Strong Nonlinear Coupling due to Induced Photon Interaction on a Si$_{3}$N$_{4}$ Chip
Sven Ramelow Alessandro Farsi Zachary Vernon Stephane Clemmen Xingchen Ji John E. Sipe Marco Liscidini Michal Lipson Alexander L. Gaeta





Corresponding Author: sven.ramelow@physik.hu-berlin.de


Institute for Physics, Humboldt University Berlin, Germany


Equally contrubuting author


Dept. of Appl. Physics and Appl. Math, Columbia University, New York,
USA



Dept. of Physics, University of Toronto, Toronto, Canada



Dept. of Inf. Tech., Ghent University, Ghent, Belgium



Dept. of Electrical Engineering, Columbia University, New York, USA



Dept. of Physics, University of Toronto, Toronto, Canada



Universit degli Studi di Pavia, Pavia, Italy



Dept. of Electrical Engineering, Columbia University, New York, USA



Dept. of Appl. Physics and Appl. Math, Columbia University, New York,
USA

Strong Nonlinear Coupling due to Induced Photon Interaction on a
Si_3N_4 Chip
    Alexander L. Gaeta
    December 30, 2023
=============================================================================

Second-order optical processes lead to a host of applications
in classical and quantum optics. With the enhancement of parametric
interactions that arise due to light confinement, on-chip implementations
promise very-large-scale photonic integration. But as yet there is
no route to a device that acts at the single photon level. Here we
exploit the q^(3) nonlinear response of a Si_3𝐍_4
microring resonator to induce a large effective q^(2).
Effective second-order upconversion (ESUP) of a seed to an idler can
be achieved with 74,000 %/W efficiency, indicating that single photon
nonlinearity is within reach of current technology. Moreover, we show
a nonlinear coupling rate of seed and idler larger than the energy
dissipation rate in the resonator, indicating a strong coupling regime.
Consequently we observe a Rabi-like splitting, for which we provide
a detailed theoretical description. This yields new insight into the
dynamics of ultrastrong effective nonlinear interactions in microresonators,
and access to novel phenomena and applications in classical and quantum
nonlinear optics.

Current strategies for implementing nonlinearity at the single photon
level <cit.>
have to make use of very complex experimental arrangements, such as
high-Q photonic crystal cavities and quantum dots, which are difficult
to implement using conventional photonic technologies. Nonetheless
they are being actively pursued, in part because of the common assumption
that for purely parametric nonlinear optical processes –
that is, those that do not involve the excitation of a resonance in
the response of a material medium – the intrinsic material
nonlinearities would be far too weak to allow for efficient nonlinear
interactions between single photons. 

This assumption was challenged by Langford et al. <cit.>,
who argued that the use of a strong pump for one of the fields of
a four-wave mixing (FWM) process in a χ^(3)-nonlinear medium
would lead to an effective χ^(2) nonlinearity for the remaining
three fields. Not directly limited by the intrinsic material properties,
this effective χ^(2) interaction can in principle be made
arbitrarily large by increasing the pump power. While theoretically
this would allow for an induced photon interaction efficient at the
single-photon level <cit.>, to date such
an effective χ^(2) has experimentally only been demonstrated
at relatively low efficiencies using a photonic crystal fiber <cit.>,
a chalcogenide nanofiber <cit.>
and a resonant χ^(3)-nonlinearity in warm Rb-vapor <cit.>.
Yet such an approach is also compatible with material platforms and
light confinement strategies characterized by strong field enhancements
and exceptionally large Q factors, which offer the promise of strong
effective second-order photon interactions. We shall see below that
this promise can be fulfilled. 

In any such scenario, the nonlinear processes that can result are
richer than those occurring in materials with a natural χ^(2),
but are analogous to them. In particular, it is possible to produce
an effective second-order upconversion (ESUP) process (see Fig. <ref>)
in which two photons are converted to one photon at a higher photon
energy, but with much less than double the original photon energy
that would characterize the analogous process of second harmonic generation
(SHG). To measure the strength of ESUP we employ the standard figure-of-merit
used for quantifying SHG, a normalized efficiency quoted in %/W.
While ultra-high Q, mm-sized resonators made from lithium niobate
(LN) can reach 300,000 %/W <cit.>,
current state-of-the-art values for integrated devices are on the
order of few thousand %/mW, such as the 5400 %/W achieved in LN
waveguides <cit.>, and the recently reported
2,500 %/W in AlN micro-ring resonators <cit.>.

