Spatio-Spectral Quantum State Estimation of Photon Pairs from Optical Fiber Using Stimulated Emission

Linearly polarized (LP) modes are the transverse spatial modes supported in a conventional cylindrically-symmetric optical fiber that satisfies the weakly guided approximation [43]. These modes are denoted as $\operatorname{LP}_{lmq}$, where $l$, $m$, and $q$ are the azimuthal, radial, and parity indices describing their modal structures [11, 43]. In this paper, we consider a few-mode polarization-maintaining fiber (PMF) that supports three LP modes: $\ket{\operatorname{LP}_{01}}=\ket{g}$, $\ket{\operatorname{LP}_{11e}}=\ket{e}$, and $\ket{\operatorname{LP}_{11o}}=\ket{o}$ (see Fig. 1(a)). As transverse-mode basis states, these three LP modes can be combined to form superposition states, e.g., $\ket{d,a}=\left(\ket{e}\pm\ket{o}\right)/\sqrt{2}$ and $\ket{r,l}=\left(\ket{e}\pm i\ket{o}\right)/\sqrt{2}$ as shown in Fig. 1(a). The modes are then further affected by the two types of birefringence in the PMF: a polarization birefringence $\Delta=n^{x}-n^{y}$ between the slow ($x$) and fast ($y$) axes of the PMF and a parity birefringence $\Delta^{p}=n^{o}-n^{e}$ between transverse modes with even ($e$) and odd ($o$) parities. Fig. 1 shows how the slow ($x$) axis of the PMF is oriented along the vertical direction and the mode $\ket{e}$ intensity lobes.

When characterizing the photon-pair generation process in a few-mode PMF, it is important to accurately describe the property of a transverse mode at a given wavelength. For this purpose, we define an effective refractive index, which takes into account the transverse geometrical effect of the optical fiber ($T_{\nu}$) as well as its material dispersion property ($\omega_{\nu}$): $n^{T_{\nu}}_{\nu}=n_{\mathrm{eff}}(\omega_{\nu},T_{\nu})$, where $\nu$ indicates pump $p$, signal $s$, or idler $i$ and $T_{\nu}$ and $\omega_{\nu}$ indicate transverse mode and frequency, respectively. Using this convention in the $xx$-$yy$ cross-polarized scheme [11, 44], in which the pump is polarized along $x$ and the signal and idler are polarized along $y$, the effective refractive indices of the $\ket{e}$ and $\ket{o}$ modes can be represented as the following: $n_{p}^{ex}=n_{p}^{e}+\Delta$, $n_{p}^{ox}=n_{p}^{ex}+\Delta^{p}$, $n_{s,i}^{ey}=n_{s,i}^{e}$, and $n_{s,i}^{oy}=n_{s,i}^{ey}+\Delta^{p}$.

Utilizing the transverse modes, the few-mode PMF can generate photon pairs correlated in transverse mode and frequency [10, 11, 12] through a nonlinear optical process called spontaneous four-wave mixing (SFWM) [45]. The SFWM process relies on the $\chi^{(3)}$ nonlinear optical susceptibility of the fiber to annihilate two pump photons ($p_{1}$, $p_{2}$) and create a signal ($s$) and an idler ($i$) photon pair. For this nonlinear process to occur, it needs to satisfy a phase-matching condition, which is determined by the energy ($\Delta\omega=0$) and momentum conservation ($\Delta k=0$) constraints, with

$$\begin{split}\Delta\omega&=\omega_{p_{1}}+\omega_{p_{2}}-\omega_{s}-\omega_{i},\\ \Delta k&=k_{p_{1}}+k_{p_{2}}-k_{s}-k_{i}-k_{NL}\\ &=n(\omega_{p_{1}},T_{p_{1}})\frac{\omega_{p_{1}}}{c}+n(\omega_{p_{2}},T_{p_{2}})\frac{\omega_{p_{2}}}{c}-n(\omega_{s},T_{s})\frac{\omega_{s}}{c}-n(\omega_{i},T_{i})\frac{\omega_{i}}{c}-k_{NL},\end{split}$$

