## Abstract

Using a Sagnac fiber loop functions as a deterministic splitter of photon pairs produced by the frequency degenerate four wave mixing, we show that the background noise of the degenerate photon pairs is contributed by both Raman scattering and frequency non-degenerate four wave mixing. To improve the purity of photon pairs in the high gain regime, in addition to suppressing the noise photons by cooling the nonlinear fiber and by optimizing the detuning between the frequencies of the pump and photon pairs, the walk-off effect of the two pulsed pump fields should be mitigated by managing the dispersion of the fiber. Our investigation is not only the first step towards the generation of multi-mode squeezed vacuum in fiber optical parametric amplifiers pumped with pulsed lights, but also contributes to improving the purity of the fiber sources of degenerate photon pairs.

© 2014 Optical Society of America

Multi-mode squeezers are essential resources in quantum information processing and quantum metrology [1, 2]. The squeezed state generated from the frequency degenerate parametric amplifiers pumped with pulsed lasers usually exhibit multi-mode characteristics in the spectral degree of freedom [3]. Besides much effort on studying the generation and characterization of the multi-mode squeezers from both the theoretical and experimental aspects [3–5], their potential application on lowing the timing noise and improving the accuracy of space-time positioning has recently been demonstrated [6]. To make the multi-mode squeezers really useful, the experimental setups, which are still fraught with complexity, should be more compact.

It is well known that the fiber based light sources have the potential of being more compact and practical because of the waveguide structure and obvious advantage—free of alignment. Moreover, similar to the *χ*^{(2)} crystal based parametric processes [1, 7], the four wave mixing (FWM) parametric process occurred in *χ*^{(3)} optical fiber also can be used to generate the nonclassical lights [8, 9]. In general, classified by the degenerate or non-degenerate in frequency, there are two kinds of co-polarized four-wave mixing in fibers. One is the frequency non-degenerate four wave mixing (NDFWM), in which two pump photons at frequencies *ω _{p}*

_{1}and

*ω*

_{p}_{2}scatter through the

*χ*

^{(3)}nonlinearity to create the signal and idler photon pairs at frequencies

*ω*and

_{s}*ω*, respectively, such that

_{i}*ω*

_{p}_{1}+

*ω*

_{p}_{2}=

*ω*+

_{s}*ω*and

_{i}*ω*≠

_{s}*ω*. The other is the frequency degenerate FWM (DFWM), in which two pump photons at the different frequencies nonlinearity to create a pair of identical photons at the

_{i}*ω*

_{p}_{1}and

*ω*

_{p}_{2}scatter through the

*χ*

^{(3)}mean frequency

*ω*, such that

_{si}*ω*

_{p}_{1}+

*ω*

_{p}_{2}=

*ω*+

_{s}*ω*and

_{i}*ω*=

_{s}*ω*=

_{i}*ω*. In the former case, the phase matching condition

_{si}*k*

_{p}_{1}+

*k*

_{p}_{2}=

*k*+

_{s}*k*with

_{i}*k*

_{p}_{1(2)}and

*k*

_{s}_{(}

_{i}_{)}respectively denoting the wave-vectors at

*ω*

_{p}_{1(2)}and

*ω*

_{s}_{(}

_{i}_{)}should be satisfied, while for the latter case, the phase matching condition

*k*

_{p}_{1}+

*k*

_{p}_{2}= 2

*k*

_{s}_{(}

_{i}_{)}should be satisfied.

In the past ten years, using the pulse pumped NDFWM, various kinds of non-classical lights in the domain of both discrete and continuous variables, have been realized, and the factors influencing the purity of photon pairs and the noise fluctuation of squeezed state have been extensively investigated [8–13]. On the other hand, with the usage of the pulse pumped DFWM, only a few experiments demonstrate the generation of frequency degenerate photon pairs in the domain of discrete variables. Right after Fan et al. proposed and experimentally demonstrated the frequency degenerate photon pairs by using the DFWM in a piece of straight fiber [14], Chen et al. demonstrated the spatial separation of this kind of photon pairs by using a Sagnac fiber loop (SFL) [15] and presented its application in realizing the quantum controlled-NOT gate [16]. So far, squeezed states have not been experimentally realized via DFWM yet. Moreover, in contrast to the NDFWM, the influence of the other nonlinear effects in fibers on the purity of the frequency degenerate photon pairs has not been investigated. In this paper, by varying the gain of DFWM and by varying the detuning between the frequencies of pump and photon pairs in a SFL, we characterize how the purity of the degenerate photon pairs is affected by Raman scattering (RS) and NDFWM. The investigation is not only the first step towards the generation of squeezed vacuum via DFWM, but also contributes to improving the purity of the fiber sources of degenerate photon pairs.

