# Reservoir computing based on a silicon microring and time multiplexing for
binary and analog operations

Massimo Borghi  corresponding author<EMAIL_ADDRESS>Stefano Biasi
Lorenzo Pavesi  Nanoscience laboratory, Department of Physics, University of
Trento, Via Sommarive 14, 38123, Trento, Italy

###### Abstract

Photonic implementations of reservoir computing (RC) promise to reach ultra-
high bandwidth of operation with moderate training efforts. Several
optoelectronic demonstrations reported state of the art performances for hard
tasks as speech recognition, object classification and time series prediction.
Scaling these systems in space and time faces challenges in control
complexity, size and power demand, which can be relieved by integrated optical
solutions. Silicon photonics can be the disruptive technology to achieve this
goal. However, the experimental demonstrations have been so far focused on
spatially distributed reservoirs, where the massive use of splitters/combiners
and the interconnection loss limits the number of nodes. Here, we propose and
validate an all optical RC scheme based on a silicon microring (MR) and time
multiplexing. The input layer is encoded in the intensity of a pump beam,
which is nonlinearly transferred to the free carrier concentration in the MR
and imprinted on a secondary probe. We harness the free carrier dynamics to
create a chain-like reservoir topology with 50 virtual nodes. We give proof of
concept demonstrations of RC by solving two nontrivial tasks: the delayed XOR
and the classification of Iris flowers. This forms the basic building block
from which larger hybrid spatio-temporal reservoirs with thousands of nodes
can be realized with a limited set of resources.

Optical neural systems, Neural networks, Nonlinear optics, Integrated optics,
Silicon microresonators

##  I Introduction

The last decade has seen a flourish of new machine learning (ML) methods and
applications, sometimes surpassing the human’s capabilities in solving hard
tasks such as object recognition [1, 2], playing board games [3] or lipreading
[4]. Among the different approaches, Reservoir Computing (RC) has attracted
growing interests due to its trade-off between performance and training
complexity [5]. RC is a machine learning paradigm inspired by Recurrent Neural
Networks (RNN) where a set of input stimuli triggers the nonlinear dynamics of
a network of thousands of nodes, typically sparsely connected by fixed random
weights [6]. The reservoir processes the information and maps the original
input space into one of increased dimension, thus mimicking the sequence of
nonlinear transformations of feed forward neural networks. In contrast with
these, only the nodes in the readout layer are trained, while the network
separability is guaranteed by the highly nonlinear dynamics of the reservoir.
The training process is thus reduced to a linear combination of the output
states of the reservoir, which can be implemented with minimal computational
resources. At the heart of RC lies the fact that loose conditions on the
topology of the reservoir are required for operation [7], which find
realization in a variety of physical systems. Among these, photonic RC
promises to be the ideal platform for hardware acceleration, due to an ultra-
high bandwidth of operation and ease of implementation [8, 9, 10, 11].
Excellent demonstrations exist, which fall into two distinct categories:
spatial or delayed-node reservoirs. The first has a close analogy with RNN, in
which nodes are spatially distributed and can be physically interconnected, as
in the network of Semiconductor Optical Amplifiers proposed in [12], or can be
software-connected, as the pixels of the spatial light modulator in [13].
Delayed node reservoirs use instead a single physical node for computation.
Here, the nodes are virtual and multiplexed in time along an optical delay
loop [10]. The network connectivity is given by the inertia of the system [9],
or by asynchronously sampling the readout nodes with respect to the time of
injection of the inputs [14, 8]. Implementations often use optoelectronic
delay loops embedding nonlinear elements as Mach Zehnders or phase modulators
[9], DFB lasers [15], VCSELs [16] or photodetectors [8]. Being built from off-
the shelf optics and electronics, these systems operate at GHz speed and
couple thousands of nodes. State of the art performances have been
demonstrated on several benchmarking tests, as chaotic time series prediction,
nonlinear channel equalization and speech recognition. However, to satisfy the
growing demand of more challenging tasks, the number of virtual nodes and
their interconnection complexity is foreseen to increase over the next years.
Scaling these parameters with bulk optoelectronic solutions will soon become
unpractical for both spatial and delay-loop architectures. This is why
numerous RC schemes based on integrated optics have been proposed, and some of
them experimentally demonstrated. Silicon photonics is the ideal platform for
very large scale integration, offering the possibility to fabricate hundreds
of individually addressable and reconfigurable optical elements in few cm2
[17]. However, the number of experimental demonstrations of RC based on
silicon photonics is still limited [18, 19, 15]. Most of the proposed schemes
focus on passive spatially distributed reservoirs made by small matrices of
waveguides [18, 20], photonic crystal cavities [21] or ring resonators [22,
23, 24]. The fading memory of these systems is given by the delay lines which
connect the nodes, or by the photon lifetime of the cavities. This memory
spans from few ps to some ns. The nonlinearity is typically provided by the
use of square law photodetectors, which exploit the coherence properties of
light propagating in these structures. The scale of the reservoir is limited
by the loss of the delay lines and by the repeated use of splitters (coherent
combining) which create the connectivity of the network [25]. The integration
of single nodes with delayed feedback eliminates the interconnection problem,
but also faces challenges, since lossy delay loops of the order of few ns are
required to store hundreds of virtual nodes with ps separation [15, 19]. It is
natural to seek the use of hybrid spatio-temporal architectures as an optimal
trade-off solution. These may adopt a small scale spatial topology with sparse
connectivity, where each physical node can accommodate few hundreds of time
multiplexed virtual nodes, similar to what is suggested in [26].
In this work, we propose and experimentally validate a key element of this
scheme. A RC architecture is realized with a silicon MR and time multiplexing.
The input space is encoded in time bins of different light intensity, which
are sequentially injected in the device to trigger a transient state in the
free carrier population. The carrier dynamics defines the connections between
the virtual nodes. In particular, we adopt a scheme in which the input
information is nonlinearly transferred from a pump to a weak continuos wave
(CW) probe laser inside the MR. This is achieved by generating carriers
through Two Photon Absorption (TPA) and then by imprinting their signature on
the probe intensity by Free Carrier Dispersion (FCD), which shifts the
resonance frequency of the cavity. The system does not require active
materials and does not rely on photodetection to induce nonlinearities, being
these naturally provided by TPA. The strength of information transfer is two
orders of magnitude higher than the one mediated by Kerr, and occurs at modest
input power thanks to the light enhancement inside the MR. In contrast to the
previously reported passive schemes [18, 23, 24, 21], where the inputs are
constrained to vary at rates above $10\,\textrm{Gbps}$ to minimize the
interconnection lengths (hence the loss), the reservoir operates at the free
carrier timescale, which in our case is $45\,\textrm{ns}$ ($\sim
20\,\textrm{MHz}$). This is beneficial for the driving electronics, since we
can achieve an high Signal to Noise Ratio (SNR) at the readout and minimize
the quantization noise on the input signal, which are both factors affecting
the performance of the device [27]. We experimentally demonstrate the RC
capabilities with both binary and analog input signals. The binary task we
address is the $1$-bit delayed XOR, where we achieved a minimum detectable Bit
Error Rate (BER) of $1.4\times 10^{-3}$ for bitrates up to $30\,\textrm{MHz}$.
The analog task we tackled is the classification of the Iris dataset [28],
which yielded $(99.3\pm 0.2)\%$ accuracy at a rate of $0.38\,\textrm{MHz}$. We
experimentally and theoretically show that the reservoir achieves its maximum
performance when the carrier and the temperature dynamics slightly mix
together, which occurs at the transition edge between a stable and a self-
pulsing regime [29]. Finally, we discuss how to scale the presented scheme,
showing that GHz operation with thousands of nodes can be possible with
limited resources.

