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Multimode squeezed states are essential resources in quantum information processing and quantum metrology with continuous variables. Here we present the experimental generation of squeezed vacuum via the degenerate four wave mixing realized by pumping a piece of dispersion shifted fiber with mode-locked ultrafast pulse trains. The noise fluctuation is lower than the shot noise limit by 1.1 ± 0.08 dB (1.95 ± 0.17 dB after correction for detection losses). The detailed investigation illustrates that the results can be further improved by suppressing Raman scattering and by reshaping the spectrum of the local oscillator to achieve the required mode-matching of the homodyne detection system. Our study is useful for developing a compact fiber source of multi-mode squeezed vacuum.
© 2016 Optical Society of America
1. Introduction
Squeezed states are important resources of quantum information processing with continuous variables [1, 2]. In recent years, there is growing interest in generating multimode squeezed states, which are promising candidates for simulating quantum network and for quantum metrology [3–5 ]. One efficient method for realizing the multi-mode squeezed state is based on the degenerate optical parametric process with a pulsed configuration [6]. Because of the broadband nature of the pulsed pump, the degenerate signal and idler twin beams is in a superposition of many amplified signal/idler frequency modes. Recently, Pinel and co-workers demonstrated the multimode quantum frequency comb using a degenerate optical parametric oscillator synchronously pumped by a mode-locked laser [7,8]. Here we present the experiment of generating multimode squeezed vacuum by using a pulse pumped fiber optical parametric amplifier.
In fact, optical fiber is one of the favorable nonlinear media for generating squeezed state because of the potential of developing into a compact fiber source and the compatibility with the existing fiber network. Right after the first demonstration of squeezed state via the four wave mixing (FWM) in atom, Shelby et al. realized the squeezed light in fiber via FWM by using a continuous wave (CW) laser as the pump [9]. Later on, various kinds of squeezed lights, such as squeezed vacuum and squeezed solitons, had been generated by exploiting the Kerr nonlinear phase shift in fibers [10–13 ]. However, because of the existence guided-acoustic-wave Brillouin scattering (GAWBS) [14] in fiber, the noise reductions via CW laser pumped near-degenerate FWM and Kerr nonlinearity were only observed at frequencies between GAWBS peaks. In 2001, Sharping and co-works demonstrated the intensity difference squeezing of twin beams by using a pulse pumped fiber optical parametric amplifier (FOPA) [15], in which the influence of GAWBS for pulse pumped FOPA is negligible because the detuning between the signal/idler field and pump field is much greater than the frequency range of GAWBS. Recently, based on the multimode quantum theory of pulse pumped FOPA, our group optimized the experimental parameter and improved intensity difference squeezing to 10.4 dB (after the correction of detection efficiency) [16,17].
In general, the co-polarized four-wave mixing in FOPAs can be divided in to two categories. One is the frequency non-degenerate four wave mixing (NDFWM), in which two pump photons at frequencies ω p1 and ω p2 are scattered through the χ (3) nonlinearity of fiber to create the signal and idler photon pairs at frequencies ωs and ωi, respectively, such that ω p1 + ω p2 = ωs + ωi and ωs ≠ ωi. The other is the frequency degenerate four wave mixing (DFWM), in which two pump photons at the different frequencies ω p1 and ω p2 are scattered through the χ (3) nonlinearity to create a pair of identical photons at the mean frequency ωsi, such that ω p1 + ω p2 = ωs + ωi and ωs = ωi = ωsi. So far, using the NDFWM in the pulsed pumped FOPA, various kinds of non-classical lights in the domain of both discrete and continuous variables, have been realized [15,16,18–21 ]. However, using the DFWM of the pulsed pumped FOPA, only the quantum correlated photon pairs in the domain of discrete variable had been demonstrated [22–24 ].
In this paper, we will demonstrate the generation of multimode squeezed vacuum by exploiting the DFWM in fiber, to the best of our knowledge, for the first time. The experiment is realized by pumping 150 m dispersion shifted fiber (DSF) with two frequency non-degenerate pump pulse trains. Moreover, we will study the factors influencing the noise reduction of the squeezed vacuum. Our results show that suppressing the Raman scattering, which accompanies the DFWM in DSF, and matching the mode of the homodyne detection by reshaping the local oscillator are crucial for improving the measurement of noise reduction.