(1)

where $\omega_{\nu}$, $T_{\nu}$, $k_{\nu}$ are the angular frequency, transverse mode, and wavenumber, respectively of $\nu=\{p_{1},p_{2},s,i\}$, $c$ is the speed of light, and $k_{NL}$ is the nonlinear contribution from self- and cross-phase modulation [44]. Among the different types of SFWM that Eq. 1 can represent, in this paper, we concentrate on the xx-yy cross-polarized birefringent phase-matching with frequency-degenerate pumps ($\omega_{p}=\omega_{p_{1}}=\omega_{p2}$) to take advantage of the reduced Raman scattering noise and the number of possible SFWM processes [44, 46, 47]. Additionally, since the effective refractive index $n(\omega_{\nu},T_{\nu})$ depends on the transverse mode ($T_{\nu}$) and frequency ($\omega_{\nu}$), the phase-matching condition in Eq. 1 will vary for different combinations of the two. This can lead to photon pairs in different transverse modes to acquire dissimilar frequencies as shall be shown in Sec. 4.

With the fundamentals of the transverse modes and SFWM introduced, we can now express the quantum state $\ket{\psi_{si}}$ of the photon pair created from the few-mode PMF. Assuming cross-polarized birefringent phase-matching and frequency-degenerate pumps, the signal-idler photon pair $\ket{\psi_{si}}$ can be generated in a superposition of $N$ distinct SFWM processes as,

$$\ket{\psi_{si}}=\sum_{j}^{N}\int d\omega_{s}\,d\omega_{i}\,c_{j}\ket{\omega_{s}\omega_{i},T_{s}T_{i},yy,...}_{j}=\sum_{j}^{N}C_{j}\otimes\ket{T_{s}T_{i}}_{j},$$

(2)

where the prefactors weighting each process $j$ are

$$c_{j}=M_{cj}\sqrt{P_{p_{1}j}P_{p_{2}j}}f_{j}(\omega_{s},\omega_{i})O_{j}(T_{p_{1}},T_{p_{2}},T_{s},T_{i}),\ C_{j}=\int d\omega_{s}\,d\omega_{i}\,c_{j}\ket{\omega_{s}\omega_{i},yy,...}_{j}.$$

(3)

Here, $\ket{\omega_{s}\omega_{i},T_{s}T_{i},yy,...}_{j}$ represents the signal-idler state from SFWM process $j$ in transverse mode, frequency, polarization, and other implicit degrees of freedom, e.g., position, time, etc. This expression is simplified as $C_{j}\otimes\ket{T_{s}T_{i}}_{j}$ to highlight the transverse-mode contribution. The prefactors $c_{j}$ and $C_{j}$, which determine the relative amplitude and phase of each SFWM process $j$, are functions of average pump power $P_{p_{1,2}j}$, joint spectral amplitude (JSA) $f_{j}(\omega_{s},\omega_{i})$, and transverse-mode overlap integral $O_{j}(T_{p_{1}},T_{p_{2}},T_{s},T_{i})$. $O_{j}$ quantifies the spatial overlap of the four transverse modes participating as defined in [11]. $M_{cj}$ is the normalization constant satisfying $\braket{\psi_{si}}{\psi_{si}}=1$.

The JSA $f_{j}(\omega_{s},\omega_{i})$, which contains information about spectral correlations between signal and idler photons for each SFWM process [11, 12, 10, 48, 13, 1], is defined and linearly approximated as [44],

$$f_{j}(\omega_{s},\omega_{i})=\int d\omega_{p}\,\alpha(\omega_{p})\alpha(\omega_{s}+\omega_{i}-\omega_{p})\phi_{j}(\omega_{s},\omega_{i})\approx\alpha(\omega_{s},\omega_{i})\phi_{j}(\omega_{s},\omega_{i}),$$

