ized JSF with a filter of proper center wavelength and bandwidth.  The results indicate that by changing the pump bandwidth and the length of SMF, the wavelength and bandwidth of pure state single photon with $h_sh_i\rightarrow 1$ can be flexibly adjusted within the gain bandwidth of FWM in DSF.


Good separation between the islands is desirable in order to filter out each island \cite{Su19,su-arxiv}. However, the relative separation in Fig. \ref{fig_NLI_photon pair}(c,d,e) do not change with $\Delta \lambda_p$ and $L_{DM}$ but will increase with the stage number $N$, as shown in Ref.[\onlinecite{Su19,su-arxiv}]. Here we see well-separated islands even with $N=3$, which is already better than the case of $N=2$\cite{Su19}.
Note that U’Ren {\it et al.}\cite{uRen2005} had previously proposed the idea of generating pure state single photon by using a sequential array of $\chi^{(2)}$-crystal gapped in between by linear dispersive media. However, since the amount of dispersion induced by each linear spacer and $\chi^{(2)}$ crystal is comparable, the approximation in Eq. (2) is not valid so that it requires a large number of $\chi^{(2)}$-crystals (large $N$ in Eq. (1)) to separate different islands in JSF similar to Fig. \ref{fig_NLI_photon pair}(c,d,e). Because of the sophisticated structure, the proposal in Ref. [\onlinecite{uRen2005}] has not been experimentally realized yet.

\begin{figure*}[htbp]
	\centering
	\includegraphics[width=0.92\textwidth]{Fig2_scheme_HOM-01.jpg}
	\caption{(a) Experimental setup of two-photon HOM interference by using two heralded single photon sources (HSPS1 and HSPS2). The shaded areas are HSPS1 and HSPS2, whose detail is given in Fig. 1(a). P1-P2, pulsed pump; NLI, nonlinear interferometer; F, filter; DBF, dual-band filter; FPC, fiber polarization controller; BS, 50/50 beam splitter; SPD, single photon detector.
	(b) Contour map of true coincidence of photon pairs as function of $\lambda_s$ and $\lambda_i$, reflecting the joint spectral intensity of photon pairs.
	}
	\label{fig_setup}
\end{figure*}

The schematic of the experimental setup is shown in Fig. 2(a) where the gray shaded frames are the heralded single photon sources (HSPS1, HSPS2). The details of each HSPS is given in Fig. 1(a). The parameters of each HSPS are the same as those in plotting Fig. 1(c). The transmission loss of NLI is mostly due to imperfect fiber splicing and is about 17$\%$. The NLI is submerged in liquid nitrogen to suppress Raman scattering  \cite{XYLi04}. The pump P$_1$ or P$_2$ is obtained by passing the output of a femoto-second fiber laser through a bandpass filter (F1) and 50/50 beam splitter (BS1). The repetition rate of laser is about 36.8 MHz. The pulse duration of the pump with FWHM of 1 nm is about 4 ps. Quarter ($\lambda/4$) and half wave plates ($\lambda/2$) placed in front of the polarization beam splitter (PBS1) are used to select the signal and idler photon pairs co-polarized with the pump and to reject the Raman-scattering (RS) cross-polarized with the pump \cite{XYLi04}.

Because the fiber nonlinearity is weak, only about 0.04 photon pair is produced within a typical 4 ps duration pump pulse containing $10^{7}$ photons. Thus, to reliably detect the correlated photon pairs, a high pump-to-signal rejection ratio is required. For HSPS1, we achieve this by passing the output of NLI through a dual-band filter (DBF1), which can separate the photon pairs from the pump with an isolation greater than $\sim 110$ dB. The DBF is realized by cascading a tunable filter and a programmable optical filter (POF, model: Finisar Waveshaper 4000S), and the spectrum of DBF in each band is rectangularly shaped. The bandwidth of pass band in both signal and idler fields can be adjusted from 0.1 nm to 1.5 nm, and the central wavelength of each band can be tuned from 1530 to 1570 nm.  The signal and idler photons passing through DBF1 are detected by single photon detectors, SPD1 and SPD2, which are operated at gated Geiger mode at a rate that is the same as the repetition rate of pump, and the dead time of the gate is set to be 10 $\mu$s.
The detection efficiency of each SPD is $\sim15\%$.  The detection events of a photon in signal (idler) band heralds the presence of a single photon in idler (signal) band with the probability of $h_{i(s)}$.






