# Quantum-Referenced Spontaneous Emission Tomography

I. I. Faruque<EMAIL_ADDRESS>Quantum Engineering Technology
Laboratories, University of Bristol, Bristol, United Kingdom  B. M. Burridge
Quantum Engineering Technology Laboratories, University of Bristol, Bristol,
United Kingdom  Quantum Engineering Centre for Doctoral Training, Centre for
Nanoscience & Quantum Information, University of Bristol, Bristol, United
Kingdom  M. Borghi  Dipartimento di Fisica, Università di Pavia, via Bassi 6,
27100, Pavia, Italy  J. G. Rarity  Quantum Engineering Technology
Laboratories, University of Bristol, Bristol, United Kingdom  J. Barreto
Quantum Engineering Technology Laboratories, University of Bristol, Bristol,
United Kingdom

###### Abstract

We present a method of tomography that measures the joint spectral phase (JSP)
of spontaneously emitted photon pairs originating from a largely
uncharacterized “target” source. We use quantum interference between our
target source and a reference source to extract the JSP with four spectrally
resolved measurements, in a process that we call quantum-referenced
spontaneous emission tomography (Q-SpET). We have demonstrated this method on
a photonic integrated circuit for a target micro-ring resonator photon-pair
source. Our results show that spontaneously emitted photon pairs from a micro-
ring resonator are distinctively different from that of stimulated emission,
and thus cannot in general be fully characterized using classical stimulated
emission tomography without detailed knowledge of the source.

††preprint: APS/123-QED

##  I Introduction

Single photons are one of the simplest and more powerful quantum information
carriers harnessed by quantum technologies. As flying qubits, single photons
are the workhorse for applications in quantum communications such as global,
entanglement-based quantum networks [1, 2, 3, 4, 5], while also enabling
direct and indirect interactions with the environment for metrology, sensing
and imaging applications [6, 7]. Moreover, integrated photonic circuits using
single photons are one of the leading candidates for performing quantum
information processing tasks [8, 9, 10]. Non-linear, parametric processes are
at the heart of all-optical photon pair generation, and are a promising means
to create heralded single photons [11]. Heralding mitigates the vacuum
contributions associated with the probabilistic generation of photons, enables
entanglement distribution protocols [2], and produces what is effectively a
single photon source. However, as a parametric process, heralding in the
presence of correlation between the photons of a pair can project the
remaining photon into a mixed state. Therefore the accurate characterization
of photon-pair sources is essential.

Photon pairs can be described by their biphoton wave function, or joint
spectral amplitude (JSA), a quantity that depends on the frequencies of the
two photons. The square magnitude of the JSA — the joint spectral intensity
(JSI), can be measured directly using single-photons and a scanning
monochromator [12] or more rapidly using stimulated emission tomography (SET)
that makes use of commercial optical characterization equipment [13, 14].
However, the JSA is a complex-valued function, requiring both amplitude and
phase information for a complete description. Typically, the assumption is
made that the JSI can be used to infer the spectral correlations of the JSA
[14], resulting in an upper bound on spectral correlation or entanglement. The
argument of the JSA — the joint spectral phase (JSP) describes the phase
correlation of photon pairs, which also contributes to their entanglement.
Therefore, the full JSA is required to fully characterize the entanglement
between photon pairs of a specific source. Moreover, it has recently been
shown that the JSP of photon pairs generated from resonant sources cannot be
extracted by using SET alone [15]. If an analytical model of photon-pair
generation inside the resonant source exists, it may be combined with SET
measurements to provide an estimate of the correct JSP. This may not be
feasible for a complex resonant source such as topological sources [16].

Figure 1: Schematic of our photonic chip. VGC structures are used as optical
input and output (O1, O2, O3) ports. The circled numbers (1, 2, 3) refer to
the kets ($|\Psi\rangle$) at specific places in the photonic circuit. The
optical pump is distributed to the spiral and the resonator element using an
MZI ($|\Psi_{1}\rangle$). On-chip AMZI filters are then used to remove the
pump from the optical channel ($|\Psi_{2}\rangle$). A phase $\theta$ can then
be applied to one of the arms using a thermal phase-shifter, before the
optical path compensation (i.e., matching) element. The photon states
generated at the spiral and resonator are brought together and interfered
using an MZI ($|\Psi_{3}\rangle$). Two additional tuneable on-chip optical
filters based on resonator structures allow for performing frequency-resolved
measurements.

There have been some attempts to remedy this separately, with methods
previously reported for both single photons, and photon pairs [17, 18] using
Electro-Optic Shearing Interferometry (EOSI), which enforces tight timing
synchronization, and phase-stabilization [19]. Experimental complexity can be
substituted for large processing overheads associated with artificially
constrained phase-retrieval algorithms [20]. Alternatively, the methodology
can be simplified under the assumption that the photons are highly correlated
[21]. Methods that use SET to overcome photon-counting limitations [22], do
not identify or address concerns raised in [15]. Nevertheless, it is possible
to extract the correct JSP using a self-referenced method [18] by implementing
Electro-Optic Shearing Interferometry as well as additional post-processing
using Fourier filtering.

Here we investigate a technique that recovers the JSP through the use of
quantum interference between our target source, or “Device-Under-Test” (DUT),
and a reference source. Our DUT is a micro-resonator and our reference is a
waveguide source. A salient and necessary feature of our method is the
utilization of two dissimilar sources, as we rely on the resulting
interference pattern to characterize our target device; in direct contrast to
most experiments reported in the literature that have been performed with
photons generated by identical sources (e.g., [23, 4, 24, 25, 26, 27, 28, 29,
30]).

Our technique is implemented entirely on-chip - showing the useability of our
method for chip-scale quantum photonic technologies. We use a micro-racetrack
resonator as a “proof of principle” because it is a conventional and well-
understood optical device and one of the most versatile, bright, and efficient
photon-pair sources in integrated quantum photonic technologies [31, 32, 2].
For our reference, we use a waveguide photon-pair source to eliminate the need
for a phase-stable, classical reference. The quantum nature of our reference
ensures inherent spectral overlap with our target, without the need to enforce
it with a complex combination of an external coherent state, and phase-
stabilization [33, 19].

Q-SpET’s constraints are that the DUT spectra must overlap well with that of
our reference, and any feature (such as chirp) in the profile of the pump
should not vary significantly across the spectrum of the DUT. This ensures
that the waveguide JSA has a flat phase profile and can be a suitable
reference for the DUT.