2. Theoretical background
Degenerate four wave mixing can be realized by launching two frequency non-degenerate pumps, respectively at the wavelength of λ p1 and λ p2, into a piece of dispersion shift fiber (DSF). Under the condition , where λ 0 is the zero dispersion wavelength (ZDW) of DSF, the phase matching condition of DFWM is satisfied. In this case, the input-output relationship of the DSF in the single frequency mode form is described as [25]
where b̂ and â are the operators of the output and input fields of DSF, respectively, μ and ν with the relation |μ|2 − |ν|2 = 1 are the gain amplitude of DFWM. According to the definition of quadrature components of the frequency degenerated twin beams X̂ = b̂ † +b̂ and Ŷ = i(b̂ † − b̂), the variance of the quadrature components are written as and where g = (|μ| + |ν|)2 > 1 is referred to as the power gain of DFWM. Comparing with the noise variance of coherent state 〈Δ2 X̂〉c = 〈Δ2 Ŷ〉c = 1, which is the so called shot noise limit (SNL), one sees that the variance of the quadrature component Ŷ, 〈Δ2 Ŷ〉, decrease with gain and is lower than SNL, indicating the output of DSF is in the squeezed vacuum state.Accompanying the DFWM, there is Raman scattering in DSF. So we need to exploit two synchronized laser pulse trains centering at λ p1 and λ p2, respectively, as the pump to obtain the high gain DFWM and to suppress the Raman scattering [24]. In this case, the output field of DSF centering at λ 0 is in a superposition of many amplified frequency modes, i.e., the squeezed vacuum is in multi-temporal modes [6, 24]. To efficiently measure the squeezed vacuum by using homodyne detection, it is crucial to match the spectra of local oscillator with the measured field [26]. In general, the following two methods can be used to improve the mode matching. One is shaping the spectrum of local oscillator with a wave shaper when the spectrum of the squeezed field is fixed. The other is varying the spectrum of the squeezed vacuum by adjusting the temporal overlap of the two pulsed pumps when the spectrum of local oscillator is fixed.
3. Experimental preparation and procedure
Our experimental setup is shown in Fig. 1. The squeezed vacuum is generated by pumping 150 m dispersion shift fiber (DSF) with two synchronized pump pulse trains, labeled as P1 and P2. The nonlinear coefficient and ZDW of the DSF are about γ ≈ 2 W−1/km and λ 0 ≈ 1552 nm (at the room temperature), respectively. The two pumps are simultaneously coupled into the DSF via a coarse wavelength division multiplexer (CWDM). The squeezed field out of the DSF is transmitted through a wavelength division multiplexing (WDM) filter to isolate the strong pumps. The CWDM has four channels, centering at 1571 nm (for pump P2), 1551 nm (for reference signal), 1531 nm (for pump P1) and 1511 nm (not used), respectively, and the 1-dB bandwidth of each channel is 16 nm. The central wavelength of the WDM is the same as the ZDW λ 0, and the full-width at half maximum (FWHM) of the WDM is about 1.2 nm.
Fig. 1 Experimental setup. G1–G4, gratings; FC1–FC2, fiber couplers; EDFA1–EDFA3, erbium-doped fiber amplifiers; F1–F3, double grating filters; FPC1–FPC6, fiber polarization controllers; PBS1–PBS3, polarization beam splitters; LO, local oscillator; Ref., reference light; P1, pump pulses centering at λ p1; P2, pump pulses centering at λ p2; CWDM, coarse wavelength division multiplexer; DSF, dispersion shift fiber; PZT, piezoelectric transducer; BS, 50/50 beam splitter; PD1–PD2, photodiode; ESA, electrical spectrum analyzer.
The squeezed vacuum is measured by the homodyne detection (HD) system, which is comprised of a 50/50 beam splitter (BS) and two photodiodes (PD, ETX-500) with circuits the same as those in [16]. The DC outputs of the two PDs are sent to two ammeters (not shown in the Fig. 1) for monitoring the power of local oscillator (LO), while the difference of AC outputs of the two PDs is analyzed by using an electrical spectrum analyzer (ESA). The total detection efficiency of our HD system is about 62% when the transmission efficiency of squeezed field 75% and quantum efficiency of PDs 83% are included. Since the noise of the squeezed vacuum depends on the relative phase between the squeezed field and local oscillator (LO), we control the phase of LO with a piezo-electronic transducer (PZT) mounted on a high reflection mirror.