(4)

where $\alpha(\omega_{s},\omega_{i})$ is the pump spectral envelope function and $\phi_{j}(\omega_{s},\omega_{i})$ is the phase-matching function specific for $j$. For degenerate pumps, the JSA can be linearly approximated to $f_{j}(\omega_{s},\omega_{i})\approx\alpha(\omega_{s},\omega_{i})\,\operatorname{sinc}(\frac{L}{2}\Delta k_{j})e^{i\frac{L}{2}\Delta k_{j}}$ where $L$ is the length of the fiber and $\Delta k_{j}$ is the phase mismatch for the process $j$ as defined in Eq. 1. For non-degenerate pumps, while the JSA can be linearly approximated in the same form as Eq. 4, it is also a function of the temporal walk-off between the two pumps [49, 50]. In our system, $\alpha(\omega_{s},\omega_{i})$ and $\phi_{j}(\omega_{s},\omega_{i})$ determine the spectral widths of the JSA peak along the diagonal and anti-diagonal directions, respectively. In this paper, we measure the joint spectral intensity (JSI) $|f_{j}(\omega_{s},\omega_{i})|^{2}$. The joint spectral phase (JSP) is defined as $\arg\{f_{j}(\omega_{s},\omega_{i})\}$. See Supplement 1 for a more comprehensive explanation of the factors in Eq. 3 and the quantum state representation of the pump.

The fiber parameters for the PMF considered here (Fibercore HB800C) are obtained through genetic algorithm analysis [10] to be: fiber core radius $r=1.74\mathrm{\ \mu m}$, numerical aperture $NA=0.17$, $\Delta=2.37\times 10^{-4}$, $\Delta^{p}=4.41\times 10^{-4}$. With these parameters and the three transverse modes ($\ket{g}$, $\ket{e}$, and $\ket{o}$) for the pump, signal, and idler, only 10 out of 15 SFWM processes satisfy orbital angular momentum (OAM) and parity conservation and therefore are experimentally realizable [10, 11]. Considering only the $\ket{e}$ and $\ket{o}$ modes, 5 of the above SFWM processes are viable with the following transverse mode combinations $\left(T_{p_{1}},T_{p_{2}},T_{s},T_{i}\right)$: A $(e,o,o,e)$, B $(o,o,o,o)$, C $(e,e,e,e)$, D $(e,o,e,o)$, and E $(o,o,e,e)$, where the labels A-E will be used throughout the paper to indicate the corresponding processes.

Using the formalism introduced earlier in Eqs. 2 and 3, these five FWM processes can be obtained with the two pumps in superpositions of $\ket{e}$ and $\ket{o}$ transverse modes, i.e., $\ket{\psi_{p}}=\ket{\psi_{p_{1}}}=\ket{\psi_{p_{2}}}=A_{e}\ket{e_{p}}+A_{o}\ket{o_{p}}$ given that we do not have individual control over the two pumps. The quantum states of the pumps $\ket{\psi_{p_{1}p_{2}}}$ and the signal-idler photon pairs $\ket{\psi_{si}}$ can be represented as

$$\begin{split}\ket{\psi_{p_{1}p_{2}}}&=\ket{\psi_{p}}^{\otimes 2}=B_{ee}\ket{e_{p_{1}}e_{p_{2}}}+2B_{eo}\ket{e_{p_{1}}o_{p_{2}}}+B_{oo}\ket{o_{p_{1}}o_{p_{2}}},\\ \ket{\psi_{si}}&=C_{ee}\ket{e_{s}e_{i}}+C_{eo}\ket{e_{s}o_{i}}+C_{oe}\ket{o_{s}e_{i}}+C_{oo}\ket{o_{s}o_{i}},\end{split}$$

(5)

where $A_{j}$ and $B_{j}$ are prefactors similar to $C_{j}$ (see Supplement 1 for details; these are different from the FWM process labels, A, B, and C). Here, $\otimes$ between $A_{j},B_{j},C_{j}$ and $\ket{...}_{j}$ are omitted for simplicity. Notice that $B_{eo}\ket{e_{p_{1}}o_{p_{2}}}=B_{oe}\ket{o_{p_{1}}e_{p_{2}}}$ is satisfied due to their indistinguishability, resulting in an extra factor of 2 before the $B_{eo}$ pump term and the corresponding $C_{eo}$ and $C_{oe}$ signal-idler terms, implicitly through $P_{p_{1,2}j}$ in Eq. 3.