##  II Implementing Spontaneous Emission Tomography

In our photonic chip, as can be seen in Fig. 1, the pump light (described by a
coherent state $|\alpha\rangle$) enters from the left through a vertical
grating coupler (VGC), and its power is then divided across two paths
according to the intensity ratio $\eta$ set by a Mach-Zehnder Interferometer
(MZI). The pump light in each channel can lead to the production of photon-
pairs through spontaneous four-wave mixing (SFWM) in the structure that
follows. The upper channel leads to a spiral waveguide (W), and the lower
channel to a micro-ring resonator (R). The quantum state exiting the MZI is
represented by the circled number 1 in Fig. 1, and the associated ket can be
expressed as

$\displaystyle|\Psi_{1}\rangle=|\sqrt{\eta}\alpha\rangle_{W}|i\sqrt{1-\eta}\alpha\rangle_{R}.$
(1)

If we use normalized biphoton wavefunctions $\Phi_{R}(\nu_{s},\nu_{i})$ and
$\Phi_{W}(\nu_{s},\nu_{i})$ to characterize the probability amplitudes for the
generation of pairs in the resonator and waveguide respectively, where
$\nu_{s}$ and $\nu_{i}$ are signal and idler frequencies, then the normalized
kets associated with these generation processes are [34]

$\displaystyle|\psi_{II}\rangle_{k}=\int
d\nu_{s}d\nu_{i}\Phi_{k}(\nu_{s},\nu_{i})a^{\dagger}_{s}(\nu_{s})a^{\dagger}_{i}(\nu_{i})|\mathrm{vac}\rangle,$
(2)

where $k=\\{R,W\\}$, $\nu_{s}$ and $\nu_{i}$ are signal and idler frequencies,
$a^{\dagger}_{s}(\nu_{s})$ and $a^{\dagger}_{i}(\nu_{i})$ are creation
operators for signal and idler photons, and $|\mathrm{vac}\rangle$ is the
vacuum state.

Figure 2: Spontaneous emission tomography. A. Four measured interferograms at
$\theta=\\{0,\pi/4,\pi/2,3\pi/4\\}$. A low-pass filter is used here to remove
high-frequency noise. B. Simulated and experimental JSP for spontaneously
emitted photon pairs generated from a micro-resonator. The superimposed white
dotted contours represent constant intensity lines of the JSI (Fig. 3B),
indicating 25%, 10%, and 1% of the maximum signal.

An important condition for implementing Q-SpET is a restriction to the weak
pump regime [35, 5], where the pump power is low enough that the probability
of more than one photon pair generation event is negligible. Then for each
photon pair, there is a probability amplitude that it is generated in the
waveguide and a probability amplitude that it is generated in the resonator.
After the photon pair generation process, we use asymmetric MZIs to filter out
the pump, resulting in a quantum state (circled number 2 in Fig. 1) consisting
mainly of the vacuum state with a small amount of the two-photon state
$|\Psi_{2}\rangle$

$\displaystyle|\Psi_{2}\rangle=N_{R}(\eta)|\psi_{II}\rangle_{R}|\mathrm{vac}\rangle_{W}+N_{W}(\eta)|\mathrm{vac}\rangle_{R}|\psi_{II}\rangle_{W},$
(3)

where the factors $N_{R}$ and $N_{W}$ depend on the intensity ratio $\eta$,
and $|N_{R}|^{2}+|N_{W}|^{2}=1$. The photon pairs from the resonator then
propagate through a path matching waveguide section, and a relative phase
($\theta$) is applied using a thermo-optic phase-shifter to this arm of the
circuit with respect to the waveguide arm.

Next, both arms of the circuit are connected to an MZI configured as a
balanced beam-splitter. The result (See Supplemental Material) is a state
which is sensitive to the interference between the two biphoton wavefunctions.

After this MZI, we use two on-chip, resonator-based tuneable optical filters
to select the frequencies of light that we divert towards a given output.
Photons filtered towards O3 and O1 are labeled signal and idler, respectively,
and the remaining photons propagate into O2. All three outputs are finally
coupled out of the chip using VGCs, and signal and idler photons are detected
using off-chip superconducting single-photon detectors (Fig. S1). The
probability of detecting both a signal and idler of specific frequency in arms
O3 and O1, given a phase angle $\theta$ and an intensity ratio $\eta$,
involves a superposition of the amplitudes $\Phi_{R}(\nu_{s},\nu_{i})$, and
$\Phi_{W}(\nu_{s},\nu_{i})$ and can be expressed as:

$\displaystyle P\left(\theta,\eta,\nu_{s},\nu_{i}\right)$
$\displaystyle=\frac{1}{4}\left|N_{R}(\eta)\Phi_{R}(\nu_{s},\nu_{i})e^{i2\theta}+N_{W}(\eta)\Phi_{W}(\nu_{s},\nu_{i})\right|^{2}.$
(4)
$\displaystyle\frac{P(\theta=\frac{3\pi}{4})-P(\theta=\frac{\pi}{4})}{P(\theta=0)-P(\theta=\frac{\pi}{2})}$
$\displaystyle=\frac{N_{R}N_{W}|\Phi_{W}|\mathrm{Im}\\{\Phi_{R}\\}}{N_{R}N_{W}|\Phi_{W}|\mathrm{Re}\\{\Phi_{R}\\}}=\frac{\mathrm{Im}\\{\Phi_{R}\\}}{\mathrm{Re}\\{\Phi_{R}\\}}=\tan(\theta_{\mathrm{JSA}}).$
(5)

Our method of tomography uses a specific set of $\theta$ values
($\\{0,\pi/4,3\pi/4,\pi/2\\}$). At each setting of $\theta$ we sweep the on-
chip filters over a range of frequencies ($\nu_{s}$ and $\nu_{i}$) and measure
the coincidences, which, once normalized, can be directly related to the
probability (Eq. 4). This process results in the four interferograms seen in
Fig. 2A. The resolution of our data is governed by the bandwidth of our on-
chip filters (10 pm), affording us a high level of precision in the single
photon regime . We can use simple algebraic manipulation (Eq. 5) on the
measured data to recover the JSP (denoted by $\theta_{\mathrm{JSA}}$) as
expressed in Eq. 5.

One significant feature of Q-SpET is that the estimated JSP does not depend on
the normalization factors ($N_{R}$, $N_{W}$), and thus the method is
independent of the intensity ratio $\eta$. This can be seen in Eq. 5, as all
the pre-factors cancel out, leaving only the real
($\mathrm{Re}\\{\Phi_{R}\\}$) and imaginary ($\mathrm{Im}\\{\Phi_{R}\\}$)
parts of the JSA.

Figure 3: Stimulated emission tomography. Simulated and experimental A \- JSP
and B \- JSI of the StFWM in a micro-resonator. The white contour lines
represent the 25%, 10%, and 1% intensity bounds as in Fig. 2B. The
experimental signal is primarily detectable within the innermost contour.
Considering very weak signal is present outside this contour, the experimental
JSPs have remarkable similarity within the outer contours. Data was not
collected in the white region in the JSI plot.

Fig 2B shows both a simulated JSP obtained following the method described
earlier [36] and the JSP that we have obtained from experimental measurements
(Fig 2A). The JSPs in Fig 2B additionally include white ‘confidence’ contours
indicating 25%, 10%, and 1% of the maximum signal. We find reasonably good
agreement between simulated and measured data, especially within the two
innermost contours (signal above 10% of the maximum). Over the frequency range
observed for the resonator JSA (25 GHz), the waveguide JSA varies so little
that it can indeed serve as a good reference [37] (See Supplemental Material).

