1. Introduction

One of the most appealing platforms for photonic quantum technolgies is integrated optics [1,2]. Since the first demonstrations more than a decade ago [3], the complexity and scale of quantum optical circuits exponentially increased over the years [4–6]. Irrespective of their application, key elements shared by every photonic circuit are photon sources. Silicon photonics architectures rely on Spontaneous Four Wave Mixing (SFWM) to probabilistically generate pairs of photons [7]. Waveguides sources constitutes the simplest example, emitting photons with high heralding efficiency but strong spectral correlations [8]. Complementarily, micro-resonators produce photons in almost pure states [9–11], but necessitates of tuning elements to ensure indistinguishability among independent sources [12,13]. In parallel to source optimization, the developement of tools for their spectral characterization has known a burst since the introduction of Stimulated Emission Tomography [14]. SET allows to make predictions on SFWM (Spontaneous Parametric Down Conversion) based on the output of its classical counterpart, i.e., Stimulated Four Wave Mixing (Difference Frequency Generation (DFG)) [15]. Being the latter orders of magnitude more intense than the corresponding quantum process, integration times have been enormously decreased, and the resolution improved compared to spectrally resolved coincidence measurements [16]. The Joint Spectral Amplitude is the function describing the spectral correlations of the photon pair [7]. So far, most of the applications of SET focused on the determination of the modulus of the JSA, referred as the Joint Spectral Intensity (JSI), since it is directly measurable from the power of the stimulated field. Examples includes silicon nanowires [17], AlGaAs ridge waveguides [18], optical fibers [19,20], All-Pass resonators [21] and coupled rings [22]. A variation of SET, which implements Sum Frequency Generation (SFG), has been used to map the JSA of an array of evanescently coupled Lithium Niobate waveguides [23]. A method for the extraction of the complex JSA in multi-port devices, which implements DFG, has been proposed based on the eigenmode expansion of single channel excitations [24]. SET has been also extended to others degrees of freedom, like space [23] and polarization [25,26]. The Joint Spectral Phase (JSP) is of the same relevance of the JSI, but historically received minor attentions. The JSI itself is, however, an incomplete picture of the quantum state, since it hides the spectral correlations encoded in the phase domain [27]. As an example, only the lower bound of the Schmidt number can be estimated if the JSP is not known [18]. Beside quantum homodyne tomography of two photon states [28,29], which suffers the same time and resolution issues of spectrally resolved coincidence measurements, phase resolved applications of SET have been so far limited to in-line sources as waveguides [27,30] or cold atomic ensembles [31]. In all these works, the Pump, the Seed and the reference beams are all carved from the same laser source to guarantee mutual coherence, and fiber based interferometers are used to extract the JSP.

Here, a novel technique which allows to measure the complex JSA of a double bus, integrated silicon ring resonator, is proposed and experimentally validated. The method exploits the on-chip interference between the stimulated field of the ring, carrying the information on the JSA, and the one of a coherently pumped reference waveguide. Outside to the hypothesis of SET, the stimulating Seed is not mimicking the asymptotic output field of the corresponding photon of the pair [32]. This makes the excitation scheme trivial, since both the Seed and the Pump lasers are directly coupled into the same port of the resonator, but comes at the expense of determining the transfer function of the device. However, the the integrated circuit is designed to to perform this task without additional complexity. Phase resolved tomography is completely implemented on a chip (an external filter is solely used to increase the spectral resolution), without the use of any auxiliary reference laser, since this is naturally provided by the broadband emission of the reference waveguide. Thermal phase shifters allow the circuit to perform different tasks, as addressing the individual JSI of the ring and of the waveguide, manipulating their interference, and probing the device transfer function. Through an Hamiltonian treatment [32], it is rigorously proved that the scheme can be applied to a single resonator with an arbitrary number of channels, even if they are not physically accessible, as the ones associated to losses [33].

2. Stimulated emission tomography on an Add-Drop resonator

The specific case of interest is of an Add-Drop (double bus) resonator with four channels, which will be labelled as Input, Through, Add and Drop (respectively I,T,A and D in Fig. 1). As a first step, the device will be considered as lossless, a condition that will be relaxed later. The geometry is that of a ring made by a single mode waveguide which has been bent on itself. The Pump and the Seed lasers are coupled to the Input port, and the intensity of the stimulated field (the Idler) is monitored in either the Drop or the Through bus waveguide. Light is coupled from the bus to the ring by evanescent coupling, a condition that is realized by bringing two waveguides very close to each other, typically of the order of few hundreds of nanometers. During each roundtrip along the ring, a fraction of light is scattered into the Drop and Through waveguides, while the rest recirculates. For particular frequencies, all the recirculating waves add in phase, this is called the resonance condition. Due to the high intensity which builds up inside the ring at resonance, nonlinear effects such as SFWM produce photon pairs in narrow spectral intervals around these frequencies. Quantum-mechanically, this process can be described using different approaches. Commonly, the system Hamiltonian $H$ is written as $H=H_{\textrm {ring}}+H_{\textrm {wg,D}}+H_{\textrm {wg,T}}+H_{\textrm {ring,T}}+H_{\textrm {ring,D}}+H_{\textrm {nl}}$, where $H_{\textrm {ring}}$, $H_{\textrm {wg,D}}$ and $H_{\textrm {wg,T}}$ are the linear Hamiltonians of the ring, the Drop and the Through waveguides respectively, each one describing the free field evolution of their own eigenmodes. The terms $H_{\textrm {ring,T}}$ and $H_{\textrm {ring,D}}$ describe the coupling of the ring modes to the waveguide, while $H_{\textrm {nl}}$ is the nonlinear Hamiltonian responsible for SFWM in the ring. In this approach, linear combinations of the eigenstates of $H_{\textrm {ring}}$ and $H_{\textrm {wg,(T,D)}}$ are built to express the evolution of photon pairs under the perturbation Hamiltonian $H_{\textrm {nl}}$ [33]. In this paper, the approach based on asymptotic fields, described in [34], is preferred due to the ease of implementation. Essentially, in place of using the eigenstates of the isolated photonic components, one constructs eigenstates of the full linear system $H_L$, that is $H_L = H_{\textrm {ring}}+H_{\textrm {wg,D}}+H_{\textrm {wg,T}}+H_{\textrm {ring,T}}+H_{\textrm {ring,D}}$, leaving $H_{\textrm {nl}}$ as a perturbation. The eigenstates of $H_L$ are called asymptotic states, and correspond to solutions of Maxwell’s equations of the entire photonic structure. A device with $M$ ports has $M$ asymptotic states. The asymptotic input state associated to port $m$ has a wave incoming into that port, and outgoing waves from all the other ports (in principle, even outgoing from $m$) as a consequence of the system’s evolution under $H_L$ (linear scattering problem). Equivalently, the asymptotic output state of port $m$ has an outgoing wave from that channel, and incoming waves in all the other (in principle, even incoming to $m$) [34]. Importantly, these states describe fields which extend over both the waveguides and the ring, as a consequence of the fact that they are eigenstates of the full linear Hamiltonian. A sketch of these states is shown in Fig. 1. One of the key hypothesis of SET is that the Seed laser, with wavevector $k_s$, must be coupled in the same asymptotic output field of the corresponding photon of the pair [14]. In this case, the amplitude $\gamma (k_i)$ of the stimulated field at wavevector $k_i$ is directly proportional to the JSA $\phi (k_s, k_i)$. The aim of this section is to derive a more general relation between $\phi$ and $\gamma (k_i)$, which holds even when the Seed is not an asymptotic output field (as in this work), and from this to define an alternative method for recovering $\phi$. The nonlinear Hamiltonian $H_{\textrm {nl}}$ responsible for SFWM is [31]:

 

Fig. 1. The four asymptotic input (blue box) and output (red box) fields of the system. Arrows indicate the direction of energy flow. The asymptotic fields are labelled according to the ports pointed by their corresponding colored arrow. The (wavelength dependent) complex amplitude associated to each arrow is labeled as $T_H$ if it refers to the Through transfer function, or $D_R$ if it refers to the Drop one. The first is obtained by seeding the input port, and by collecting the spectra of the complex field at the output of the Through port. Analogously, the Drop transfer function is obtained by collecting the spectra of the complex field at the output of the Drop port. For asymptotic output states, the absence of an outgoing arrow in some ports indicates that the field is vanishing due to destructive interference.

