states,
photonic quadrature lattices enable the understanding of rich multimode
systems from the perspective of squeezing and deterministic entanglement
generation. Notably, Hermitian quadratic bosonic lattices translate to non-
Hermitian lattices in quadrature space. Thus, the rich physics of non-
Hermitian lattices, which includes a plethora of effects such as optical
exceptional points [15], the non-Hermitian skin effect [16], and beyond,
manifests in quadrature space with novel implications for squeezing and
entanglement [16, 17, 2, 4].

In contrast to traditional non-Hermitian optical lattices, where the interplay
between gain and loss is governed by photon absorption, quadrature lattice
dynamics are governed by parametric processes (i.e., energy-conserving) such
as nonlinear optical pair creation and annihilation. Thus, the realization of
a bosonic quadrature lattice requires both multimode parametric gain and loss,
and the coupling or transfer of bosons between lattice sites. In recent
experiments, non-Hermitian lattice effects in quadrature lattices were
observed in optomechanics [2, 18, 3] by coupling several parametrically pumped
opto-mechanical resonator modes together. In photonics, a fruitful approach
for observing non-Hermitian lattice physics is along the synthetic frequency
dimension of a single optical cavity, in which the optical modes are coupled
by modulation at the free spectral range (FSR) frequency [19, 20, 21]. Such
lattices can also be realized in a nonlinear environment [22], and even in the
non-dissipative quantum regime by coupling photons generated across multiple
frequency modes by second-order pair production with electro-optic modulation
[8]. However, observing quadrature lattices requires detecting the multimode
quadrature states with a local oscillator [23, 10, 11, 24], and probing the
non-Hermitian lattice effects, which has not been achieved so far.

Remarkably, Kerr combs [13] (including dissipative Kerr solitons [25] and
soliton crystals [26]) generated by third-order nonlinear resonators,
naturally provide both pair production [27, 28] and coupling between the
frequency modes through the process of Bragg scattering [5, 29, 30, 31]. These
properties make Kerr combs a unique system to study multimode quantum
phenomena [7, 32] and squeezing [33]. Interestingly, these Kerr combs also
function as a multi-frequency pump (in contrast to a single pump [9]) which
produces transport and amplification of the quantum fluctuations in the
resonator, giving rise to “below-threshold” phase-locked combs [7]. The
complexity of these squeezed combs, and their quadratic bosonic Hamiltonian,
which includes parametric gain and coupling terms, has only recently begun to
be treated theoretically [34, 33]. However, to date, such below-threshold
multimode quadratures have not been measured directly, and it was not clear
how these combs are related to quadrature lattices and the non-Hermitian
lattice phenomena that govern them. Additionally the major development of
dissipative Kerr microcombs, which has become the focal point of vast research
efforts, offers new possibilities in shaping quantum light, but also pose new
limitations. Thus, studying the multimode quantum squeezing and anti-squeezing
generated by them is essential for both utilizing their quantum properties and
gaining a better understanding of the quantum origin of their formation and
their quantum noise limitations.

In this work, we experimentally study non-Hermitian lattice physics in
photonic quadratures for the first time. Our quadrature lattice is formed by a
tunable dissipative Kerr comb that couples different quadratures along the
frequency dimension. We show theoretically and experimentally the relationship
between dispersion symmetries and the squeezing properties of the multi-
frequency lattice states of the below-threshold comb. Our work explores Kerr
combs exhibiting multimode squeezed states with robust single peak frequency
structure and double peak structure, thereby revealing fundamental properties
related to the quantum nature of Kerr comb formation and quantum noise. We
then observe such multimode states experimentally by measuring the anti-
squeezing and photon statistics of the below-threshold Kerr comb. Our work
demonstrates the relationship between non-Hermitian lattice physics and
squeezed Kerr combs. To this end, we use the framework of non-Hermitian
lattice physics and draw parallels with seminal classical non-Hermitian
observations such as power oscillations caused by parity-time symmetry (PT
symmetry) [15], and PT symmetric lattices [35, 36, 37]. This advancement will
enable the generation of novel quantum optical resources on-chip, manipulation
of quantum noise, and enhance the understanding of the quantum origin of Kerr
comb formation.

##  2 Results

We consider a 2-FSR Kerr comb where every other cavity mode (every even-
numbered cavity mode) is predominantly populated by coherent light, and light
is pair-generated by the Kerr comb on all the other cavity modes (Fig. 1(a)).
The loss in the odd cavity modes is greater than the parametric gain, such
that these modes are populated by below-threshold light, i.e. a squeezed
vacuum state. Each odd cavity mode is also directly coupled to neighboring odd
modes (but not to the even modes) by Bragg scattering which originates
predominantly from Kerr comb teeth separated in frequency by 2-FSR, creating a
1D lattice of frequency sites populated by vacuum states of light.

Figure 1:  Squeezed quadrature lattices induced by Kerr combs. a. A 2-FSR Kerr
comb (red) and a generated sub-threshold multimode state with squeezing
frequency structure (green). b. General scheme of the system: A Kerr ring
resonator is pumped by CW light, generating a 2-FSR bright comb (red circles).
This comb generates a sub-threshold comb (green circles), which has the
connectivity of a 1D lattice by pair generation and Bragg scattering. c. A
unit cell of the 1D lattice in the bosonic creation and annihilation operator
basis, longer range couplings decay with frequency distance.

Such a scenario can be evoked either by injecting a comb, as depicted in Fig.
1a into a Kerr resonator, or by pumping the resonator with continuous-wave
(CW) light, causing the other comb teeth to emerge spontaneously via the
symmetry-breaking phenomenon of Turing rolls [38] (Fig. 1(b)). The dynamics of
the below-threshold light at the odd modes (centered around the pump at
$\mu=0$) is given by the following Hamiltonian[7]

$\
\hat{H}=\sum_{\mu}-\Delta\omega_{\mu}\hat{a}_{\mu}^{\dagger}\hat{a}_{\mu}-\frac{g_{0}}{2}\sum_{\mu,\nu,j,k}\delta\left[\mu+\nu-
j-k\right]\left(A_{\mu}A_{\nu}\hat{a}_{j}^{\dagger}\hat{a}_{k}^{\dagger}+2A_{k}^{*}A_{\nu}a_{j}^{\dagger}a_{\mu}+\text{h.c}\right)$
(1)

where $A_{\mu}$ are above-threshold amplitudes (the Kerr comb) in even cavity
modes $\mu$, $\hat{a}_{\mu}^{\dagger}\left(\hat{a}_{\mu}\right)$ are the
creation (annihilation) operators of quantum light in odd cavity mode $\mu$,
which are described in the rotating frame of reference of the Kerr comb
$U=exp(i\hat{R}t),\;\hat{R}=\sum_{\mu}\left({\omega_{p}+\Delta\Omega\mu}\right)a^{\dagger}_{\mu}a_{\mu}$
where $\Delta\Omega$ is the frequency spacing of the Kerr-comb.
$\Delta\omega_{\mu}$ is the frequency detuning of the cavity-mode $\mu$ from
the Kerr comb frame of reference, $g_{0}$ is the nonlinear coefficient, the
Kronecker $\delta$ reflects photon azimuthal wave number conservation in the
system, and h.c stands for Hermitian conjugate.