Very generally, once the JSP is determined in this way, and the JSI identified
by direct detection of the frequency dependence of the emitted signal and
idler photons, the full JSA can be constructed.

##  III Stimulated Vs Spontaneous Emission Tomography of a resonant process

Of course, SET can be used to determine the JSI much more easily than direct
detection of the spontaneous process [13, 15]. We have done this for our
integrated circuit (Figs. S1, S6) using a seed at signal frequencies and
recording the stimulated intensity at the idler frequencies. The results are
shown in Fig. 3B together with a simulation; again there is reasonably good
agreement between the measured and simulated data. The resolution of our SET
data, 125 MHz (1 pm) and 20 MHz (0.16 pm) for the seed and idler frequencies
respectively, is at least an order of magnitude better than the existing
literature [14].

Interestingly, just as interferograms were constructed in the redQ-SpET
experiment, we can construct stimulated FWM (StFWM) interferograms, here by
sweeping the wavelength of an external seed laser and recording the spectra of
the stimulated idler photons at O2 (see Fig. 1). By using the interferograms
at specific values of $\theta$, we can then extract a JSP of the StFWM
process, (See Supplemental Material) analogously to what was done for the
spontaneous process (Fig 2B). And again, there is a reasonably good agreement
between the measured data and the simulation (Fig 3A).

However, while the JSI determined by the spontaneous and stimulated processes
should agree – that, of course, being the basis of SET – the JSP obtained from
the two processes are in general different. This illustrates how a measured
JSI can deviate from that of simulation (Fig. 2a, Supplemental Material,
[38]).In our case, the spontaneous FWM JSP (Fig 2B) is dominated by an energy-
conservation trend, angle correlation with $\Delta\nu_{i}$-$\Delta\nu_{s}$ and
anti-correlation with $\Delta\nu_{s}$+$\Delta\nu_{i}$, the JSP of stimulated
FWM is dominated by the value of $\Delta\nu_{s}$, with very little noticeable
change arising from $\Delta\nu_{i}$. This follows from the full expressions
for the JSAs for the spontaneous and stimulated processes in a ring resonator.
The JSA of the SFWM process, $\Phi_{R-Sp}$, and the JSA for the stimulated
process $\Phi_{R-St}$, are given by

$\displaystyle\Phi_{R-Sp}(\nu_{s},\nu_{i})=$ $\displaystyle N\int
d\nu_{p1}d\nu_{p2}\alpha(\nu_{p1})\alpha(\nu_{p2})F_{p-}(\nu_{p1})F_{p-}(\nu_{p2})F^{*}_{s+}(\nu_{s})F^{*}_{i+}(\nu_{i})\delta(\nu_{p1}+\nu_{p2}-\nu_{s}-\nu_{i}),$
(6) $\displaystyle\Phi_{R-St}(\bar{\nu}_{s},\nu_{i})=$ $\displaystyle
N^{\prime}\int
d\nu_{p1}d\nu_{p2}\alpha(\nu_{p1})\alpha(\nu_{p2})F_{p-}(\nu_{p1})F_{p-}(\nu_{p2})F^{*}_{s-}(\bar{\nu}_{s})F^{*}_{i+}(\nu_{i})\delta(\nu_{p1}+\nu_{p2}-\bar{\nu}_{s}-\nu_{i}),$
(7)

where the subscripts $p$, $s$ and $i$ represent pump, signal, and idler
photons respectively, and $\alpha(\nu_{p})$ is the pump field amplitude. In
the first of these equations $\nu_{s}$ and $\nu_{i}$ are the frequencies of
the spontaneously emitted signal and idler, while in the second
$\bar{\nu}_{s}$ is the CW seed frequency and $\nu_{i}$ is the frequency of the
stimulated idler; $N$ and $N^{\prime}$ are normalization constants, and energy
conservation is expressed by the Dirac delta function, $\delta(\nu)$. The
resonant field enhancement is characterized by $F_{k}(\nu)$,

$\displaystyle
F_{k\pm}(\nu)=\frac{1}{\sqrt{L}}\frac{\gamma_{k}^{*}}{2\pi(\nu_{k}-\nu)\pm
i\bar{\Gamma}_{k}},$ (8)

where $k$ can be pump ($p$), signal ($s$) or idler ($i$), $\gamma_{k}$ is a
coupling coefficient, $\bar{\Gamma}_{k}$ corresponds to the half-width half-
maximum of the line-width of the resonances, and $L$ is the geometrical length
of the resonator [36].

Eqs. 6 and 7 are very similar; indeed, the JSIs that follow from them are
identical but the JSPs are different. According to the asymptotic field model
[36], the complex amplitude of the SET signal will be proportional to the JSA
only when the seed field excites the system in one of its asymptotic output
states. For our all-pass resonator, this would require engineering the seed in
such a way that the input and the loss channel of the ring are both excited,
so that the seed photons can only leave the interaction region from the
through port [15]. However, in the absence of direct physical access to the
loss channel for our all-pass resonator we can only seed the input port; hence
we are not able to excite the resonator into one of its asymptotic output
state. The direct consequence is that the phase of the stimulated JSA in Eq. 7
differs from the phase of the spontaneous JSA in Eq. 6 as shown in Figs. 3(A)
and 3(B). Mathematically, the resonant field enhancement factors for signal
($F^{*}_{s+}(\nu_{s})$) in spontaneous case and seed in the stimulated case
($F^{*}_{s-}(\bar{\nu}_{s})$) are different, leading to different JSPs. This
confirms that spontaneous emission cannot be imitated completely by the
corresponding stimulated emission process [15, 38]. If the structure (i.e.,
source) generating the photon pairs is not well known and a detailed
analytical model does not exist, then we require a redQ-SpET method to extract
the information from the spontaneous FWM process.

Among all the demonstrations of JSP, the most direct experimental method is by
Jizan et al. [33] which uses the phase-sensitive nature of the SET method,
alongside fast and high-resolution classical interferometers that reduce the
complexity posed by the other methods [22, 20]. Q-SpET’s approach results in
surprisingly similar expressions for the stimulated and spontaneous processes
(Eqs. 4 and 5), exploiting quantum interference to directly measure the JSP as
in phase-sensitive SET [33]. This allows us to extract the correct JSP at the
expense of the lower detection rate imposed by the use of single-photon
detectors. Our method also allows us to eliminate any complex phase retrieval
algorithms, Fourier filtering, or time-resolved single-photon counting
spectrometers (TRSPSs) [22, 39, 20, 21]. Our interferometry also benefits from
the inherent phase stability of the integrated photonic platform, and due to
our implementation, does not require any fast modulators [18].