Download Full Size | PDF

The result of Eq. (6) tells that the JSAs are actually the kernels which relate the amplitude of the stimulated field to the ones of the asymptotic-output fields of the Seed. If the latter is exactly mimicking one of them, i.e., $\gamma _{sy}(k_{s0})=0$ and $\gamma _{sy'}(k_{s0})\neq 0$ with $y \neq y'$, then the amplitude of the stimulated field in channel $x$ is directly proportional to the wavefunction $\phi _{xy'}$. However, in most of the cases it is impractical to engineer the Seed such that this condition is realized. Even in the simple example of a lossless All-Pass ring, the reconstruction of the output state associated to the Through port requires to inject the Seed laser at the Input with a wavelength dependent amplitude $T_H(\lambda _s)^*$, where $T_H$ is the complex transfer function of the Through port of the resonator [32]. A way to overcome this complexity is to seed only one port, as the Input, and to exploit the relations in Eq. (6). If the two coupling regions are equal, the phase matching functions $S_{xy}$ do not depend on the channel combination, so that $\phi _{xy}=\phi$. It is easier to express the state of the Seed in terms of asymptotic inputs, using the general input-output relations derived in [32], which give (see Appendix A):

It is important to stress the assumptions behind Eq. (8), to show to what extent this approach can be considered as universal. In general, as shown in Appendix A, the amplitude of the stimulated field $\gamma$ depends on a linear combination of the different JSA of the device. If port X is seeded, and the amplitude $\gamma _{Y, k_{i}}$ of the stimulated field in port Y (at wavevector $k_{i}$) is measured, this can be expressed by (see Eq. (51) in Appendix A):

2.1 JSA reconstruction

In order to determine the JSA outside the hypothesis of SET, the complex amplitude $\gamma _{iT(D)}(k_{si})$ of the stimulated field and the phase $\theta _{\textrm {FE}}$ of the field enhancement have to be measured. The strategy that will be adopted in the following is to coherently excite the resonator and a reference waveguide, and to make their stimulated fields to interfere in order to address their relative phase. The reference source should have an almost flat amplitude profile over the bandwidth of the resonator, such that their relative phase is, up to an overall constant factor, following the same variations of the one of the ring. Provided that the Pump does not carry significant chirp, a waveguide source meets this requirement, since its FWM bandwidth can be easily made to extend by more than $30 \,\textrm {nm}$ [42]. Using a similar strategy, $\theta _{\textrm {FE}}$ can be extracted. Since the field circulating in the resonator can not be directly accessed, this is circumvented by measuring the complex transfer function of the Through port, and by using the TCMT relation $\textrm {FE}=-i\sqrt {\frac {\tau _e}{2\tau _{\textrm {rt}}}}(T_H(\lambda )-1)$ (where $\tau _e$ is the extrinsic photon lifetime associated to loss into the bus waveguide and $\tau _{\textrm {rt}}$ is the roundtrip time of light into the cavity) to extract the complex field enhancement. The modulus of $T_H$ is given by the intensity of the light transmitted by the resonator, while the phase is measured relative to the one of the reference source. In the most general case described by Eq. (9), in order to reconstruct the JSAs it would be necessary to measure the amplitudes of the stimulated fields for all the ports of the device, and for all the seeding conditions. This could be implemented by placing a beam splitter in each of the ports of the device to let its stimulated field to interfere with the one of the reference spiral. The latter can be obtained from a single waveguide source, or by feeding in parallel different spirals using the same Pump and Seed lasers. The principle of inferring the JSAs through a series of interferograms formed by the stimulated fields of the device and of a reference spiral (with a flat JSP) is then quite general, and can be applied to any structure, even if they are effectively “black boxes”. However, the phase recovery algorithm can be simplified if some prior information (e.g. symmetries, device geometry or energy conservation relations) on the system is provided, as in this case.

3. Device and experimental setup

The setup for the JSA reconstruction is sketched in Fig. 2(a). The Pump is a femtosecond pulsed laser (Pritel), tuned at the resonance wavelength $\lambda _{m_p} = 1553.5\,\textrm {nm}$ and with a repetition rate of $50\,\textrm {MHz}$. The spectral width is set to $250\,\textrm {pm}$ by using a variable bandwidth tunable filter (Yenista XTA-50). The Pump is combined to a CW laser (Yenista TS100-HP) using a $200\,\textrm {GHz}$ commercial Dense Wavelength Division Multiplexing module (DWDM, Opneti), which also cleans the background noise of the laser. The polarization is set to TE by using Fiber Polarization Controllers. Light is injected and collected to and from the chip using a $16$ channel Fiber Array and grating couplers. After loss calibration, an average Pump power of $-8.5\,\textrm {dBm}$ and a Seed power of $-4.5\,\textrm {dBm}$ have been estimated at the Input waveguide of the circuit. The device, sketched in Fig. 2(b), is patterned on a $220$ nm SOI wafer using Electron-Beam lithography (EBL) from the Applied Nanotools foundry [43]. Single mode waveguides have a cross section of $500\times 220\,\textrm {nm}^2$ and lie on a $2\,\mu \textrm {m}$ thick Buried Oxide (BOX) layer. A $2.2\,\mu \textrm {m}$ thick Silica layer is deposited on the top of the waveguides, which provides optical isolation from the heater layer. All the thermal phase shifters can be simultaneously controlled by an external multi-channel current driver.

 

Fig. 2. (a) Sketch of the experimental setup. The Pump and the Seed lasers are respectively indicated with red and blue colors, while the stimulated field is indicated in green. PLS: Pulsed Laser Source, TLS: Tunable Laser Source, BPF: Band Pass Filter, FPC: Fiber Polarization Controller. (b) Layout of the chip. Waveguides are shown in black, while heaters are indicated in yellow.

Download Full Size | PDF

The circuit can be divided into three stages. In the first, depending on the choice of the phases $\theta _{1(2)}$, the Pump and the Seed lasers can be directed to the upper and/or lower arms of the device. The routing is achieved by asymmetric Mach-Zehnder interferometers (aMZI) with an FSR of $800\,\textrm {GHz}$ and an Extinction Ratio (ER) of $-35\,\textrm {dB}$. In this way, in the second stage, stimulated FWM can isolately occur in the ring resonator, or in the reference spiral, or simultaneously in both of them. The resonator source is a double bus ring of mean radius $13.87\,\mu \textrm {m}$, a measured linewidth of $250\,\textrm {nm}$ (quality factor $Q=6200$), FSR of $800\,\textrm {GHz}$ and ER of $-13\,\textrm {dB}$. The reference source is a spiral waveguide with a length of $L=2.35\,\textrm {mm}$, which has been engineered to have a FWM bandwidth of more than $30\,\textrm {nm}$ and a comparable brightness to the one of the resonator. A spiral has been added after the resonator to compensate the path length mismatch between the upper and lower arm of the circuit. In this way, the stimulated fields, which are manipulated in the third stage, experience the same optical path from the sources to the final beamsplitter (based on a MultiMode Interference device). Through the tuning of $\Delta \theta = \theta _4-\theta _5$, the relative phase of the two stimulated fields can be varied. An on-chip filter for the stimulated radiation is used, which is based on an Add-Drop ring resonator. This has an FSR of $1200\,\textrm {GHz}$, and due to unexpectedly high bending losses, the ER is only $-2\,\textrm {dB}$ and the linewidth $140\,\textrm {pm}$. The phase $\theta _6$ sweeps the filter wavelength across the Idler resonance order $m_i$. The stimulated field at the output of the chip is isolated from the Pump and the Seed laser by using a DWDM, and directed to a Superconducting Nanowire Single Photon Detector (SNSPD), operating at $85\%$ detection efficiency and with a dark count level $<\;200\,\textrm {Hz}$. Optionally, light can be directed to an off-chip tunable filter (Yenista XTA-50) to increase the spectral resolution to $50\,\textrm {pm}$ (Full With at Half Maximum (FWHM)).

4. Measurement of the JSI using the on-chip filter

The JSI of the resonator and of the spiral are measured by monitoring the output power of the stimulated field when the upper or the lower arm of the device are excited. These are shown in Figs. 3(a)–3(c), while Figs. 3(b)–3(d) are simulations which uses the same parameters of the experiment. The resolution of the Seed wavelength $\lambda _s$ is $20\,\textrm {pm}$, while the one on the stimulated field is $140\,\textrm {pm}$, and is limited by the linewidth of the on-chip filter. Add-Drop filters with FWHM $<\;40\,\textrm {pm}$ are routinely available in SOI [44], which can potentially increase the resolution. The calculated fidelities with the simulation are $\textrm {F}=(96.7\pm 0.2)$ for the spiral and $\textrm {F}=(91.5\pm 0.4)$ for the resonator. Errorbars are computed through Monte Carlo simulations assuming poissonian distribution of the data. The experimental JSI of the resonator reveals to be much broader than expected. This has probably to be attributed to the stimulated radiation generated in the spirals located before (of length $0.65\,\textrm {mm}$) and after (of length $2.5\,\textrm {mm}$) the resonator.

 

Fig. 3. (a) Experimental JSI of the resonator. (b) Simulation of the JSI of the resonator. (c) Experimental JSI of the spiral. (d) Simulation of the JSI o the spiral. (d) Experimental interference between the stimulated fields generated by the spiral and the resonator. (e) Simulation of the interference between the stimulated fields.