##  II Principle of operation

The architecture implements a single MR in the Add-Drop configuration as a
reservoir, in which virtual nodes are defined by time multiplexing [30]. The
way we construct and process the input and output layers is schematically
shown in Fig.(1). The input signal $u(t)$, which can be a binary bit-sequence
or of analog nature, is encoded in the intensity of a pump laser which is
resonantly coupled to the input port of the MR. In contrast with common
delayed-loop architectures [10, 14], where the time of flight of light within
the feedback loop sets the rate at which samples are injected and read, here
there is no external feedback, so the bit duration $T=N_{v}\Delta$ is
determined by our choice of the number of virtual nodes $N_{v}$ and by their
time separation $\Delta$. The number $N_{v}$ is chosen as a trade-off between
the required complexity of information processing, which generally increases
with $N_{v}$ [15], and the speed of operation. According to a sample-and-hold
technique [10], each of the $N$-dimensional input sample
$\mathbf{x}^{(n)}=\\{x_{1}^{(n)},x_{2}^{(n)},...,x_{N}^{(n)}\\}$ at time
$t_{n}=nT$ is kept constant during the time interval $[t_{n},t_{n+1}]$, and is
spread over all virtual nodes through an input connectivity matrix
$\mathbf{W_{\textup{in}}}$, which defines the mask on the data. The injected
pump intensity at time $t=t_{n,i}(\sigma)=nT+i\Delta+\sigma$, where
$\sigma\in[0,\Delta)$ and $i=\\{1,...,N_{v}\\}$, is given by
$u(t_{n,i}(\sigma))=\alpha\sum_{k=1}^{N}W_{ik}x_{k}^{(n)}\theta_{n,i}(t_{n,i}(\sigma))+u_{0}$,
in which $\theta_{n,i}(t_{n,i}(\sigma))$ is a window function of duration
$\Delta$ that is equal to $1$ for $t\in[t_{n}+i\Delta,t_{n}+(i+1)\Delta)$ and
$0$ elsewhere. The coefficients $\alpha$ and $u_{0}$ are a scale factor and an
offset which respectively set the average pump power and make $u$ positive
valued. In a more compact form, if we define
$\mathbf{X}_{\textup{in}}=[\mathbf{x}^{(1)},...,\mathbf{x}^{(M)}]$ as the
matrix of the input samples, $u(t_{n,i})$ are the entries of the matrix
$\alpha(\mathbf{W}_{\textup{in}}\mathbf{X}_{\textup{in}}+u_{0}\mathbf{1})^{T}$.

Figure 1: Process flow of the encoding of the input signal. $M$ input samples
$\\{\mathbf{x}^{(1)},...,\mathbf{x}^{(M)}\\}$ of dimension $N$ are queued on
the columns of a matrix $\mathbf{X}_{\textup{in}}$. The dimension of each
sample is then increased to $N_{v}$ using a connectivity matrix
$\mathbf{W}_{\textup{in}}$. A global offset $u_{0}$ is applied to
$\mathbf{W}_{\textup{in}}\mathbf{X}_{\textup{in}}$ to remove the negative
values, and a multiplicative scale factor $\alpha$ is applied. The resulting
column values represent the input pump power (red curve) $u$ of each sample,
which are sequentially injected at times $t_{n}=n(N_{v}\Delta)$ at the input
port of the MR (central inset). Similarly, the values of the probe power
$u_{pr}(t_{n,i})$ at times $t_{n,i}=n(N_{v}\Delta)+i\Delta$, with
$i=\\{1,...,N_{v}\\}$, define the virtual nodes at the output of the Drop port
of the MR.

The pump wavelength $\lambda_{p}$ is set with a small detuning
$\Delta\lambda_{p}$ with respect to one of the resonance orders
$\lambda_{p,0}$ (pump resonance) of the MR. A second probe laser, at a
constant power $P_{pr}\ll P_{p}$, is injected into the same port, and is tuned
at a wavelength $\lambda_{pr}=\lambda_{pr,0}+\Delta\lambda_{pr}$, with a
detuning $\Delta\lambda_{pr}$ with respect to the probe resonance wavelength
$\lambda_{pr,0}$. We used $|\lambda_{pr,0}-\lambda_{p,0}|=\textup{FSR}$, where
FSR is the Free Spectral Range of the MR. In the rest of this paper, we refer
to $\lambda_{p,0}$($\lambda_{pr,0}$) as the ”cold” pump(probe) resonance, that
is the resonance wavelength of the MR in the linear operation regime, where
all the nonlinear effects, which might affect the resonance frequency, are
neglected. As a consequence of light confinement in the MR, the pump intensity
builds up, and TPA promotes free carriers from the valence to the conduction
band. This changes the refractive index of the material through FCD, inducing
a rigid blue shift of all the eigenfrequencies of the MR [29]. Concurrently,
the round-trip loss is increased by Free Carrier Absorption (FCA). Since $u$
varies in time, the magnitude of the FCD shift changes as well, and its
dynamics is imprinted on the probe laser intensity at the output of the MR.
The latter incoherently transfers the input information from the pump to the
probe beam. It is worth to note that the FCD shift caused by TPA is two orders
of magnitude higher than the Kerr shift (see Appendix B), and acts as an
effective intensity induced nonlinearity. The carrier lifetime
$\tau_{\textup{fc}}$ is much smaller than the thermal constant
$\tau_{\textup{th}}$ of the MR [31]. Therefore, when
$\Delta\sim\tau_{\textup{fc}}$, the MR temperature variations $\Delta T$
cannot follow the temporal profile of the carrier concentration. Self heating
effects due to carrier relaxation are smoothed to yield a steady state value
$\overline{\Delta T}$. In this regime, the MR temperature does not participate
to the reservoir dynamics, and only changes the effective detuning
$\Delta\lambda_{p}$($\Delta\lambda_{pr}$) of the pump(probe) resonance.
However, as discussed in Section III.III.3, this holds only at low input
power. When the average power is raised above a certain threshold, the MR
starts to self-pulse [29], and the temperature dynamics impacts the processing
capabilities of the nodes.