##  IV Conclusion and Outlook

In summary, Q-SpET is a precise and resource-efficient technique in both
detection and processing overhead, that recovers the joint spectral phase of a
biphoton wave function without relying on an analytical model of the photon
source, by using quantum interference between biphoton wave functions from two
different sources. We have demonstrated this method using a photonic chip
consisting of an optical resonator structure and a reference waveguide
structure — the source of our flat JSP. The interference we observe is
intrinsically phase stable and, beyond typical spectral characteristics, does
not require any other knowledge of the pump. Our method shows for the first
time a clear distinction between the spontaneous and stimulated four-wave
mixing processes within the same photonic device, and over the same wavelength
range, by comparing the measured JSPs.

This concept can be further applied to any unknown source, as long as its
biphoton wavefunction can be interfered with that of a well-characterized
reference. Furthermore, this concept is not restricted to photon-pair sources
and may be used for any quantum state generated in superposition using, for
instance, the weak pump regime. Additionally, JSPs of the spontaneous emission
of a source could be used as signatures for different sources, effectively
enabling certification protocols. Overall, the method presented here opens up
possibilities in future metrology and tomographic methods, applications in
calibration protocols and the definition of better standards for quantum
information applications.

## References

  * [1] Artur K. Ekert.  Quantum cryptography based on bell’s theorem.  Physical Review Letter, 67:661–663, Aug 1991.
  * [2] Juan Yin, Yuan Cao, Yu-Huai Li, Sheng-Kai Liao, Liang Zhang, Ji-Gang Ren, Wen-Qi Cai, Wei-Yue Liu, Bo Li, Hui Dai, et al.  Satellite-based entanglement distribution over 1200 kilometers.  Science, 356(6343):1140–1144, 2017.
  * [3] Siddarth Koduru Joshi, Djeylan Aktas, Sören Wengerowsky, Martin Lončarić, Sebastian Philipp Neumann, Bo Liu, Thomas Scheidl, Guillermo Currás Lorenzo, Željko Samec, Laurent Kling, Alex Qiu, Mohsen Razavi, Mario Stipčević, John G. Rarity, and Rupert Ursin.  A trusted node-free eight-user metropolitan quantum communication network.  Science Advances, 6(36):eaba0959, 2020.
  * [4] Jianwei Wang, Stefano Paesani, Raffaele Ding, Yunhong Santagati, Paul Skrzypczyk, Alexia Salavrakos, Jordi Tura, Remigiusz Augusiak, Laura Mančinska, Davide Bacco, Damien Bonneau, Joshua W. Silverstone, Qihuang Gong, Antonio Acín, Karsten Rottwitt, Jeremy L. Oxenlowe, Leif K.and O’Brien, Anthony Laing, and Mark G. Thompson.  Multidimensional quantum entanglement with large-scale integrated optics.  Science, 360:285–291, 2018.
  * [5] 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.  Silicon-on-insulator integrated source of polarization-entangled photons.  Nature Photonics, 8:104–108, 2014.
  * [6] Paul-Antoine Moreau, Ermes Toninelli, Thomas Gregory, and Miles J Padgett.  Imaging with quantum states of light.  Nature Reviews Physics, 1:367–380, 2019.
  * [7] Gabriela Barreto Lemos, Victoria Borish, Garrett D. Cole, Sven Ramelow, Radek Lapkiewicz, and Anton Zeilinger.  Quantum imaging with undetected photons.  Nature, 512:409–412, 2014.
  * [8] E. Knill, R. Laflamme, and G. J. Milburn.  A scheme for efficient quantum computation with linear optics.  Nature, 409:46–52, 2001.
  * [9] Terry Rudolph.  Why I am optimistic about the photonic route to quantum computing.  APL Photonics, 2:030901, 2017.
  * [10] Antonio A. Gentile, Brian Flynn, Sebastian Knauer, Nathan Wiebe, Stefano Paesani, John G. Granade, Christopher E.and Rarity, Raffaele Santagati, and Anthony Laing.  A scheme for efficient quantum computation with linear optics.  Nature Physics, 17:837–843, 2021.
  * [11] TB Pittman, BC Jacobs, and JD Franson.  Heralding single photons from pulsed parametric down-conversion.  Optics communications, 246(4-6):545–550, 2005.
  * [12] Yoon-Ho Kim and Warren P. Grice.  Measurement of the spectral properties of the two-photon state generated via type ii spontaneous parametric downconversion.  Opt. Lett., 30(8):908–910, Apr 2005.
  * [13] M. Liscidini and J. E. Sipe.  Stimulated emission tomography.  Physical Review Letters, 111:193602, 2013.
  * [14] Davide Grassani, Angelica Simbula, Stefano Pirotta, Matteo Galli, Matteo Menotti, Nicholas C Harris, Tom Baehr-Jones, Michael Hochberg, Christophe Galland, Marco Liscidini, and Daniele Bajoni.  Energy correlations of photon pairs generated by a silicon microring resonator probed by Stimulated Four Wave Mixing.  Scientific reports, 6(March):23564, 2016.
  * [15] Massimo Borghi.  Phase-resolved joint spectra tomography of a ring resonator photon pair source using a silicon photonic chip.  Optics Express, 28(5):7442–7462, Mar 2020.
  * [16] Sunil Mittal, Venkata Vikram Orre, Elizabeth A. Goldschmidt, and Mohammad Hafezi.  Tunable quantum interference using a topological source of indistinguishable photon pairs.  Nature Photonics, 15(7):542–548, 2021.
  * [17] Alex OC Davis, Valérian Thiel, Michał Karpiński, and Brian J Smith.  Measuring the single-photon temporal-spectral wave function.  Physical review letters, 121(8):083602, 2018.
  * [18] Alex O. C. Davis, Valérian Thiel, and Brian J. Smith.  Measuring the quantum state of a photon pair entangled in frequency and time.  Optica, 7(10):1317–1322, Oct 2020.
  * [19] Federica A Beduini, Joanna A Zielińska, Vito G Lucivero, Yannick A de Icaza Astiz, and Morgan W Mitchell.  Interferometric measurement of the biphoton wave function.  Physical review letters, 113(18):183602, 2014.
  * [20] Jean-Philippe W. MacLean, Sacha Schwarz, and Kevin J. Resch.  Reconstructing ultrafast energy-time-entangled two-photon pulses.  Phys. Rev. A, 100:033834, Sep 2019.
  * [21] N. Tischler, A. Büse, L. G. Helt, M. L. Juan, N. Piro, J. Ghosh, M. J. Steel, and G. Molina-Terriza.  Measurement and shaping of biphoton spectral wave functions.  Phys. Rev. Lett., 115:193602, Nov 2015.
  * [22] G. S. Thekkadath, B. A. Bell, R. B. Patel, M. S. Kim, and I. A. Walmsley.  Measuring the joint spectral mode of photon pairs using intensity interferometry.  Phys. Rev. Lett., 128:023601, Jan 2022.
  * [23] C. K. Hong, Z. Y. Ou, and L. Mandel.  Measurement of subpicosecond time intervals between two photons by interference.  Physical Review Letter, 59:2044–2046, Nov 1987.
  * [24] Xi-lin Wang, Yi-han Luo, He-liang Huang, Ming-Cheng Chen, Zu-en Su, Chang Liu, Chao Chen, Wei Li, Yu-Qiang Fang, Xiao Jiang, Jun Zhang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan.  18-Qubit Entanglement with Six Photons’ Three Degrees of Freedom.  Physical Review Letters, 120:260502, 2018.
  * [25] Artur K. Ekert, John G. Rarity, Paul R. Tapster, and G. Massimo Palma.  Practical quantum cryptography based on two-photon interferometry.  Physical Review Letters, 69:1293, 1992.
  * [26] Pieter Kok, Hwang Lee, and Jonathan P. Dowling.  Creation of large-photon-number path entanglement conditioned on photodetection.  Physical Review A, 65:052104, Apr 2002.
  * [27] Hiroshi Fukuda, Koji Yamada, Tetsufumi Shoji, Mitsutoshi Takahashi, Tai Tsuchizawa, Toshifumi Watanabe, Jun ichi Takahashi, and Sei ichi Itabashi.  Four-wave mixing in silicon wire waveguides.  Optics Express, 13(12):4629–4637, Jun 2005.
  * [28] Evan Jeffrey, Nicholas A Peters, and Paul G Kwiat.  Towards a periodic deterministic source of arbitrary single-photon states.  New Journal of Physics, 6:100–100, jul 2004.
  * [29] A. L. Migdall, D. Branning, and S. Castelletto.  Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source.  Physical Review A - Atomic, Molecular, and Optical Physics, 66(5):4, 2002.
  * [30] Imad I. Faruque, Gary F. Sinclair, Damien Bonneau, John G. Rarity, and Mark G. Thompson.  On-chip quantum interference with heralded photons from two independent micro-ring resonator sources in silicon photonics.  Optics Express, 26(16):20379–20395, 2018.
  * [31] Daniel Llewellyn, Yunhong Ding, Imad I. Faruque, Stefano Paesani, Davide Bacco, Raffaele Santagati, Yan-Jun Qian, Yan Li, Yun-Feng Xiao, Marcus Huber, Mehul Malik, Gary F. Sinclair, Xiaoqi Zhou, Karsten Rottwitt, Jeremy L. O’Brien, John G. Rarity, Qihuang Gong, Leif K. Oxenlowe, Jianwei Wang, and Mark G. Thompson.  Chip-to-chip quantum teleportation and multi-photon entanglement in silicon.  Nature Physics, 16:148–153, 2020.
  * [32] Y Zhang, M Menotti, K Tan, VD Vaidya, DH Mahler, LG Helt, L Zatti, M Liscidini, B Morrison, and Z Vernon.  Squeezed light from a nanophotonic molecule.  Nature communications, 12(1):1–6, 2021.
  * [33] Iman Jizan, Bryn Bell, L. G. Helt, Alvaro Casas Bedoya, Chunle Xiong, and Benjamin J. Eggleton.  Phase-sensitive tomography of the joint spectral amplitude of photon pair sources.  Optics Letter, 41(20):4803–4806, Oct 2016.
  * [34] Zhenshan Yang, Marco Liscidini, and J. E. Sipe.  Spontaneous parametric down-conversion in waveguides: A backward heisenberg picture approach.  Physical Review A, 77:033808, Mar 2008.
  * [35] Laurent Olislager, Jassem Safioui, Stéphane Clemmen, Kien Phan Huy, Wim Bogaerts, Roel Baets, Philippe Emplit, and Serge Massar.  Silicon-on-insulator integrated source of polarization-entangled photons.  Optics Letter, 38(11):1960–1962, Jun 2013.
  * [36] Milica Banic, Luca Zatti, Marco Liscidini, and J. E. Sipe.  Modeling nonlinear optics in lossy microring systems: two strategies.  arXiv:2111.14711, 2021.
  * [37] Imad I. Faruque, Gary F. Sinclair, Damien Bonneau, Takafumi Ono, Christine Silberhorn, Mark G. Thompson, and John G. Rarity.  Estimating the Indistinguishability of Heralded Single Photons Using Second-Order Correlation.  Physical Review Applied, 12:054029, 2019.
  * [38] Lee A. Rozema, Chao Wang, Dylan H. Mahler, Alex Hayat, Aephraim M. Steinberg, John E. Sipe, and Marco Liscidini.  Characterizing an entangled-photon source with classical detectors and measurements.  Optica, 2(5):430–433, May 2015.
  * [39] Jean-Philippe W. MacLean, John M. Donohue, and Kevin J. Resch.  Direct characterization of ultrafast energy-time entangled photon pairs.  Phys. Rev. Lett., 120:053601, Jan 2018.
  * [40] C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien.  60 db high-extinction auto-configured mach–zehnder interferometer.  Optics Letter, 41(22):5318–5321, Nov 2016.
  * [41] D.F. Walls and Gerard J. Milburn.  Quantum Optics.  Springer Link, 2008.
  * [42] Z. Vernon and J. E. Sipe.  Spontaneous four-wave mixing in lossy microring resonators.  Phys. Rev. A, 91:053802, May 2015.