Download Full Size | PDF

Importantly, in the region where the JSI is more intense, the spurious contribution from the waveguide after the ring is negligible, since both the Pump and the Seed fields are filtered from the resonator. The use of an aMZI filter for pump rejection, after the resonator, could be used to completely suppress this background field. The effect of the relative phase between the resonator and the spiral emerges from Figs. 3(e)–3(f), which show the interference of the stimulated fields when both sources are excited. The fidelity with the simulation is $\textrm {F}=(95.51\pm 0.07)$. In the central region, a gradual suppression of the intensity is observed. The relative phase $\delta$ between the stimulated field of the resonator and the spiral can be obtained from the relation:

 

Fig. 4. (a) Measure of the phase $|\delta |$ obtained using the on-chip filter. (b) Measure of the phase $|\delta |$ obtained using the off-chip filter set to $50\,\textrm {pm}$ of resolution. The Seed laser is scanned across the resonance order $m_s = m_p+1$ (c) Same as in panel (b), but the Seed laser is scanned across the resonance order $m_s = m_p-1$. (d) Simulation of $|\delta |$ taking into account the spectral resolution of the on-chip filter. (e) Simulation of $|\delta |$ taking into account the spectral resolution of the off-chip filter. The Seed laser is scanned across the resonance order $m_s = m_p+1$. (f) Same as in panel (e), but the Seed laser is scanned across the resonance order $m_s = m_p-1$.

Download Full Size | PDF

Figure 4(a) reveals that $|\delta |$ is not constant, but has a maximum when $\lambda _s=\lambda _{m_{s}}$ and $\lambda _i = \lambda _{m_i}$, where $\lambda _{m_{s, i}}$ are the resonance wavelengths of order $m_s$ (Seed) and $m_i$ (Idler). According to Eq. (8), the phase of the stimulated field should be dependent on the seeded resonance order, since $\theta _{\textrm {FE}}(\lambda _{s})$ differs from $\theta _{\textrm {FE}}(\lambda _{i})$. This is verified by the swapping the Seed laser wavelength from the resonance order $m_s=m_p-1$ to $m_s = m_p+1$, obtaining the phase profiles shown in Figs. 4(b)–4(c). The external tunable filter is used to increase the resolution. The results are in good agreement with the simulations in Figs. 4(e)–4(f). The main discrepancies lie outside the main diagonal, and arise from the low counts available in these spectral regions. A comparison between Fig. 4(b) and Fig. 4(c) clearly shows that $|\delta |$ is not symmetric with respect to the exchange of the seeded resonance, as it would be if $|\delta | \propto \textrm {JSP}$. Even if not shown in Fig. 4, the same JSI profile is measured in both Figs. 4(b)–4(c), proving that $I_{\textrm {res}}\propto \textrm {JSI}$. This is a remarkable result, since the fact that the system is not seeded in an asymptotic field can be only detected through a phase resolved measurement.

5. Measurement of the JSP

The phase retrieval method described in Section 4. does not allow to determine the sign of $\delta$. To this goal, both the resonator and the spiral are coherently excited, and for each combination of $(\lambda _s,\lambda _i)$, the phase $\Delta \theta = \theta _{4}-\theta _{5}$ is scanned. To extract $\delta$, the fringes of their interference pattern $I_{\textrm {int}}(\lambda _s,\lambda _i,\Delta \theta )$ are fitted using the relation:

 

Fig. 5. (a) Experimental JSI of the resonator in the spectral region where the JSP shown in Fig. 6(a) has been evaluated. Points labeled by A,B and C refers to the combinations of $(\lambda _s,\lambda _i)$ at which the fringes shown in Fig. 5(b) have been recorded. (b) Interference of the stimulated fields of the resonator and the spiral source as a function of the phase $\Delta \theta$. The labels A, B and C refer to the points in the JSI space shown in Fig. 5(a). Scatters are from experimental data, solid lines from a fit which uses Eq. (12). (c) Modulus (black scatters) and phase (blue scatters) of the Through transfer function of the resonator. Solid lines are fit of the experimental data. (d) Modulus square (black scatters) and phase $\theta _{\textrm {FE}}$ (blue scatters) of the field enhancement of the resonator. Solid lines are fit of the experimental data.

Download Full Size | PDF

 

Fig. 6. (a) Measured JSP of the resonator source. (b) Simulated JSP of the resonator source. (c) Experimental phase $\delta$ of the stimulated field. In the white regions, counts were too low for determining the value of $\delta$.

Download Full Size | PDF

6. Conclusions

In this paper, it is proposed and experimentally demonstrated a method for reconstructing the complex JSA of an integrated silicon double bus resonator. It is the for the first time, to our knowledge, that the JSA of a resonating source is measured. The approach is based on Stimulated Emission, but removes the need of seeding the system in one of its asymptotic output fields. This is made possible by measuring the complex transfer function of the device, and by exploiting the similarities between the JSAs associated to different output channels. The technique can be extended to single resonators with an arbitrary number of ports, even if these are associated to loss and hence not physically accessible for the reconstruction of the asymptotic output field. The JSI and the JSP are entirely measured on a chip, harnessing the compactness, the reconfigurability and the high stability of the optical paths. The number of external resources is minimized, since the scheme eliminates the need of an external reference laser for phase retrieval, being the latter coherently generated on the same chip. The resolution and speed of the measurement could be greatly improved by respectively adopting higher quality factor filters and more efficient grating couplers. Due to the growing interest in the optimization of integrated resonator sources, either for enhancing their purity or their heralding efficiency, the flexibility allowed by this scheme is expected to become a valuable tool for phase sensitive tomography of a wide class of future devices.

Appendix A: Hamiltonian treatment of stimulated FWM in double bus resonators

Fields and Hamiltonian

In this appendix, an Hamiltonian treatment of stimulated FWM is applied to a double bus (Add-Drop) ring resonator to derive Eq. (8) in the main text. The starting point is the FWM Hamiltonian of Eq. (1), with the creation and annihilation operators associated to the asymptotic fields sketched in Fig. 1.

Following [45,46], the state of the system $\left |\psi (t)\right \rangle$ at any time $t$ is governed by the following equation of motion:

(13)$$i\hbar\frac{d\left|\psi(t)\right\rangle}{dt}=U(t)\left|\psi(t)\right\rangle$$

where the evolution operator $U(t)$ is given by:

(14)$$U(t) = -\sum_{xy}\int S_{xy}(k_{p1}, k_{p2}, k_{s}, k_{i})a_{I, k_{p1}}a_{I, k_{p2}}b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}e^{i\bar{\omega}t}dk_{p1}dk_{p2}dk_{s}dk_{i}+\textrm{h.c.}$$

where $\bar {\omega } = \omega _{p1}+\omega _{p2}-\omega _{s}-\omega _{i}$. The state $\left |\psi (t)\right \rangle$ is taken of the form:

(15)$$\left|\psi(t)\right\rangle =D_{s}(t)D_{p}(t)D_{i}(t)\left|\textrm{mod}(t)\right\rangle$$

where $D_{j}$ is the displacement operator for the $j^{th}$ coherent state. At $t\rightarrow -\infty$, the state is represented by Pump and Seed pulses which are travelling towards the ring from port I, while at $t\rightarrow \infty$ a coherent Idler field (as well as the spontaneously generated photon pairs), stimulated from the Pump and the Seed, is outgoing from ports D and T. The expressions for the displacement operators are given in Eq. (3).

At $t\rightarrow -\infty$, the nonlinear interaction has not yet occured, so it is possible to take as initial conditions $\alpha _{I, k_{p}}(-\infty )=\bar {\alpha }_{I, k_{p}}$, $\beta _{j, k_{s}}(-\infty )=\bar {\beta }_{j, k_{s}}$, $\gamma _{j, k_{i}}(-\infty )=0$ and $\left |\textrm {mod}(-\infty )\right \rangle =\left |0\right \rangle$.

The equation of motion for $\left |\textrm {mod}(t)\right \rangle$ is obtained by differentiating both sides of Eq. (15):

(16)$$\frac{d\left|\textrm{mod}(t)\right\rangle }{dt}= \left(O_{s}D_{s}D_{i}D_{p}+D_{s}O_{i}D_{i}D_{p}+D_{s}D_{i}O_{p}D_{p}\right)\left|\textrm{mod}(t)\right\rangle+ D_{s}D_{i}D_{p}\frac{d\left|\textrm{mod}(t)\right\rangle }{dt}$$

where $O_{p}=\left (\int \frac {d\alpha _{I, k_{p}}(t)}{dt}a_{I, k_{p}}^{\dagger }dk_{p}-h.c\right )D_{p}$ and $O_{s},O_{i}$ are similarly defined. Using Eq. (13), it is possible to rewrite Eq. (16) in the form:

(17)$$\frac{d\left|\textrm{mod}(t)\right\rangle }{dt}=H_{\textrm{eff}}(t)\left|\textrm{mod}(t)\right\rangle$$

where the effective Hamiltonian $H_{\textrm {eff}}$ is given by:

(18)$$H_{\textrm{eff}}= \frac{1}{i\hbar}D_{s}^{\dagger}D_{i}^{\dagger}D_{p}^{\dagger}UD_{s}D_{i}D_{p}-D_{s}^{\dagger}D_{i}^{\dagger}D_{p}^{\dagger} \left(O_{s}D_{s}D_{i}D_{p} \right. \left. D_{s}O_{i}D_{i}D_{p}+D_{s}D_{i}O_{p}D_{p} \right)$$

If the the Pump, the Signal and the Idler resonances are confined into three not-overlapping frequency intervals, the different displacement operators commute between each other, i.e., $[D_{j},D_{k}^{\dagger }]_{j\neq k}=0$ and $[D_{j},O_{k}]_{j\neq k}=0$.