Here we utilize a high-Q Si3N4 microresonator
to generate a strong effective χ^(2). With a pump field at
1590 nm, we induce ESUP of two seed photons at 1400 nm to an
idler photon at 1260 nm. All photon energies are at microring resonances,
strongly enhancing the conversion. Changing the seed power at 1400
nm over 5 orders of magnitude, and measuring the resulting idler power,
we verify the quadratic scaling expected of a second-order process,
and measure a normalized efficiency of 74,000%/W. Intriguingly, at
higher seed powers of only a few hundreds of microwatts at 1400
nm, where our total conversion efficiency is saturating at around
5%, the nonlinear coupling rate in the resonator is comparable with
resonator damping rate. This fulfills a strong coupling condition
between the nonlinearly interacting modes, and consequently there
occurs a Rabi-like splitting of the modes, which we observe and study
both experimentally and theoretically. 

For the theoretical description, we consider the resonant nonlinear
interaction of the three fields schematically indicated in Fig. <ref>a:
There is a strong pump at ω_P, a weak
seed at ω_S, and an idler generated at ω_I=2ω_S-ω_P.
We focus on the effective second-order nonlinear interaction between
the seed and the idler (see <ref>b), which can
be described by the equations 


    dβ_S/dt    =     -Γ̅_Sβ_S+2iΛ_2(t)β_S^*β_I-iγ_S^*α_Se^-iΔ_St
    dβ_I/dt    =     -Γ̅_Iβ_I+iΛ_2^*(t)β_S^2

where β_S,I are the slowly varying seed and idler fields
amplitudes in the resonator, Γ̅_S,I are the total resonator
damping rates at ω_S,I, α_S the amplitude of the
seed driving field in the bus waveguide, Δ_S is the detuning
away from resonance, and γ_S characterizes the coupling
between the bus waveguide and the ring (see Fig. <ref>b).
The coefficient Λ_2(t), which describes the strength of
the effective χ^(2), is given by Λ_2(t)=Λ_3β_P(t),
with Λ_3≈ħω v_g^2γ_NL/(2π R)
the third order coupling coefficient, where γ_NL is the
usual nonlinear parameter, v_g the group velocity in the ring
at ω_P, R the ring radius, and ω^2=√(ω_Pω_S^2ω_I);
β_P(t) is the pump amplitude in the ring. We assume that
β_P(t) is undepleted and that the effects of self- and cross-phase
modulation are taken into account as corrections to the resonant frequencies
ω_S,I,P, which is valid for slowly varying field envelope
functions; we restrict our analysis to this regime. See Methods for
details.



The use of a high-Q resonator gives rise to dramatic enhancements
of both the circulating pump power available and the nonlinear interaction
between the three fields <cit.>.
The Si_3N_4 material was chosen to avoid
two-photon absorption, which typically plagues silicon systems. This
enables very large unit power ESUP conversion efficiencies, η_ESUP,
to be achieved. This figure of merit can be theoretically calculated
as 


    η_ESUP=64R^2γ_NL^2(Δλ_FSR)^4/π^2λ^4P_P^inQ^4(Q/Q^ext)^4

in steady state, where Δλ_FSR is the wavelength free
spectral range, P_P^in is the pump input power in the bus waveguide,
Q is the loaded quality factor and Q^ext the extrinsic quality
factor due to the coupling of the resonator to the bus waveguide.
Corrections to this simplified formula, including the fact that the
quality factors and wavelengths λ are different at different
frequencies, are described in the Methods. Importantly, the scaling
of η_ESUP with Q^4 is one power larger than its χ^(2)analogue
- the normailzed SHG efficiency, which in a resonator only scales
with Q^3. 