Our experimental setup is shown in Fig. 1. The SFL consists of 300-m-long dispersion shifted fiber (DSF), two pieces of standard single mode fibers (SMFs), SMF1 and SMF2, a fiber polarization controller (FPC), and a 50/50 fiber coupler (FC1). The DSF with nonlinear coefficient of about 2 (W·km)^{−1} is cooled in liquid nitrogen (77K) to suppress the RS serves as the nonlinear medium of DFWM, in which two pump photons at the wavelengths *λ _{p}*

_{1}and

*λ*

_{p}_{2}are scattered into a pair of co-polarized signal and idler photons having the identical mean wavelength

*λ*= 1544.5 nm, which is the same as the experimentally measured zero dispersion wavelength of the DSF. SMF1 and SMF2 function as the dispersive elements of photon pairs. Each linearly polarized pump field,

_{si}*E*(i=1,2), injected into the SFL is split into two pump waves traversing in a counter-propagating manner by FC1, and the two pump fields in the clockwise (CW) and counter-clockwise (CCW) directions respectively produce degenerate photon pairs, |2〉

_{pi}*and |2〉*

_{c}*, where the footnotes*

_{d}*c*and

*d*denote the propagation direction. The SFL is then preceded by a circulator (Cir), which redirects the SFL reflected photons to a separate spatial mode, and the two output modes of SFL are labeled as ”

*a*” and ”

*b*”.

Ideally, the output state of the SFL is written as [17]

*ψ*

_{2}〉 = |1〉

*|1〉*

_{a}*, and the footnotes*

_{b}*a*and

*b*denote the spatial modes of the photons coming out of the SFL. Since the degenerate signal and idler photon pairs inherit the phase of the two pump photons, in Eq. (1), we have

*ϕ*=

*ϕ*

_{p}_{1}+

*ϕ*

_{p}_{2}+

*ϕ*with

_{d}*ϕ*= (

_{d}*k′*

_{p}_{1}+

*k′*

_{p}_{2}− 2

*k′*)

_{si}*L*

_{2}− (

*k*

_{p}_{1}+

*k*

_{p}_{2}− 2

*k*)

_{si}*L*

_{1}denoting the dispersion induced phase difference, where

*ϕ*(

_{pi}*i*= 1, 2) is the phase difference of the pump

*E*(

_{pi}*i*= 1, 2) propagating in the CW and CCW directions,

*L*

_{1}(

*L*

_{2}) is the length of SMF1 (SMF2),

*k*(

_{pi}*k′*) and

_{pi}*k′*(

_{si}*k*) are the wave-vectors of the fields

_{si}*E*and scattered photons at

_{pi}*λ*in SMF1 (SMF2), respectively. Equation (1) shows that we can obtain the state |

_{si}*ψ*〉 = |

*ψ*

_{1}〉 for

*ϕ*= 2

*nπ*, and |

*ψ*〉 = |

*ψ*

_{2}〉 for

*ϕ*= (2

*n*+ 1)

*π*, where

*n*is an integer. In the former case, both the signal and idler photons of a pair come out in the same spatial mode, they are both simultaneously in either mode

*a*or mode

*b*; in the latter case, the signal and idler photons of a pair have different spatial modes, i.e., if one photon emerges in mode

*a*then the other one is in mode

*b*, or vice versa. Moreover, to ensure the deterministic separation of photon pairs is immune to the detuning between the central frequencies of the photon pairs and pump, the dispersion induced phase difference

*ϕ*should equal to 0 [17]. We achieve this by properly selecting the dispersion elements SMF1 and SMF2 to ensure their dispersion properties and lengths are the same.