## Methods

Experimental setup: Figure 1 shows the photonic chip and Fig S1 shows the
optical setup. Our chip is designed and fabricated on a conventional silicon-
on-insulator (SOI) platform with waveguide cross-section 220 nm $\times$ 500
nm for single-mode operation. Photon-pairs are generated through SFWM, where
two pump photons are converted into a pair of photons with different
frequencies, commonly denoted as signal and idler photons. We use a pulsed
tuneable pump laser centered at a wavelength of 1550 nm (PriTel FFL 50 MHz, 15
ps, 6 mW) and pre-filtered with two broadband filters (BFs). This is a crucial
step to reduce the pump laser’s background contribution at SFWM signal-idler
wavelengths of interest with nearly 120 dB attenuation. These broadband
filters are commercially available Dense Wavelength Division Multiplexers from
Opneti. The intensity and polarization of the pump are then adjusted using a
variable optical attenuator (VOA) and the polarization controller. The
polarization controller orients the light to a TE polarization for optimal
coupling to the onc-hip vertical grating coupler (VGC). The measured coupling
loss of these VGC is -4.5 dB. The pump is then distributed across two separate
on-chip photonic SFWM sources (spiral length of 2.8 mm, resonator geometrical
length of 355 $\mu$m), and the residual pump filtered out by asymmetric MZIs
[40]. The generated photon-pairs propagate through the circuit and are
interfered using an on-chip Mach-Zehnder Interferometer (MZI) structure. The
output is then coupled back into a single-mode fiber (SMF28) using VGCs, and
filtered using BFs, before being collected by single-photon detectors. The
detectors are connected to a time-interval analyzer to record the coincidence
counts used to produce the interferograms.