To illustrate the quadrature lattice geometry, we plot the frequency axis in a
“folded” form in which the pump is located at the edge (Fig. 1(b-c)). Thus a
unit-cell in this lattice consists of four bosonic field operators
$a_{\mu},a^{\dagger}_{\mu},a_{-\mu},a^{\dagger}_{-\mu}$, which map to four
field quadratures (see Methods 4.1 for the derivation of the quadrature
lattice model). The simplified quadratic bosonic lattice model can be used
both to understand the fundamental sub-threshold comb behavior and to engineer
it. The sub-threshold quadrature dynamics of the Kerr comb constitutes a non-
Hermitian 1D lattice [35, 36, 39, 37], which can be either APT symmetric
(pseudo Anti-PT symmetric [16, 40]) or non-symmetric around the central point
$\mu=0$. In the APT symmetric case all the eigenvalues will be purely real or
purely imaginary, while in the non-symmetric case they are complex. In a
realistic Kerr micro-comb the APT symmetry is mainly eliminated by a mismatch
between the Kerr comb frequency spacing ($\Delta\Omega)$ and the FSR of the
cavity. To understand the impact of the APT symmetry on the quadrature lattice
we reduce the number of parameters by writing the Hamiltonian for the odd
cavity modes in a unitless form. Only two parameters need to be considered for
the minimal model: $\alpha$ which defines simplified comb amplitudes by:
$A_{\mu}/A_{0}=10^{-\mathcal{R}\cdot\left|\mu\right|/10}\equiv\alpha,\mathcal{R}>0$,
and $\Delta\tilde{\omega}_{\mu}$ which is the unitless frequency detuning of
the cavity modes from a rotating frame of reference aligned with the Kerr
comb. To the first order of $\alpha$ we obtain the following form for the
lattice Hamiltonian:

$\ \hat{H}_{\rm lat}=-g_{0}A_{0}^{2}\sum_{\mu\in\rm
odd,\mu>0}\left[\frac{1}{2}\left(\Delta\tilde{\omega}_{\mu}+2\right)\hat{c}_{\mu}^{\dagger}\hat{c}_{\mu}+2\alpha\hat{c}_{\mu}^{\dagger}\hat{c}_{\mu-2}\right]\\\
+\left[\frac{1}{2}\left(\Delta\tilde{\omega}_{-\mu}+2\right)\hat{d}_{\mu}^{\dagger}\hat{d}_{\mu}+2\alpha\hat{d}_{\mu}^{\dagger}\hat{d}_{\mu-2}\right]+\left[\hat{c}_{\mu}^{\dagger}\hat{d}_{\mu}^{\dagger}+2\alpha\hat{c}_{\mu}^{\dagger}\left(\hat{d}_{\mu-2}^{\dagger}+\hat{d}_{2+\mu}^{\dagger}\right)\right]+\text{h.c}+\mathcal{O}\left(\alpha^{2}\right)\\\
$ (2)

where $c_{\mu}\equiv a_{\mu},d_{\mu}\equiv a_{-\mu}$ for $\mu>0$, and
$d_{-1}\equiv c_{1},c_{-1}\equiv d_{1}$. We note that the higher order terms
$\mathcal{O}\left(\alpha^{n}\right)$ add Bragg coupling and pair generation
terms to $n$-nearest neighbours thereby not breaking the lattice structure. In
this “folded” representation it is possible to show that if the detuning
$\Delta\tilde{\omega}_{\mu}$ is symmetric around $\mu=0$, the bosonic
quadratic lattice in Eq. LABEL:eq2 becomes a degenerate pseudo APT symmetric
lattice which results in a eigenspectrum with eigenvalues that are purely real
or purely imaginary (Methods 4.1).

To describe realistic squeezing behavior, the bosonic quadrature lattice of
Eq. LABEL:eq2 must be finite, which naturally occurs due to group velocity
dispersion (GVD) reducing the Bragg coupling coefficient for modes away from
the center. If we define
$\Delta\tilde{\omega}_{\mu}=\Delta\tilde{\omega}_{0}+\tilde{D}^{\prime}_{1}\mu+\tilde{D}_{2}\mu^{2}/2+\mathcal{O}(\mu^{3})$,
then $\Delta\tilde{\omega}_{\mu}$ is symmetric if all the odd terms are zero.
Note that ${D}^{\prime}_{1}(\tilde{D}^{\prime}_{1})$ is the (unitless)
frequency difference between the cavity FSR and the Kerr comb’s frequency
spacing. To illustrate the implication of different $D^{\prime}_{1}$ values,
we calculate the quadrature eigenspectrum of the sub-threshold light generated
by a 2-FSR Kerr comb with fixed $A_{\mu}$ profile and varying
$D^{\prime}_{1}$. For concreteness, we use $\mathcal{R}$ and $D_{2}$ taken
from our experimental system (see Methods 4.2). Figure 2(a) shows the
symmetric case in which $D^{\prime}_{1}=0$, while Figs. 2b(c) are non-
symmetric with $D^{\prime}_{1}/2\pi=3(8)MHz$. We adjust the overall comb power
in the plots in Fig. 2 to be close to threshold for detuning
$\Delta{\omega}_{0}=2g\sum_{\mu}\left|A_{\mu}\right|^{2}$. To compute the
quadrature eigenspectrum we write Eq. LABEL:eq2 in the quadrature basis
$(p_{1}...p_{N},q_{1}...q_{N})$ where $p_{i}$ and $q_{i}$ are the quadratures
of mode $i$, and $N$ is the number of participating cavity modes. Since the
transformation to quadratures is not a unitary operation, we obtain a non-
Hermitian Hamiltonian $\mathcal{M}$. To study threshold behavior we introduce
$\Gamma=\kappa/2I_{2N\times 2N}$, where $\kappa$ is the uniform intensity loss
rate.

In Fig. 2 we analyze the eigenspectrum of $\mathcal{M}-\Gamma$ while varying
the parameter $D^{\prime}_{1}$. We observe three distinct regimes. For
$D^{\prime}_{1}=0$ (Fig. 2(a)), the eigenvalues are either purely real or have
a real part that equals $-\kappa/2$ resulting from their initial purely real
eigenvalue shifted uniformly by $\kappa/2$. Interestingly, due to the
multimode nature of the system, even when the symmetry is removed (i.e. when
$D^{\prime}_{1}\neq 0$) some states close to threshold remain with imaginary
values of 0 (i.e the partially symmetric case, see Fig. 2(b)). This gradual
decrease in the number of purely real eigenvalues or robustness is analyzed in
Methods 4.2. For the highest $D^{\prime}_{1}$ in Fig. 2(c), the modes close to
threshold all have a double peak structure (i.e., imaginary part is non zero).

Figure 2:  Dispersion symmetry and the properties of supermode squeezed
states. a.-c. Eigenvalues of the multimode states of a 2-FSR comb with
different values of $D^{\prime}_{1}$: 0, 3 and 8 MHz respectively. For larger
$D^{\prime}_{1}$ the purely real eigenvalues gradually become complex.
Highlighted quadrature supermodes are those close to threshold. d.-g The
frequency($\omega$)-dependent squeezing spectrum of the supermodes close to
threshold (highlighted). Each line represents the supermode spectrum for a
linearly swept intra cavity power, and the supermodes can be seen to narrow as
they approach threshold. The supermode in f. does not narrow asymptotically
because e. reaches threshold first. h.,i. The supermode composition as a
function of cavity modes of the multimode squeezed states, for the supermodes
in e. and f. respectively.

Each eigenvalue in Fig. 2(a-c) represents a multimode squeezed state, residing
on the odd modes. As depicted in green in Fig. 1(a) the squeezing spectrum of
each supermode uniformly evolves around each cavity mode [33]. The spectral
structure around each cavity mode is frequency dependent squeezing, a concept
which recently started to be utilized in metrology application through noise
spectrum engineering [41].

To compute the multimode frequency-dependent squeezing structure, we calculate
the supermodes of $\mathcal{M}-\Gamma$, using the Bloch-Messiah decomposition
(details in Methods 4.2). The structure of the supermodes with the highest
real part (gain) in Fig. 2(a-c) is presented in Fig. 2(d-g). Supermodes with
the highest gain will dominate in total photon number over other supermodes
and have the narrowest peaks, being closest to the optical parametric
oscillator (OPO) threshold. The OPO threshold occurs when $Re(\lambda)=0$,
where $\lambda$ denotes the eigenvalue of the supermode with the highest real
part. The frequency spacing in the double peak structure is determined by the
imaginary value of the corresponding supermode. Therefore, supermodes with
imaginary value zero, exhibit a single peak structure (“un-detuned” from the
frame of reference of the Kerr comb), while supermodes with imaginary value
that is not zero exhibit double-peak spectrum (“detuned” from the frame of
reference of the Kerr comb).