Equations of motion

The effective Hamiltonian in Eq. (18) contains the time derivative of the frequency distributions of the asymptotic input and output fields. In this section their equation of motion are derived. The starting point is the the Heisenberg equation of motion of the Pump operator $a_{I, k}^{\dagger }(t)$, given by [47] :

(19)$$i\hbar\frac{da_{I, k}^{\dagger}(t)}{dt}=[a_{I, k}^{\dagger},V(t)]$$

where the operator $V$ is defined as:

(20)$$V= -\sum_{xy}\int S_{xy}(k_{p1}, k_{p2}, k_{s}, k_{i})a_{I, k_{p1}}(t)a_{I, k_{p2}}(t)b_{x, k_{s}}^{\dagger}(t)b_{y, k_{i}}^{\dagger}(t)e^{i\bar{\omega}t}dk_{p1}dk_{p2}dk_{s}dk_{i}+\textrm{h.c.}$$

Working out the commutator in Eq. (19) gives:

(21)$$i\hbar\frac{da_{I, k_{p1}}^{\dagger}(t)}{dt}= -2\sum_{xy}\int S_{xy}(k_{p1}, k_{p2}, k_{s}, k_{i})e^{i\bar{\omega}t}a_{I, k_{p2}}(t)b_{x, k_{s}}^{\dagger}(t)b_{y, k_{i}}^{\dagger}dk_{p2}dk_{s}dk_{i}$$

Similarly, the equation of motion for $b_{j, k_{i}}$ (with $j=\{T,D\}$) can be written as:

(22)$$\begin{aligned}i\hbar\frac{db_{j, k_{i}}(t)}{dt}= &-\sum_{xy}\int e^{i\bar{\omega}t}a_{I, k_{p1}}(t)a_{I, k_{p2}}(t)\left(S_{yx}(\mathbf{k})b_{y, k_{s}}^{\dagger}(t)\delta_{jx} \right. \\ &\left. +S_{xy}(\mathbf{k})b_{x, k_{s}}^{\dagger}(t)\delta_{jy}\right)dk_{p1}dk_{p2}dk_{s} \end{aligned}$$

where the fact that $\mathbf {k} = (k_{p1}, k_{p2}, k_s, k_i)$ , $[b_{j, k}, b_{x, k'}^{\dagger }b_{y, k''}^{\dagger }]=\delta _{jx}\delta (k-k')b_{y, k''}^{\dagger }+\delta _{jy}\delta (k-k'')b_{x, k'}^{\dagger }$ and $S_{xy}(k_{p1}, k_{p2}, k_{s}, k_{i})=S_{yx}(k_{p1}, k_{p2}, k_{i}, k_{s})$ have been used. An analogous equation for the operators $b_{j, k_{s}}$ can be obtained by replacing $k_{i}$ with $k_{s}$ in Eq. (22). Equations (21)–(22), which refer to operators, translates to classical equations of motion for the functions $\alpha _{I, k}(t)$, $\beta _{j, k}(t)$ and $\gamma _{j, k}(t)$ as [47]:

(23)$$i\hbar\frac{d\alpha_{I, k_{p}}}{dt}=2\sum_{xy}\int S_{xy}^{*}(\mathbf{k})e^{-i\bar{\omega}t}\alpha_{I, k_{p2}}^{*}(t)\beta_{x, k_{s}}(t)\beta_{y, k_{i}}dk_{p2}dk_{s}dk_{i}$$

(24)$$\begin{aligned}i\hbar\frac{d\beta_{j, k_{s}}}{dt}= &-\frac{1}{2}\sum_{xy}\int e^{i\bar{\omega}t}\alpha_{I, k_{p1}}(t)\alpha_{I, k_{p2}}(t)\left(S_{xy}(\mathbf{k})\delta_{px}\gamma_{y, k_{i}}^{*}(t)\right. \\ &\left. +S_{yx}(k\mathbf{k})\delta_{jy}\gamma_{x, k_{i}}^{*}(t)\right)dk_{p1}dk_{j2}dk_{i} \end{aligned}$$

(25)$$\begin{aligned}i\hbar\frac{d\gamma_{j, k_{i}}}{dt}= &-\frac{1}{2}\sum_{xy}\int e^{i\bar{\omega}t}\alpha_{I, k_{p1}}(t)\alpha_{I, k_{p2}}(t)\left(S_{yx}(\mathbf{k})\delta_{jx}\beta_{y, k_{s}}^{*}(t) \right. \\ &\left. +S_{xy}(\mathbf{k})\delta_{jy}\beta_{x, k_{s}}^{*}(t)\right)dk_{p1}dk_{p2}dk_{s} \end{aligned}$$

The factor of $2$ in Eq. (23) comes from the fact that the two pumps are degenerate. With Eqs. (23)–(25) in hand, it is possible to work out the different elements which appear on the right hand side of Eq. (18), obtaining an explicit expression for $H_{\textrm {eff}}$. After some algebra, the following equalities can be derived:

(26)$$\begin{aligned}&D_{s}^{\dagger}D_{i}^{\dagger}D_{p}^{\dagger}\left(O_{s}D_{s}D_{i}D_{p}+D_{s}O_{i}D_{i}D_{p}\right) = \\ &-\frac{1}{i\hbar}\sum\int S_{xy}(\mathbf{k})e^{i\bar{\omega}t}\alpha_{I, k_{p1}}\alpha_{k_{p2}}(\beta_{x, k_{s}}^{*}\gamma_{y, k_{i}}^{*}+b_{x, k_{s}}^{\dagger}\gamma_{y, k_{i}}^{*}+b_{x, k_{i}}^{\dagger}\beta_{y, k_{s}}^{*})d\mathbf{k}- \textrm{h.c.} \end{aligned}$$

(27)$$D_{s}^{\dagger}D_{i}^{\dagger}D_{p}^{\dagger}D_{s}D_{i}O_{p}D_{p}= \frac{2}{i\hbar}\sum\int S_{xy}^{*}(\mathbf{k})e^{-i\bar{\omega}t}\beta_{x, k_{s}}\gamma_{y, k_{i}}(\alpha_{I, k_{p1}}^{*}\alpha_{I, k_{p2}}^{*}+\alpha_{I, k_{p1}}^{*}a_{I, k_{p1}}^{\dagger})d\mathbf{k}-\textrm{h.c.}$$

(28)$$\begin{aligned}D_{s}^{\dagger}D_{i}^{\dagger}D_{p}^{\dagger}UD_{s}D_{i}D_{p} &= -\sum_{xy}\int S_{xy}(k_{p1}, k_{p2}, k_{s}, k_{i})e^{i\bar{\omega}t}(\alpha_{I, k_{p1}}+a_{I, k_{p1}}) (\alpha_{I, k_{p2}}+a_{I, k_{p2}}) \\ &(\beta_{x, k_{s}}^{*}+b_{x, k_{s}}^{\dagger})(\gamma_{y, k_{i}}^{*}+b_{y, k_{i}}^{\dagger})d\mathbf{k}-\textrm{h.c.} \end{aligned}$$

where the fact that $D_{j}^{\dagger }a_{k}D_{j}=(\alpha _{k}+a_{k})$ has been used. The explicit form of $H_{\textrm {eff}}$ is then:

(29)$$\begin{aligned}H_{\textrm{eff}}(t)= &-\frac{1}{i\hbar}\sum_{xy}\int S_{xy}(k_{p1}, k_{p2}, k_s, k_i)e^{i\bar{\omega}t} \left ( \alpha_{I, k_{p1}}\gamma_{y, k_{i}}^{*}b_{x, k_{s}}^{\dagger}a_{I, k_{p2}} \right. + \\ &+\alpha_{I, k_{p2}}\gamma_{y, k_{i}}^{*}b_{x, k_{s}}^{\dagger}a_{I, k_{p1}}+a_{I, k_{p1}}a_{I, k_{p2}}\beta_{x, k_{s}}^{*}\gamma_{y, k_{i}}^{*} +\gamma_{y, k_{i}}^{*}b_{x, k_{s}}^{\dagger}a_{I, k_{p1}}a_{I, k_{p2}}+ \\ &+\alpha_{I, k_{p1}}\alpha_{I, k_{p2}}b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger} +\alpha_{I, k_{p1}}\beta_{x, k_{s}}^{*}b_{y, k_{i}}^{\dagger}a_{I, k_{p2}}+\gamma_{y, k_{i}}^{*}b_{x, k_{s}}^{\dagger}a_{I, k_{p1}}a_{I, k_{p2}} + \\ &+\alpha_{I, k_{p1}}\alpha_{I, k_{p2}}b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}+\alpha_{I, k_{p1}}\beta_{x, k_{s}}^{*}b_{y, k_{i}}^{\dagger}a_{I, k_{p2}} +\alpha_{I, k_{p1}}b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}a_{I, k_{p2}}+ \\ &+\alpha_{I, k_{p2}}\beta_{x, k_{s}}^{*}b_{y, k_{i}}^{\dagger}a_{I, k_{p1}} +\alpha_{I, k_{p2}}b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}a_{I, k_{p1}}+\beta_{x, k_{s}}^{*}b_{y, k_{i}}^{\dagger}a_{I, k_{p1}}a_{I, k_{p2}} + \\ &\left. +b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}a_{I, k_{p1}}a_{I, k_{p2}}+f(t))\right)d\mathbf{k}+\textrm{h.c.} \end{aligned}$$

with $f(t)=-2\sum _{xy}\alpha _{I, k_{p1}}(t)\alpha _{I, k_{p2}}(t)\beta _{x, k_{s}}^{*}(t)\beta _{y, k_{i}}^{*}(t)$. Equation (29) generalizes the effective Hamiltonian in [45] to multiple channels.