Figure 2: (a) Sketch of the experimental setup. TLS = Tunable Laser Source,
AWG = Arbitrary Waveform Generator, EO mod. = Electro Optic modulator, FPC =
Fiber Polarization Controller, VOA = Variable Optical Attenuator, PD =
Photodiode, EDFA = Erbium Doped Optical Amplifier, BS = Beam Splitter, GC =
Grating Coupler, BPF = BandPass Filter, OSC = Oscilloscope. b) Device layout
and logical information flow. The intensity modulated pump (red) and the CW
probe (blue) are injected into the input GC. The incoherent transfer of
information from the pump to the probe occurs within the resonator
(reservoir), where the different virtual nodes (yellow dots) interact and
process the input data. The probe exits from the Drop port, carrying the
result of the computation. Virtual nodes are sampled and sent into several
linear classifiers, each trained to recognize a specific class. The decision
making process is based on a winner takes all scheme.

We can get an insight on the internal structure of the reservoir by assuming
small perturbations of the input power $u(t)$ with respect to a reference
value $\overline{u(t)}$. Here, small means that the corresponding free carrier
change $\Delta N-\overline{\Delta N}$ around the reference value
$\overline{\Delta N}$ induces variations of the transmitted pump/probe
intensity which are linear with $\Delta N$. As derived in Appendix A, within
this approximation the probe power $u_{pr}$ at the Drop port of the resonator
is given by:

$\begin{split}u_{pr}(t)&=c_{0}+c_{1}\int_{-\infty}^{t}e^{-\left(\frac{t-\xi}{\tau_{\textup{fc}}}\right)}u^{2}(\xi)d\xi+\\\
{}&+c_{2}\int_{-\infty}^{t}e^{-\left(\frac{t-\xi}{\tau_{\textup{fc}}}\right)}u^{2}(\xi)u_{pr}(\xi)d\xi,\end{split}$
(1)

where the definitions of $c_{0}$, $c_{1}$ and $c_{2}$ are given in Appendix A.
The first contribution to the virtual node state at time $t$ is quadratic in
the pump power $u$, and depends on the past inputs with weights that
exponentially decrease with time. This short term memory is of the order of
$\sim 3\,\tau_{\textup{fc}}$, i.e., the time required by free carriers to
reach a steady state after an abrupt change of the pump power. The memory of
this system is thus intrinsically nonlinear. This is because the input
information $\mathbf{x}^{(n)}$ is instantaneously (fs scale) transferred to
carriers through nonlinear absorption, temporarily processed from a slower
(ns) dynamics, and subsequently imprinted to the output probe. Similarly to
what is done in [30], one can approximate the integral in the first term with
a discrete sum, and find the recursive virtual node relation
$u_{pr}(t_{n,i})=\Delta
c_{1}\sum_{k=0}^{m}\left(\eta^{k}u(t_{n,i-k})^{2}\right)+\eta^{m}u_{pr}(t_{n,i-m})$
where $\eta=e^{-\left(\frac{\Delta}{\tau_{\textup{fc}}}\right)}$ (for
simplicity, the second term in Eq.(1) has been neglected, but this does not
alter our conclusions). This highlights the type of connectivity between the
virtual nodes of the reservoir, which are organized in a chain-like topology:
node $u_{pr}(t_{n,i})$ is coupled with its adjacent $u_{pr}(t_{n,i-1})$ with a
coupling strength $\eta$. The second term in Eq.(1) is a nonlinear coupling
between virtual nodes. As shown in Appendix A, this term arises from the the
resonance shift induced by FCD. This shift modifies the pump power circulating
in the MR, which in turn affects the carrier generation rate. As a result, a
recursive relation between these quantities is settled, for which the second
term in Eq.(1) accounts for the leading term. When the change in $u_{pr}$ is
no more linear with $\Delta N-\overline{\Delta N}$ due to large FCD induced
resonance shifts, Eq.(1) loses validity and the virtual node interaction
becomes increasingly complex. The general structure of the network is however
maintained, with TPA and inter-node coupling providing nonlinearity, while
carrier recombination a short term memory to the reservoir. Virtual nodes
$u_{pr}(t_{n,i})$ are sampled at the Drop port, and arranged into a state
matrix $\mathbf{X}=[\mathbf{u}_{pr}^{(1)},...,\mathbf{u}_{pr}^{(M)}]^{T}$,
where $\mathbf{u}_{pr}^{(k)}$ is the column vector collecting the $N_{v}$
virtual nodes corresponding to the $k^{\textup{th}}$ input sample
$\mathbf{x}^{(k)}$. The action of the reservoir is to project the initial set
of predictors $\mathbf{x}^{(k)}$ with corresponding observables
$\mathbf{y}^{(k)}$ (in general, a Q-dimensional vector) to a higher
dimensional space $\mathbf{x}^{(k)}\rightarrow\mathbf{u}_{pr}^{(k)}$ where
they ideally should be linearly separable. Then, the aim is to find a
$(N_{v}\times Q)$ weight matrix $\mathbf{W_{\textup{out}}}$ which realizes
$\mathbf{Y}=\mathbf{X}\mathbf{W_{\textup{out}}}$, in which the observables are
arranged in the $M\times Q$ matrix $\mathbf{Y}$. We accomplish this task by
regularized least squares (Ridge regression), where the regularization
parameter $\lambda$ is determined by a $5$-fold cross validation [32]. The
output of this procedure is a matrix $\mathbf{\tilde{W}_{\textup{out}}}$ which
minimizes the regularized least square error
$\sum_{k=1}^{M}||(\mathbf{Y}-\mathbf{\tilde{W}_{\textup{out}}}\mathbf{X})||^{2}+\lambda^{2}||\mathbf{\tilde{W_{\textup{out}}}}||^{2}$.
In case of supervised learning, as we deal in this paper, $\mathbf{y}^{(k)}$
is a $Q$-dimensional binary variable, with $1$s identifying the assignment
category of $\mathbf{x}^{(k)}$ and $0$s elsewhere. Given
$\mathbf{\tilde{y}}^{(k)}$ and $\mathbf{y}^{k}$, the way decisions are handled
depends on the task, and will be treated separately in Sections III.1 and
III.2.