Our resonator (Fig. 1) resembles a racetrack resonator except the coupling
region, which usually is implemented by a directional coupler. Instead, this
resonator has a Mach-Zehnder interferometer which acts as a variable coupling
region. By applying a voltage to the phase element in this interferometer, we
can vary the resonance linewidth. This is shown in Fig. S2, where the
resonance can be tuned from over-coupled to critically-coupled to under-
coupled regions. We have chosen an over-coupled resonance with a reasonable
quality factor. This allows relatively high resonance enhancement while
allowing broad enough linewidth (FWHM 60 pm) to collect enough data points
within the resonance using our on-chip filters.

For SET, we use a narrow linewidth (125 MHz or 1 pm) tuneable continuous wave
laser (Yenista) to seed to the FWM process at the signal wavelength. The idler
photons generated by StFWM are interfered, coupled out of the chip and
filtered (BF) to remove any background pump. We use a Finisar Optica Spectrum
Analyzer (OSA) with 20 MHz or 0.16 pm resolution to record the spectra of the
idler.

Data processing: The experimentally recorded interferograms for spontaneous
emission tomography have been passed through Gaussian and moving average
filters before using them for estimating the JSP. This filtering essentially
removes any high frequency noise present in the interferograms.

## Data availability

The data used to produce the plots within this article are available at …

## Code availability

The MATLAB program used for processing the data is nevertheless available from
the corresponding author on reasonable request.

## Acknowledgements

## Author contributions

Authors I.I.F and B.M.B contributed equally to this work. I.I.F. conceived the
idea and method of the spontaneous emission tomography. I.I.F. and M.B.
designed the photonic circuit. B.M.B. and I.I.F. performed the experiment.
B.M.B., I.I.F. and J.B. analyzed the data and performed the simulation. J.B.
and J.G.R supervised the project. All the authors contributed in writing the
manuscript.

## Funding

This work was supported by the Engineering and Physical Sciences Research
Council (EPSRC) under the grant EP/L0240201. B.M.B. acknowledges the support
of the EPSRC training grant EP/LO15730/1. The authors also acknowledge EPSRC’s
additional support of the work via grants EP/M024458/1, EP/K033085/1 and
EP/N015126/1.

## Competing interests

The authors declare no competing interests.

## Supplemental Material: Spontaneous emission tomography

##  V Experimental setup

As shown in Fig. S1, we use a photonic chip for our experiment with an off-
chip pump laser and two off-chip single-photon detectors. The details of our
design parameters, simulation parameters and experimental values are listed in
Tab. S1. The chosen linewidth, Full-Width at Half-Maximum (FWHM), of the
resonator is 7.5 GHz (or 60 pm), which is an over-coupled configuration as
shown in Fig. S2. Subsequently its four-wave mixing bandwidth is considerably
narrower than that of the waveguide source ($>$10 nm) [37]. This can be seen
from the JSA comparisons in Fig. S3 B.

Figure S1: Experimental setup. A Spontaneous emission tomography. Schematic of the experimental setup for spontaneous emission tomography. Signal and idler photons are collected from the on-chip filters at O1 and O3 and filtered again using off chip broadband filters (BFs) to remove background pump photons and detected using single-photon detectors (D1 and D2) for coincidence counts. B Stimulated emission tomography. Schematic of the experimental setup used for stimulated emission tomography. Similarly to the set-up presented in Fig 2 a pulsed laser is used as a pump and a second (seed) laser is also added. In this case an Optical Spectrum Analyzer (OSA) is used to measure the stimulated idler spectra.  Parameters |  Values
---|---
Waveguide dimension |  220 nm $\times$ 500 nm
Waveguide length |  2.8 mm
Waveguide loss |  2.5 dB/cm, 0.01 dB/bend
Compact model, [$n_{1}$, $n_{2}$, $n_{3}$] |  [2.4473, -1.1327, -0.0440]
Group velocity, ng |  4.181 at 1.55$\mu$m
Resonator linewidth |  7.44 GHz (59.6 pm)
Resonator geometrical length |  363 $\mu$m
On-chip filter linewidth |  1.25 GHz (10 pm)
Pump wavelength |  1546.23 nm
Pump repetition rate |  50 MHz
Pump linewidth |  15 ps
Signal wavelength |  1538.55 nm
Idler wavelength |  1553.98 nm
Table S1: Compact model:
$n_{eff}=n_{1}+n_{2}(\lambda-\lambda_{0})+n_{3}(\lambda-\lambda_{0})^{2}$,
where $\lambda$ is wavelength in $\mu$m. Figure S2: Tuneable coupling of our
resonator. A. Resonant lineshapes as a function of applied voltage in the
coupling region. The applied voltage tunes the fraction of light that is being
coupled to the resonator. Thus, the coupling can be tuned from over-coupling
to critical-coupling to under-coupling. B. Extracted parameters of the
resonances as a function of applied voltage. i) Quality factor (Q) and full-
width at half-maximum (FWHM) of the linewidth vs voltage. ii) extinction ratio
(ER) vs voltage. The highest extinction configurations are the critically-
coupled configurations. We chose 60 pm linewidth—an over-coupled
configuration.

##  VI Quantum interference between two different sources

Prior experiments show that near-unity visibility (fidelity) can be achieved
when photons from the same source or identical sources are interfered [35, 5],
as signal and idler photons have the same JSA. However, our two sources have
different JSAs, leading to measurable visibility differences following the
interference.

Figure S3: Quantum interference between two dissimilar sources. Using the
experimental setup of the redQ-SpET we can observe the following interaction
among the photon-pairs from the waveguide and the photon-pairs. A. The plot
shows reference classical interference fringe (black), and quantum
interference data (blue) and a fitted fringe (blue). The position of the on-
chip spectral filter is marked by a white dot in B. The quantum interference
fringe has a distinctive doubling of the period and a visibility of $~{}88\%$.
B. Simulated JSAs of the waveguide (left) and the resonator (right) on-chip
sources based on the measured parameters. The white contours lines show
constant amplitude of JSA. From innermost to outer contours, the amplitudes
are 50%, 75% and 90%. This plot shows that withing the range of the JSA of a
resonator, the waveguide JSA varies very little.  Figure S4: Joint spectral
intensity from spontaneous photon-pair emission. We have only pumped the
micro-resonator in our circuit and used the setup of Fig. S1A to collect the
signal and idler photons to construct the JSI.

Fig. S3 A, shows a quantum interference fringe between our sources. A
classical interference fringes is also plotted here for comparison. The white
dot, labeled as “F”, in Fig. S3 B, represents the position of the on-chip
tuneable filter (10 pm bandwidth) while collecting the data of this quantum
interference fringe. As expected, the quantum interference has the distinctive
period-doubling compared to the classical case, with a measurable visibility
of about 88%, which is remarkable considering these are two different sources.