The partially symmetric case exhibits both double-peak and single peak
evolution for different supermodes with high gain simultaneously, with
frequency spacing between the peaks which is generally smaller than in the
completely non-symmetric case. The different plotted lines in each spectral
shape in Fig 2(d-g) represent the squeezing for different distances from
threshold (by changing the overall power of the Kerr comb). Together with an
intra-modal frequency-dependent structure, each supermode has a spectral
structure in the cavity mode basis, shown for the partially symmetric case in
Fig. 2(h-i). The bandwidth of these supermodes is predominantly determined by
the values of $D^{\prime}_{1}$ and $D_{2}$.

Next, we aim to experimentally observe the frequency-dependent quadrature
variance directly for the different cases described above using integrated
microring resonators [42]. In this work, we utilize 4H-silicon carbide-on-
insulator [43], which offers low losses and high third-order nonlinearity,
enabling low-power operation. We excite a localized 2-FSR Kerr comb by
injecting CW light near an avoided mode crossing, generating strong local
anomalous dispersion. The comb is naturally phase-locked via cascaded
parametric processes and centered within the telecommunications c-band (1545
nm). The characterization of the device parameters, including dispersion, is
presented in Methods 4.3 and Methods 4.5).

We begin by characterizing the modal structure of the generated 2-FSR comb in
its native basis. Using a custom-built single-photon optical spectrum analyzer
(SPOSA)[7], which provides $>100$ dB of dynamic range with single-photon
sensitivity, we simultaneously capture the photon populations above- and
below-threshold, as shown in Fig. 3(a). Mapping out the two-photon correlation
matrix of the below-threshold state can reveal the inter-modal connectivity in
the Hamiltonian, and provide initial indications of the formation of the
supermodes introduced in Fig. 1(b,c). We perform pairwise second-order photon
correlation measurements,
$g^{(2)}_{ij}(\tau)=\langle\hat{a}_{i}^{\dagger}(0)\hat{a}_{j}^{\dagger}(\tau)\hat{a}_{j}(\tau)\hat{a}_{i}(0)\rangle/{\langle\hat{a}_{i}^{\dagger}(0)\hat{a}_{i}(0)\rangle\langle\hat{a}_{j}^{\dagger}(\tau)\hat{a}_{j}(\tau)\rangle}$
on each pair of below-threshold modes in the comb, and the resulting matrix is
shown in Fig. 3(b). The signature of supermodes is seen as pair-wise
correlations between modes close in their distance from the center pump mode
(consistent with the folded picture in Fig. 1(b)).

We proceed to demonstrate the transition between the non-symmetric regime and
the symmetric regime through continuous tuning of the resonator dispersion
properties. Gradually heating the microring changes the mode splitting, and
therefore the threshold point, allowing us to vary $D^{\prime}_{1}$ (see
Methods 4.3). Tuning our below-threshold comb above threshold, we observe
through two-tone homodyne detection that adjusting the voltage of the heater
changes the secondary comb frequency structure above threshold discontinuously
(Fig. 3(c) and Methods 4.4). Figure 3(d-g) shows the above-threshold peaks for
different applied voltages. We observe three distinct voltage regimes: In the
first, the threshold presents a double peak structure with the peak separation
of approximately 300 MHz (Fig. 3(d)), significantly larger than the cavity
linewidth of 100 MHz, thus corresponding to the strongly non-symmetric regime
of Fig. 2(c,g). Further tuning the voltage, we transition discontinuously to a
regime in which the detuning is small, around 30 MHz, which corresponds to the
partially symmetric regime of Fig. 2(b,e,f) followed by a transition to a
single peak (corresponding to the symmetric case threshold crossing of Fig.
2(a,d)). The discontinuous change of the frequency spectrum with the voltage
occurs because each regime in Fig. 2 corresponds to a different supermode
crossing threshold, which is not a continuous process. Afterward, the system
abruptly transitions to the partially-symmetric state again (Fig. 3(g)). This
matches completely to the regimes described in Fig. 2.

Figure 3: 2-FSR Kerr comb and observation of the secondary threshold at
different peak separation. a. Single-photon optical spectrum analyzer (SPOSA)
measurement of photon populations in above threshold (red) and below threshold
(green) modes of the 2-FSR state. b. The two-photon correlation ($g^{(2)}$)
matrix of the below threshold light in the odd modes, showing the correlations
generated by Bragg scattering and pair-generation in the quadrature lattice.
c. Illustration of the homodyne measurement of the quadrature lattice, the LO
is generated by an electro-optic modulator (EOM) and does not have the same
spacing as the Kerr comb. d.-g. Homodyne measurements for four different
applied heater voltages, showing the different regimes from symmetric to non-
symmetric. The double peak is also observed with a single tone LO and with a
LO located at other modes as well.

The above-threshold light shown in Fig. 3(d-g) presents only a partial picture
of the eigenspectrum evolution, since it shows the single mode that reaches
threshold first, without the below-threshold quadrature variance structure. To
observe the competing multimode processes in the quantum regime, we also
measure the below-threshold quadrature vacuum fluctuations (anti-squeezing),
whose evolution precedes the formation of the above-threshold light shown in
Fig. 3(d-g). To observe the below-threshold light, we reduce shot noise using
on-chip and off-chip filters (Methods 4.3). Furthermore, we tune the power
ratio of our two-tone LO to a ratio of $LO_{1}=10\cdot LO_{-1}$, which we
found to maximize the overlap with the amplified quadrature supermode (Fig.
4(a)). Figure 4(b) shows our 2-FSR Kerr comb at the soliton crystal stage,
with symmetry corresponding to the partially symmetric case (Fig. 2(b,e,f)).
Below threshold, we observe both the single peak and double peak modes of Fig.
2(b,e,f) simultaneously. By further red-tuning the input pump frequency, we
observe that while the detuned supermode is amplified and narrows further
until it crosses threshold, the non-detuned supermode does not (Fig. 4(b)).

Focusing on the non-detuned state (Fig. 4(d,e)), we observe the oscillations
(induced by the phase sweep) dip to shot noise. Squeezing is not observed due
to out-coupling losses and incomplete overlap of the double tone LO with the
supermode, but anti-squeezing is visible. The phase-dependent quadrature
variance is consistent with the picture presented in Fig. 2(b,e,f), of the
partially symmetric state, visible only through below-threshold quadrature
measurements. In principle, the fine structure of the anti-squeezing is
equivalent to that of the squeezing but more robust to losses making it useful
for exploration. Using these states as a squeezed quantum resource would
require overcoupling to the resonator and utilizing a LO with higher
bandwidth, along with very low out-coupling losses.

Tuning more toward the symmetric state, we are able to measure the sub-
threshold spectrum of the non-detuned supermode. We perform the same
measurement as in Fig. 4(a) but for every two adjacent odd cavity modes
simultaneously using a separate laser (Fig. 4(f)). The separate laser is not
phased-locked to the pump; therefore, it does not show the quadrature phase
dependency but does provides the quadrature mean variance value and spectrum.
With this approach, we can resolve 3 separate peaks in each measurement (Fig.
4(g)). Two broad peaks represent the quadrature variance signals of two odd
cavity modes, with comb leakage in between, which is narrow (since it is
classical above-threshold light). Thus, retrieving each pair of neighbouring
odd modes allows us to measure the general non detuned quadrature supermode
structure, as shown in Fig. 4(h). The structure is similar in width to the
observed correlations in Fig. 3(b). Similar analysis was performed on the
detuned supermode giving similar multi-mode bandwidth since it is dictated by
$D_{2}$ (Methods 4.6).