Stimulated emission provides an efficient way to characterize the state in Eq. 5 using either the signal or idler as a seed. By controlling the transverse modes and frequencies of the pump and the seed (idler), individual FWM process(es), and thus the photon-pair state in Eq. 5 can be selectively excited (see Fig. 1(b)). This is equivalent to applying a projection operator $\ket{\omega_{i},T_{i},y}_{j_{0}}\bra{\omega_{i},T_{i},y}_{j_{0}}$, which describes the idler of process $j_{0}$, to $\ket{\psi_{si}}$ in Eq. 5 to obtain the stimulated photon state $\ket{\omega_{s},T_{s},y}_{j_{0}}$. This process is highly efficient [14, 34] when used in conjunction with a classical seed beam since the stimulated photon number is linearly proportional to the seed photon number [25, 26]. This requires sufficiently well-defined pump ($T_{p_{1}}$, $T_{p_{2}}$) and seed ($T_{i}$) transverse modes as well as a narrow spectral bandwidth seed ($\omega_{i}$), as will become clear in the following sections.

Using the methods described in Sec. 3, we measure transverse-mode-resolved JSIs. Figure 3 shows the measured JSI plots of the photon pairs and the transverse-mode images of the stimulated signal photons. The signal transverse modes are imaged with exposure times of 200 ms for Fig. 3(a-b) and 400 ms for Fig. 3(c). Narrow signal spectral filters are used to isolate individual FWM processes. Note that the states of the stimulated signal ($st$) and the seed ($d$) will reflect those of the spontaneously generated signal ($s$, around 680 nm) and the idler ($i$, around 570 nm) photons, respectively.

We vary the seed transverse modes to $\ket{e}$, $\ket{o}$, and $\ket{d}$ (see Fig. 3(a-c)) while keeping the pump mode at $\ket{d}=(\ket{e}+\ket{o})/\sqrt{2}$. This choice of seed transverse mode as well as its wavelength changes the JSI and the stimulated signal transverse mode, which helps isolate different FWM processes. Specifically, only the FWM processes that involve the given idler (seed) transverse mode and wavelength manifest in the JSI plot and the signal images. For example, with $\ket{e}$ ($\ket{o}$) seed and $\ket{d}$ pump, only the A and C (B and D) processes are stimulated, as shown in Fig. 3(a) (Fig. 3(b)). On the other hand, with the seed in superposition state $\ket{d}$, all of the four FWM processes are stimulated, as shown in Fig. 3(c) (process E is outside our spectral range of interest and expected to appear at $(\lambda_{s},\lambda_{i})_{E}\sim(730,540)_{E}\mathrm{\ nm}$). Through further measurements with $\ket{e}$ and $\ket{o}$ pumps to resolve the remaining ambiguity in the pump transverse modes, we can conclude that each JSI lobe is associated with a $\left(T_{p_{1}},T_{p_{2}},T_{s},T_{i}\right)$ FWM process as labeled in Fig. 3: A $(e,o,o,e)$, B $(o,o,o,o)$, C $(e,e,e,e)$, and D $(e,o,e,o)$. To determine the center signal and idler wavelengths $(\lambda_{s},\lambda_{i})_{j}$ of the corresponding process, we fit each JSI lobe with a Gaussian function giving: $(680.7,568.1)_{A}\textrm{\ nm}$, $(678.7,570.0)_{B}\textrm{\ nm}$, $(677.2,571.6)_{C}\textrm{\ nm}$, and $(675.3,573.3)_{D}\textrm{\ nm}$. Remarkably, this characterization is also possible in real time (see Visualization 1 and 2), similar to [31] with free-space nonlinear crystals. Supplement 1 provides more information on the characterization efficiency and the relative intensities of the JSI lobes and their relation to the transverse-mode overlap integral $O_{j}$.