_{d}We create the pulsed pump fields *E*_{p1} and *E*_{p2}, which are respectively centering at the wavelengths *λ*_{p1} and *λ*_{p2}, by taking the 40 MHz train of 150-fs pulses centering at 1550 nm from a mode-locked fiber laser with bandwidth of about 60 nm, dispersing them with the gratings, *G*_{1}, *G*_{2} and *G*_{3}, and then spectrally filtering them to obtain two synchronous beams with a tunable wavelength separation of about 10–30 nm. To achieve the required power and to avoid the cross-talk between the two pumps, we first send the two pump pulses with a time delay of about 10 ns into FC2 from its two input ports, then feed one output port of FC2 into the erbium-doped fiber amplifier (EDFA). Photons at the wavelength of *λ _{si}* = 1544.5 nm from the laser that leak through the spectral-dispersion optics and from the amplified spontaneous emission of the EDFA are suppressed by passing the amplified pump pulses through a dual-band filter F1, whose full-width-half-maximum (FWHM) in each band is about 0.6 nm. F1 is a double grating filter [8], hence, both the path delays and central wavelengths of the pump fields

*E*

_{p1}and

*E*

_{p2}are adjustable. The gain of DFWM in DSF is maximized by matching the optical paths traversed by the two co-polarized input pump pulses.

To reliably detect the photon pairs, an isolation to pump in excess of 100 dB is required because the efficiency of the scattered photons at *λ _{si}* = 1544.5 nm is low. To achieve this, we send the photons emerging in the spatial modes

*a*and

*b*through the filters F2 and F3, respectively. F2 and F3, providing a more than 100 dB rejection to pump, are realized by cascading two WDM filters (Santec, WDM-15), whose central wavelength and FWHM are about 1544.5 nm and 1.1 nm, respectively. To suppress the cross-polarized RS, F2 (F3) is then followed by FPC3 (FPC4) and PBS1 (PBS2) [10]. In mode

*b*, the co-polarized photons passing through F3 are measured by SPD3; while in mode

*a*, co-polarized photons passing through F2 are split by FC4, and the two outputs of FC4 are respectively detected by SPD1 and SPD2. All the three SPDs are operated in the gated Geiger mode. The 2.5 ns gate pulses arrive at a rate of about 2.58 MHz, which is 1/16 of the repetition rate of the pump pulses, and the dead time of the gate is set to 10

*μ*s.

For the output state of SFL (see Eq. (1)), the counting probability of the individual SPDs in either the spatial mode *a* or *b* is constant, while the two-fold coincidence counting rates of photon pairs respectively in the same and different spatial modes *C ^{s}* (SPD1 and SPD2) and

*C*(SPD2 and SPD3) are given by [18, 19]

^{d}*N*(

_{Di}*i*= 1, 2, 3) refers to the individual counting rates of SPDi,

*η*

_{1}and

*η*

_{3}are the total detection efficiency in the channels of SPD1 and SPD3, respectively, and

*ξ*determined by the bandwidths of pump and photon pairs is the collection efficiency of photon pairs [19]. The term ${g}^{(2)}=1+\frac{1}{\sqrt{1+\frac{{\sigma}_{\mathit{si}}^{2}}{2{\sigma}_{p}^{2}}}}$ in Eq. (2) is the intensity correlation of the field centering at

_{si}*λ*, where the spectra of both pump and photon pairs are assumed to be Gaussian shaped with

_{si}*σ*and

_{p}*σ*respectively denoting the FWHM of pump photons and photon pairs. Usually, the term

_{si}*N*

_{D}_{2}

*η*

_{3}

*ξ*in Eq. (3), originated from the quantum correlation of photon pairs, is called the true coincidences. Defining the ratio between the true coincidences and accidental-coincidences as ${\mathit{CAR}}^{(t)}=\frac{{\eta}_{3}{\xi}_{\mathit{si}}}{{N}_{D3}}$, we then obtain the expressions of the coincidences to accidental-coincidences ratio (CAR) of photon pairs in the same and different spatial modes:

_{si}*ψ*〉 = |

*ψ*

_{2}〉 (

*ϕ*=

*π*), the ratio

*CAR*

^{(d)}is maximized, while the ratio

*CAR*

^{(s)}is minimized. Additionally, we note that in contrast to the case of the frequency non-degenerate photon pairs [17, 20], the minimized value of

*C*in Eq. (2) does not equal to the accidental coincidence rate due to the bunching effect [21].