We set out to investigate the ESUP experimentally. The chip-integrated
microring resonators utilized in this work were made from a high quality
Si3N4 film on an SiO2
substrate, grown via low-pressure chemical vapor deposition and electron
beam lithography (see <cit.> for details).
The waveguide dimensions (Fig.<ref>b) of 730 nm
x 910 nm (height x width) are carefully chosen: With a zero group
velocity dispersion (GVD) at 1410 nm, the waveguide exhibits phase-matching
between a strong pump at 1590 nm, the seed field at 1400 nm,
and the idler field at 1260 nm. Moreover, the large normal GVD at
ω_P strongly inhibits any undesired parametric process (e.g.,
spontaneous FWM) from the strong pump . The microring resonator (radius
45 μ m) with a free spectral range of 500 GHz is coupled
to the bus waveguide with a gap of 450 nm, and exhibits loaded
Q-factors of 250,000 for the pump (over-coupled), 1.2 million for
the seed (close to critical coupling), and about 3.5 million for the
idler (under-coupled). Using these parameters we theoretically expect
a unit power ESUP efficiency of 87,000 ± 17,000 %/W, where the
error interval is estimated from the uncertainty on the experimental
parameters (e.g. coupled power, quality factors, and propagation losses).
This corresponds to a χ_eff^(2)≃1 pm/V (see
Methods section), which is comparable in magnitude to the
second-order nonlinear susceptibility used in LN WGM-resonators. And
although this is two orders of magnitude smaller than the nonlinear
susceptibility of semiconductors such as GaAs or GaN, the upconversion
efficiency here is among the highest ever reported.



Our measurement scheme is shown in Fig. <ref>a). Pump
and seedignal light are generated by tunable external cavity lasers.
The pump is amplified with a high power EDFA to mW and combined with
the seed with wavelength division multiplexers (WDMs). They are injected
into the chip (and collected) via lensed fibers with a total insertion
loss of 8.5 dB, resulting in 85 mW of on-chip pump power at 1590
nm. The output is separated into the three wavelength bands by cascaded
WDMs to be monitored separately on InGaAs photodiodes. The idler output
at 1260 nm is further strongly filtered by a low-pass filter yielding
total losses of 6.3 dB, including the WDMs. When measuring very
low intensities, we optionally filter the idler with a grating monochromator
(0.55 nm filter bandwidth), and then detect with superconducting
single-photon detectors. We enclose the chip and stabilize the temperature
to reduce thermal and mechanical effects.

For our measurements of the idler output power at 1260 nm as a
function of the 1400 nm seed wavelength, first the 1590 pump
wavelength is tuned into resonance from the short wavelength side
to follow the thermal red-shift and achieve thermal locking <cit.>.
We then measure the power scaling characteristics of the ESUP process.
To verify the quadratic scaling of ESUP process with respect to
seed power, we vary it over almost five orders of magnitude and we
measure the generated idler power levels; they vary over eight orders
of magnitude. Starting from 350μ W seed power in the bus waveguide
and successively reducing it to nanowatt level, the from the quadratically scaling idler
powers inferred effective conversion efficiencies (linear scaling) are plotted in Fig. <ref>. From this we infer an on-chip normalized
η_ESUP = 74,000 %/W, in excellent agreement with the theoretically
predicted value.






We turn now to seed powers larger than a few tens of μ W. Given
the lack of two-photon absorption, the saturation of the generated
idler observed in the main panel of Figure <ref> suggests
that at such seed powers the nonlinear coupling between the modes
can no longer be treated perturbatively. To further investigate this
we scan the seed laser across the resonance wavelengths while the
idler power is monitored. Typical scans are shown in Fig. <ref>
for seed power P_seed= 50 nW (left) and P_seed= 300μW(right).
We see a gradual broadening of the seed peak as the seed power is
increased, until two dips can be clearly resolved. This is in excellent
agreement with the theoretical curves calculated starting from Eqs.
(1)-(2) (see Methods section) and using the nominal parameters
of the system without the need for any fitting. 