The preparation of the light fields, including the two pulsed pumps (P1, P2) and LO, is shown in the box with dotted frame in Fig. 1. We first disperse the 36 MHz train of 100-fs pulses centered at 1560 nm from a mode-locked fiber laser with gratings (G1–G4) and spectrally filter them to obtain three beams, whose wavelengths are selected at 1538.5 nm (for P1), 1566.8 nm (for P2), and 1552.5 nm (for LO) when the DSF is placed at the room temperature (300K). Under this situation, the phase matching condition of DFWM in DSF is satisfied. To achieve the required pump powers, we then feed the beams at wavelengths of 1538.5 and 1566.8 nm into erbium-doped fiber amplifiers EDFA1 and EDFA3, respectively. The outputs of EDFA1 and EDFA3 are further cleaned up with the filters F1 and F3, respectively. To ensure the polarization and power adjustment of the pump P1/P2, the output of F1/F3 is then propagated through the fiber polarization controller FPC1/FPC3 and polarization beam splitter PBS1/PBS3. The beam at the wavelength of 1552.5 nm is split by the 50/50 fiber coupler (FC1). One output of FC1, being amplified by EDFA2 and sequentially propagating through the filter F2, FPC2 and PBS2, functions as the LO of HD. In our experiments, the bandwidths (pulse widths) of P1/P2 and LO are 0.58 nm (6 ps) and 0.73 nm (5 ps), respectively.
Because the noise variance of the squeezed vacuum is closely related to the gain of DFWM, we characterize the gain by simultaneously coupling the two pulsed pump fields and the reference light into the DSF via CWDM. The reference signal light (Ref.) is obtained by attenuating the one output of FC1 (see Fig. 1) to about 1 μW. In the experiment, the powers of the pumps P1 and P2 are set to equal. In order to optimize the gain, the temporal modes of the fields (P1, P2 and Ref.) involved in the DFWM are matched by adjusting their relative optical delay for pulse overlapping, while their polarization modes are matched by tuning the polarization of P1 and Ref. with FPC4 and FPC5, respectively. Moreover, since the amplification process of DFWM with the non-zero injection signal is phase sensitive, phase locking is necessary. Therefore, at the output of DSF, we split the amplified signal with a 50/50 FC (not shown in Fig. 1) and send one output of the FC to a locking system to lock the relative phase of P1, P2 and the Ref. light. After locking the phase for obtaining the maximized gain, we then measure the power gain by analyzing the other output of the FC with an optical spectrum analyzer (OSA) when the total pump power is set at different level [27]. The gain is the defined as the ratio between the powers of the amplified signal and injection signal. The result (circles) is shown in Fig. 2. One sees the gain of DFWM increases exponentially with the total pump power. When the total pump power is about 1.6 mW, the observed gain is about 24. Additionally, we also characterize the gain of DFWM at room temperature when the delay between the two pumps P1 and P2 is 5 ps (see later for details). The result (diamonds in Fig. 2) is measured after the arrival time of the reference signal pulses is adjusted for achieving the optimized amplification of reference signal. It is obvious that for a given pump power, the gain of DFWM for P1 and P2 with 5 ps delay is reductive because the temporal overlap between two pumps is decreased. We note that the Ref. light is only used for characterizing the gain feature, it is blocked when the noise variance is measured by using the HD system.
Fig. 2 Classical gain of DFWM in DSF versus the total power of the pumps P1+P2. The circles represent the data for P1 and P2 well overlapped to achieve the maximized gain, the diamonds represent data for P1 and P2 with delay of 5 ps. In the measurement, the DSF is placed at room temperature (300 K).
4. Experimental results
We first study the noise variance of the squeezed field when the two pumps P1 and P2 are well overlapped to achieve the maximized gain. Figure 3(a) shows the results when the powers of both P1 and P2 are 0.2 mW. In the measurement, the power of LO is fixed at 0.7 mW, and the phase of LO is scanned. The SNL (trace (ii)) measured by blocking the output of DSF is about 9.5 dB higher than the electronic noise (trace (i)) of the HD system. The noise variance of the squeezed vacuum (trace (iii)) changes with the phase of LO, and the minimum noise is lower than the SNL by 0.42 ± 0.07 dB (0.47 ± 0.08 dB after being corrected with the electronic noise of HD system). We also measure the noise variance of the squeezed field by varying the total power of the two pumps and scanning the phase of LO. Figure 3(b) shows the minimum relative noise as a function of the total pump power. One sees there is a turning point at 0.4 mW. When the total pump power is smaller than 0.4 mW, the relative noise decreases with the increase of pump; when the total pump power is larger than 0.4 mW, the relative noise starts to increase with the pump.