^{s}In fact, when the two pulsed pumps are simultaneously launched into the SFL, the photons centering at the wavelength 1544.5 nm are scattered from the five kinds of nonlinear processes taking place in DSF in both the CW and CCW directions. In addition to DFWM originated from the combination of two pump fields *E*_{p1} and *E*_{p2}, there are also RS and NDFWM originated from the individual pumps *E*_{p1} and *E*_{p2}, respectively. Therefore, to improve the purity of the degenerated photon pairs, it is necessary to study how to enhance the desired DFWM and to suppress the other deleterious nonlinear processes.

During the experiment, the SFL is properly set so that the output state is |*ψ*〉 = |*ψ*_{2}〉 = |1〉* _{a}* |1〉

*. To realize the required phase difference*

_{b}*ϕ*=

*ϕ*

_{p}_{1}+

*ϕ*

_{p}_{2}=

*π*and mode matching, we first adjust FPC2 to ensure the SFL functions as a 50/50 power splitter of the pump fields

*E*

_{p1}and

*E*

_{p2}, then adjust FPC1 to guarantee the two-fold coincidence rate of SPD1 and SPD2 is minimized [20]. After recording the coincidence and accidental-coincidence rates for the detected photons originated from the same and adjacent pulses, we find the minimized value of

*CAR*

^{(s)}is about 1.8. The measured value of

*CAR*

^{(s)}is slightly higher than that calculated by using Eq. (2). This is because the spectra of the filters F2 and F3 are super-Gaussian shaped, which are different from the assumptions used to deduce the expression of

*g*

^{(2)}in Eq. (2). Hence, the result implies that the coincidence rate of the photons emerging from the two output ports of FC4 is originated from the photon bunching effect but not the quantum correlation of photon pairs [18]. In this situation, the simultaneously created signal and idler photons, having identical mode [22], are deterministically separated into the different spatial modes of SFL.

We start with characterizing the influence of RS and NDFWM upon the purity of degenerate photon pairs. In the experiment, the central wavelengths of the pump fields *E*_{p1} and *E*_{p2} are *λ*_{p1} = 1539.5 nm and *λ*_{p2} = 1549.5 nm, respectively. We record the counting rate of SPD3 as a function of the pump power for three cases: only the individual pump field of (i) *E*_{p1} and (ii) *E*_{p2} is launched into SFL, and (iii) both *E*_{p1} and *E*_{p2} with equal power are launched into SFL. For cases (i) ((ii)), we fit the measured data with the second order polynomial function
${N}_{s(a)}={s}_{1}^{s(a)}{P}_{1(2)}+{s}_{2}^{s(a)}{P}_{1(2)}^{2}$, as shown in Fig. 2(a) (Fig. 2(b)), where *P*_{1(2)} is the average power of the field *E*_{p1} (*E*_{p2}), and the fitting coefficients of the linear and quadratic terms,
${s}_{1}^{s(a)}$ and
${s}_{2}^{s(a)}$ are respectively determined by the strengths of RS and NDFWM of the pump field *E*_{p1} (*E*_{p2}) [8, 23]. From Figs. 2(a) and 2(b), one clearly sees that photons originated from the RS and NDFWM of the individual pump of *E*_{p1} and *E*_{p2} also contribute to the detected photons. For the RS at 1544.5 nm, the Bose population factor of optical phonon for *E*_{p2} is less than that for *E*_{p1} [23]. Therefore, it is straightforward to understand that the intensity of the detected photons contributed by RS in Fig. 2(a) is greater than that in Fig. 2(b) for the individual pump field with a given power. On the other hand, the phase matching condition of NDFWM should not be satisfied for the pump having the central wavelength *λ*_{p1} = 1539.5 nm, which is in the normal dispersion regime of DSF. However, the intensity of the detected photons contributed by NDFWM of the field *E*_{p1} with a given power is unexpectedly greater than that of *E*_{p2}. We think this is because the dispersion property of the DSF is inhomogeneous and the measured value of zero dispersion wavelength ∼1544.5 nm is actually an averaged result of the 300-m-long DSF.