The splitting indicates a regime of ESUP in which the nonlinear coupling
rate Λ_3|β_Pβ_S| between the modes is comparable
to the energy loss rate of the resonator, thus resulting in
strong nonlinear coupling. Consequently a Rabi-like splitting of
the modes occurs. which is analogous to that theoretically investigated
by Carusotto and La Rocca <cit.>, who
focused on the second-order nonlinear interaction between two fields
oscillating at ω and 2ω in a doubly resonant system.
They predict a Rabi-like resonance splitting at the fundamental frequency
when the nonlinear coupling rate exceeds the resonator damping rates. 



In the ring resonator, the steady state solutions for the mode amplitudes
predict the onset of this saturation as well as the presence of the
Rabi-like splitting of the resonance at ω_S (see Methods section). In particular, one can observe the splitting when the
“resolvability” R, i.e., the ratio of the splitting to the line
width, is about unity. This quantity depends on the ring parameters
as well as the pump and seed powers according to 


    R=|Λβ_Sβ_P/Γ̅_S|≈8Λ/ω_S√(P_PP_S/ħω_S^2ħω_P^2)Q_P/√(Q_P^ext)Q_S^2/√(Q_S^ext),

where P_P,S are the pump and seed input powers and Q_P,S^ext
are the quality factor of the resonators determined solely by the
external coupling. The last term of the equation is obtained by assuming
a linear solution for the pump and the seed powers and is strictly
valid for R<1, and yet it allows one to have a rough estimate of
when the strong nonlinear coupling regime can be achieved. 






In Figure <ref> we plot the seed transmission
as a function of detuning for several seed powers. It should be noticed
that unlike recent studies in which Rabi oscillations have been predicted
<cit.> and Rabi splittings have been experimentally
observed <cit.> for frequency conversion in the strong
linear-coupling regime, here the strength of the coupling between
the fields at ω_S and ω_I depends on the
intensity of the seed field itself, and in particular is linear in
β_S. Finally, we stress that this condition has been achieved
because our effective second-order nonlinear interaction is controlled
by the pump intensity, and occurs in a structure characterized by
an exceptionally large field enhancement and Q-factor. It is the combination
of these two elements that has allowed the first observation of a
nonlinear strong-coupling. 

The strength of this second-order nonlinear response raises the question
of whether the field associated with just a single pair of photons
at ω_S would be sufficient for a complete pair conversion
to a photon at ω_I in our system. In such a regime, new
phenomena are expected given the energy quantization of the optical
field <cit.>. So far, much of the research has
been focused on atomic-like systems, either quantum dots or with atomic
clouds, while non-resonant photon-photon interactions in bulk nonlinear
materials have been far from approaching unit efficiency. For instance,
SHG driven by photon pairs has been demonstrated in LN waveguides,
but only yielding a very low probability p_2→1 of a
two-photon conversion <cit.>. To further
increase p_2→1 one can take advantage of light confinement
in micro resonators, either directly exploiting a natural second-order
material nonlinearity or a strong effective second-order response,
the latter of which we have done in this work. In Fig. <ref>
we summarize several nonlinear resonator systems and show p_2→1
as a function of the resonator quality factor Q, heuristically
defined as p_2→1=η2E_sp/τ_cavity=4ηħω^2/Q
(where 2E_sp is the energy of two photons to be converted, τ_cavity
the cavity dwelling time, and η is the efficiency of the corresponding
parametric process). We stress that one would expect quantization
effects to appear when p_2→1 approaches a significant
fraction of unity. Currently, our system is characterized by p_2→1=4.7×10^-7
(star in Fig. <ref>), which is the largest value ever
reported. This is despite the fact that our device has not yet been
optimized towards optimal coupling for all wavelength, with strong
over- and undercoupling for the pump and idler resonances, respectively.
Achieving critical coupling for all wavelength at the same intrinsic
Q and moderately increasing the pump power to 500mW would directly
yield an enhancement of p_2→1 by 2 orders of magnitude.
Moreover, because η scales with the 4th power of Q (see equation
<ref>) it follows, that p_2→1 scales with
Q^3. Thus a 30 times higher Q would result heuristically already
in p_2→1≃1.

This is challenging, for it requires bus-ring coupling optimization
at different wavelengths and the control and compensation of nonlinear
phase matching at higher pump powers, but these have individually
been shown already on the Si_3N_4 platform,
and even more encouragingly Q factors close to 70 million have
been recently obtained <cit.>. Based on these
simplified arguments we thus conclude that approaching the p_2→1≃1
regime is within reach of current technology.