This characterization capability is instrumental in assessing the degree of spectral overlap among different FWM processes, which is essential for creating transverse-mode entanglement as we investigate in Sec. 4.2. The FWM processes are spectrally separated in the JSI due to different phase matching conditions and effective refractive indices of transverse modes, as explained in Sec. 2. Consequently, this means that the quantum state of a signal-idler photon pair expressed in the transverse-spectral-mode basis, $\ket{\psi_{si}}=\sum_{j}\int d\lambda_{s}\,d\lambda_{i}\,c_{j}\ket{T_{s}T_{i},\lambda_{s}\lambda_{i}}_{j}$ where $j$ denotes a FWM process, becomes a mixed state in the transverse-mode basis with the spectral DOF is traced out, i.e., a reduced density matrix, $\rho_{si}^{T}=\operatorname{tr}_{\lambda}(\rho_{si})=\sum_{j}C_{j}\ket{T_{s}T_{i}}_{j}\bra{T_{s}T_{i}}_{j}$.

We now turn our attention to the processes B and C. We discovered that in order to explain the spectral separation between B and C, apparent in Fig. 3(c) and consistently observed in previous studies [54, 55], a new correction parameter called parity birefringence dispersion $\delta$ needs to be introduced. Without such correction, the numerical simulation may incorrectly predict the two FWM processes to completely overlap in JSI (see Fig. 4(a)), which is instrumental for enabling transverse-mode entanglement [40, 54, 55]. For simplicity, here we assume that $\delta$ is a constant describing the difference between the signal and the idler parity birefringences, i.e., $\delta=\Delta^{p}_{s}-\Delta^{p}_{i}$. With numerical simulation, we vary $\delta$ and observe the change in JSI, as shown in Fig. 4. We find the non-zero dispersion of $\delta\sim 3\times 10^{-5}$ (see Fig. 4(c)) best explains the experimental results presented in Fig. 3(c). Although $\delta$ is an order of magnitude smaller than $\Delta^{p}$ ($\sim 10^{-4}$), it contributes significantly to the spectral distinguishability between B and C processes, as shown in Fig. 4. The remaining discrepancies between the experimental data and numerical simulation arise due to imperfect estimation of the fiber parameters. Therefore, more accurate analysis using a full genetic algorithm calculation [11, 10] along with precise measurement of $\delta$ as a function of wavelength may help improve the agreement.

Building upon the characterization results described in Sec. 4.1, we apply stimulated emission to characterize a fiber source that generates photon pairs with partial transverse-mode entanglement. This characterization allows us to numerically estimate the quantum state of the photon pairs created in the transverse-mode basis. Through this process, we identify potential factors that can degrade the transverse-mode entanglement and find ways to optimize the source accordingly.

To create a maximally entangled transverse-mode Bell state in our system [54, 55], $\ket{\psi_{si}}=\left(\ket{e_{s}e_{i}}+\ket{o_{s}o_{i}}\right)/\sqrt{2}$, indistinguishabilities between $\ket{e_{s}e_{i}}$ and $\ket{o_{s}o_{i}}$ in all other degrees of freedom are necessary, i.e., $C_{ee}=mC_{oo}$ (see Eq. 3), where $m$ is a constant. Consequently, the JSIs (Eq. 4) need to satisfy the following overlap condition at the desired frequencies, $\omega_{s}$ and $\omega_{i}$: $\left|f_{ee}(\omega_{s},\omega_{i})\right|^{2}=\left|f_{oo}(\omega_{s},\omega_{i})\right|^{2}$. For this, we employ a shorter cross-spliced PMF ($2.5\mathrm{\ cm}\times 2$, HB800C) with the same experimental setup used in Sec. 3. The smaller the fiber length $L$, the wider the spectral bandwidth of the phase-matching function $\phi(\omega_{s},\omega_{i})$ (see Sec. 2.3) and thus the more spectral overlap arise along the anti-diagonal direction in the JSI. Cross-splicing, where we fusion splice two 2.5 cm-long PMFs such that the second fiber’s slow axis is aligned along the first fiber’s fast axis, helps compensate for temporal walk-off between the $\ket{e}$ and $\ket{o}$ modes [56].