##  III Experimental realization

Our experimental apparatus is sketched in Fig.2(a). The pump is a C-band, CW
tunable laser (Pure Photonics) which is intensity modulated by an electro-
optic IQ modulator (IxBlue MXIQ-LN-$30$) to realize the desired input pump
waveform $u(t)$. The modulator is driven by a $65\,\textrm{Gs}$ Arbitrary
Waveform Generator (AWG) from Keysight, whose output is amplified by a high
bandwidth amplification stage (IxBlue DR-AN-$28$-MO), providing the necessary
voltage swing of $V_{\pi}\sim 7\,\textrm{V}$ to exploit the full dynamic range
of the modulator. In order to claim that the nonlinear transformation on the
input samples is solely coming from the MR, we did a pre-compensation of the
signal out of the AWG, which eliminates the $\cos^{2}$ dependence of the
modulator intensity response with the applied voltage. A tap of $10\%$ is
placed at the output of the modulator to monitor the input pump, which is
detected by a fast photodiode (Thorlabs DXM20AF). The pump is amplified to a
fixed level of $20\,\textrm{dBm}$ by an Erbium Doped Optical Amplifier (EDFA,
IPG Photonics) and the power regulated by an electronic Variable Optical
Attenuation (VIAVI mVOA-C1). A second CW probe laser (Pure Photonics) is
combined with the pump before being injected into the input port of the MR
using a single mode fiber and a Grating Coupler ($\sim\,3.5\,\textrm{dB}$
loss). The MR under test has a racestrack shape, with a radius of
$7\,\mu\textrm{m}$ and a waveguide cross section of $450\,\textrm{nm}\times
220\,\textrm{nm}$. Two bus waveguides are coupled to the straight sections of
the MR, with a gap of $250\,\textrm{nm}$ and a coupling length of
$3\,\mu\textrm{m}$. The device is described in great details in [31]. From the
low-power transmission spectra recorded at the Through port, we extracted an
intrinsic quality factor (Q) of $Q_{\textup{i}}=1.11(8)\times 10^{5}$ and a
loaded Q of $Q_{\textup{L}}=6.5(2)\times 10^{3}$. We use two adjacent
resonance orders to resonantly couple the pump and probe laser, which have
respectively a cold resonance wavelength of $\sim 1549\,\textrm{nm}$ and $\sim
1538\,\textrm{nm}$. Light at the output of the Drop port is amplified by a
second EDFA (Thorlabs), and a tunable band-pass filter ($0.8\,\textrm{nm}$
bandwidth) removes the spontaneous emission and directs the probe or the pump
laser to the output photodiode (Thorlabs D400FC, $1\,\textrm{GHz}$ bandwidth).
A $4\times 40\,\textrm{Gs}$ oscilloscope (LeCroy SDA 816Zi-A) records the
input pump waveform $u(t)$ and the output probe $u_{pr}(t)$. Virtual nodes are
sampled from $u_{pr}(t)$ at times $t_{n,i}$ and queued to form the state
matrix $\mathbf{X}$. Training and validation are performed off-line using
Matlab.

###  III.1 Binary input: 1-bit delayed XOR

The first task we tested is the 1-bit delayed XOR [18]. Given the virtual node
vector $\mathbf{u}_{pr}^{(k)}$ at time $t_{k}$, the goal is to predict the
result of the XOR operation between the bits $x^{(k)}$ and $x^{(k-1)}$.

Figure 3: (a) Examples of waveforms processed during the XOR task. The pump
laser driving the input port of the MR is shown in black, while the pump and
probe outputs from the Drop port are respectively shown in red and blue. (b)
Bit Error Rate (BER) as a function of the bitrate for the 1-bit delayed XOR
task. Black dots use a predictor matrix $\mathbf{X}$ whose entries are sampled
from the input pump power. Red and blue dots use respectively predictors
sampled from the pump and the probe traces at the Drop port of the MR. In all
the three cases, the average pump power is set to $3\,\textrm{dBm}$. (c) Bit
Error Rate as a function of the average pump power for a fixed bitrate of
$20\,\textrm{Mbps}$. The insets show details of the probe waveform at the pump
powers $-10\,\textrm{dBm}$ and $4\,\textrm{dBm}$.

The input variable can assume two values, $1$ (full transmission of the
modulator) and $0$ (full extinction). In this task, we use $N_{v}=3$ virtual
nodes, and the connectivity mask is simply a $3\times 1$ column vector of
$1$s. This implies that no mask is applied to the input data, so that
$u(t_{n,i}(\sigma))=x^{(n)}\theta(t_{n,i}(\sigma))$. Despite the apparent
simplicity, this task unveils most of the features of the reservoir
architecture: the nonlinear transformation of the inputs, the presence of a
fading memory, and the ability to deal with binary inputs. In this experiment,
we fix the average power of the pump at the input to $3\,\textrm{dBm}$, which
is sufficient to trigger TPA and carrier generation, and its detuning to
$\Delta\lambda_{p}=60\,\textrm{pm}$. The pump intensity is modulated with
$\sim 18\,\textrm{dB}$ of extinction, and the input is a Pseudo Random Binary
Sequence (PRBS) of bitrate $B$, which we varied from $10\,\textrm{Mbps}$ to
$80\,\textrm{Mbps}$. The probe power is set to $P_{pr}=-3\,\textrm{dBm}$,
which is sufficiently low to not alter the resonance shift imparted by the
pump, and its detuning to $\Delta\lambda_{pr}=60\,\textrm{pm}$. An example of
waveforms at the input and output of the MR for a bitrate of
$30\,\textrm{Mbps}$ are shown in Fig.3(a). Whenever the input pump changes
from a low to an high state, a transient exponential decay is observed at the
Drop port, which is due to the blue shift of the resonance as a consequence of
carrier generation and accumulation. The cold cavity detuning
$\Delta\lambda_{p(pr)}$ is increased by FCD, determining a decrease of the
transmittance, which also occurs for the probe output. The opposite trend is
observed, in the probe trace, during the transition from a high to a low
state. Here, carrier recombines and the detuning $\Delta\lambda_{p}(pr)$
decreases, pulling the resonance back to the cold position. All the transient
phenomena occur at the characteristic timescale of
$\tau_{\textup{fc}}=45\,\textrm{ns}$, i.e., the value of the carrier lifetime
we measured in one of our previous works [31].
Virtual nodes are obtained by sampling these traces at $10\,\textrm{Gbps}$,
smoothing them with a moving average filter ($10$ points), and downsampling
them at times $\Delta=(N_{v}B)^{-1}$. After smoothing, the Signal to Noise
ratio (SNR) increases from $18\,\textrm{dB}$ (from the raw data acquisition)
to $\sim 20\,\textrm{dB}$. Figure 3(b) shows the Bit Error Rate (BER) as a
function of the bitrate for different predictor matrices $\mathbf{X}$, whose
entries are respectively populated by sampling the virtual node values from
the input pump intensity (black), the output pump power (red) and the output
probe power (blue). To extract the BER, $6500$ bits are used for training and
$3500$ for validation. Errorbars are obtained by partitioning the validation
set into $5$ non-overlapping batches, over which the BER is evaluated. The
error is estimated as the standard deviation over the different batches. The
output of the linear classifier $\tilde{\mathbf{Y}}$ is digitized by applying
a threshold, which optimizes the BER at each bitrate. The XOR task is never
solved by sampling virtual nodes from the bare input pump. The average BER is
$\sim 30\%$ for all the bitrates. This agrees with the fact that the task is
not linearly separable and requires at least one bit of memory to be
performed. Virtual nodes sampled from the pump at the output of the Drop port
improve the BER, which is however still bounded above $\sim 5\%$. The reason
behind this is that bits which are zero do not carry optical power, so these
virtual nodes are sampled from the background noise. This issue is solved by
using the probe signal. Indeed, error free operation is achieved for bitrates
lower than $25\,\textrm{Mbps}$. Since we use $700$ bits for each partition,
error free means that the actual BER is lower than $1.4\times 10^{-3}$. At
higher rates, carriers are no more able to react sufficiently fast to follow
the pump power variations, so the BER increases. The ability to solve the
delayed XOR demonstrates the presence of memory and nonlinearity in the
reservoir. This also illustrates that the use of a probe beam is beneficial
for tasks where the input is binary and with full modulation. It helps to
improve the separability of the reservoir [23]. To investigate the influence
of the input power on the BER, we fixed the bitrate to $20\,\textrm{Mbps}$,
and we changed the average input power of the pump $P_{p}$. The calculated BER
is shown in Fig.3(c). The error rate is almost power-insensitive and equal to
$\sim 25\%$ for $P_{p}<-4\,\textrm{dBm}$, while it drops below $1\%$ at
$P_{p}=0\,\textrm{dBm}$. At $P_{p}>2\,\textrm{dBm}$, error-free operation is
achieved. The low and high power insets shown in Fig.3(c) explain why the BER
improves by increasing power. At $P_{p}=-10\,\textrm{dBm}$, few carriers are
generated, so the probe is weakly perturbed from its original CW state and
virtual nodes acquire a noisy flat distribution of values. At
$P_{p}=4\,\textrm{dBm}$, the FCD shift is source of more variability in the
virtual nodes states, improving the input separability of the reservoir. We
also averaged several traces at low power in order to compare the performance
at the same SNR of the ones at high power, but we only observed a minor
improvement of the BER. This confirms that the poor performance measured at
low power is due to the reduced virtual node variability.