In conclusion, we demonstrated induced photon interactions
based on effective second-order upconversion with a normalized efficiency
of 74,000 %/W, and experimentally reached the regime of strong nonlinear
coupling for on-chip powers as low as few microwatts, in excellent
agreement with our theoretical model. The Si_3N_4
platform is a mature and highly promising technology for obtaining
two-photon interaction at the single-pair level. 

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§ METHODS


To understand the dynamics of the nonlinear induced photon interaction
and ESUP process, we use a quantum mechanical Hamiltonian formalism
that fully captures the behavior of the system both in the few-photon
limit, and in the classical regime of strong coherent beams. A detailed
development of the formalism we employ can be found elsewhere in the
literature <cit.>; here we present a summary.

For the pump, signal, and idler, we identify intra-resonator annihilation
operators b_J with J∈{ P,S,I} for the three
resonances that participate in the ESUP process. The Hamiltonian H_ring
for the isolated resonant modes in the ring is then given by 

    H_ring=∑_Jħω_Jb_J^†b_J-ħΛ_3(b_P^†b_I^†b_Sb_S+H.c.),

in which we have retained only those nonlinear terms that participate
in the ESUP process. In particular, this Hamiltonian does not include
terms that represent self- and cross-phase modulation between the
three modes. However, for slowly-varying field envelope functions,
the effect of these processes is well-described by an effective shift
in the resonant frequencies ω_J <cit.>.
Since in this work we focus on CW input beams, this approximation
is well justified, and we assume the frequencies ω_J in
(<ref>) already contain these corrections. The nonlinear
coupling constant Λ_3 is related to the third-order nonlinearity
of the waveguide material, and is estimated by 

    Λ_3≈ħ(ω_Pω_S^2ω_I)^1/4v_g^2γ_NL/2π R,

where v_g is the group velocity in the ring waveguide, R the
ring radius, and γ_NL the nonlinear parameter; for the ring
used in these experiments we find Λ_3≈10 s^-1.

For each ring mode operator b_J we also introduce a corresponding
input and output channel field operator ψ_J<(t) and ψ_J>(t)
at the ring-channel coupling point, normalized such that v_J⟨ψ_J<^†(t)ψ_J<(t)⟩
(v_g⟨ψ_J>^†(t)ψ_J>(t)⟩) is the
mean incoming (outgoing) photon flux in the side channel at frequencies
near ω_J, with v_J the corresponding group velocity.
The time dependencies of operators are understood in the usual Heisenberg
picture context. The coupling of these fields to the ring can be described
using standard cavity input-output theory <cit.>.
Defining ring-channel coupling coefficients γ_J to each
channel field/ring mode pair, the input-output relation for these
field operators is given by 

    ψ_J>(t)=ψ_J<(t)-iγ_J/v_Jb_J(t),

with b_J(t) the time-dependent ring mode operator in the Heisenberg
picture.

The equations of motion in the classical limit for the ring mode amplitudes
can be found by calculating the full Heisenberg equations of motion
and replacing each operator by its expectation value. In what follows
we remove the fast optical time dependence and consider slowly-varying
quantities only. This gives 

    dβ_S/dt    =     -Γ_Sβ_S-iγ_S^*α_Se^-iΔ_St+2iΛ_2(t)β_S^*β_Ie^iΔ_rest,
    dβ_I/dt    =     -Γ_Iβ_I+iΛ_2^*(t)β_S^2e^-iΔ_rest

for the seed and idler modes, where β_J is the classical
ring mode amplitude for mode J, and Γ_J is
the full damping rate for mode J (including both coupling to the
channel and to scattering from the resonator), related to the full
loaded quality factor Q_J for that mode via Γ_J=ω_J/(2Q_J).
Here we have assumed that the seed is driven by a coherent beam with
amplitude α_Se^-iΔ_St in the channel, corresponding
to a seed input power P_S=ħω_Sv_S|α_S|^2
and detuning Δ_S from resonance. The detuning Δ_res=2ω_S-ω_T-ω_P
accounts for any deviation of the resonances from perfectly equal
spacing in frequency. The effective second-order coupling coefficient
is defined as Λ_2(t)=Λ_3β_P(t), with β_P(t)
the pump amplitude in the resonator.