Figure 4: Below-threshold quadrature measurements: partially symmetric regime
a. Scheme of the below-threshold homodyne measurement. Unlike the measurement
in Fig. 3(d-g) here the ratio of LO tones is optimized and shot noise is
filtered. b. Below-threshold quadrature oscillations induced by the phase
shifter, showing the co-existence of two quadrature supermodes - non-detuned
and detuned simultaneously, showing the partially symmetric regime directly.
c. Tuning closer to resonance for the state in (b), we observe that the
detuned state is increasing in gain while the non-detuned state remains weak,
proving that there are two separate supermodes competing. d.-e. Zooming in on
the frequency axis on the quadrature anti-squeezing oscillations of the un-
detuned below threshold state. f.-g. Scheme and a measurement of mapping the
non-detuned quadrature supermode by measuring 3 neighbouring modes with an
additional laser. Each measurement produces two below-threshold quadrature
variance together with a weak filtered comb line h. Quadrature variance of the
non-detuned peaks, showing the profile of the non-detuned variance of the
quadrature supermode.

The co-existense of quadrature supermodes and the robustness of single-peaked
supermodes have significant implications for the process of comb formation and
evolution: The two-peaked sub-threshold gain profile dictates the secondary
comb frequency structure. It is revealing to understand how the different
types of anti-squeezing structures affect the threshold from primary 2-FSR
comb to a 1-FSR secondary comb. Before presenting the experimental results, we
wish to gain further insights from the Lugiato–Lefever equations (LLE) [44]
and simulate the classical non-linear evolution at the threshold (see LLE
simulation details in Methods 4.5). By tuning the quality factors and
dispersion in the simulation to impose a small $D^{\prime}_{1}$, we can
simulate a non-detuned threshold to a secondary comb (Fig. 5(a-b)). In this
type of threshold, the 2-FSR comb turns into a 1-FSR comb but does not form an
rf beatnote as is generally the case [45]. Therefore, only after the first
threshold to an un-detuned 1-FSR comb, a second threshold of a detuned comb
occurs between the 1-FSR comb and an additional 1-FSR comb, which in-turn
leads to an rf beatnote and transmission fluctuations Fig. 5(b).

Figure 5: Primary to secondary threshold behavior of the quadrature supermodes
for different regimes. a-b. LLE simulation of a non detuned supermode that
cross threshold and becomes a secondary 1-FSR comb. The simulation shows that
it is followed by a second threshold in which a detuned state crosses
threshold, which is then followed by the generation of light that causes an rf
beatnote and transmission oscillations. c-d. The experimentally observed
evolution of a non-detuned supermode into a secondary comb in the symmetric
regime; c. shows the first threshold of a, d. is the second threshold that
follows c and is detuned. e-f. A direct detuned threshold corresponding to the
completely non-symmetric case. Unlike a-d there is only a single threshold
before transmission oscillations. Plots e and f show the same threshold at
different modes (+1 and +3).

Returning to our experiment, we follow the sub-threshold light with the same
homodyne measurement as in Fig. 4(a). We take multiple snapshots of the
quadrature structure during the process of threshold crossing. In Fig. 5(c),
we observe how the non detuned sub-threshold anti-squeezed light crosses the
threshold. The first threshold crossing results in a 1-FSR comb, and no rf
beatnote. This is followed by a second quadrature supermode, this time
detuned, crossing the threshold and giving rise to sub-combs (Fig. 5(d)). One
can clearly see that after the threshold crossing of the un-detuned mode, the
detuned mode still exists and keeps growing until it also crosses the
threshold. Thus, our measurements reveal the mechanism for creating a 1-FSR
comb from a 2-FSR primary comb without formation of an rf beatnote. The case
of the un-detuned threshold contrasts with the case in which the detuned
quadrature supermode reaches threshold first. In that case, as we show in two
different measurements of the same quadrature supermode in Fig. 5(e,f), a comb
with an rf beatnote can appear immediately after the first threshold. The two
measurements are single tone homodyne measurements at different modes
($\mu=1,\mu=3$). Additionally, we present five more measurements in Methods
4.6 to show the uniformity of the quadrature variance structure along
different cavity modes. This demonstrates how both types of quadrature
supermodes result in different types of threshold behavior - one threshold vs
two consecutive thresholds.

##  3 Discussion

In this work we demonstrated non-Hermitian lattice effects in photonic
multimode quantum fluctuations described as photonic quadrature lattices. We
used Kerr combs that generate photonic quadrature lattices to experimentally
explore their non-Hermitian properties for the first time. By controlling the
resonator dispersion via an integrated microheater, we were able to observe
the three theoretically predicted regimes corresponding to different non-
Hermitian lattice behavior. Namely, we observed the symmetric case, partially
symmetric, and non-symmetric. Kerr combs allow scalability and rich complexity
on a compact on-chip platform. By measuring the anti-squeezing of 2-mode
projection of the quadrature supermode, we revealed the existence of detuned
and robust non-detuned squeezed supermodes and explained how these form in the
context of lattice symmetries. Exploring such fundamental properties paves the
way for designing more complex lattice phenomena by injecting or probing
different above-threshold states. Furthermore, understanding comb formation
and the quantum noise of Kerr combs with our approach can lead to improved
comb technology, particularly for combs that are governed by quantum noise
which dictates the ultimate limit on comb stability. The concepts studied here
form a bridge that will enable the exploitation of the rich non-Hermitian
lattice physics in quadrature bosonic lattices to engineer complex quantum
resources on-chip. The complex transport of quadratures offered by the Kerr
effect and demonstrated in this work will lead to lattice phenomena such as
non-Hermitian skin effect, Anderson localization, edge-states and topological
phases. These phenomena would manifest as quadrature modes offering new
opportunities to control quantum light.

##  4 Methods

###  4.1 Band model for quadrature lattice induced by a Kerr comb

As shown in Eq. LABEL:eq2 in the main text, 2-FSR Kerr combs generate 1D
quadrature lattices on the frequency axis. The lattice geometry allows
reducing the complexity of modeling the physics from a general network of
quadratures to a 1D lattice geometry. The goal of this section is to explain
the physics of the quadrature lattice by deriving an analytical expression for
the band model of the quadrature lattice and its Brillouin zone. First, we
transform the Hamiltonian in Eq. LABEL:eq2 into the quadrature basis
$\vec{R}=(p_{-N},...,p_{N},q_{-N}...,q_{N})^{T}$ where
$p_{\mu}=\frac{1}{\sqrt{2}}\left(a_{\mu}^{\dagger}+a_{\mu}\right),q_{\mu}=\frac{i}{\sqrt{2}}\left(a_{\mu}^{\dagger}-a_{\mu}\right)$
and write the quadrature non-Hermitian Hamiltonian $\mathcal{M}$ [46], which
satisfies the Heisenberg equation:

$\ \frac{\partial\vec{R}}{\partial t}=\mathcal{M}\vec{R}$ (3)

We obtain translation symmetry of the lattice by assuming periodic boundary
conditions and flat dispersion $\Delta\tilde{\omega}_{\mu}$ in the folded
lattice, meaning that
$\Delta\tilde{\omega}_{\mu}=\tilde{\delta}_{A}\Theta(\omega-\mu)+\tilde{\delta}_{B}\Theta(\omega+\mu)$
where $\Theta$ is the step function and $\tilde{\delta}_{A(B)}$ are real
numbers. We again use the operators $\hat{c}_{\mu},\hat{d}_{\mu}$ and their
appropriate quadratures:
$\hat{p}_{c,\mu},\hat{q}_{c,\mu},\hat{p}_{d,\mu},\hat{q}_{d,\mu}$ . The
difference $\tilde{\delta}_{A}-\tilde{\delta}_{B}$ creates an asymmetry in the
dispersion $\Delta\tilde{\omega}_{\mu}$ which removes the anti-parity
symmetry, thus it functions as a simplified representation of the mismatch
$D^{\prime}_{1}$. By using the translation symmetry, we transition to the
reciprocal space defined by the good quantum number $k$, obtaining a $4\times
4$ Hamiltonian:

$\ \mathcal{M}_{k}=\left(\begin{array}[]{cccc}0&g_{A,k}&0&f_{k}\\\
-g_{A,k}&0&f_{k}&0\\\ 0&f_{k}&0&g_{B,k}\\\
f_{k}&0&-g_{B,k}&0\end{array}\right)\left(\begin{array}[]{c}\hat{p}_{d,k}\\\
\hat{q}_{d,k}\\\ \hat{p}_{c,k}\\\ \hat{q}_{c,k}\end{array}\right)$ (4)

where the variables $g_{A(B),k}=\left(\tilde{\delta}_{A(B)}+2\right)+2\alpha
e^{ik}+2\alpha e^{-ik}$ originate from the dispersion, Bragg scattering, and
self phase modulation, while $f_{k}=-1-2\alpha e^{ik}-2\alpha e^{-ik}$ are
pair generation terms, and
$\mathcal{M}=\sum_{k}{\mathcal{M}_{k}+\mathcal{O}(\alpha^{2})}$. The
Hamiltonian $\mathcal{M}_{k}$ has pseudo anti-parity symmetry (implying that
the eigenvalues of $\mathcal{M}_{k}$ are either purely real or purely
imaginary) if $\tilde{\delta}_{A}=\tilde{\delta}_{B}$, which shows why there
exist a detuned threshold when $\Delta\tilde{\omega}_{\mu}$ is not symmetric
around $\mu=0$.

Figure 6: 2-band model of a quadrature lattice induced by a 2-FSR Kerr comb.
a. Illustration of the above threshold comb (red) and below threshold comb
(green) in the model, and of the different terms - pair generation and Bragg
scattering that couple the different modes. b. The 1D lattice unit cell, in
the generalized quadrature basis
$\left(\tilde{p}^{\prime}_{\mu},\tilde{q}^{\prime}_{\mu}\right)$. c-e.
Different band topologies (different number of exceptional points) of the
Brilloin zone of the lattice in b, plot c corresponds to $A_{\pm
2}/A_{0}=0.25,\Delta\tilde{\omega}=2$, plot d corresponds to
$A_{2}/A_{0}=0.25,A_{-2}/A_{0}=0.2,\Delta\tilde{\omega}=2$ and plot e to
$A_{\pm 2}/A_{0}=0.25,\Delta\tilde{\omega}=0$. f-i. The number of exceptional
points for different 2-FSR Kerr combs on different slices of the 3D parameter
space. The parameter space is spanned by the two ratios of the sidebands and
the overall detuning

Therefore, by considering the case in which
$\tilde{\delta}_{A}=\tilde{\delta}_{B}\equiv\tilde{\delta}$ the quadrature
Hamiltonian $\mathcal{M}_{k}$ becomes degenerate. We simplify the treatment
further by transitioning to a new quadrature basis in the unit cell according
to $\vec{R^{\prime}}=U\vec{R}$ where
$\vec{R^{\prime}}=\left(p^{\prime}_{1,k},q^{\prime}_{1,k},p^{\prime}_{2,k},q^{\prime}_{2,k}\right)$
is a generalized quadrature basis, and the unitary is:

$\ U=\frac{1}{4}\left(\begin{array}[]{cccc}1&1&-1&1\\\ -1&1&-1&-1\\\
1&1&1&-1\\\ -1&1&1&1\end{array}\right)$ (5)

Applying the change of coordinates with $U$ leads to the following 2 by 2
Hamiltonian:

$\ \mathcal{M^{\prime}}_{k}=\left(\begin{array}[]{cc}1+2\alpha e^{ik}+2\alpha
e^{-ik}&\left(\tilde{\delta}+2\right)+2\alpha e^{ik}+2\alpha e^{-ik}\\\
-\left(\tilde{\delta}+2\right)-2\alpha e^{ik}-2\alpha e^{-ik}&-1-2\alpha
e^{ik}-2\alpha e^{-ik}\end{array}\right)$ (6)

which is related to $\mathcal{M}_{k}$ of Eq. 4 by
$\mathcal{M}_{k}=U^{\dagger}\mathbf{I}_{2,2}\otimes\mathcal{M}^{\prime}_{k}U$.
The 2-band model of Eq. 6 represents the most basic lattice quadrature physics
and is pseudo Hermitian with APT symmetry. This means that its eigenvalues are
either purely real or purely imaginary and it has exceptional points for
appropriate lattice parameters. To illustrate the APT symmetry consider the
operators $\mathcal{P}=\sigma_{x}$ and $\mathcal{T}=K$, where $\sigma_{x}$ is
the x Pauli matrix, and $K$ is the conjugation operator. These operators
$\mathcal{PT}$ and $\mathcal{M^{\prime}}_{k}$ are anti-commuting:
$\\{\mathcal{PT},\mathcal{M^{\prime}}_{k}\\}=0$ which shows the APT symmetry.

Due to the degeneracy, we use only two variables:
$\left(p^{\prime}_{k},q^{\prime}_{k}\right)$, allowing us to explore some of
the fundamental physics of our lattices. Figure 6 analyzes the 2-band model of
Eq. 6. Figure 6(a) shows different non-linear terms and the structure of the
comb (red) and sub-comb(green), while Fig. 6(b) presents the folded
representation where translation symmetry exists.

Next, we analyze the parameter space of the comb shown in Fig. 6(a) where
$\alpha<0.5$. In this parameter sub-space there can exist 0,2 or 4 exceptional
points in the Brillouin zone (Fig. 6(c-d)). We map the 3D parameter space for
combs (also for combs with $A_{2}\neq A_{-2}$) presenting the different band
topologies (Fig. 6(f-i)). We note that having 4 exceptional points is not an
inherent maximum; beyond the presented parameter space, more exceptional
points can exist. Additionally, the relative phases between the comb tones can
influence the number of exceptional points when more than two pumping
amplitudes are considered. Therefore, while we provide the basic mechanisms
governing such lattices, they can become significantly more complex. The
analysis of the 2-band or 4-band models sheds light on the eigen-vector and
eigen-value structures. However, the infinite lattice cannot reach threshold
by definition and is therefore not suitable for studying multi-mode squeezing
phenomena on its own, and must be truncated for that purpose. More details on
the truncation and its implication on the results of the 2-band model will be
discussed in the next section.

###  4.2 The finite model and analysis of the robustness of non-detuned
quadrature lattice modes

In this section, we continue with the theoretical analysis akin to Methods 4.1
focusing on finite lattices. Similar to structures in physical space, a
realistic lattice in the frequency domain is inherently finite. Naturally,
dispersion is never completely flat (i.e group velocity dispersion or higher-
order terms are never zero), implying that as the lattice extends towards
higher and lower frequencies away from the pump, the detuning between the cold
cavity dispersion and the quadrature lattice modes increases. This effect
reduces the coupling efficiency of Bragg scattering and possibly pair
generation, leading to “soft edges” and resulting in a gradual decrease in
transport between frequencies. In contrast, edges can also be made abrupt
through engineering of mode crossing or other type of frequency dependent
defects [20]. These sharp edges can be desirable as they may host confined
edge-states in the synthetic frequency dimension [47]. While Kerr resonators
can accommodate both options, our focus here is on naturally occurring edges
due to non-zero group velocity dispersion.