###  III.2 Analog input: Iris species recognition

Figure 4: (a) Examples of waveforms of the input pump (top panel, in black)
and of the probe at the output of the MR (bottom panel, in blue) for
$N_{v}=50$ and a bitrate of $20\,\textrm{Mbps}$. Different background colours
highlight the subspecies of the flower. A representative picture of them is
shown in the insets of the top panel. Virtual nodes sampled from the probe
trace are indicated with red dots on the central band. (b) Maps of the
(normalized) intensity of the different virtual nodes ($N_{v}=50$, bitrate
$20\,\textrm{Mbps}$) sampled from the output probe, acquired at the pump
powers of $4\,\textrm{dBm}$ (left), $7\,\textrm{dBm}$ (center), and
$9\,\textrm{dBm}$ (right). These are $189$ flower samples vertically stacked
in each map. Even if they are randomly injected at the input port during the
training and test phase, they are shown grouped together in the three
subspecies, as indicated by the labels on the left. This highlights the
distinction between the classes. (c) Classification rate as a function of the
average input power of the pump for $N_{v}=50$ and bitrates of
$20\,\textrm{Mbps}$ (left) and $40\,\textrm{Mbps}$ (right). Black scatters use
virtual nodes sampled from the output probe while red scatters from the input
pump. The dashed blue line is the lower bound in the classification rate
obtained by feeding the input samples into a linear classifier, without
electro-optic conversion. (d) Confusion charts for a MR reservoir which uses
$10$ (left), $25$ (center), and $50$ (right) virtual nodes. The bitrate is
$20\,\textrm{Mbps}$.

The second task we tested is the classification of the Iris flowers, a well
known dataset [28] in the field of machine learning. Here, the goal is to
classify a given Iris flower into one of the three possible subspecies:
_Setosa_ , _Versicolor_ and _Verginica_. The classification is based on four
real inputs x, physically corresponding to the length and width of the petals
and sepals of the flower. All the four inputs are required to solve the task,
since the flowers can not be linearly classified on the basis of a single
feature. Moreover, even by using all the four inputs, only one species
(_Setosa_) is linearly separable from the others, making the task nontrivial.
As described in Section II, the dimension of the original input space is
increased from $4$ to $N_{v}$ by implementing the transformation
$\mathbf{x}\rightarrow\mathbf{W}_{\textup{in}}\mathbf{x}$, where the
connectivity matrix $\mathbf{W}_{\textup{in}}$ has dimension $N_{v}\times 4$.
An offset is applied to the transformed input to remove the negative values,
and the result is scaled to reach the desired target average power $P_{p}$.
The sequence is then encoded in the pump intensity, and sequentially injected
to the input port of the MR. We tested different numbers of virtual nodes
$N_{v}=\\{10,25,50\\}$, and two values of time separation
$\Delta=\\{25,50\\}\,\textrm{ns}$. Each flower sample is sequentially encoded
in the pump intensity $u(t)$ and injected into the input port of the MR.
Different flowers are separated by a delay of $\delta=100\,\textrm{ns}$,
during which the pump power is turned off to quench the carrier dynamics. This
avoids undesired crosstalk of information between neighbouring flowers. The
original dataset contains $150$ flowers, with an equal number of
representatives of each class. Each instance is injected into the chip is
randomly extracted from the dataset. Figure 4(a)(top panel) shows an example
of pump waveform at the input of the MR for three flower instances of
different subspecies. Each is made by $50$ steps, which corresponds to the
virtual nodes. The low speed of operation allows to use the full resolution of
the digital to analog converter of the AWG ($8$ bit), imprinting on the
optical power trace the fine details of each flower sample. The associated
probe waveform, after being processed by the MR reservoir, is shown in
Fig.4(a)(bottom panel). With respect to the XOR task, where the input is
binary, the multilevel excitation imparts a richer dynamics on the output
probe, which results in an enhanced variability between the virtual nodes. The
reservoir response to different flower samples is represented in the maps of
Fig.4(b), which shows the intensities of the $50$ virtual nodes for $189$
flower instances (vertically stacked). In Fig.4(c) we report the
classification rate for $20\,\textrm{Mbps}$ and $40\,\textrm{Mbps}$ as a
function of the average input power. Categories are assigned by training
multiple linear classifiers, one for each subspecies, and decisions are made
on the basis of a winner takes all scheme [9]. The best performance is
obtained for $20\,\textrm{Mbps}$ and a pump power of $P_{p}=7\,\textrm{dBm}$,
yelding a classification rate of $(99.3\pm 0.2)\%$. In comparison, the
classification rate obtained by sampling the virtual nodes from the input pump
is below $87\%$. This value is slightly above the theoretical limit of
$84.66\%$, obtained by directly feeding the input predictor matrix
$\mathbf{W}_{\textup{in}}\mathbf{X}_{\textup{in}}$ into a linear classifier.
This is probably due to some residual nonlinear distortion in the electro-
optic conversion. The rate of classification obtained by a single MR exceeds
the accuracy achieved by a more complex integrated system (three layered feed
forward neural network) presented in a recent work [33], and has competitive
performance with software based classification algorithms [34]. The
classification rate in Fig.4(c) increases from low to high power, is maximum
at $7\,\textrm{dBm}$ and then decreases for higher powers. We can understand
this trend by observing the virtual node intensity in the maps of Fig.4(b). At
low powers ($P_{p}=4\,\textrm{dBm}$ in Fig.4(b)), the MR reservoir shows poor
separability of the inputs, as witnessed by the quite flat virtual node
intensity distribution observed both within the same flower (horizontal lines)
and between different subspecies (vertical lines). As discussed for the XOR
task, this is related to the weak perturbations imprinted on the CW probe from
the pump intensity at low powers. At the highest classification rate of
$P_{p}=7\,\textrm{dBm}$ (Fig.4(b)), the virtual node distribution is richer,
with a relative variation that can exceed $70\%$ (see Fig.4(b) at
$P_{p}=7\,\textrm{dBm}$). We also observe the creation of two bands, where the
virtual node intensity is around the $35\%$ of its maximum value. As discussed
in Section III.3, these form as a consequence of the interplay between thermal
and free carrier dispersion. At high power ($P_{p}=9\,\textrm{dBm}$ in
Fig.4(b)), multiple bands of low transmittivity are formed, which lack
synchronization with the rate of flower injection. As discussed in Section
III.3, within the bands the MR is off resonance, hence the information can not
be efficiently imprinted from the pump to the probe, effectively ”burning”
these virtual nodes from the reservoir. This explains the drop in the
classification accuracy at high powers.
The net flower classification rate can be calculated as
$(N_{v}\Delta+\delta)^{-1}$, which for $N_{v}=50$ and
$\Delta=50\,\textrm{ns}$, i.e., the configuration showing the best accuracy,
is $0.38\,\textrm{MHz}$. We tried to scale down to $N_{v}=25$ and $N_{v}=10$
in order to improve the speed of operation, but as shown by the confusion
charts in Fig.4(d), the classification accuracy drops to $(93.6\pm 0.3)\%$ for
$N_{v}=25$ and $(89.2\pm 0.2)\%$) for $N_{v}=10$. We did not observe any
significant improvement by scaling up to $N_{v}=100$.