The inset of Fig. 2(c) shows the measured total counting rates *N _{t}* of SPD3 versus the power of two pump fields

*P*

_{1}+

*P*

_{2}for case (iii). The main plot of Fig. 2(c), obtained by subtracting the background noise

*N*and

_{s}*N*of individual pump fields from

_{a}*N*, shows the counting rate of detected photons originated from DFWM

_{t}*N*versus pump power. Moreover,

_{d}*N*fits the function

_{d}*N*=

_{d}*s*

_{2}(

*P*

_{1}+

*P*

_{2})

^{2}very well, where the fitting coefficient

*s*

_{2}is proportional to the gain of DFWM. Comparing Fig. 2(c) with Figs. 2(a) and 2(b), one sees that in the low (high) gain regime of DFWM, the background of photon pairs is mainly contributed by RS (NDFWM), because the noise photons originated from RS (NDFWM) are greater than that from NDFWM (RS).

To further characterize the influence of RS and NDFWM, we then repeat the measurement of Fig. 2 by adjusting G1, G2, G3 and F1 to vary the detuning Ω between the central frequencies of pump and photon pairs (see Fig. 1). Fig. 3(a) (Fig. 3(b)) shows the fitting coefficients of the RS and NDFWM originated from the individual pump field *E*_{p1} (*E*_{p2}) as a function of
$\mathrm{\Omega}=\frac{c({\lambda}_{\mathit{si}}-{\lambda}_{p1})}{{\lambda}_{\mathit{si}}^{2}}(\mathrm{\Omega}=\frac{c({\lambda}_{p2}-{\lambda}_{\mathit{si}})}{{\lambda}_{\mathit{si}}^{2}})$, where *c* is the speed of light in vacuum. As a comparison, the fitting coefficient proportional to the gain of DFWM *s*_{2} versus the detuning
$\mathrm{\Omega}=\frac{c({\lambda}_{p2}-{\lambda}_{p1})}{2{\lambda}_{\mathit{si}}^{2}}$ is shown in Fig. 3(c). One sees that the coefficients of RS in the Stokes band
${s}_{1}^{s}$ increases with detuning, while that in the anti-Stokes band
${s}_{1}^{a}$ increases with detuning for |*λ*_{p2} − *λ _{si}*| < 10 nm but will start to decrease slightly for |

*λ*

_{p2}−

*λ*| > 10 nm [23]. On the other hand, the fitting coefficients of NDFWM, ${s}_{2}^{s}$ and ${s}_{2}^{a}$, decrease rapidly with the increase of detuning; while the fitting coefficient

_{si}*s*

_{2}decreases gradually with the increase of detuning due to the walk-off effect of the two pulsed pump fields. Obviously,

*s*

_{2}in Fig. 3(c) is greater than ${s}_{2}^{s}$ and ${s}_{2}^{a}$ in Figs. 3(a) and 3(b), because the satisfaction of the phase matching condition for DFWM is better than that for NDFWM. However, it is worth noting that the gain coefficients of NDFWM in both Figs. 3(a) and 3(b) are still remarkable for the case of $\frac{{\lambda}_{p2}-{\lambda}_{p1}}{2}<5\hspace{0.17em}\text{nm}$ (Ω < 0.63 THz). Therefore, under the condition of high pump powers, the suppression of the background noise of degenerate photon pairs can not be ensured by decreasing the detuning and cooling the DSF, because the noise photons might be mainly contributed by NDFWM. This is in contrast to the case of frequency non-degenerate photon pairs in DSF [10, 11].