First order solution to the equations of motion

To first order, the solution of Eq. (17) is:

(30)$$\left|\textrm{mod}(\infty)\right\rangle =\left|0\right\rangle +\intop_{-\infty}^{\infty}H_{\textrm{eff}}(t')\left|0\right\rangle dt'$$

Since the only term in $H_{\textrm {eff}}$ which does not involve at least one annihilation operator on the right is $\alpha _{I, k_{p1}}\alpha _{I, k_{p2}}b_{x, k_{s}}^{\dagger }b_{y, k_{i}}^{\dagger }$, it follows that:

(31)$$\left|\textrm{mod}(\infty)\right\rangle =\left|0\right\rangle +\sqrt{p_{\textrm{tot}}}\sum_{xy}\int\sqrt{\frac{p_{xy}}{p_{\textrm{tot}}}}\phi_{xy}(k_{s}, k_{i})b_{x, k_{s}}^{\dagger}b_{y, k_{i}}^{\dagger}\left|0\right\rangle dk_{s}dk_{i}$$

where the normalized biphoton wavefunctions $\phi _{xy}$:

(32)$$\phi_{xy}(k_{s}, k_{i})=\frac{2\pi i}{\hbar\sqrt{p_{xy}}}\int S_{xy}(k_{s}+k_{i}-k_{p_{2}}, k_{p_{2}}, k_{s}, k_{i})dk_{p_{2}}$$

have been introduced. The quantity $|\phi _{xy}(k_{s}, k_{i})|^{2}dk_{s}dk_{i}$ has then to be interpreted as the probability of finding a photon with wavevector $k_{s}$ in the asymptotic output state $x$ and a photon with wavevector $k_{i}$ in the asymptotic output state $y$. The real numbers $p_{xy}$ and $p_{\textrm {tot}}$ are respectively the probability of generating a photon pair in the channel combination $xy$ and the overall probability of generating a pair ($p_{\textrm {tot}}=\sum _{xy}p_{xy}$). At first order, the result of Eq. (30) does not depend on the presence of an initial Seed beam which stimulates the process. Hence, the final state, both in the stimulated and in the spontaneous case, is given by:

(33)$$\left|\psi(\infty)\right\rangle =D_{s}(\infty)D_{p}(\infty)D_{i}(\infty)(\left|0\right\rangle +\sqrt{p_{\textrm{tot}}}\left|II\right\rangle )$$

with $\left |II\right \rangle$ the two-photon state in Eq. (31). In the absence of a Seed, the solution of Eq. (25) is $\gamma _{j, k_{i}}(\infty )=0$, so there is no stimulated coherent Idler field at the output of the resonator, but only pairs generated by spontaneous FWM. In case that a Seed field is applied, the undepleted Pump approximation gives:

(34)$$\begin{aligned}\gamma_{j, k_{i}}(\infty)= &\frac{i\pi}{\hbar}\sum_{xy}\int\bar{\alpha}_{k_{s}+k_{i}-k_{p2}}\bar{\alpha}_{k_{p2}}\left(S_{yx}(k_{s}+k_{i}-k_{p2}, k_{p2}, k_{s}, k_{i})\right. \\ &\delta_{jx}\bar{\beta}_{y, k_{s}}^{*} \left. +S_{xy}(k_{s}+k_{i}-k_{p2}, k_{p2}, k_{s}, k_{i})\delta_{jy}\bar{\beta}_{x, k_{s}}^{*}\right)dk_{p2}dk_{s} \end{aligned}$$

If the Signal is assumed to be monochromatic at the wavevector $k_{s0}$, the final expressions for the frequency distributions of the Idler asymptotic output states are:

(35)$$\gamma_{T, k_{i}}=\sqrt{p_{TT}}\phi_{TT}(k_{s0}, k_{i})\beta_{T, k_{s0}}^{*}+\sqrt{p_{TD}}\phi_{TD}(k_{s0}, k_{i})\beta_{D, k_{s0}}^{*}$$

(36)$$\gamma_{D, k_{i}}=\sqrt{p_{DD}}\phi_{DD}(k_{s0}, k_{i})\beta_{D, k_{s0}}^{*}+\sqrt{p_{DT}}\phi_{DT}(k_{s0}, k_{i})\beta_{T, k_{s0}}^{*}$$

where the definitions in Eq. (32) have been used. Equations (35)–(36) are formally equivalent to Eq. (6) in the main text.

Resonator with an arbitrary number of channels

The expressions in Eqs. (35)–(36) are given in terms of the asymptotic output states of the Seed $\beta _{j, k_{s}}$, while it is much easier to define its initial state in terms of asymptotic inputs. By using the same formalism in [34], the asymptotic input states $E_{n, k}^{in}$ can be expressed in terms of the asymptotic output ones (and vice-versa) by:

(37)$$E_{n, k}^{in}=\sum_{n'}H_{nn'}^{out}(k)E_{n'k}^{out}$$

(38)$$E_{n, k}^{out}=\sum_{n'}H_{nn'}^{in}(k)E_{n'k}^{in}$$

To calculate $H_{nn'}^{in}(k)$, we use the fact that [34]:

(39)$$E_{n, k}^{in}=E_{n, k}^{iso}+\sum H_{nn'}^{out}(k)E_{n'(-k)}^{iso}$$

where $E_{n'(-k)}^{iso}$ is an outgoing wave from the channel $n'$ with wavevector $k$. By looking at Fig. 1, it is evident that:

(40)$$H_{D, A}^{out}=T_{H};\:H_{D, I}^{out}=D_{R};\:H_{T, I}^{out}=T_{H};\:H_{T, A}^{out}=D_{R}$$

where $D_{R}$ and $T_{H}$ are the Drop and Through transfer functions of the Add-Drop resonator. By using the fact that $H_{nn'}^{in}(k)=\left (H_{nn'}^{out}(k)\right )^{*}$ [34]:

(41)$$E_{T, k}^{out}=H_{T, I}^{out}(k){}^{*}E_{I, k}^{in}+H_{T, A}^{out}(k){}^{*};\:E_{A, k}^{in}=T_{H}^{*}E_{I, k}^{in}+D_{R}^{*}E_{A, k}^{in}$$

(42)$$E_{D, k}^{out}=H_{D, I}^{out}(k){}^{*}E_{I, k}^{in}+H_{D, A}^{out}(k){}^{*};\:E_{A, k}^{in}=D_{R}^{*}E_{I, k}^{in}+T_{H}^{*}E_{A, k}^{in}$$

It is worth to note that the asymptotic output states in Eqs. (41)–(42) are associated to annihilation operators [34], while the $\beta _{j, k_{s}}$ in Eq. (36) refer to creation operators (see Eq. (6)). It is possible to shift from one set to the other by a complex conjugation. By using Eqs. (41)–(42) into Eqs. (35)–(36) and considering that in the experiment of this paper $E_{A, k}^{in}=0$:

(43)$$\gamma_{T, k_{i}} =\left(\sqrt{p_{TT}}\phi_{TT}(k_{s0}, k_{i})T_{H}^{*}(k_{s0})+\sqrt{p_{TD}}\phi_{TD}(k_{s0}, k_{i})D_{R}^{*}(k_{s0})\right)\beta_{I, in}^{*}$$

(44)$$\gamma_{D, k_{i}}=\left(\sqrt{p_{DD}}\phi_{DD}(k_{s0}, k_{i})D_{R}^{*}(k_{s0})+\sqrt{p_{DT}}\phi_{DT}(k_{s0}, k_{i})T_{H}^{*}(k_{s0})\right)\beta_{I, in}^{*}$$

The phase matching functions $S_{xy}$ have as explicit expression [34]:

(45)$$S_{xy}=\mathbb{N}\int E_{I, k_{p1}}^{in}(\mathbf{r})E_{I, k_{p2}}^{in}(\mathbf{r})\left(E_{x, k_{s}}^{out}(\mathbf{r})\right)^{*}\left(E_{y, k_{i}}^{out}(\mathbf{r})\right)^{*}e^{i\Delta k z}d\mathbf{r_{t}}dz$$

where $\mathbf {r}=(\mathbf {r_{t}}, z)$ indicates the spatial coordinates of the integration ($\mathbf {r_{t}}$ is the transverse coordinate along the radial direction of the ring, while $z$ runs along the ring circumference), $\mathbb {N}$ is a constant which is proportional to the nonlinear susceptibility of the material and $\Delta k = k_{p1}+k_{p2}-k_{s}-k_{i}$. The integration is performed only within the ring due to the higher intensity with respect to the bus waveguides, so $E_{nk}^{in}\sim \textrm {FE}_{nk}$, where $\textrm {FE}_{nk}$ denotes the internal field enhancement when the resonator is excited at port $n$. The relation $E_{nk}^{in}=\left (E_{nk}^{out}\right )^{*}$ [34] can be used to express $S_{xy}$ only in terms of the asymptotic input fields, so as $S_{xy}\propto E_{I, k_{p1}}^{in}(\mathbf {r})E_{I, k_{p2}}^{in}(\mathbf {r})E_{x, k_{s}}^{in}(\mathbf {r})E_{y, k_{i}}^{in}(\mathbf {r})$. When the device is symmetric, $\textrm {FE}_{I, k}=\textrm {FE}_{j, k}=\textrm {FE}$ with $j=\{T, D\}$ and consequently $\phi _{xy}=\phi$ where:

(46)$$\phi(k_{s}, k_{i})=\frac{2\pi i}{\sqrt{p}\hbar}\mathbb{N}' \textrm{FE}(k_{s})\textrm{FE}(k_{i})\int \textrm{FE}(k_{s}+k_{i}-k_{p_{2}})\textrm{FE}(k_{p_{2}})dk_{p_{2}}$$

in which $\mathbb {N}'$ includes the result of the spatial integration of the asymptotic fields (which gives the inverse of the effective volume) and $e^{i(k_{p1}+k_{p2}-k_{s}-k_{i})z}\sim 1$. In Eq. (46), $p$ represents any of the probabilities $p_{xy}$, since they all coincide. Equation 43 reduces to:

(47)$$\gamma_{T, k_{i}}=\sqrt{p}\beta_{I, in}^{*}\phi(k_{s}, k_{i})(T_{H}^{*}(k_{s})+D_{R}^{*}(k_{s}))$$

From Temporal Coupled Mode Theory (TCMT) in weak coupling regime [36] it is possible to show that $D_{R}=i\sqrt {\frac {2}{\tau _{e}}}\textrm {FE}$ and $T_{H}=1+i\sqrt {\frac {1}{\tau _{e}}}\textrm {FE}$, with:

(48)$$\textrm{FE}=\frac{i\sqrt{\frac{2}{\tau_{e}}}}{\sqrt{\tau_{\textrm{rt}}}\left ( \frac{1}{\tau_{\textrm{tot}}}-i(\omega-\omega_{j}) \right )}$$

where $\tau _{e}$ is the extrinsic photon lifetime due to the coupling with the bus waveguide, $\tau _{\textrm {tot}}=2/\tau _e$ is the total photon lifetime, $\tau _{\textrm {rt}}$ is the cavity round-trip time and $\omega _{j}$ is one of the eigenfrequencies of the resonator (in our case, $j=p, s, i$). The sum $(T_{H}+D_{R})^{*}$ gives:

(49)$$(T_{H}+D_{R})^{*}=-\frac{\frac{2}{\tau_{e}}-i(\omega-\omega_{j})}{\frac{2}{\tau_{e}}+i(\omega-\omega_{j})}=\frac{\textrm{FE}^{*}}{\textrm{FE}}$$

which inserted into Eq. (47) gives:

(50)$$\gamma_{T, k_{i}}=\sqrt{p}\beta_{I, in}^{*}\phi(k_{s}, k_{i})\left(\frac{\textrm{FE}^{*}}{\textrm{FE}}\right)=\sqrt{p}\beta_{I, in}^{*}|\phi|\exp\left(i(\theta_{\phi}-2\theta_{\textrm{FE}})\right)$$

that is the same expression in Eq. (8) of the main manuscript. In case of a resonator with $M$ channels, in which only one is seeded (arbitrarily this channel can be called the Input), Eq. (35) generalizes to:

(51)$$\gamma_{T, k_{i}}=\sum_{m=1}^{M}\sqrt{p_{Tm}}\phi_{Tm}(k_{s0}, k_{i})H_{m, I}^{out*}(k_{s0})\beta_{I, in}^{*}$$

From Fig. 1, it follows that $H_{m, I}^{out}=T_{H}$ if $m=T$, and $H_{m, I}^{out}=D_{R}^{(m)}$ if $m\neq T$. The function $D_{R}^{(m)}$ is the Drop transfer function seen from the Input port of the resonator when the device is excited from the $m^{th}$ port, and is given by $D_{R}^{(m)}=i\sqrt {\frac {2}{\tau _{e, I}}}\textrm {FE}_{m}$, where $\tau _{e, I}$ is the extrinsic photon lifetime associated to the coupling with the Input port. The different field enhancements $\textrm {FE}_{m}$ can all be expressed relative to the one of the Input port $\textrm {FE}_{I}$ by $\textrm {FE}_{m}=\sqrt {\frac {\tau _{e, I}}{\tau _{e, m}}}\textrm {FE}_{I}=\sqrt {\frac {\tau _{e, I}}{\tau _{e, m}}}\textrm {FE}_{T}$. Equations (45)–(46) implies that $\phi _{Tm}=\sqrt {\frac {p_{TT}}{p_{Tm}}}\sqrt {\frac {\tau _{e, i}}{\tau _{e, m}}}\phi _{TT}$ so that Eq. (51) becomes:

(52)$$\gamma_{T, k_{i}} = \sqrt{p_{TT}}\phi_{TT}(k_{s0}, k_{i})\beta_{I, in}^{*}\left(1+i\textrm{FE}_{T}\left(\sqrt{\frac{2}{\tau_{e, I}}}+\sum_{m\neq T}i\frac{\sqrt{2\tau_{e, I}}}{\tau_{e, m}}\right)\right)^{*}$$

Replacing $\textrm {FE}_{T}$ with the expression in Eq. (48), yields:

(53)$$\gamma_{T, k_{i}} = \sqrt{p_{TT}}\phi_{TT}(k_{s0}, k_{i})\beta_{I, in}^{*} \left(\frac{-i(\omega-\omega_{s0})+\frac{1}{\tau_{\textrm{tot}}}-\left(\frac{2}{\tau_{e, I}}+\sum_{m\neq T}\frac{2}{\tau_{e, m}}\right)}{-i(\omega-\omega_{s0})+\frac{1}{\tau_{\textrm{tot}}}}\right)^{*}$$

In the numerator of Eq. (53), the term $\frac {2}{\tau _{e, I}}+\sum _{m\neq T}\frac {2}{\tau _{e, m}}$ equals $\frac {2}{\tau _{\textrm {tot}}}$, so that the expression in the parenthesis reduces to the right hand side of Eq. (49), and Eq. (53) gets the same formal expression of the two-channel case in Eq. (50). Naturally, one could recover a similar result in a general channel $m$ by replacing $\phi _{TT}$ in Eq. (53) with $\phi _{mm}$. The information on the number of channels is stored inside the field enhancement factors. In all this derivation, no assumptions are made on the nature of the channels, so Eq. (53) holds even in presence of linear loss. Indeed, as shown in [48], they can be included by adding a phantom channel in the Hamiltonian, which behaves exactly as any of the other physical channels.

Appendix B: Temporal coupled mode theory

In this section, the expression for the amplitude of the stimulated field in the Add-Drop resonator is derived using TCMT, showing that it agrees with the result of the Hamiltonian treatment. The Pump and the Seed fields are modeled as $A_{p}(t)=a_{p}(t)e^{-i\omega _{p}t}$ and $A_{s}(t)=a_{s}(t)e^{-i\omega _{s}t}$, where $a_{p(s)}$ are slowly varying envelopes compared to the carrier frequencies $\omega _{p(s)}$. The equations which govern the (slowly varying) energy amplitudes $u(t)$ inside the resonator are [49]:

(54)$$\begin{aligned}\frac{du_{p}}{dt} = &\left( i(\omega_{p}-\omega_{p0})-\frac{1}{\tau_{p, \textrm{tot}}} \right) u_{p}+i\sqrt{\frac{2}{\tau_{p, e}}}a_{p}(t) \\ \frac{du_{s}}{dt} = &\left( i(\omega_{s}-\omega_{s0})-\frac{1}{\tau_{s, \textrm{tot}}}\right)u_{s}+i\sqrt{\frac{2}{\tau_{s, e}}}a_{s}(t) \\ \frac{du_{i}}{dt}= &\left[i(\omega_{i}-\omega_{i0})-\frac{1}{\tau_{i, \textrm{tot}}}\right]u_{i}+\gamma u_{p}(t)^{2}u_{s}(t)^{*} \end{aligned}$$

where the subscripts $(p, s, i)$ label the Pump, the Seed and the Idler. The nonlinear coupling parameter $\gamma$ is related to the more familiar nonlinear coefficient $\gamma _{\textrm {nl}}=\frac {\omega n_{2}}{cA_{\textrm {eff}}}$ ($n_2$ is the nonlinear refractive index while $A_{\textrm {eff}}$ the effective area of the waveguide) by $\gamma =\frac {\gamma _{\textrm {nl}}L_{\textrm {res}}}{\tau _{\textrm {rt}}^{2}}$, in which $L_{\textrm {res}}$ the resonator perimeter. Consistently with the Hamiltonian treatment, all the parasitic nonlinearities, except FWM, are neglected. Casting the Pump and Seed equations into the Fourier domain and solving for the energy amplitudes gives:

(55)$$U_{p(s)}(\omega)= \sqrt{\tau_{rt}}\textrm{FE}_{p(s)}(\omega)A_{p(s)}(\omega) = \frac{i\sqrt{\frac{2}{\tau_{p(s), e}}}}{\frac{1}{\tau_{p(s), \textrm{tot}}}-i(\omega-\omega_{p(s)0})}A_{p(s)}(\omega)$$

where $U_{p(s)}(\omega )=u_{p(s)}(\omega -\omega _{p(s)})$. Inserting Eq. (55) into Eq. (54) yields:

(56)$$\begin{aligned}&U_{i}(\omega) = \gamma\tau_{\textrm{rt}}^{2}\frac{\textrm{FE}_{i}(\omega)}{i\sqrt{\frac{\tau_{i, e}}{2}}}\int \textrm{FE}_{p}(\omega-\omega'-\omega^{\prime\prime}-\omega_{i}+\omega_{p})\times \\ &\times\textrm{FE}_{p}(\omega^{\prime\prime}+\omega_{p})\textrm{FE}_{s}^{*}(\omega'+\omega_{s}) A_{p}(\omega-\omega'-\omega^{\prime\prime}-\omega_{i}+\omega_{p})A_{p}(\omega^{\prime\prime}+\omega_{p})A_{s}^{*}(\omega'+\omega_{s})d\omega'd\omega^{\prime\prime} \end{aligned}$$

where the convolution property of the Fourier transform is recursively used. The Seed field is assumed to be monochromatic and at frequency $\bar {\omega _{s}}$. In this case, the power amplitude per unit frequency $P_{i, \textrm {res}}=i\sqrt {\frac {\tau _{i, e}}{2}}U_{i}$ of the Idler into the Through (or Drop) waveguide is:

(57)$$\begin{aligned}P_{i, \textrm{res}}(\omega) &=\gamma_{\textrm{nl}}L_{\textrm{res}}^{2}\left(\frac{\textrm{FE}(\bar{\omega_{s}})^{*}}{\textrm{FE}(\bar{\omega_{s}})}\right)A_{s}^{*}\textrm{FE}_{i}(\omega)\textrm{FE}_{s}(\bar{\omega_{s}}) \\ &\int \textrm{FE}_{p}(\omega+\bar{\omega_{s}}-\omega')\textrm{FE}_{p}(\omega')A_{p}(\omega+\bar{\omega_{s}}-\omega')A_{p}(\omega')d\omega' \end{aligned}$$

in which it is used the fact that, from energy conservation, $\omega _{i}=2\omega _{p}-\omega _{s}$. The fully quantum mechanical calculation for the resonator JSA, $\phi _{\textrm {res}}$, gives [9]:

(58)$$\phi(\omega, \bar{\omega}_{s})_{\textrm{res}} = \mathbb{N}_{\textrm{res}} \textrm{FE}_{s}(\bar{\omega_{s}})\textrm{FE}_{i}(\omega) \int \textrm{FE}_{p}(\omega+\bar{\omega_{s}}-\omega')\textrm{FE}_{p}(\omega')A_{p}(\omega+\bar{\omega_{s}}-\omega')A_{p}(\omega')d\omega'$$

where $\mathbb {N}_{\textrm {res}}$ is a normalization constant. From a comparison between Eq. (58) and Eq. (57), the following identity holds:

(59)$$P_{i, \textrm{res}}(\omega)=\mathbb{N}_{\textrm{res}}'\left(\frac{\textrm{FE}(\bar{\omega_{s}})^{*}}{\textrm{FE}(\bar{\omega_{s}})}\right)\phi_{\textrm{res}}(\omega, \bar{\omega_{s}})$$

in which $\mathbb {N}_{\textrm {res}}'$ includes all the pre-factors. The result in Eq. (59) is formally equivalent to Eq. (8) in the main text and to Eq. (50), which has been derived by an Hamiltonian treatment.

Appendix C: Simulation of the JSA of the spiral and of the resonator

The resonator JSA is modeled using Eq. (58). The JSI shown in Fig. 3(b) of the main text takes into account the effect of the finite resolution of the filter for SET. The final expression thus reads:

(60)$$|\phi_{\textrm{res}}(\omega_s, \omega_i)|^2 = \int |G(\omega, \omega_i)|^2|\phi_{\textrm{res}}(\omega_s, \omega)|^2 d\omega$$

where $G(\omega )$ is the spectral response of the filter. This is modeled as:

(61)$$|G(\omega, \omega_i)|^2=\begin{cases} \frac{\left(\frac{\Gamma}{2}\right)^{2}}{\left(\frac{\Gamma}{2}\right)^{2}+(\omega-\omega_{i})} &\textrm{on-chip filter}\\ \textrm{rect}\left(\omega\right) &\textrm{off-chip filter} \end{cases}$$

where $\Gamma$ is the FWHM of the filter, $\omega _i$ is its center frequency and $\textrm {rect}$ is a box-like function. The choice of a Lorentzian lineshape for the on-chip filter comes from the fact that this is implemented using an Add-Drop resonator. The off-chip filter is a tunable fiber Bragg grating, whose measured spectral response well approximates a box-like function. The JSA of the spiral is calculated using the following expression [47]:

(62)$$\phi_{\textrm{spi}}(\omega_s, \omega_i) = \mathbb{N}_{\textrm{spi}}\int e^{i\Delta kL/2} \textrm{sinc}(L_{\textrm{spi}}/L_c) A_p(\omega_s+\omega_i-\omega')A_p(\omega')d\omega'$$

where $\mathbb {N}_{\textrm {spi}}$ is a normalization constant, $L_{\textrm {spi}}$ is the spiral length, $\Delta k = k(\omega _s+\omega _i-\omega ')+k(\omega ')-k(\omega _s)-k(\omega _i)$ is the wavevector mismatch and $L_c=2\pi /\Delta k$ is the coherence length. In the same way of the resonator, the JSI of the spiral plotted in Fig. 3(d) in the main text has been convoluted by the spectral response of the filter. Table 1 lists all the parameters used in the simulation.

Table 1. List of all the parameters used to simulate the JSA of the resonator and of the spiral. FEM = Finite Element Method

View Table

Funding

Engineering and Physical Sciences Research Council (EP/L024020/1).

Acknowledgments

The author would like to acknowledge prof. M. Liscidini for the support and the useful discussions, I. Faruque, W. McCutcheon, G. Sinclair and S. Paesani for all the fruitful conversations. The work was done at the Quantum Engineering and Technology Labs of the University of Bristol, funded by the EPSRC Programme Grant EP/L024020/1.

Disclosures

The author declare no conflicts of interest.

References

1. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, “Integrated photonic quantum technologies,” Nat. Photonics pp. 1–12 (2019).

2. T. Rudolph, “Why i am optimistic about the silicon-photonic route to quantum computing,” APL Photonics 2(3), 030901 (2017). [CrossRef]  

3. A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320(5876), 646–649 (2008). [CrossRef]  

4. N. C. Harris, D. Bunandar, M. Pant, G. R. Steinbrecher, J. Mower, M. Prabhu, T. Baehr-Jones, M. Hochberg, and D. Englund, “Large-scale quantum photonic circuits in silicon,” Nanophotonics 5(3), 456–468 (2016). [CrossRef]  

5. J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, K. L. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360(6386), 285–291 (2018). [CrossRef]  

6. J. C. Adcock, C. Vigliar, R. Santagati, J. W. Silverstone, and M. G. Thompson, “Programmable four-photon graph states on a silicon chip,” Nat. Commun. 10(1), 3528 (2019). [CrossRef]  

7. L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, “Integrated sources of photon quantum states based on nonlinear optics,” Light: Sci. Appl. 6(11), e17100 (2017). [CrossRef]  

8. J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, “On-chip quantum interference between silicon photon-pair sources,” Nat. Photonics 8(2), 104–108 (2014). [CrossRef]  

9. Z. Vernon, M. Menotti, C. Tison, J. Steidle, M. Fanto, P. Thomas, S. Preble, A. Smith, P. Alsing, M. Liscidini, and J. E. Sipe, “Truly unentangled photon pairs without spectral filtering,” Opt. Lett. 42(18), 3638–3641 (2017). [CrossRef]  

10. D. Grassani, S. Azzini, M. Liscidini, M. Galli, M. J. Strain, M. Sorel, J. Sipe, and D. Bajoni, “Micrometer-scale integrated silicon source of time-energy entangled photons,” Optica 2(2), 88–94 (2015). [CrossRef]  

11. C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351(6278), 1176–1180 (2016). [CrossRef]  

12. I. I. Faruque, G. F. Sinclair, D. Bonneau, J. G. Rarity, and M. G. Thompson, “On-chip quantum interference with heralded photons from two independent micro-ring resonator sources in silicon photonics,” Opt. Express 26(16), 20379–20395 (2018). [CrossRef]  