We first verify the spectral property of photon pairs generated by the three-stage NLI. In the experiment, the pump power of P$_1$ is 50 $\mu$W, the bandwidth of DBF in both signal and idler bands is 0.1 nm. The two outputs of DBF1 (idler1 and signal1 fields) are directly coupled into SPD1 and SPD2, respectively. We measure both the two-fold coincidence counts of signal and idler photons originated from the same pump pulse and adjacent pulses, $C^{c}$ and $C^{acc}$, when the central wavelengths of DBF in signal (idler) channel are scanned from 1552.5 (1545) to 1558 (1539.5) nm with a step of 0.1 nm.
Then, we deduce the true coincidence counts of photon pairs $C^{T}$ by subtracting the measured $C^{acc}$ from $C^{c}$. Fig. 2(b) plots the contour map of $C^{T}$ in the wavelength coordinates of $\lambda_s$ and $\lambda_i$, which reflects the JSI of  photon pairs. One sees that the contour map exhibits “islands” pattern and agrees well with the theoretical prediction in Fig. 1(c).

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.48\textwidth]{Fig3_single_coin2.pdf}
	\caption{(a) and (b) are the counting rates of SPD1 for individual idler band versus the average pump power $P_{ave}$ for the three-stage NLI and non-NLI (single piece 450-m long DSF) cases, respectively. The second-order polynomials  $N_{i}=s_1 P_{ave} + s_2 P_{ave}^2$ (solid curves) are used to fit the data in plots (a) and (b). The linear and quadratic terms of the fitting functions are represented by the dashed and dotted lines, respectively. (c) The true coincidence rate versus the counting rate $N'_i=s_2 P_{ave}^2$ of idler photons originated from FWM in two cases.}
	\label{fig-single_coin}
\end{figure}


We then measure the achievable heralding efficiency of the HSPS1. Note that in Fig. 2(b), the shape of island centering at 1543.8 and 1553.7 nm is round, indicating the spectral factorable property. By properly adjusting the DBF1, we select out this round shaped island to efficiently obtain uncorrelated photon pairs. In this experiment, the bandwidth of each band is adjusted to be 1.5 nm (see the dashed lines in Fig. 2(b)), and the overall detection efficiencies (including the efficiencies of SPD and DBF1) in signal and idler bands are $\eta_s = 4.1\%$ and $\eta_i = 4.3\%$, respectively.
During the measurement, we record the single counts of individual signal and idler fields and the two-fold coincidence counts of two fields under different pump power. For each pump power level, we subtract $C^{acc}$ to obtain the true coincidence counting rate $C^{T}=C^{c}-C^{acc}$. The measured data of single counts in signal (idler) band is fitted with $N_{s(i)}=s_1 P_{ave} + s_2 P_{ave}^2$, where $P_{ave}$ is the average pump power, $s_1$ and $s_2$ are the fitting parameters. $s_1 P_{ave}$ and $s_2 P_{ave}^2$ respectively correspond to the intensities of spontaneous Raman scattering and FWM in NLI. The results are shown in Fig. \ref{fig-single_coin}(a) where the dashed and  dotted lines are $s_1 P_{ave}$ and $s_2 P_{ave}^2$, respectively.
We then deduce the collection efficiency of signal (idler) photons according to $h_{s(i)}= \frac{C^{T}}{\eta_{i(s)}N'_{i(s)}}$, where $N'_{s(i)}=s_2 P_{ave}^2$