Furthermore, we will numerically demonstrate that the non-detuned multimode
states persist even when dispersion is non-flat and are robust against the
breaking of dispersion symmetry around the pump frequency $\mu=0$. Starting
from Eq. LABEL:eq2, we assume
$\Delta\tilde{\omega}_{\mu}=\omega_{p}+D^{\prime}_{1}\mu+D_{2}\mu^{2}$, where
$D^{\prime}_{1}=\Delta\Omega-D_{1}$ and $D_{1}$ represents the FSR. When
$D_{2}\neq 0$ and/or $D^{\prime}_{1}\neq 0$ the multimode states localize,
allowing us to analyze them using a finite Hamiltonian $\mathcal{M}$. For this
analysis, we assume equal quality factors for all resonances
$\Gamma=\kappa/2\textbf{I}_{2N\times 2N}$. Since $\Gamma$ is real and
proportional to the unit matrix, it only translates the real part of the
eigenspectrum of $\mathcal{M}-\Gamma$, and does not influence the imaginary
part of the eigenvalues or the peak separation of the squeezed multimode
states. Following the conventions in [46] the input-output relation of the
system and its Fourier transform are given by:

$\
\frac{d\mathbf{R}_{out}}{dt}=\left(-\Gamma+\mathcal{M}\right)\mathbf{R}\left(t\right)+\sqrt{2\Gamma}\mathbf{R}_{in}\left(t\right)$
(7) $\
\textbf{R}_{out}\left(\omega\right)=S\left(\omega\right)\mathbf{R}_{in}\left(\omega\right)$
(8)

The linear response matrix $S\left(\omega\right)$ describes in our case how
the system amplifies and squeezes the quantum fluctuations in the quadrature
lattice and is given by:

$\
S\left(\omega\right)=\sqrt{2\Gamma}\left(i\omega\mathbf{I}+\Gamma-\mathcal{M}\right)^{-1}\sqrt{2\Gamma}-\mathbf{I}$
(9)

The matrix $S\left(\omega\right)$ can be decomposed by the Bloch-Messiah
decomposition which leads to:

$\
S\left(\omega\right)=U\left(\omega\right)D\left(\omega\right)V^{\dagger}\left(\omega\right)$
(10)

where the matrices $U$ and $V$ are unitary and
$D\left(\omega\right)=\text{diag}\left\\{d_{1}\left(\omega\right),...,d_{N}\left(\omega\right)|d_{1}^{-1}\left(\omega\right),...,d_{N}^{-1}\left(\omega\right)\right\\}$
corresponds to squeezing and anti-squeezing values of the different quadrature
supermodes. Next we aim to show more broadly the impact of group velocity
dispersion and asymmetry in dispersion relative to the comb on the multi-mode
states. In Fig. 7(a-b) we present a typical complex eigenvalue band structure
for a quadrature lattice with $D_{2}\neq 0$, setting $D_{2}/2\pi=1.2~{}MHz$,
$\mathcal{R}=0.225$, and $Q_{i}=1.74\cdot 10^{6}$, which are experimental
parameters also used in Fig. 2 of the main text. Clearly, dispersion
compresses the spectrum bandwidth of the quadrature supermode and the number
of participating supermodes. By focusing on the most relevant supermodes, we
observe that APT symmetry persists: Supermodes are either purely real (un-
detuned) or purely imaginary (detuned), with the purely real ones exhibiting
higher gain.

Figure 7: Existence of non-detuned state in the presence of non-zero $D_{2}$
and $D^{\prime}_{1}$ a. Eigenvalue solution of the quadrature Hamiltonian
$\mathcal{M}$ for a 2-FSR Kerr comb with side band drop ratio of
$\mathcal{R}=0.225$, and $D_{2}/2\pi=1.2~{}MHz$. b. Zoom-in on the quadrature
supermodes with higher-gain showing that all of them are undetuned (imaginary
value is 0.) c-d. Same as a. and b. only with $D^{\prime}_{1}/2\pi=2.8~{}MHz$.
e.-f. Numerically counting the un-detuned states as a function of underlying
GVD (D2) and detuning of the comb for the symmetric case of $D^{\prime}_{1}=0$
and the non-symmetric case $D^{\prime}_{1}/2\pi=2.8~{}MHz$

Introducing a mismatch between the cavity FSR and the Kerr-comb frequency
spacing eliminates lattice symmetry, yielding more general complex eigenvalue
solutions for the $\mathcal{M}$ matrix. When the mismatch is sufficiently
small, as depicted in Fig. 7(c-d), we observe that some supermodes remain with
purely real eigenvalues while others do not. Figure 7(e,f) shows the existence
of un-detuned (imaginary eigenvalue is 0) supermodes as a function of $D_{2}$
and constant pump detuning. Figure 7(e) enumerates supermodes with purely real
eigenvalues for $D^{\prime}_{1}=0$, while Fig. 7(f) counts the supermodes when
$D^{\prime}_{1}/2\pi=2.8~{}MHz$.

###  4.3 Additional data on fabricated device and experimental setup

Our devices are etched from a 550 nm thick 4H-SiC-on-Insulator [43] and capped
with SiO2. Microheaters are patterned via liftoff of 100 nm platinum with a 5
nm titanium adhesion layer. Coupling to and from the chip is achieved using
lensed fibers, with collection efficiency of up to 50%.

Figure 8: Micro-ring resonator: fabrication and dispersion. a. Optical image
of the Kerr ring resonator alongside the smaller on-chip filter ring and
heaters. b. Integrated dispersion of the TE mode of the Kerr ring resonator
(blue) and fit to $D_{2}=1.2~{}MHz,D_{1}=152.7$ GHz (red). Dispersion defects
are avoided mode crossings with other mode families.

The ring resonators exhibit intrinsic quality factors of up to 8 million, with
typical values around 1.5 million. The resonators exhibit a $D_{2}/2\pi$ of
1.2MHz, and an FSR of 153 GHz. Additionally, the ring is multimode, causing
mode splitting through modal interactions (Fig. 8(b), which allows a broad
range of narrow-bandwidth states to be observed in the devices, such as 2-FSR
primary combs (Fig. 9(a-b)), as well as the transition to 1-FSR combs and
soliton crystals. The heaters modify the mode-splitting properties, thereby
altering the detuning at threshold and consequently the Kerr comb’s frequency
spacing $\Delta\Omega$. Figure 9 presents the experimental evolution of the
comb for the symmetric case. The comb starts as a 2-FSR primary comb (Fig.
9(a-b)) and undergoes an undetuned (single peak) threshold to a 1-FSR phase
locked comb (Fig. 9(c)). Following the threshold, a second detuned (double
peak) threshold occurs followed by additonal thresholds and an rf beatnote in
the comb (Fig. 9(d,e)). Finally, the comb abruptly turns to a 2-FSR soliton
crystal (Fig. 9(f)).

Figure 9: Kerr comb for different pump detuning - a single evolution. a.-b.  A
2-FSR primary comb, where b. is closer to the secondary threshold (the pump is
at a longer wavelength) than a. c. Tuning the pump to longer wavelengths, the
2-FSR comb evolves to a single phase locked 1-FSR comb corresponding with Fig.
5(c-d). d.-e. Additional thresholds creating additional 1-FSR combs with
subcombs corresponding with Fig. 5(d). f. Red tuning the pump further results
in a 2-FSR soliton crystal (corresponding with the measurements in Fig. 4)

Next, we provide a more detailed description of the experimental setup (Fig.
10(a-b)). We utilize two tunable CW telecom lasers named CW laser 1 (Velocity,
1520-1570 nm, TLB-6728) and CW laser 2 (Toptica CTL1500: 1460-1570nm). The
lasers are split into two channels. The first channel (upper channel in Fig.
10(b)) receives light from laser 1 and after amplification through an EDFA
excites the Kerr resonator. The Kerr resonator outputs two channels: the first
carries the quantum signal with the filtered Kerr comb directly to the
homodyne unit, while the second channel contains the Kerr comb passing
thorough the filter ring to a photodiode.

The second channel (bottom in Fig. 10(b)), receives light either from CW laser
1 or CW laser 2 or both. CW laser 1 is used for measuring modes -1 and 1,
while CW laser 2 allows measuring other modes but is not phase-locked to CW
laser 1.