To quantitatively study the purity of the degenerate photon pairs, we define the ratio between the detected photons contributed by DFWM and the sum of RS and NDFWM, i.e.,

Under a given pump power, the higher the ratio*R*, the better the purity of the photon pairs is. By substituting the fitting functions of

*N*,

_{d}*N*and

_{s}*N*(see the caption of Fig. 2) into Eq. (6), we arrive at

_{a}*P*

_{1}=

*P*

_{2}, which is necessary for maximizing the gain of DFWM, Eq. (7) is rewritten as where the coefficients ${\zeta}_{R}=\frac{{s}_{1}^{s}+{s}_{1}^{a}}{2{s}_{2}({P}_{1}+{P}_{2})}$ and ${\zeta}_{F}=\frac{{s}_{2}^{s}+{s}_{2}^{a}}{4{s}_{2}}$ respectively denote the influence of RS and NDFWM on the purity of photon pairs. Clearly, the ratio

*R*depends on the pump power. When the power (

*P*

_{1}+

*P*

_{2}) is low (high), the ratio

*R*is mainly determined by the value of

*ζ*(

_{R}*ζ*): the smaller the value of

_{F}*ζ*or ${s}_{1}^{s}+{s}_{1}^{a}$ (

_{R}*ζ*or ${s}_{2}^{s}+{s}_{2}^{a}$), the higher the ratio

_{F}*R*is.

According to Eq. (6), we calculate the ratio *R* versus the detunings under different pump power levels *P*_{1} + *P*_{2} by using our experimentally measured results *N _{s}*,

*N*and

_{a}*N*, as shown in Fig. 4(a). For the low pump power of

_{d}*P*

_{1}+

*P*

_{2}= 0.05 mW, the value of

*R*is the highest for the small detuning of Ω = 0.63 THz ( $\frac{{\lambda}_{p2}-{\lambda}_{p1}}{2}=5\hspace{0.17em}\text{nm}$). In this case, although the gain of NDFWM is relatively high, the noise of photon pairs is still mainly contributed by RS. With the increase of pump power, one sees the detuning corresponds to the highest value

*R*increase. For example, for the pump power of

*P*

_{1}+

*P*

_{2}= 0.34 mW, the highest

*R*corresponds to Ω = 0.88 THz ( $\frac{{\lambda}_{p2}-{\lambda}_{p1}}{2}=7\hspace{0.17em}\text{nm}$). The results agree with the prediction of Eqs. (6)–(8). However, when the power

*P*

_{1}+

*P*

_{2}is increased to 0.55 mW, the highest

*R*still corresponds to Ω = 0.88 THz, and the value of

*R*starts to decrease with the increase of detuning. The results indicate that when the detuning is greater than a certain value, which is determined by the dispersion of DSF, the ratio

*R*will decrease with the increase of detuning because the walk-off effect of the two pump fields results in a decreased gain of DFWM (see Fig. 3(c)).

Finally, considering the purity of photon pairs is often characterized by the CAR of the photon pairs [10], we then measure the value of CAR by recording the coincidence rates of SPD1 and SPD3 for the photons produced by the same and adjacent pump pulses, respectively, when the pump power *P*_{1} + *P*_{2} is varied. The results in Fig. 4(b) are obtained for the detuning Ω of about 0.63 and 0.88 THz, respectively. One sees that for a given pump in the low power regime, although the pair production rate for detuning of 0.63 THz is slightly higher than that for detuning of 0.88 THz (see Fig. 3(c)), the value of CAR for the detuning of 0.63 THz is higher. With the increase of pump power, the difference between the values of CAR of the two kinds of detunings decreases for *P*_{1} + *P*_{2} less than 0.15 mW. However, when *P*_{1} + *P*_{2} is greater than 0.15 mW, for a given pump power, the value of CAR for the detuning of 0.88 THz is always higher because the noise photons contributed by NDFWM is less than that for detuning of 0.63 THz. The results are in consistent with Eqs. (6)–(8).