With a shorter fiber for spectral indistinguishability and cross-splicing for temporal indistinguishability, we measure transverse-mode-resolved JSIs as in Sec. 3. To identify the FWM processes, similar to Fig. 3, we conduct a series of measurements with 5 different pump-seed transverse-mode combinations ($e$-$e$, $o$-$o$, $d$-$e$, $d$-$o$, and $d$-$a$). Figure 5(a) shows the measured JSI with the pump in $\ket{d}$ and the seed in $\ket{a}$, where we have labeled the spectral peaks with the associated transverse modes $(T_{p_{1}},T_{p_{2}},T_{s},T_{i})$ and FWM processes as before. The solid curves in Fig. 5(a) again represent the $1/e^{2}$ contours of the two-dimensional Gaussian curve fittings (goodness of fit $R^{2}\approx 0.9$). The B and C contours exhibit some spectral overlap, promising some transverse-mode entanglement at the intersection. Compared to Fig. 3(a-c), in Fig. 5(a), the four processes A, B, C, and D are positioned much closer despite considering the fiber length change effect – A and D are located fully inside B and C. Here, we attribute this deviation to the difference in fiber parameters. Even for the same type of fiber, HB800C, the fiber parameters can vary spool to spool, which may result in different FWM peak positions. Here, given that the fibers used for Figs. 3(a-c) and  5(a) are from different spools, we identify that this shorter PMF likely has a smaller parity birefringence $\Delta^{p}\sim 1\times 10^{-4}$ compared to that used in Sec. 4.1.

Before estimating the signal-idler quantum state with our stimulated emission approach, for reference, we measure the state using a conventional transverse-mode quantum state tomography (QST) [15]. Specifically, we project the signal-idler transverse-mode states into six mutually unbiased measurement basis states ($e,o,d,a,r,l$) and conduct 36 coincidence measurements ($ee,eo,...,lr,ll$) [15, 38, 57, 58, 59, 55] by installing an additional SLM, single-mode fibers, single-photon detectors, and a coincidence counter in the detection part of the setup in Fig. 2 (see Supplement 1 for more details). Figure 5(b) shows the measured density matrix $\rho_{QST}$ with partial transverse-mode entanglement quantified by concurrence $=0.27\pm 0.03$. It has a fidelity, or closeness, to the target Bell state of $0.48\pm 0.02$ and a purity of $0.52\pm 0.01$. As with typical quantum state tomography results, we can only roughly ascribe the low concurrence, fidelity, and purity to low $\ket{ee}\bra{oo}$ and $\ket{oo}\bra{ee}$ off-diagonal cross terms and non-zero $\ket{oe}\bra{oe}$ and $\ket{eo}\bra{eo}$ on-diagonal components each describing the coherence and purity of $\ket{ee}$ and $\ket{oo}$ states, respectively. We can hypothesize the origins of such contributions, but it will be challenging to trace and verify them experimentally without conducting additional measurements. The errors presented here are computed from $10^{2}$ randomly sampled density matrices assuming Poissonian noise in the coincidence counts for QST.