In this work we assume a CW pump, so that the magnitude of Λ_2
is constant; The squared pump magnitude |β_P|^2 (corresponding
to the mean pump photon number in the resonator) in this regime is
given by 

    |β_P|^2=4P_PQ_P^2u_P/ħω_P^2Q_P^ext,

where P_P is the input pump power in the channel, and Q_P^ext
is the extrinsic quality factor of the pump associated with the channel-ring
coupling only, given by Q_P^ext=ω_Pv_P/|γ_P|^2.
This expression assumes that the pump is sufficiently strong to render
any back-action or depletion effects on the pump mode negligible.
The degradation factor u_P=(1+4l_P^2)^-1 ranges from 0
to 1, with l_P the pump detuning from resonance in units of the
pump resonance full-width-half-max linewidth, l_P=Δ_P/(2Γ_P)
with Δ_P the pump detuning from resonance. We define the
steady amplitude Λ_2 via Λ_2(t)=Λ_2e^-iΔ_Pt.

We seek a self-consistent stationary solution of Eqs. (<ref>,<ref>).
Thus we search for a steady state solution of the form β_S(t)=β_Se^-iΔ_St,
β_I(t)=β_Ie^-iΔ_It, with the overbar
denoting constant quantities. By substitution it directly follows
that this is only possible when the idler field oscillates with detuning
Δ_I=Δ_res-Δ_P+2Δ_S; for what
remains we estimate Δ_res=Δ_P=0 so Δ_I=2Δ_S.
The constant idler amplitude can then be written in terms of the seed
amplitude as 

    β_I=iΛ_2^*β_S^2/-iΔ_I+Γ_I.

Inserting this back into the equation for the the seed, we obtain

    (-iΔ_S+Γ_S+2|Λ_2|^2|β_S|^2/-iΔ_I+Γ_I)β_S=-iγ_S^*α_S.

To solve for β_S we first need a value for |β_S|^2.
This can be obtained by taking the modulus squared of (<ref>),
which yields an implicit cubic equation for |β_S|^2≡ N_S:

    a_3N_S^3+a_2N_S^2+a_1N_S+a_0=0,

where a_3=4|Λ_2|^4, a_2=4|Λ_2|^2(Γ_SΓ_I-Δ_SΔ_I),
a_1=(Γ_SΓ_I-Δ_SΔ_I)^2+(Δ_IΓ_S+Δ_SΓ_I)^2,
and a_0=(Γ_I^2+Δ_I^2)P_S/(ħ Q_S^ext).
Note the explicit dependence of a_0 on the extrinsic quality
factor for the seed resonance, making the coupling condition (under,
over, or critically coupled) of the seed a relevant parameter that
influences the conversion dynamics; indeed, the predictions of our
theory were found to depend quite sensitively on this coupling condition.

For a given set of ring parameters a positive real root N_S of
Eq. (<ref>) can be found for each seed power and detuning,
representing a steady intra-resonator solution for the seed and idler
amplitudes. The seed amplitude can be calculated from (<ref>),
and can be substituted in (<ref>) to find the idler amplitude.
The input-output relations (<ref>) can then be used
to derive the output powers of the seed and the idler into the channel.
The transmission ratio of the seed is given by 




    T=|1-2Γ_S-iΔ_I+Γ_I/(-iΔ_S+Γ_S)(-iΔ_I+Γ_I)+2|Λ_2|^2N_S|^2,




and the corresponding generated idler power output to the channel
can be calculated from P_gen=2ħω_IΓ_I|β_I|^2.
In this expression and (<ref>) the quantities Γ_S
and Γ_I are the channel-ring coupling rates for the seed
and idler resonances, related to the corresponding extrinsic quality
factors via Γ_J=ω_J/(2Q_J^ext).

Viewed as a function of Δ_S, the transmission ratio T
represents a seed transmission spectrum; this was used to produce
the theoretical curves in Fig. <ref> in the
main body. The corresponding generated power in the idler at each
point in these curves can be calculated, forming the theoretical spectral
curves for the generated idler in Fig. <ref>,
as well as the power scaling in Fig. <ref>.