13. J. W. Silverstone, R. Santagati, D. Bonneau, M. J. Strain, M. Sorel, J. L. O’Brien, and M. G. Thompson, “Qubit entanglement between ring-resonator photon-pair sources on a silicon chip,” Nat. Commun. 6(1), 7948 (2015). [CrossRef]  

14. M. Liscidini and J. Sipe, “Stimulated emission tomography,” Phys. Rev. Lett. 111(19), 193602 (2013). [CrossRef]  

15. L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B 29(8), 2199–2212 (2012). [CrossRef]  

16. K. Zielnicki, K. Garay-Palmett, D. Cruz-Delgado, H. Cruz-Ramirez, M. F. O’Boyle, B. Fang, V. O. Lorenz, A. B. UŔen, and P. G. Kwiat, “Joint spectral characterization of photon-pair sources,” J. Mod. Opt. 65(10), 1141–1160 (2018). [CrossRef]  

17. I. Jizan, L. Helt, C. Xiong, M. J. Collins, D.-Y. Choi, C. J. Chae, M. Liscidini, M. Steel, B. J. Eggleton, and A. S. Clark, “Bi-photon spectral correlation measurements from a silicon nanowire in the quantum and classical regimes,” Sci. Rep. 5(1), 12557 (2015). [CrossRef]  

18. A. Eckstein, G. Boucher, A. Lemaître, P. Filloux, I. Favero, G. Leo, J. E. Sipe, M. Liscidini, and S. Ducci, “High-resolution spectral characterization of two photon states via classical measurements,” Laser Photonics Rev. 8(5), L76–L80 (2014). [CrossRef]  

19. B. Fang, O. Cohen, M. Liscidini, J. E. Sipe, and V. O. Lorenz, “Fast and highly resolved capture of the joint spectral density of photon pairs,” Optica 1(5), 281–284 (2014). [CrossRef]  

20. J. Erskine, D. England, C. Kupchak, and B. Sussman, “Real-time spectral characterization of a photon pair source using a chirped supercontinuum seed,” Opt. Lett. 43(4), 907–910 (2018). [CrossRef]  

21. D. Grassani, A. Simbula, S. Pirotta, M. Galli, M. Menotti, N. C. Harris, T. Baehr-Jones, M. Hochberg, C. Galland, and M. Liscidini et al., “Energy correlations of photon pairs generated by a silicon microring resonator probed by stimulated four wave mixing,” Sci. Rep. 6(1), 23564 (2016). [CrossRef]  

22. R. Kumar, J. R. Ong, M. Savanier, and S. Mookherjea, “Controlling the spectrum of photons generated on a silicon nanophotonic chip,” Nat. Commun. 5(1), 5489 (2014). [CrossRef]  

23. F. Lenzini, A. N. Poddubny, J. Titchener, P. Fisher, A. Boes, S. Kasture, B. Haylock, M. Villa, A. Mitchell, A. S. Solntsev, A. A. Sukhorukov, and M. Lobino, “Direct characterization of a nonlinear photonic circuits wave function with laser light,” Light: Sci. Appl. 7(1), 17143 (2018). [CrossRef]  

24. J. G. Titchener, A. S. Solntsev, and A. A. Sukhorukov, “Generation of photons with all-optically-reconfigurable entanglement in integrated nonlinear waveguides,” Phys. Rev. A 92(3), 033819 (2015). [CrossRef]  

25. B. Fang, M. Liscidini, J. Sipe, and V. Lorenz, “Multidimensional characterization of an entangled photon-pair source via stimulated emission tomography,” Opt. Express 24(9), 10013–10019 (2016). [CrossRef]  

26. L. A. Rozema, C. Wang, D. H. Mahler, A. Hayat, A. M. Steinberg, J. E. Sipe, and M. Liscidini, “Characterizing an entangled-photon source with classical detectors and measurements,” Optica 2(5), 430–433 (2015). [CrossRef]  

27. I. Jizan, B. Bell, L. Helt, A. C. Bedoya, C. Xiong, and B. J. Eggleton, “Phase-sensitive tomography of the joint spectral amplitude of photon pair sources,” Opt. Lett. 41(20), 4803–4806 (2016). [CrossRef]  

28. C. Ren and H. F. Hofmann, “Analysis of the time-energy entanglement of down-converted photon pairs by correlated single-photon interference,” Phys. Rev. A 86(4), 043823 (2012). [CrossRef]  

29. F. A. Beduini, J. A. Zielińska, V. G. Lucivero, Y. A. de Icaza Astiz, and M. W. Mitchell, “Interferometric measurement of the biphoton wave function,” Phys. Rev. Lett. 113(18), 183602 (2014). [CrossRef]  

30. M. Avenhaus, B. Brecht, K. Laiho, and C. Silberhorn, “Time-frequency quantum process tomography of parametric down-conversion,” arXiv preprint arXiv:1406.4252 (2014).

31. K.-K. Park, J.-H. Kim, T.-M. Zhao, Y.-W. Cho, and Y.-H. Kim, “Measuring the frequency-time two-photon wavefunction of narrowband entangled photons from cold atoms via stimulated emission,” Optica 4(10), 1293–1297 (2017). [CrossRef]  

32. M. Liscidini, L. Helt, and J. Sipe, “Asymptotic fields for a Hamiltonian treatment of nonlinear electromagnetic phenomena,” Phys. Rev. A 85(1), 013833 (2012). [CrossRef]  

33. Z. Vernon and J. Sipe, “Spontaneous four-wave mixing in lossy microring resonators,” Phys. Rev. A 91(5), 053802 (2015). [CrossRef]  

34. M. Liscidini, L. Helt, and J. Sipe, “Asymptotic fields for a hamiltonian treatment of nonlinear electromagnetic phenomena,” Phys. Rev. A 85(1), 013833 (2012). [CrossRef]  

35. Z. Yang, M. Liscidini, and J. Sipe, “Spontaneous parametric down-conversion in waveguides: a backward Heisenberg picture approach,” Phys. Rev. A 77(3), 033808 (2008). [CrossRef]  

36. A. Li, T. Van Vaerenbergh, P. De Heyn, P. Bienstman, and W. Bogaerts, “Backscattering in silicon microring resonators: a quantitative analysis,” Laser Photonics Rev. 10(3), 420–431 (2016). [CrossRef]  

37. C. Reimer, L. Caspani, M. Clerici, M. Ferrera, M. Kues, M. Peccianti, A. Pasquazi, L. Razzari, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Integrated frequency comb source of heralded single photons,” Opt. Express 22(6), 6535–6546 (2014). [CrossRef]  

38. C. Tison, J. Steidle, M. Fanto, Z. Wang, N. Mogent, A. Rizzo, S. Preble, and P. Alsing, “Path to increasing the coincidence efficiency of integrated resonant photon sources,” Opt. Express 25(26), 33088–33096 (2017). [CrossRef]  

39. Y. Liu, C. Wu, X. Gu, Y. Kong, X. Yu, R. Ge, X. Cai, X. Qiang, J. Wu, X. Yang, and X. Ping, “High-spectral-purity photon generation from a dual-interferometer-coupled silicon microring,” Opt. Lett. 45(1), 73–76 (2020). [CrossRef]  

40. S. Azzini, D. Grassani, M. Galli, D. Gerace, M. Patrini, M. Liscidini, P. Velha, and D. Bajoni, “Stimulated and spontaneous four-wave mixing in silicon-on-insulator coupled photonic wire nano-cavities,” Appl. Phys. Lett. 103(3), 031117 (2013). [CrossRef]  

41. R. Kumar, M. Savanier, J. R. Ong, and S. Mookherjea, “Entanglement measurement of a coupled silicon microring photon pair source,” Opt. Express 23(15), 19318–19327 (2015). [CrossRef]  

42. M. Borghi, C. Castellan, S. Signorini, A. Trenti, and L. Pavesi, “Nonlinear silicon photonics,” J. Opt. 19(9), 093002 (2017). [CrossRef]  

43. https://www.appliednt.com.

44. M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, and L. Pavesi, “High-frequency electro-optic measurement of strained silicon racetrack resonators,” Opt. Lett. 40(22), 5287–5290 (2015). [CrossRef]  

45. K.-K. Park, J.-H. Kim, T.-M. Zhao, Y.-W. Cho, and Y.-H. Kim, “Measuring the frequency-time two-photon wavefunction of narrowband entangled photons from cold atoms via stimulated emission,” Optica 4(10), 1293–1297 (2017). [CrossRef]  

46. M. Liscidini and J. Sipe, “Stimulated emission tomography,” Phys. Rev. Lett. 111(19), 193602 (2013). [CrossRef]  

47. L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B 29(8), 2199–2212 (2012). [CrossRef]  

48. Z. Vernon and J. Sipe, “Spontaneous four-wave mixing in lossy microring resonators,” Phys. Rev. A 91(5), 053802 (2015). [CrossRef]  

49. M. Borghi, A. Trenti, and L. Pavesi, “Four wave mixing control in a photonic molecule made by silicon microring resonators,” Sci. Rep. 9(1), 408 (2019). [CrossRef]