##  VII Tomography model and effect of loss

This section elaborates Eqs. 1-5 and the analytical model used for estimating
the JSP.

Neglecting scattering loss for the moment, we can write the quantum state of
the system after the pump filters as

$\displaystyle|\Psi_{2}\rangle=\int
d\nu_{s}d\nu_{i}\left(N_{R}\Phi_{R}\left(\nu_{s},\nu_{i}\right)c^{\dagger}_{s}(\nu_{s})c^{\dagger}_{i}(\nu_{i})e^{j2\theta}+N_{W}\Phi_{W}\left(\nu_{s},\nu_{i}\right)d^{\dagger}_{s}(\nu_{s})d^{\dagger}_{i}(\nu_{i})\right)|\mathrm{vac}\rangle.$
(S1)

The operators $c_{k}(\nu_{k})$ and $d_{k}(\nu_{k})$ refer to the parts of the
circuit associated with the resonant and reference sources, respectively (see
Fig. S5). During the interference at the final 50:50 beam splitter (Fig. S5),
these operators transform according to

$\displaystyle
c_{k}^{\dagger}(\nu_{k})=\frac{1}{\sqrt{2}}\left(e^{\dagger}_{k}(\nu_{k})+if^{\dagger}_{k}(\nu_{k})\right),$
(S2) $\displaystyle
d_{k}^{\dagger}(\nu_{k})=\frac{1}{\sqrt{2}}\left(ie^{\dagger}_{k}(\nu_{k})+f^{\dagger}_{k}(\nu_{k})\right),$
(S3)

where ($k=\\{s,i\\}$) and the operators $e_{k}(\nu_{k})$ and $f_{k}(\nu_{k})$
refer to the output arms of the interferometer [41].

Figure S5: Effect of loss represented by a phantom channel. A. Beam splitter
transformation operators at the final 50:50 beam splitter in the circuit. B.
Phantom channels representing loss during the photon-pair generation process
in the resonator. The relevant operators for different parts of the circuit
are indicated at the relevant parts of the circuit.

Using $\hat{n}_{s}$ and $\hat{n}_{i}$ to denote the number operators for
signal and idler photons in the “$f$” arm of the interferometer, the
coincidence counts in this arm are expressed as

$\displaystyle
P\left(\theta,\eta,\nu_{s},\nu_{i}\right)=\langle\hat{n}_{s}\hat{n}_{i}\rangle=\frac{1}{4}\left|N_{R}\Phi_{R}\left(\nu_{s},\nu_{i}\right)e^{i2\theta}+N_{W}\Phi_{W}\left(\nu_{s},\nu_{i}\right)\right|^{2}.$
(S4)

where, $\langle\hat{x}\rangle$ represents the expectation value with respect
to the ket the ket $|\Psi_{3}\rangle$, the state after the interference. For
$\theta=0$, we get

$\displaystyle P(\theta=0)$
$\displaystyle=\frac{1}{4}\left|N_{R}\Phi_{R}+N_{W}\Phi_{W}\right|^{2}$
$\displaystyle=\frac{1}{4}\left(N_{R}\Phi_{R}+N_{W}\Phi_{W}\right)\left(N_{R}\Phi_{R}^{*}+N_{W}\Phi_{W}^{*}\right)$
$\displaystyle=\frac{1}{4}\left(\left|N_{R}\Phi_{R}\right|^{2}+\left|N_{W}\Phi_{W}\right|^{2}+N_{R}N_{W}|\Phi_{W}|\left(\Phi_{R}+\Phi_{R}^{*}\right)\right)$
$\displaystyle=\frac{1}{4}\left(\left|N_{R}\Phi_{R}\right|^{2}+\left|N_{W}\Phi_{W}\right|^{2}+2N_{R}N_{W}|\Phi_{W}|\mathrm{Re}\\{\Phi_{R}\\}\right).$
(S5)

Here, we considered that the reference waveguide JSA varies very little over
the JSA range of the resonator (Fig. S3). However, in the presence of the
phase profile in the pump, or due to the reference structure (waveguide), if
there is any feature in the reference JSP, it can be included explicitly at
this point to improve the accuracy of the model. In fact, flat-phase profile
is not a necessary condition for our tomography. Any structure can be used as
a reference if the JSP of that structure is well characterized. In our
experiment, for the ease of measurement, we have used a specific length of the
waveguide [37] that gives us a flat phase profile. Without any loss of
generality we can assume that the $\Phi_{W}$ has only real part; any constant
phase in $\Phi_{W}$ can be included in $\theta$, so we can use

$\displaystyle\Phi_{W}=\Phi_{W}^{*}=|\Phi_{W}|$ (S6)

and

$\displaystyle\Phi_{R}+\Phi_{R}^{*}=2\times\mathrm{Re}\\{\Phi_{R}\\},$ (S7)
$\displaystyle\Phi_{R}-\Phi_{R}^{*}=2i\times\mathrm{Im}\\{\Phi_{R}\\},$ (S8)

where, Re$\\{\\}$ and Im$\\{\\}$ indicate real and imaginary parts
respectively. Similarly, we get

$\displaystyle P(\theta=\frac{\pi}{2})$
$\displaystyle=\frac{1}{4}\left(\left|N_{R}\Phi_{R}\right|^{2}+\left|N_{W}\Phi_{W}\right|^{2}-2N_{R}N_{W}|\Phi_{W}|\mathrm{Re}\\{\Phi_{R}\\}\right),$
(S9) $\displaystyle P(\theta=\frac{3\pi}{4})$
$\displaystyle=\frac{1}{4}\left(\left|N_{R}\Phi_{R}\right|^{2}+\left|N_{W}\Phi_{W}\right|^{2}+2N_{R}N_{W}|\Phi_{W}|\mathrm{Im}\\{\Phi_{R}\\}\right),$
(S10) $\displaystyle P(\theta=\frac{\pi}{4})$
$\displaystyle=\frac{1}{4}\left(\left|N_{R}\Phi_{R}\right|^{2}+\left|N_{W}\Phi_{W}\right|^{2}-2N_{R}N_{W}|\Phi_{W}|\mathrm{Im}\\{\Phi_{R}\\}\right).$
(S11)

From these equations, we get

$\displaystyle
P(\theta=0)-P(\theta=\frac{\pi}{2})=N_{R}N_{W}|\Phi_{W}|\mathrm{Re}\\{\Phi_{R}\\},$
(S12) $\displaystyle
P(\theta=\frac{3\pi}{4})-P(\theta=\frac{\pi}{4})=N_{R}N_{W}|\Phi_{W}|\mathrm{Im}\\{\Phi_{R}\\}.$
(S13)

These equations give us Eq. 5.

We now consider the effects of scattering losses in the resonant source and
verify that even if the loss of the resonator included, the final result (Eq.
5) would stay the same. We model the lossy resonator following [36],
introducing a phantom channel to account for the scattered photons [42]; in
Fig S5 we sketch the system with and without the loss represented in this way.
The waveguide is sufficiently short that we can continue to model it as
lossless.