Figure 10: Principle components of the full experimental setup for quadrature
variance measurements. a. Realistic illustration of the table top setup. b.
Schematic diagram of the setup presented in a. the abbreviations are: EDFA-
erbium doped amplifier, EOM- electro-optic modulator, WS- waveshaper, PD-
photodiode, pre-amp- low-power EDFA,

To span a 3-FSR range with our LO, we employ an electro-optic modulator
(EOSpace LiNbO3 20GHz phase modulator). Subsequently, a programmable
WaveShaper (4000A multiport optical processor) transmits specific sidebands
that overlap with the modes of the Kerr comb, filtering out all other lines.
The frequency tones that overlap with the classical comb are mixed with the
teeth that pass through the filter ring in order to track the frequency
difference between the Kerr comb and the LO (center of the diagram in Fig.
10(b)). The frequency tones of the LO that overlap with the quantum signal are
passed through a piezo fiber stretcher that stretches the fiber to change the
phase of the LO. This continuous phase ramp is imprinted on the homodyne
signal, allowing to distinguish it from other non-phase dependent
fluctuations. The signal is then amplified with a low input power EDFA
(OptiLab-MSA Pre-Amp EDFA Module, C-band) and then mixed with the quantum
signal from the Kerr comb and fed into a balanced photodetector pair (WL-
BPD1GA 1 GHz Dual-Balanced InGaAs Low Noise), to perform the homodyne
measurement.

###  4.4 Multimode Homodyne detection

To measure the maximal squeezing and anti-squeezing of the multimode below-
threshold light, balanced homodyne detection must be performed with a multi-
tone local oscillator (LO). The intensity and relative phase of each line of
the LO should match the maximally-squeezed supermode of the squeezed vacuum
[10]. In this work, the multi-tone LO is generated from a 20 GHz EOM comb [48]
driven by the same pump laser. The low bandwidth of a single phase modulator
(or a few cascaded phase modulators) requires either working with relatively
large micro-resonators or only partial overlap with the maximally squeezed
supermode. Large micro-resonators pose additional challenges, including higher
operation powers (for the same intrinsic quality factor) and a lower $D_{2}$
for the same resonator cross-section. The theory and experimental validation
we present in this work does not overlap optimally with the squeezed
supermode, but is sufficient to capture the frequency-dependence of the most
squeezed states. Therefore we are able to use a two-tone local oscillator
despite the fact that the connectivity expands beyond two modes. We found that
the two-tone local oscillator measures significantly stronger quadrature
variance signal than a single-tone local oscillator by improving the overlap
with the squeezed mode. Additionally, the two-tone symmetric LO may measure a
time-independent phase relative to the signal in the absence of active phase
locking of the EOM comb to the repetition rate of the Kerr comb [49].

We now provide further mathematical details on the multi-tone homodyne
measurement.

To calculate the noise variance spectrum for a multi-tone local oscillator at
the repetition rate of the comb [33], we can define a supermode decay operator

$\ \hat{L}(t)=\sum_{j}\alpha_{j}\hat{a}_{j}(t)$ (11)

whose Hermitian conjugate creates a photon in a superposition of azimuthal
modes $\hat{a}_{j}(t)$ across the frequency comb.
$\alpha_{j}=|\alpha_{j}|e^{i\phi_{j}}$ defines the amplitude and phase
composition of the supermode. Its associated supermode output operator is

$\ \hat{L}_{\text{out}}(t)=\sum_{j}\alpha_{j}\hat{b}_{\text{out},j}(t)$ (12)

where $\hat{b}_{\text{out},j}(t)$ is the output operator of mode
$\hat{a}_{j}(t)$. The corresponding Hermitian quadrature operators of
$\hat{L}_{\text{out}}$ are:

$\
\hat{L}_{\text{out}}^{(+)}(t)=\frac{1}{\sqrt{2}}\big{(}\hat{L}_{\text{out}}^{\dagger}(t)+\hat{L}_{\text{out}}(t)\big{)}$
(13)

$\
\hat{L}_{\text{out}}^{(-)}(t)=\frac{i}{\sqrt{2}}\big{(}\hat{L}_{\text{out}}^{\dagger}(t)-\hat{L}_{\text{out}}(t)\big{)}$
(14)

The noise variance is then defined as

$\
V^{(\pm)}(\omega)=\int_{-\infty}^{\infty}d\tau\langle\hat{L}_{\text{out}}^{(\pm)}(t)\hat{L}_{\text{out}}^{(\pm)}(t+\tau)\rangle
e^{i\omega\tau}.$ (15)

The maximally-squeezed supermodes can be obtained from the columns of
$U(\omega)$ (Eq. 10), which defines the linear combination of quadratures
$(q_{1}(t),...,q_{n}(t)|p_{1}(t),...,p_{n}(t))^{T}$ for a given supermode.
When $U(\omega)$ is real, then the composition for the $k$th
maximally-(anti)squeezed supermode at $\omega$ may be written:

$\ |\alpha_{j}|=\sqrt{U_{j,k}(\omega)^{2}+U_{j+n,k}(\omega)^{2}}$ (16)

$\ \phi_{j}=\operatorname{atan2}{(U_{j+n,k}(\omega),U_{j,k}(\omega))}$ (17)

The quantum (anti)-squeezing is therefore significant when the local
oscillator composition overlaps significantly with a multimode state that has
large value $d_{i}\left(\omega\right)$. In this approach maximum quadrature
variance can be obtained through this method for the undetuned squeezing.
Since the undetuned squeezing occurs at $\omega=0$, the value of its
eigenvector is real $U\left(0\right)\in\mathbb{R}$ and a time-independent
local oscillator (in the frame rotating at the repetition rate of the comb)
will measure maximal (anti)-squeezing. In the case of detuned squeezing,
$U\left(\omega\right)$ is complex and a stationary local oscillator will not
necessarily measure maximal squeezing.

However, when measuring a symmetric double peak configuration one may obtain
maximal squeezing [49]: when a local oscillator is equidistant between a
nondegenerate squeezed state, measurement at the sideband frequency probes
both modes. If the double peak is symmetric, we are summing two peaks with
opposite Fourier components. Of course, as shown in Methods 4.6 the double
peak quadrature variance spectrum can be asymmetric.

###  4.5 Additional data on LLE simulation

In order to support the theoretical conclusions and experimental observations,
we preformed LLE simulations of a 2-FSR Kerr comb. The LLE simulates the
classical comb dynamics [44], and allows us to focus on specific aspects, some
not previously addressed in the experiment. Particularly, the squeezing
structures, both detuned and non-detuned, significantly influence the
formation of secondary combs. Hence, studying the primary comb to secondary
comb transitions in a classical framework is valuable in the context of the
findings presented in this work.

We start by simulating a perturbed dispersion yielding a comb similar to that
of our experiment. The dispersion perturbation in our setup changes in
response to heating. For the same reason, the perturbation also changes as a
function of power in the cavity (i.e it is changing during the evolution of
the comb). However, since the effects we describe do not require this
behavior, we can observe the main features under constant dispersion. We use
experimentally extracted parameters of $D_{2}/2\pi=1.2MHz$ and extract the
perturbation parameters from the measured dispersion. For simplicity, we only
use the perturbed modes near the pump modes $-2,-1,0,1,2$. For quality
factors, we use a constant value of 1.83 million, which we reduce at modes
distant 17 FSR away from the pump to avoid pair generation there. The
normalized pump power is $f=5.93$ where
$f^{2}=8g_{0}\kappa_{c}P_{in}/\kappa^{3}\hbar\omega_{0}$ and $P_{in}$ is the
input power, $\kappa_{c}$ is the outcoupling power loss, and $\omega_{0}$ is
the pump angular frequency.