In conclusion, using the SFL functions as a deterministic splitter of the degenerate photon pairs, we have demonstrated that the generation of photon pairs via DFWM in DSF is inevitably accompanied by background noise originated from RS and NDFWM of the individual pump field *E*_{p1} and *E*_{p2}. When the total power of the two pump fields are low, the purity of photon pairs are mainly influenced by RS. However, when the total power of the two pump fields are high, which is the necessary condition for generating squeezed vacuum in DSF [24], the purity is also influenced by the NDFWM, which can not be suppressed by cooling the DSF. Our investigation shows that increasing the detuning helps to suppress the noise photons via the NDFWM with a narrow gain bandwidth, but often results in a decreased gain of DFWM due to walk-off effect of two pump fields. Therefore, to improve the purity of photon pairs or to reduce the noise fluctuation of squeezed vacuum, the optimization of the detuning in the high gain regime of DFWM should take the intensity of the photons contributed by both RS and NDFWM into account. Moreover, by properly managing the dispersion property of the DSF or by using DSF with a shorter length, the influence of walk-off effect upon the gain of DFWM can be mitigated.

## Acknowledgments

This work was supported in part by the State Key Development Program for Basic Research of China (Grant No. 2010CB923101), the National Natural Science Foundation of China (Grant No. 11074186), and the Innovation Fund of China Academy of Space Technology (Grant No. CAST201234).

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**18. **Equations (2) and (3) can be derived by utilizing the method used in Ref. [19] and using the Hamiltonian of the DFWM ${H}_{I}=\alpha {\chi}^{(3)}\int dV({E}_{p1}^{+}{E}_{p2}^{+}{\widehat{E}}_{\mathit{si}}^{-}{\widehat{E}}_{\mathit{si}}^{-}+H.c.)$, where α is the constant determined by experimental details. In the expression of Hamiltonian H_{I}, ${E}_{pj}^{+}\propto {e}^{-i\gamma {P}_{pj}z}\int d{\omega}_{pj}{e}^{-{({\omega}_{pj}-{\omega}_{p0j})}^{2}/2{\sigma}_{pj}^{2}}{e}^{i{k}_{pj}z-i{\omega}_{pj}t}$(j= 1, 2) denotes the strong pump pulse, where P_{pj}, ω_{p0j} and σ_{p0j} are the peak power, central frequency and bandwidth of the pump field ${E}_{pj}^{+}$, respectively; ${\widehat{E}}_{\mathit{si}}^{-}=\int d{\omega}_{\mathit{si}}\sqrt{\frac{\overline{h}{\omega}_{\mathit{si}}}{2{\epsilon}_{0}{V}_{Q}}}\frac{{\widehat{a}}^{+}({\omega}_{\mathit{si}})}{n({\omega}_{\mathit{si}})}{e}^{-i({k}_{\mathit{si}}z-{\omega}_{\mathit{si}}t)}$represents the quantized electromagnetic signal (idler) field expanded in multi-mode, where ε_{0}, V_{Q} and n(ω_{si}) are the vacuum permittivity, the quantization volume and the refractive index of the fiber, respectively, and â^{+}(ω_{si}) is the creation operator of the field at frequency ω_{si}.

**19. **L. Yang, X. Ma, X. Guo, L. Cui, and X. Li, “Characterization of a fiber-based source of heralded single photons,” Phys. Rev. A **83**, 053843 (2011). [CrossRef]

**20. **X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu, “All fiber source of frequency-entangled photon pairs,” Phys. Rev. A **79**, 033817 (2009). [CrossRef]

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**22. **X. Ma, X. Li, L. Cui, X. Guo, and L. Yang, “Effect of chromatic-dispersion-induced chirp on the temporal coherence properties of individual beams from spontaneous four-wave mixing,” Phys. Rev. A **84**, 023829 (2011). [CrossRef]

**23. **X. Li, P. Voss, J. Chen, K. Lee, and P. Kumar, “Measurement of co- and cross-polarized raman spectra in silica fiber for small detunings,” Opt. Express **13**, 2236–2244 (2005). [CrossRef] [PubMed]

**24. **C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron. **18**, 958–969 (2012). [CrossRef]