###  III.3 Enhancing the reservoir performance by mixing thermal and free
carrier dynamics

In order to explain the behavior of the classification rate as a function of
the input pump power in Fig.4(c), we performed numerical simulations. They are
based on the theoretical model of [31]. Specifically, we computed the time
evolution of the complex field of the MR, including TPA, FCA, FCD and the
thermo optic dispersion (TOE). Whenever possibile, the parameters are taken
from the experiment. A more comprehensive description of the model can be
found in [31]. We fix the number of virtual nodes to $50$, the bit rate to
$20\,\textrm{MHz}$ and the quenching time to $\delta=100\,\textrm{ns}$.

Figure 5: (a) Top: numerical simulation of the Iris dataset classification
rate as a function of the input power. Bottom: probability distribution of the
virtual node detuning $\Delta\lambda$ as function of the input pump power. (b)
Simulated probe power at the Drop port of the MR (top), temperature variation
(middle) and free carrier concentration for $P_{p}=9\,\textrm{dBm}$. The red
dashed lines mark the injection times of three different flower samples. The
color code indicates the instantaneous detuning of the probe, whose relation
with the Drop intensity is shown in panel (c). (c) Normalized intensity at the
Drop port of the MR as a function of wavelength. The dashed line marks the
wavelength of the probe laser. The color code is used to label the
instantaneous detuning of the time traces in panel (b).

The top panel in Fig.5(a) shows the simulated classification rate as a
function of the input power, estimated from the whole Iris dataset ($150$
flowers). We add white noise to the simulated output of the MR to set the SNR
to $18\,\textrm{dB}$, which is comparable to the one of the experiment. Each
point is the result of an average over $1000$ realizations. We notice a very
good agreement with the experimental data in Fig. 4(c), with a maximum of
classification occurring around $7\textrm{dBm}$. We investigate the origin of
this trend starting from the color map reported in the bottom panel of Fig.
5(a). Here, for each input power, we calculate the probability that a virtual
node is sampled at the probe resonance detuning $\Delta\lambda$ from the cold
cavity condition. At low power, all virtual nodes are sampled around the cold
resonance detuning, which in our case lies at $\Delta\lambda=60\,\textrm{pm}$.
As we raise the input power up to about $5\,\textrm{dBm}$, FCD shifts the
resonance frequency towards the blue, further increasing the initial detuning.
Starting from $6\,\textrm{dBm}$, we observe that the variance increases, and
that there is a small probability to sample virtual nodes at negative
detunings. For $P_{p}>6\,\textrm{dBm}$, the distribution is bimodal, i.e,
nodes are concentrated around two bands: a narrow one at positive
$\Delta\lambda$ and a broad one at negative $\Delta\lambda$. This is a
signature that the temporal evolution is not only related to free carriers,
where the temperature acts as a constant background. In fact, at these powers,
we are entering a regime where carriers and temperature dynamics mix together.
This is shown in the panels of Fig.5(b), which refer to
$P_{p}=9\,\textrm{dBm}$. Here we report, as a function of time, the output
power of the probe (top), the temperature difference between the MR and its
surroundings (middle), and the free carrier concentration (bottom) for three
consecutive flower samples (the injection times are marked with red dashed
lines). The color code indicates the instantaneous detuning of the probe,
whose relation with the Drop intensity is shown in Fig.5(c). When a flower
sample is injected, the temperature of the MR increases, but at low powers TOE
is counteracted by FCD. At high power TOE prevails, and the wavelength
detuning of the cavity can eventually become negative. When this occurs, a
positive feedback is established between the decrease of the carrier
concentration and the decrease of MR energy. This corresponds to the
transition from the high power stability branch of the carrier bistability
curve to the one at low power [35]. After the transition, the MR lies off
resonance [29]. The detuning is now negative due to the residual thermo optic
shift. This originates the bands of low transmittivity in Fig.4(b) and in
Fig.5(b). Virtual nodes sampled within these bands form the second broad peak
in the probability distribution in Fig.5(a). As the MR cools down, the
resonance moves towards the pump(probe) wavelength, gradually increasing the
probe signal. A transition from the low power stability branch of the carrier
bistability curve to the one at high power restores the initial conditions, so
that virtual nodes are no more sampled within the band. This interplay of
thermal and free carrier effects would generate cavity self-pulsing in case of
CW input powers [29]. It turns out that, for a limited range of powers
centered around $7\,\textrm{dBm}$, the presence of the bands increases the
classification rate. This finds explanation in the enhanced variability that
the virtual nodes values assume. Indeed, in this regime, they efficiently
sample all the probe resonance. Moreover, quenching the MR for
$\delta=100\,\textrm{ns}$ allows to reset the initial conditions for the
temperature and the carrier population between consecutive flower samples,
creating bands always in the same time slot (see Fig.4(b),
$P_{p}=7\,\textrm{dBm})$. If the power is increased above $7\,\textrm{dBm}$,
bands get wider in time and many of them form within the same flower sample.
In these conditions, a large number of virtual nodes are sampled within bands,
where the cavity lies off resonance and hence no information can be
transferred from the pump to the probe beam. Furthermore, the quenching time
is no more sufficient to reset the initial conditions, so as bands form at
random times, turning the probe output to chaotic (see Fig.4(b),
$P_{p}=9\,\textrm{dBm}$) [36].