Fig. 3 Observed noise levels when the pumps P1 and P2 are well overlapped for achieving the maximized gain. (a) Measurement of the noise variance when the pump powers of both P1 and P2 are 0.2 mW. (b) The minimum relative noise of the squeezed vacuum as a function of the total pump power P1+P2. Trace (i), the electronic noise of HD system; trace (ii), the shot noise limit (SNL); trace (iii), the noise variance of the squeezed vacuum for the pumps with total power of 0.4 mW. In the measurement, the temperature of DFS is 300 K; the central frequency, RBW and VBW of the ESA at zero span are set to 5MHz, 1MHz and 1kHz, respectively; and the data in plot (b) has been corrected by the electronic noise of HD system.
As we have pointed out in Sec. 2, the mode matching of HD system plays an important role in obtaining a good noise reduction. Because the spectrum of the LO is fixed by the spectrum of filter F2, we try to improve the mode matching by adjusting the relative delay between the pumps P1 and P2, and between the squeezed field and LO. The delay of LO is always set to maximally overlap the amplified reference signal, so that the optimized noise reduction is achievable. We find that when the delay between P1 and P2 is about 5 ps, we obtain the optimized noise reduction. Figure 4(a) shows the results for the total pump power of 0.8 mW. One sees that the noise reduction (trace (iii)) lower than the SNL (trace (ii)) by 0.55 ± 0.05 dB [0.63 ± 0.06 dB after being corrected by the electronic noise (trace (i))] is obtained. Under this condition, we also measure the noise reduction by varying the total pump power. Figure 4(b) shows the minimum relative noise as a function of the total pump power. Similar to Fig. 3(b), there is a turning point in Fig. 4(b) as well. But the power corresponds to the turning point is about 0.8 mW, at which the gain of DFWM is about 2.6.
Fig. 4 Observed noise levels when the delay between the pumps P1 and P2 is 5 ps. (a) Measurement of the noise variance when the pump powers of both P1 and P2 are 0.4 mW. (b) The minimum relative noise of the squeezed vacuum as a function of the total pump power P1+P2. Trace (i), the electronic noise of HD system; trace (ii), the shot noise limit (SNL); trace (iii), the noise variance of the squeezed vacuum for the pumps P1 and P2 with total power of 0.8 mW. In the measurement, the temperature of DSF is 300 K; the central frequency, RBW and VBW of the ESA at zero span are set to 5MHz, 1MHz and 1kHz, respectively; and the data in plot (b) has been corrected by the electronic noise of the HD system.
Comparing the results in Figs. 3(b) and 4(b), one sees that for a fixed gain of DFWM (see Fig. 2), the noise reduction in Fig. 4(b) is better than that in Fig. 3(b). For example, when the measured gain is about 5 (corresponding to the power of 0.8 and 1.2 mW in Figs. 3(b) and 4(b), respectively), the noise reduction in Fig. 4(b) is lower than that in Fig. 3(b) since the mode matching of HD system is improved by introducing the delay between P1 and P2 to change the spectrum of squeezed field. Moreover, when the pump power is greater than a certain value, which corresponds to the turning point, the relative noise starts to increase with the pump in both Figs. 3(b) and 4(b). The phenomenon is in conflict with the prediction of single mode theory (see Eq. (3)) but can be explained by the detection theory of multi-mode squeezed states [6, 26]. In our experiment, although the mode matching of HD system in Fig. 4 is better than that in Fig. 3, the mode of LO is not perfectly matched with any decomposed Schmidt temporal mode of the squeezed field. The measurement result of the HD systems is contributed by all the modes non-orthogonal to the mode of local oscillators. The noise contributed by the higher order decomposed Schmidt temporal modes, whose phases are different from the fundamental mode, may exponentially increase with pump power. Hence, when the pump power is higher than a certain level, the noise will start to increase with the pump power.
5. Influence of Raman scattering
To further investigate the factors influencing the measured degree of squeezing, in addition to measuring the noise reduction via DFWM in DSF, we measure the noise originated from the individual pump P1/P2 by blocking P2/P1. We find the noise produced by the individual P1/P2 is insensitive to the phase of LO and is always higher than the SNL. Figures 5(a) and 5(b) show the measured excess noise versus the power of individual pump when the delay between P1 and P2 is the same as that for obtaining the data in Fig. 3 and Fig. 4, respectively. For a fixed pump power of P1/P2, the measured noise in Fig. 5(a) is about 20% higher than that in Fig. 5(b). This is because the delay of LO is always set to well match the mode of the amplified reference light, so the mode matching between the excess noise and LO in Fig. 5(a) is better than that in Fig. 5(b). However, it is worth pointing out that for the optimized noise reduction obtained under a given gain of DFWM, the excess noise in Fig. 4 is higher than that in Fig. 3 due to the gain reductive feature induced by the delay between P1 and P2.