To compare with the QST, we estimate the transverse-mode density matrix $\rho_{SE}$ from the transverse-mode-resolved JSI, which can provide additional information about the sources of low concurrence, fidelity and purity. We start the estimation procedure by characterizing the transverse-mode density matrix $\rho_{tot}(\lambda_{s},\lambda_{i})$ ($\rho_{tot}(\omega_{s},\omega_{i})$) in the signal-idler wavelength (frequency) space. Using spectral decomposition [60], the full density matrix at given signal and idler wavelengths can be represented as a linear combination of density matrices $\rho_{j}$ as $\rho_{tot}(\lambda_{s},\lambda_{i})=\sum_{j}m_{j}\rho_{j}(\lambda_{s},\lambda_{i})$, where each $\rho_{j}=\ket{\psi_{j}}\bra{\psi_{j}}$ describes a pure quantum state $\ket{\psi_{j}}$ of a FWM process $j$ satisfying the normalization condition $\sum_{j}m_{j}=1$ with $m_{j}\geq 0$. For example, $\rho_{B}$, $\rho_{C}$, and $\rho_{B\cap C}$ each represents a pure signal-idler state within the boundary of the process B, C, and the intersection of B and C, respectively, where $\ket{\psi_{B}}=C_{B}\otimes\ket{oo}_{B}$, $\ket{\psi_{C}}=C_{C}\otimes\ket{ee}_{C}$, and $\ket{\psi_{B\cap C}}=C_{C}\otimes\ket{ee}_{C}+C_{B}\otimes\ket{oo}_{B}$. In general, if $N$-FWM processes exist, there will be $2^{N}-1$ binary combinations of $\rho_{j}$’s that specify whether a given signal-idler wavelength coordinate lies inside or outside of a certain FWM process. Representing each FWM process with a two-dimensional Gaussian fitting function as shown in Fig. 5(a) simplifies the density matrix calculation at a given signal-idler wavelength domain and thus the estimation procedure. Then, we integrate all these point-wise density matrices in the given signal-idler spectral range experimentally defined by interference filters, thereby producing a single transverse-mode density matrix we name $\rho_{SE}$. In other words, $\rho_{SE}$ is a reduced density matrix $\rho_{tot}^{T}$ in the transverse-mode domain, where the spectral DOF is traced out: $\rho_{SE}=\rho_{tot}^{T}=tr_{\lambda}(\rho_{tot}(\lambda_{s},\lambda_{i}))=\int d\lambda_{s}\,d\lambda_{i}\,\rho_{tot}(\lambda_{s},\lambda_{i})$.

Figure 5(c) shows the stimulated-emission estimated density matrix $\rho_{SE}$ using this calculation accounting for the interference filters used in the QST measurement (the full range shown in Fig. 5(a); $\lambda_{i}=[567.5,574.5]\mathrm{\ nm},\lambda_{s}=[673.0,681.0]\mathrm{\ nm}$). $\rho_{SE}$ exhibits concurrence $=0.00$, Bell fidelity $=0.48$, and purity $=0.40$. Except for the relative amplitude of $\ket{ee}\bra{ee}$ and $\ket{eo}\bra{eo}$ elements, overall, $\rho_{SE}$ shows a similar trend as the quantum state tomography result, $\rho_{QST}$ – high $\ket{ee}\bra{ee}$ and $\ket{oo}\bra{oo}$, non-zero $\ket{ee}\bra{oo}$ and $\ket{oo}\bra{ee}$ coherent interaction elements, and other residual elements. Quantitatively, the fidelity $F$ that describes the degree of similarity between the QST and stimulated-emission estimated states is $F(\rho_{QST},\rho_{SE})=0.73$. $\rho_{SE}$ provides a closer estimation if the phase information can be ignored, i.e., $F(|\rho_{QST}|,\rho_{SE})=0.85$. This is related to the current phase measurement limitation of our method as shall be discussed later in Sec. 4.3.