Importantly, the formation of our 2-FSR comb is not a direct result of the
anomalous dispersion, but it is formed locally due to the dispersion defect
(see Methods 4.3). This means that the initial comb formation may be governed
by adjusting the dispersion of sites $-2,0,2$. The detuning value of the
threshold to a 1-FSR comb depends on the dispersion of the odd modes and the
perturbation of modes $-1,1$. By adjusting these parameters, we are able to
simulate a comb evolution that is qualitatively similar to what we observe in
the experiment.

Figures 5(a-b) and 11(a) show the comb evolution in the simulation, which is
similar to our experiment (Fig. 9). The process begins with a 2-FSR primary
comb which transitions to a 1-FSR secondary comb. Figure 11 demonstrates the
transitions from one rolling Turing pattern to another. Since this transition
is un-detuned, the first 1-FSR comb is a single phase locked comb and does not
have sub combs.

Figure 11: Intracavity field in real space for different regimes in the comb
formation - LLE with perturbed dispersion - undetuned threshold. a.
Transmission as a function of continuous pump detuning (toward longer
wavelengths). This plot is the same as in Fig. 5(a) but the shaded areas mark
different regimes plotted in (b-d). b. Intracavity field in real space for the
transition between the primary comb (2-FSR) and the secondary comb (1-FSR)
corresponding to the shaded red region in a. c. Same as b but for the
transition from the secondary comb (1-FSR) to a second detuned 1-FSR comb
which oscillates (shaded green region). d. The region of a 2-FSR soliton
crystal (shaded blue region).

The next transition involves the threshold crossing of a detuned 1-FSR comb,
which produces an rf beatnote with the existing 1-FSR comb. The oscillations
in this state are evident in both the transmission and the intra-cavity field
(Fig. 11(c)). Finally the system transitions to a 2-FSR soliton crystal (Fig.
11(d)). To better understand the unique symmetric transition from a 2-FSR comb
to a single phase locked 1-FSR comb, we explore the comb state at a pump
detuning close to this threshold at 0.115 GHz.

Figure 12: Examination of the simulated threshold between an un-detuned 2-FSR
comb to an un-detuned 1-FSR comb at 0.115 GHz detuning. a. Spectrum of the
2-FSR primary Kerr comb (red) and light in odd modes (green). b. Intracavity
field in real space (rolling turing) c. $g^{(2)}$ correlations of the below
threshold light. d. Eigen-spectrum of the comb showing that an un-detuned
supermode crosses threshold, and that the spectrum corresponds to the
partially symmetric case. e. The squeezing structure, showing the single peak
which correspond with the un-detuned supermode that crosses threshold in d. f.
The spectrum of the most squeezed state

Figure 12(a-b) shows the comb from Fig. 11 for the detuning of 0.115 GHz in
both the frequency (mode) space and real physical space respectively. Using
the obtained comb amplitudes and the hot-cavity dispersion, we calculate the
$g^{(2)}$ correlations of each cavity-mode $\mu$ with each cavity-mode $\nu$,
which resembles the experimental $g^{(2)}$ in Fig. 3(b). Next, we calculate
the quadrature Hamiltonian $M$ associated with the 2-FSR comb at 0.115GHz. The
eigenspectrum is plotted in Fig. 12(d), showing that the non-detuned state
(purely real eigenvalue) is approaching threshold. Correspondingly, the
squeezing structure is a single peak (Fig. 12(e)), localized near the pump
(Fig. 12(f)).

In addition to the perturbed dispersion case, we wish to explore in simulation
the case of un-perturbed dispersion with a $D_{2}>0$ term. Throughout this
work, a perturbation in dispersion somewhat detached the general dispersion of
the cavity from the frequency spacing of the 2-FSR Kerr comb, allowing for
more possible behaviors.

Figure 13: Simulation of a 2-FSR primary comb transition to a 1-FSR secondary
comb in a unperturbed dispersion. a. Transmission vs pump-detuning of the
2-FSR comb without dispersion perturbation. b. spectrum of the 2-FSR Kerr comb
(red) in an unperturbed dispersion. c. Eigen-spectrum showing 40 supermodes.
In this case all the modes are detuned (purely imaginary eigenvalues) while
the Hamiltonian is symmetric. d. Zoom-in on the spectrum in c. e. The
squeezing structure, showing the double peak which correspond with the detuned
state that crosses threshold in c,d f.. The spectrum of the most squeezed
state

However, when a comb is formed without perturbation, the general dispersion
and the comb’s frequency spacing are more inherently linked, reducing the
number of possible outcomes. While this case is conceptually simpler than the
perturbed dispersion case, it can be more challenging to realize due to the
high anomalous dispersion that is required for the 2-FSR comb to form
(together with a relatively small FSR). Additionally, high anomalous
dispersion causes the squeezed supermodes to localize in four modes, reducing
the relevance of the lattice model description (Fig. 6). Nevertheless, this
case does highlight some aspects which are relevant to the main case studied
in this work. To generate a 2-FSR comb, we use high anomalous dispersion of
$D_{2}/2\pi=86.4~{}MHz$, an intrinsic Q of 3.66 million and coupling Q of 4.25
million. These parameters result in a 2-FSR primary comb, which we pump at
12.55mW with CW light at 1545nm. This comb reaches threshold at a pump
detuning of $-0.04~{}GHz$ (Fig. 13(a)). At this pump detuning value, even
though the dispersion is symmetric and $D^{\prime}_{1}=0$, the threshold is
still detuned (double peak). The symmetry implies that the eigenvalues are
purely real or purely imaginary. Thus, if all the states are purely imaginary,
there are no un-detuned states in existence, leading to a double peak
threshold. Correspondingly, the 2-FSR comb in Fig. 13b, for the detuning
marked by the orange dot in Fig. 13(a), has a fully detuned spectrum (Fig.
13(c,d)). Such a spectrum is associated with a detuned threshold, i.e, a
squeezing spectrum that peaks away from zero. The localization that high
$D_{2}$ dispersion causes can be seen in the spectrum of the supermode in Fig.
13(f). The localization occurs for all supermodes, as Bragg scattering is not
efficient away from the center of the comb.

###  4.6 Single supermode dominance

As we demonstrate in the theoretical and experimental measurements of this
work, the quantum noise in the different cavity modes is dominated by one or
two supermodes. Additionally, the squeezing frequency spectrum is uniform for
each supermode, meaning that the below threshold light, has similar anti-
squeezing in all cavity modes in our experiment. In this section we show
experimentally how a single supermode dominates the spectrum close to
threshold. We focus on the non-symmetric case, which shows distinct two peaks,
and perform a single-tone homodyne measurement in each cavity mode. The
results are presented in Fig. 14, which shows how similar two peaks cross the
threshold for each cavity mode.

Figure 14: Threshold crossing of detuned supermode by individually homodyning
modes -5 to 7. a.-g. Extension of the data in Fig. 5(e,f), showing additional
5 homodyne measurements of the threshold crossing of a double peak (detuned)
threshold, showing the dominance of a single supermode in all modes

Note that the two peaks appear both in the below threshold light (lower line
plots in each sub-figure) and above threshold light (upper line-plots).

Acknowledgements We gratefully acknowledge discussions with Edwin Ng, Ryotatsu
Yanagimoto, and Kiyoul Yang. This work is funded by the Defense Advanced
Research Projects Agency under the QuICC program and by the NSF QuSeC-TAQS
Award ID 2326792, as well as the Vannevar Bush Faculty Fellowship from the US
Department of Defense. E.L acknowledges the Yad Hanadiv Rothschild fellowship,
and the Zuckerman institute Zuckerman fellowship. Part of this work was
performed at the Stanford Nanofabrication Facility (SNF) and the Stanford Nano
Shared Facilities (SNSF). S. F. acknowledge the support of a MURI project for
the U. S. Air Force Office of Scientific Research (Grant No.
FA9550-22-1-0339). We thank NGK Insulators, Ltd. for 4H-SiCOI substrates used
to fabricate devices in this work.

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