Here we are dealing with a high-finesse resonator point-coupled to a channel
waveguide in a weak pump regime, and our experiment relies on the measurement
of coincidences. Under these conditions, the scattering loss is expected to
affect only the linewidth of the JSA [36, 42]; the above analysis should be
valid even in the presence of scattering losses, provided the resonator’s full
linewidth is used in simulations. We verify this by comparing the derivation
of Eq. 5 with and without the part of the state describing the scattered
photons included.

With the scattered photons accounted for, the state of the system before the
interference is,

$\displaystyle|\Psi_{2}\rangle=$ $\displaystyle\int
d\nu_{s}d\nu_{i}\Bigg{(}N_{R}\bigg{(}\Phi_{R}^{rr}\left(\nu_{s},\nu_{i}\right)e^{i2\theta}c^{\dagger}_{sr}\left(\nu_{s}\right)c^{\dagger}_{ir}\left(\nu_{i}\right)+\Phi_{R}^{rl}\left(\nu_{s},\nu_{i}\right)e^{i\theta}c^{\dagger}_{sr}\left(\nu_{s}\right)c^{\dagger}_{il}\left(\nu_{i}\right)$
$\displaystyle+\Phi_{R}^{lr}\left(\nu_{s},\nu_{i}\right)e^{i\theta}c^{\dagger}_{sl}\left(\nu_{s}\right)c^{\dagger}_{ir}\left(\nu_{i}\right)+\Phi_{R}^{ll}\left(\nu_{s},\nu_{i}\right)c^{\dagger}_{sl}\left(\nu_{s}\right)c^{\dagger}_{il}\left(\nu_{i}\right)\bigg{)}+N_{W}\Phi_{W}\left(\nu_{s},\nu_{i}\right)d^{\dagger}_{s}\left(\nu_{s}\right)d^{\dagger}_{i}\left(\nu_{i}\right)\Bigg{)}|\mathrm{vac}\rangle$
(S14)

If the scattered photons were traced out, we would recover the form of Eq. S1.
We again model the beamsplitter interaction with the transformations in Eqs.
S2 and S3. In this case, only the annihilation operators associated with a
real channel waveguide (labeled with the subscript $r$) will be transformed to
form the interferograms,

$\displaystyle
c_{kr}^{\dagger}(\nu_{k})=\frac{1}{\sqrt{2}}\left(e^{\dagger}_{k}(\nu_{k})+if^{\dagger}_{k}(\nu_{k})\right)$
(S15)

In Eq. S14 only the parts of the JSA that have at least one photon in the real
channel will contribute to the observed interferograms; the terms in the last
set of brackets will not contribute. In principle, the second and third set of
brackets could contribute to the interferograms, but because we seek
coincidences, these ‘broken pair’ terms will have no effect (see Eq. S4). The
only terms in S14 which contribute to the interferograms are those in the
first set of brackets, which has the same form as Eq. S1. We can then see that
the recorded interferogram will be similar to Eq. 4,

$\displaystyle
P\left(\theta,\eta,\nu_{s},\nu_{i}\right)=\frac{1}{4}\left|N_{R}\Phi_{R}^{rr}\left(\nu_{s},\nu_{i}\right)e^{i2\theta}+N_{W}\Phi_{W}^{rr}\left(\nu_{s},\nu_{i}\right)\right|^{2}.$
(S16)

The function $\Phi_{R}^{rr}$ is identical to $\Phi_{R}$ in Eq. S1, up to a
constant factor [36]. As mentioned in the main text, our method does not
require that $N_{R}=N_{W}$, so the presence of loss should have no effect, in
principle.

The above result relies on the fact that we always measure photon coincidences
for our experiment. The above equation shows that the coincidence counts act
as a noise filter in the weak pump regime; as long as two sources do not
produce photon-pairs at the same time, the broken pairs from the resonant
sources will not contribute to the noise.

##  VIII SET using Q-SpET circuit

Following [13], [15] and [33], we can reconfigure our photonic circuit for SET
measurements. If the amplitude of the seed laser is $A_{seed}$ and the JSA of
the stimulated emission process is $F(\nu_{s},\nu_{i})$, then the amplitude of
the stimulated idler can be written as blue

$\displaystyle A_{n}^{SET}\propto A^{*}_{seed}\times F_{n}(\nu_{s},\nu_{i})$
(S17)

where, n={W,R} represents waveguide and resonator respectively. In this
context, Eq. S17 illustrates how the seed $A^{*}_{seed}$ imprints phase
characteristics to the photon-pairs generated ($A_{n}^{SET}$) which would be
otherwise absent in the non-stimulated process $F_{n}(\nu_{s},\nu_{i})$. Now
the arm containing the resonator source has a thermal phase element,
introducing $\theta$ phase on that arm. At the final beam splitter

$\displaystyle\mathbf{A}_{out}$
$\displaystyle=\mathrm{\mathbf{M}_{BS}}\times\mathrm{\mathbf{M}_{\theta}}\times\mathbf{A}_{in},$
(S18) $\displaystyle\begin{bmatrix}A_{o2}\\\ A_{o2}\end{bmatrix}$
$\displaystyle=\frac{e^{i\pi/2}}{\sqrt{2}}\begin{bmatrix}1&i\\\
i&1\end{bmatrix}\times\begin{bmatrix}e^{i\theta}&0\\\
0&1\end{bmatrix}\times\begin{bmatrix}A_{R}^{SET}\\\ A_{W}^{SET}\end{bmatrix}.$
(S19)

Here, $\mathrm{M_{BS}}$ and $\mathrm{M_{\theta}}$ are transfer matrices of the
50:50 beam-splitter and the phase element respectively. Therefore, choosing
the upper arm of the interferometer ($A_{o1}$), the interference between
stimulated idlers from the waveguide and the resonator can be expresses as

$\displaystyle
I(\theta,\nu_{s},\nu_{i})\propto\left|N^{{}^{\prime}}_{R}F_{R}\left(\nu_{s},\nu_{i}\right)e^{i\theta}+iN^{{}^{\prime}}_{W}F_{W}\left(\nu_{s},\nu_{i}\right)\right|^{2}.$
(S20)

Here, $N^{{}^{\prime}}_{n}$ are constants of proportionality. This equation is
very similar to Eq. S4 except the phase accumulated by the resonator arm is
half that of the Eq. S4. Therefore, following the same calculations, we have
to double the values of those 4 specific values of phase to retrieve the
resonator’s stimulated JSA.

Figure S6: Stimulated emission tomography (SET). Four measured interferograms
at four values of $\theta=\\{0,\pi/2,\pi,3\pi/2\\}$, which are distinctly
different from the $\theta$ values of redQ-SpET due to the absence of the
phase doubling (e.g. $2\theta$) effect of the biphoton state.