In the limit of a weak seed input we can ignore the back-action on
the seed amplitude from its coupling to the idler, which enables a
simple analytic formula to be derived for the generated idler power.
Dropping the nonlinear term in the first of ((<ref>) and solving
for |β_I|^2, we find (after using the input-output
relation (<ref>)) for the generated idler power output
in the channel 



    P_gen    =(256Λ_3^2/ħ^2ω_S^4ω_P^2)(P_S)^2P_P(Q_PQ_IQ_S^2)Q_P/Q_P^extQ_I/Q_I^ext(Q_S/Q_S^ext)^2u_Pu_net,




where u_net=(1+4l_net^2)^-1 is the degradation
factor between 0 and 1 associated with the net detuning l_net
(in units of idler resonance FWHM linewidth) associated with both
the pump and Δ_res. For our experiment both u_P
and u_net were estimated to be approximately unity,
corresponding to a perfectly resonant system. This expression for
the generated power demonstrates the quadratic scaling of the generated
idler power with the seed input, and can be used to predict the unit
power ESUP conversion efficiency η_ESUP as defined in the
main body (eq. <ref>), 

    η_ESUP=P_gen/P_S^2.

Our formula takes into account different frequencies, coupling ratios
and quality factors for each resonance; the rough estimate of η_ESUP
in the main body is found by taking these all to be roughly equal
across the different resonances.

A few subtleties should be noted: first, the steady state solutions
described in this section for our experimental parameters are stable,
as verified by a first-order fluctuation analysis. However, unstable
regimes can be reached with realistic physical parameters, the presence
of which were found to sensitively depend on the seed coupling ratio.
Second, in order to obtain good agreement with the experimentally
measured values, it was necessary to include the effects of back-scattering
of the seed resonance in the resonator; this effect was strong enough
to be directly measured in our experiments. In particular, the presence
of back-scattering changes the procedure by which the coupling ratios
are extracted from measured linear transmission spectra of the microring
resonator. We modeled this phenomenologically with the inclusion of
a term H_BS=ħ q(b_Sb_S-^†+H.c.)
in the ring Hamiltonian, with b_S-^† the creation operater
for the counter-propagating mode at the seed frequency in the ring
that couples to the channel in the usual way, and where q is a
backscattering strength parameter.

Both of these effects highlight the care which must be taken when
carrying out a mathematical analysis of nonlinear optics in high-Q
resonators, especially when considering strongly driven phenomena
like ESUP. 

Finally, to calculate the effective equivalent second-order nonlinearity
χ_eff^(2) for our system, we study the nonlinear
part of the ring Hamiltonian that would describe a true degenerate
SHG/SPDC interaction, 

    H_SHG=-ħΛ_SHG(b_Fb_Fb_SH^†+H.c.),

 where F describes the fundamental and SH the second harmonic
mode. The constant Λ_SH describes the strength of the second-order
nonlinearity for this interaction, and is given by 

    Λ_SHG=ħ^1/2ω_F^3/2χ^(2)/2n^3ϵ_0^1/2(LA)^1/2,

 where ω_F is the frequency of the fundamental, χ^(2)
the relevant component of the second-order nonlinear tensor, n
the material refractive index (assumed not to vary between the fundamental
and second harmonic frequencies), L the ring length, and A the
effective area.

The Hamiltonian (<ref>) is formally identical to
the Hamiltonian (<ref>) that describes the ESUP process
in the case of a CW pump. Thus we can identify 

    |Λ_3β_P|=Λ_SHG.

 By calculating the pump amplitude |β_P| and the coupling
strength |Λ_3| in our experiment, using the the expression
(<ref>) for Λ_SHG, we can extract
the magnitude of χ^(2) that would be required in this system
to produce a comparable unit power conversion efficiency; this is
precisely the effective equivalent second-order nonlinearity χ_eff^(2).
We find for a pump power of 85 mW, using Λ_3=10 s^-1,
that 

    χ_eff^(2)≈0.6 pm/V.