Fig. 5 The excess noise of the field generated by individual pumps P1 and P2. Plot (a) is obtained when the P1 and P2 are well overlapped for achieving the maximized gain. Plots (b) and (c) are obtained when the delay between P1 and P2 is 5 ps. In the measurement, the delay between LO and the measured fields is always set for obtaining the optimized noise reduction.
To understand the source of the excess noise, we also study the nonlinear effect by launching individual pump P1/P2 and reference signal into DSF and monitoring the output of DSF with the OSA (not shown in Fig. 1). In this experiment, the delay between P1 and P2 is set to optimize the gain of DFWM (the same as in Fig. 3). We find the Raman gain/loss at the wavelength of reference light is observable [16], but the four wave mixing for amplifying the reference light by the individual pump P1/P2, i.e., the NDFWM, is unobservable because the detuning between the reference signal and pump is too large to satisfy the phase matching condition of NDFWM [24]. The results indicate that the excess noise is originated from Raman scattering of individual P1 and P2.
The Raman scattering can be suppressed by cooling the DSF [28, 29]. So we try to improve the noise reduction by cooling the DSF with liquid nitrogen (77 K). In this experiment, we replace the WDM filter with the one centering at 1548.4 nm and adjust the central wavelength of LO to 1548.4 nm since the zero dispersion wavelength of the DSF at 77 K is about 1548 nm. The delay between the pumps P1 and P2 is still about 5 ps. However, to ensure phase matching condition of DFWM is still satisfied, the central wavelengths of P1 and P2 are tuned to 1534.4 and 1562.7 nm, respectively. We first measure the excess noise produced by Raman scattering of the individual pump P1/P2, as shown in Fig. 5(c). As expected, for a given pump power, the excess noise is obvious lower than that in Fig. 5(b). We then measure the noise of squeezed field by varying the total pump power. Figure 6 shows the minimum relative noise as a function of the total pump power. Comparing Fig. 6 with Fig. 4(b), it is obvious that the maximized noise reduction is improved to 1.1 ± 0.08 dB (1.95 ± 0.17 dB after efficiency correction). Although the pump power corresponding to the turning point in Fig. 6 is increased to 1.6 mW, the gain of DFWM for obtaining the maximized noise reduction 3.6 is only slightly higher than that in Fig. 4(b). We believe the reduced excess noise is also responsible for the shift of turning point. Moreover, we note that the loss induced by the cooling is only about 2%, the deteriorating gain feature in cooled DSF is likely caused by the extra birefringence and inhomogeneity due to the cooling induced stress.
Fig. 6 The minimum relative noise of the squeezed vacuum as a function of the total pump power P1+P2. In the measurement, the delay between P1 and P2 is 5 ps, and the temperature of DSF is 77K.
6. Summary and discussion
In summary, we have demonstrate the generation of multimode squeezed vacuum via the pulse pumped degenerate four wave mixing process in DSF. The noise of the squeezed vacuum is not influenced by guided-acoustic-wave Brillouin scattering. When the gain of the DFWM is about 3.6, the measured noise reduction is about 1.1 ± 0.08 dB (1.95 ± 0.17 dB after efficiency correction). Our investigation illustrates that Raman scattering pulsed pumps and mode mismatching of the homodyne detection system prevent the noise of squeezed field from further reduction. Comparing with the squeezing obtained by non-degenerate four wave mixing in fibers [15,16], we think that at the current stage, the degree of squeezing is mostly affected by the mode mismatching between the spectra of local oscillator and squeezed vacuum. It is straightforward to improve the detection efficiency of our experiment by using WDM filters and photodiodes with higher efficiency. By figuring out the spectral correlation property of the squeezed field and by reshaping the spectrum of local oscillator, a further reduction of the measured noise and a demonstration of squeezing in different temporal modes are achievable [26,30]. We believe the fiber source of multimode squeezed vacuum will be a compact and useful tool for studying the quantum simulation and quantum metrology [8].
Acknowledgments
This work was supported in part by the State Key Development Program for Basic Research of China (No. 2014CB340103), the National NSF of China (No. 11527808, No. 11504262), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120032110055), the Natural Science Foundation of Tianjin (No. 14JCQNJC02300), PCSIRT and 111 Project B07014.
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