##  IV Discussion

In the above sections, we demonstrated the use of a single integrated MR as a
reservoir, in which the core computing power is provided by the complex
coupled dynamics of free carriers, TPA and temperature. The architecture is
simple, and as such faces limitations which can be however mitigated by
different engineering solutions. While a data processing speed of few MHz
could be of interest for some specific tasks as speech recognition [37], event
detection or remote control [38], it is not as competitive as electronic
solutions. In fact, the optical implementation of machine learning is not only
of interest for the low power consumption and minimal latency, but especially
for the large amount of information that can be processed, at rates in excess
of Gbps [39]. We envisage that there are at least three ways to raise the
performances of our scheme. First, the MR can be used to form more advanced
spatial topologies, i.e. matrices/arrays of MR. As an example, using
resonators with a moderately high Q factor of $6.5\times 10^{4}$, a network
made by $10$ MR can accomodate $500$ virtual nodes and operates with
$5\,\textrm{mW}$ of power. Moreover, small sequences of coupled resonators
have shown very complex dynamics under thermal and free carrier nonlinearities
[36]. In principle, this suggests that the same performance could be obtained
with less virtual nodes, hence increasing the speed of operation.
Additionally, one could multiplex several spatial inputs [40] and coherently
combine them on chip using linear optical elements, such as arrays of Mach-
Zehnder interferometers arranged in a universal scheme [41]. This hybrid
spatio-temporal architecture combines a simple hardware implementation,
typical of delayed nodes RC, with the compactness of a silicon photonic chip.
Second, to increase the speed of operation, the intrinsic time response of the
RC can be improved by reducing the carrier lifetime. Indeed, our devices have
unusually long recombination times compared to what is reported in literature
[42]. A control of the lifetime can be obtained by embedding carrier depletion
modulators within the MR [43]. Third, by exploiting different resonance
orders, we can implement wavelength-division multiplexing [44, 45]. It is
worth to note that this is not equivalent to operate several independent
reservoirs in parallel. Indeed, since the carrier population interacts with
all the optical modes, they become coupled (second term in Eq.(1)) [46].
Moreover, additional virtual node interaction would arise from Cross Photon
Absorption.
Another aspect of concern is the memory capacity. Since our device lacks of
any delay loop, the memory is provided by carrier relaxation, and is of short
term. As shown in Section II, the memory is of the order of $\sim
3\,\tau_{\textup{fc}}$, which only extends over the first neighboring virtual
nodes. Moreover, the terms in Eq.(1) indicate that the memory on the past
inputs of the pump ($\propto u_{p}^{2}$) and of the virtual nodes ($\propto
u_{pr}u_{p}$) is nonlinear. There is a clear unbalance between memory and
nonlinearity, the second being preponderant, which is known to be sub-optimal
for a certain class of tasks [47]. However, we believe that this is not a
severe limitation. It has been already shown and experimentally validated that
by providing past inputs to the input layer of a memory-less reservoir, the
latter performs as well as the ones with delayed feedback loops. [48, 19].

##  V Conclusions

In this work, we have experimentally shown a time multiplexed architecture
which uses a single integrated MR as a substrate for RC. We encode the input
layer in the temporal evolution of the intensity of a pump laser. When this is
resonantly coupled to the MR, the information is transferred to a secondary
probe laser, which is tuned on an adjacent resonance order. The transfer is
enabled by TPA, which generates a free carrier dynamics that is imprinted on
the probe laser by FCD. We train the reservoir to perform some proof of
concept tasks by using regularized linear regression. We experimentally
demonstrate that the system can handle problems with binary and analog inputs.
We have shown that the MR solves the 1-bit delayed XOR task with a minimum BER
of $1.4\times 10^{-3}$ for bitrates up to $30\,\textrm{MHz}$. This result is
obtained by only exploiting the nonlinear free carrier dynamics. As a second
step, we have shown that the system can solve the analog task of the
classification of the Iris dataset.
We demonstrated a maximum classification rate of $(99.3\pm 0.2)\%$ at a rate
of $0.38\,\textrm{MHz}$, using an average power of $7\,\textrm{dBm}$ at the
input of the MR. In this regime, the thermal and the free carrier dynamics are
strongly coupled. This feature enhances the separability properties of the
reservoir, by adding more variability to the virtual node distribution. The
optimal performance is found across the edge between a stable and a self-
pulsing regime, which agrees with the predictions of numerical simulations.
To the best of the author’s knowledge, this is the first experimental
demonstration of RC using a silicon MR. We envisaged several future
developments and engineering solutions, all leveraging on silicon photonics.
This will open the doors to the realization of a wide range of hybrid spatio-
temporal reservoirs offering increased computational complexity.

## Appendix A: Derivation of the reservoir dynamical equation

Here we derive Eq.(1) of the main text, which describes the time evolution of
the probe at the output of the Drop port of the resonator. The starting point
is the equation governing the free carrier dynamics, which is given by [29]:

$\frac{d\Delta N}{dt}=-\frac{\Delta
N}{\tau_{\textup{fc}}}+g_{\textup{tpa}}U^{2},$ (2)

where $g_{\textup{tpa}}$ is the free carrier generation rate per unit energy
induced by TPA and $U$ is the internal energy of the resonator. Equation (2)
has the formal solution:

$\Delta
N(t)=g_{\textup{tpa}}\int_{-\infty}^{t}e^{-\frac{t-\xi}{\tau_{\textup{fc}}}}U^{2}(\xi)d\xi.$
(3)

The total energy is given by $U=U_{p}+U_{pr}$, i.e., the sum of the energies
of the pump ($U_{p}$) and probe ($U_{pr}$) lasers. Since the probe is weaker
than the pump, we let $U\sim U_{p}$. From temporal coupled mode theory, the
expression for $U_{p}$ is given by $U_{p}=f_{p}P_{p}$, where $f_{p}$ is
defined as [49]:

$f_{p}=\frac{\gamma_{e}}{\left(\omega_{p,0}(1+\delta\omega)-\omega_{p}\right)^{2}+\gamma_{\textup{tot}}^{2}},$
(4)

in which $\gamma_{\textup{tot}}$ is the total loss rate of photons from the
cavity, $\gamma_{e}$ the extrinsic loss in the bus waveguide, and
$\delta\omega$ the normalized frequency shift imparted by thermal and free
carrier dispersion. The latter is given by
$\delta\omega=-\frac{\Gamma}{n}\left(\sigma_{\textup{FCD}}\Delta
N+\sigma_{\textup{TOE}}\Delta T\right)$, where $\Gamma$ is the modal
confinement factor, $n$ the refractive index of the waveguide core material,
$\sigma_{\textup{FCD}}$ the FCD coefficient, $\sigma_{\textup{TOE}}$ the
thermo optic coefficient and $\Delta T$ the differential temperature of the MR
with respect to the cold cavity condition. An similar expression holds for
$f_{pr}$, which defines $U_{pr}=f_{pr}P_{pr}$. Note that Eq.(4) assumes that
the MR internal energy adiabatically follows the input power variations
$P_{p}$. In our case, this holds since $P_{p}$ varies on the timescale of the
free carrier lifetime $\tau_{\textup{fc}}=45\,\textrm{ns}$, while the photon
lifetime is $\frac{2}{\gamma_{\textup{tot}}}\sim 10\,\textrm{ps}$. Moreover,
in our experiment the pump power varies much faster than the thermal decay
constant of the MR [31], so that after an initial transient, the temperature
reaches an equilibrium value $\overline{\Delta T}$. As discussed in Section
3.C, for our MR this approximation holds up to $P_{p}=6\,\textrm{dBm}$, after
which thermal effects drive the cavity in an unstable regime and have to be
accounted for in the dynamics. We now expand $f_{p}^{2}$ to the first order in
$\Delta N$ around a reference value $\overline{\Delta N}$ as
$f_{p}^{2}\sim\overline{f_{p}^{2}}+\frac{df_{p}^{2}}{d\Delta N}\left(\Delta
N-\overline{\Delta N}\right),$ where:

$\frac{df_{p}^{2}}{d\Delta
N}=\frac{4\frac{\Gamma\sigma_{\textup{FCD}}}{n}\omega_{p,0}\gamma_{e}^{2}\left[\left(1+\delta\omega\right)\omega_{p,0}-\omega_{p}\right]^{2}}{\left[\left(\left(1+\delta\omega\right)\omega_{p,0}-\omega_{p}\right)^{2}+\gamma_{\textup{tot}}^{2}\right]^{3}}.$
(5)

In deriving Eq.(5), we neglected the dependence of $\gamma_{\textup{tot}}$ on
both $\Delta N$ and $U$, which is due respectively to FCA and TPA, since their
contribution is small compared to FCD. If the pump power variations around the
reference value $\overline{P}_{p,0}$, giving the equilibrium condition $\Delta
N=\overline{\Delta N}$, are sufficiently small to justify the first order
expansion of $f_{p}^{2}$, we can insert it into Eq.(3) to give:

$\Delta
N(t)=g_{\textup{tpa}}\int_{-\infty}^{t}e^{\frac{t-\xi}{\tau_{\textup{fc}}}}P_{p}^{2}(\xi)\left(\overline{f_{p}^{2}}+\frac{df_{p}^{2}}{d\Delta
N}\left(\Delta N(\xi)-\overline{\Delta N}\right)\right)d\xi.$ (6)

Since the probe and the pump lasers have similar detunings and linewidth
$\gamma_{\textup{tot}}^{-1}$ with the corresponding MR resonances, we can also
expand $f_{pr}$ to the first order in $\Delta N$. This allows to relate
$\Delta N$ to the probe energy $U_{pr}$ as:

$\Delta N-\overline{\Delta N}=\left(P_{pr}\frac{df_{pr}}{d\Delta
N}\right)^{-1}\left(U_{pr}-\overline{U_{pr}}\right).$ (7)

By substituting Eq.(7) into Eq.(6) and rearranging terms, we obtain Eq.(1) of
the main text, with the definitions:

$c_{0}=\gamma_{e}\left(\overline{U_{pr}}-\frac{df_{pr}}{d\Delta
N}\overline{\Delta N}P_{pr}\right),$ (8) $c_{1}=\frac{df_{pr}}{d\Delta
N}g_{\textup{tpa}}\gamma_{e}\overline{f_{p}^{2}}P_{pr}\left(1-\frac{1}{\overline{f_{p}^{2}}}\left(\frac{df_{p}^{2}}{d\Delta
N}\right)\left(\frac{df_{pr}}{d\Delta
N}\right)^{-1}\frac{\overline{U_{pr}}}{P_{pr}}\right),$ (9)
$c_{2}=g_{\textup{tpa}}\gamma_{e}\frac{df_{p}^{2}}{d\Delta N}.$ (10)

By taking the values of the varius coefficients from [31], and by using
$P_{pr}=0.5\,\textrm{mW}$, one finds $c_{1}\sim 2430\,\textrm{fJ}^{-1}$ and
$c_{2}\overline{U_{pr}}\sim-c_{1}$.

## Appendix B. Relative magnitude between Kerr and Free Carrier effects

Using the same notation of Appendix A, the normalized resonance shift
$\delta\omega$ imparted by Kerr and FCD are given respectively by
$\delta\omega_{\textup{FCD}}=-\frac{\Gamma}{n}\sigma_{\textup{FCD}}\Delta N$
and $\delta\omega_{\textup{Kerr}}=-\frac{cn_{2}U_{p}}{n^{2}V}$ [50], where
$n_{2}$ is the nonlinear refractive index of silicon, $c$ the speed of light
and $V$ the volume of the MR. We can calculate $\Delta N$ as the steady state
solution of Eq.(2) as $\Delta N=\tau_{\textup{fc}}g_{\textup{tpa}}U_{p}^{2}$,
that combined to Eq.(4) yields the following expression for
$\delta\omega_{\textup{fc}}$:

$\delta\omega_{\textup{fc}}=-\frac{\sigma_{\textup{FCD}}\tau_{\textup{fc}}g_{\textup{tpa}}q^{2}(\omega_{p},\omega_{p,0})Q^{2}P_{p}^{2}}{16\omega_{p}^{2}n},$
(11)

where $q(\omega_{p},\omega_{p,0})$ is defined as:

$q(\omega_{p},\omega_{p,0})=\left(\left(\frac{\omega_{p,0}}{\gamma_{\textup{tot}}}\left(1+\delta\omega-\frac{\omega_{p}}{\omega_{p,0}}\right)\right)^{2}+1\right)^{-1}.$
(12)

For simplicity, we assumed $\gamma_{\textup{tot}}\sim 2\gamma_{e}$, which in
our case is justified by the fact that $Q_{i}=\frac{\omega}{\gamma_{i}}\gg
Q_{e}$ (see Section 3 of the main text). Similarly, the expression for
$\delta\omega_{\textup{Kerr}}$ is given by:

$\delta\omega_{\textup{Kerr}}=-\frac{cn_{2}q(\omega_{p},\omega_{p,0})QP_{p}}{4\omega_{p}n^{2}V}.$
(13)

We can now calculate the ratio between the FCD and the Kerr frequency shift
as:

$\frac{\delta\omega_{\textup{fc}}}{\delta\omega_{\textup{Kerr}}}=\frac{\sigma_{\textup{FCD}}\tau_{\textup{fc}}g_{\textup{tpa}}q(\omega_{p},\omega_{p,0})nVQP_{p}}{4\omega_{p}cn_{2}}.$
(14)

By inserting the values of the parameters in [31], and using
$\sigma_{\textup{FCD}}=-4\times 10^{-27}\textrm{m}^{3}$ [51],
$P_{p}=5\,\textrm{mW}$ and $q=0.95$ (which corresponds to the experimental
pump detuning of $-7.5\,\textrm{GHz})$ we obtain
$\frac{\delta\omega_{\textup{fc}}}{\delta\omega_{\textup{Kerr}}}\sim 200$. We
expect the actual ratio to be slightly lower due to the degradation of the
quality factor caused by TPA and FCA, which however does not alter the claim
that FCD imparts a resonance shift which is two orders of magnitude higher
than the one of the Kerr.

## Acknowledgements

The authors acknowledge Dr. Mattia Mancinelli and Mr. Davide Bazzanella for
the technical support in the experiment and the fruitful discussions.

## Funding

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 788793, BACKUP), and from the MIUR under the project PRIN
PELM (20177 PSCKT).

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