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_p1 and ω_p2 scatter through the χ⁽³⁾ nonlinearity to create the signal and idler photon pairs at frequencies ω_s and ω_i, respectively, such that ω_p1 + ω_p2 = ω_s + ω_i and ω_s ≠ ω_i. 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 ω_p1 and ω_p2 scatter through the χ⁽³⁾ mean frequency ω_si, such that ω_p1 + ω_p2 = ω_s + ω_i and ω_s = ω_i = ω_si. In the former case, the phase matching condition k_p1 + k_p2 = k_s + k_i with k_p1(2) and k_s(i) respectively denoting the wave-vectors at ω_p1(2) and ω_s(i) should be satisfied, while for the latter case, the phase matching condition k_p1 + k_p2 = 2k_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_p1 and λ_p2 are scattered into a pair of co-polarized signal and idler photons having the identical mean wavelength λ_si = 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, E_pi (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〉_c and |2〉_d, where the footnotes 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”.

 

Fig. 1 Experimental setup. DSF, dispersion shifted fiber; SMF, single mode fiber; Cir, circulator; G, grating; FPC, fiber polarization controller; F, filters; FC, 50/50 fiber coupler; PBS, polarization beam splitter; SPD, singe photon detector. Inset, spectra of two pumps.

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

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, G₁, G₂ and G₃, 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 μs.

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^d (SPD2 and SPD3) are given by [18, 19]

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〉 = |ψ₂〉 = |1〉_a |1〉_b. To realize the required phase difference ϕ = ϕ_p1 + ϕ_p2 = 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 P₁₍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.

 

Fig. 2 Counting rate of SPD3 versus the pump power for the case of only the individual pump field (a) E_p1 and (b) E_p2 is launched into SFL. The second-order polynomials $N_s=s_1^s{P}_1+s_2^s{P}_1^2$ and $N_a=s_1^a{P}_2+s_2^a{P}_2^2$ (solid curves) are used to fit the data in plots (a) and (b), respectively. The linear and quadratic terms of the fitting functions are represented by the dashed and dotted lines, respectively. (c) The difference, N_d, between the total counting rate of SPD3 N_t and the sum of N_s and N_a versus the total power of the two pump fields P₁ + P₂. The solid curve is the fitting of the function N_d = s₂(P₁ + P₂)². The inset plots the total counting rate N_t versus the power P₁ + P₂. The fitting coefficients are $s_1^s=0.00318$, $s_2^s=0.01348$, $s_1^a=0.00127$, $s_2^a=0.01383$ and s₂ = 0.02298.

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The inset of Fig. 2(c) shows the measured total counting rates N_t of SPD3 versus the power of two pump fields P₁ + P₂ for case (iii). The main plot of Fig. 2(c), obtained by subtracting the background noise N_s and N_a of individual pump fields from N_t, shows the counting rate of detected photons originated from DFWM N_d versus pump power. Moreover, N_d fits the function N_d = s₂(P₁ + P₂)² very well, where the fitting coefficient s₂ 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 s₂ 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_si| > 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 s₂ decreases gradually with the increase of detuning due to the walk-off effect of the two pulsed pump fields. Obviously, s₂ 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].

 

Fig. 3 The fitting coefficients of the counting rates versus the detuning $\mathrm{\Omega }=\frac{c({\lambda }_{\mathit{si}}-{\lambda }_{p1})}{{\lambda }_{\mathit{si}}^2}$ and $\mathrm{\Omega }=\frac{c({\lambda }_{p2}-{\lambda }_{\mathit{si}})}{{\lambda }_{\mathit{si}}^2}$ for the SFL pumped with the individual field (a) E_p1 and (b) E_p2, respectively. (c) The fitting coefficient s₂ versus the detuning $\mathrm{\Omega }=\frac{c({\lambda }_{p2}-{\lambda }_{p1})}{2{\lambda }_{\mathit{si}}^2}$. $s_1^s$ and $s_1^a$ are proportional to the gain of RS. $s_2^s$ and $s_2^a$ are proportional to the gain of NDFWM, s₂ is proportional to the gain of DFWM.

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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.,

According to Eq. (6), we calculate the ratio R versus the detunings under different pump power levels P₁ + P₂ by using our experimentally measured results N_s, N_a and N_d, as shown in Fig. 4(a). For the low pump power of P₁ + P₂ = 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 P₁ + P₂ = 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 P₁ + P₂ 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)).

 

Fig. 4 (a) The ratio R versus the detuning $\mathrm{\Omega }=\frac{c({\lambda }_{p2}-{\lambda }_{p1})}{2{\lambda }_{\mathit{si}}^2}$ for pump with different power levels. (b) The value of CAR versus average power of two pump fields P₁ + P₂ when the detuning Ω is 0.63 THz and 0.88 THz, respectively.

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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 P₁ + P₂ 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 P₁ + P₂ less than 0.15 mW. However, when P₁ + P₂ 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.