Despite the remaining discrepancies, $\rho_{SE}$ can still provide sufficient information to deduce potential factors leading to low transverse-mode entanglement, namely, the presence of spectral distinguishabilities among the FWM processes. Within the given spectral window in Fig. 5(a), imperfect spectral overlap between the processes B and C can be observed, as well as the presence of other processes A and D. Based on the previous discussions, we can realize that in the transverse-mode basis, spectral overlap between the two processes gives coherence (off-diagonal components in the density matrix), whereas spectral separation gives incoherence (no off-diagonals, leading to a mixed state). Therefore, between $\ket{oo}$ (B) and $\ket{ee}$ (C), we can logically predict that the density matrix will have slight off-diagonal coherence from the B-C overlap in the JSI, as well as the mostly on-diagonal incoherence from the remaining spectrally non-overlapping regions. Similarly, examining the JSI plot in Fig. 5(a), we can infer that while $\ket{eo}$ (D) and $\ket{oe}$ (A) will not have off-diagonal elements, they will have non-zero off-diagonal values with $\ket{oo}$ (B) and $\ket{ee}$ (C), respectively (see Fig. 5(c)). Ultimately, all these factors contribute to low transverse-mode entanglement. As such, the stimulated-emission method can help probe spectral distinguishabilities that are challenging to assess solely based on the transverse-mode QST result.

Considering the spectral origin, we may now think about tailored strategies to optimize the source for higher transverse-mode entanglement, that is, lowering the spectral indistinguishability. In principle, we can do so by choosing a narrower spectral window that focuses on the B-C intersection area at the cost of reduced counts. Since we can choose an arbitrary spectral window when calculating $\rho_{SE}$, we can easily simulate to determine the optimal filtering strategy. For example, making a $1.5\mathrm{\ cm}\times 2$ fiber source out of the Sec. 4.1 fiber and using a square-shaped 1 nm-wide spectral window centered at B-C intersection can produce a photon-pair state with concurrence = 0.82, Bell fidelity = 0.91, and purity = 0.84. More fundamentally, we may engineer the phase-matching  [61] to increase the B-C overlap while decreasing the A and D contributions.

The aforementioned difference in quantitative measures (concurrence, Bell fidelity, and purity) between $\rho_{QST}$ and $\rho_{SE}$ can be explained by the following assumptions made in the calculation, which may improve with adequate treatments. First, we assume a Gaussian phase-matching function instead of the more accurate sinc (degenerate pumps) and complex error functions (non-degenerate pumps) described in Sec. 2.3. A Gaussian function can underestimate the spectral overlap that may originate from the tails of the sinc and non-degenerate pump functions [44], reducing the overall entanglement. As a resolution, we may fit each FWM process with an accurate phase-matching function model accounting for the pump degeneracy (B, C: degenerate, A, D: non-degenerate). Alternatively, we may directly use a non-fitted raw JSI data that could be measured by exciting only one FWM process at a time using a precise transverse-mode control. Second, we assume a flat joint spectral phase (JSP) for all the FWM processes involved. This lack of phase information can explain why the imaginary part of the $\rho_{SE}$ is zero even though that of the $\rho_{QST}$ is not. This can be addressed by measuring the JSP experimentally as in [28, 1, 62], which will determine the phase of a corresponding FWM process through $c_{j}\propto f_{j}\propto e^{i(JSP)}$ (see Eq. 3 and Supplement 1 for more details). Third, we assume the distinguishabilities that can undermine the transverse-mode entanglement do not exist outside the spatial and spectral DOFs. Although the linear polarizers and a cross-spliced fiber are employed to compensate any residual polarization and temporal distinguishabilities in the system (the second PMF in the cross-spliced PMF corrects for polarization- and transverse-mode parity-dependent temporal walk-offs introduced in the first PMF [56]), there is a chance that some unaccounted distinguishabilities still remain. As a remedy, extending the technique to other DOFs, e.g., polarization and time-resolved characterization, may be helpful [14, 63]. Alternatively, we may also consider a full frequency-resolved transverse-mode stimulated emission tomography, whose projection measurements naturally accounts for all possible distinguishabilities. This may produce better estimation, albeit will lack the information on the source of distinguishability if it exists outside the spatial and spectral domains. Ultimately, all the resolutions presented here can lead to potential improvements in the numerical model used in previous works [10, 11, 12] to better predict the quantum state.