1. Introduction

Quantum correlation shared by multiple quantum correlated beams is an interesting topic [1,2] and has many potential applications in quantum information science [3] such as discrete variable (DV) [4,5] and continuous variable (CV) [6–8] quantum systems. Recent progresses in the DV regime include multi-photon entanglement [9,10], three-photon energy-time entanglement [5], the generation of quantum correlated photon triplets [11] as well as the experimental demonstration of topological error correction [4]. Alternatively, the CV regime offers the advantages of deterministic generation of entangled state and high efficiency quantum detection. The most well-established technique for generating multiple quantum correlated beams in the CV domain is based on mixing squeezed states on linear beam splitter network [7,8,12,13]. Another type of promising scheme is to integrate all the nonlinear processes into only one nonlinear interaction process, which can avoid using the linear beam splitter network, thus reduce the complexity and ensure the scalability [14,15]. In this aspect, it has been demonstrated that ultra-large-scale CV cluster state can be generated in both the frequency domain [16,17] and the time domain [18–20]. All these schemes are based on the nonlinear crystal put in a cavity.

Almost a decade ago, strong intensity-difference squeezing and Einstein-Podolsky-Rosen entangled state have been generated from four-wave mixing (FWM) process with a double-$\Lambda$ energy level configuration in $^{85}Rb$ vapor cell [21,22]. This system has been proved to be a promising candidate for many applications in quantum information and precision measurement, such as the tunable delay of EPR entangled states [23], the enhancement of intensity-difference squeezing [24], the realization of an SU(1,1) interferometer [25–28] and the generation of high purity narrow-bandwidth single photons [29]. Based on this double-$\Lambda$ FWM, several schemes have been utilized to generate multi-beam quantum correlation [30–34]. Among them, two-beam pumped cascaded FWM (CFWM) is proved to be an effective way to generate multi-beam quantum correlation in a single step [31–34]. Therefore, it is important to investigate how to maximize both the number and the quantum correlation degree of the generated multiple beams from this two-beam pumped CFWM system. In this article, we experimentally study how these quantities depend on the system parameters, including the angle between the two pump beams, one-photon detuning and two-photon detuning. These detailed studies boost the number of the emitted quantum correlated beams to 14 from the result of 10 in our previous work [32].

2. Experiment

Our detailed experimental setup for the two-beam pumped CFWM scheme is shown in Fig. 1(a). The system is based on the two-beam pumped CFWM process in a double-$\Lambda$ configuration in which two pump photons can convert to one signal photon and one idler photon as shown in Fig. 1(b). We use a cavity stabilized Ti: Sapphire laser as the main source of the setup. The wavelength of the laser is about 795 nm, and the bandwidth of the laser is about 60 KHz. Its frequency is blue detuned from the $^{85}Rb$ D1 line (5S$_{\textrm {1/2}}$, F = 2 $\rightarrow$ 5P$_{\textrm {1/2}}$), which is called one-photon detuning ($\Delta$). A polarization beam splitter (PBS) is used to divide the laser into two beams. One beam is sent to a spatial light modulator (SLM) to generate two parallel pump beams (Both of two pump beams have the power of about 100 mW), the other beam is passed through an acousto-optic modulator (AOM) to obtain the signal beam which is red or blue detuned from the $^{85}Rb$ ground-state hyperfine splitting of 3.036 GHz and this causes a two-photon detuning ($\delta$). The power of the signal beam is about 40 $\mu$W. The $^{85}Rb$ vapor cell is 12 mm long and the temperature of the $^{85}Rb$ vapor cell is stabilized at 117 $^{\circ }$C. At the center of the vapor cell, the waists of two pump beams are about 600 $\mu$m, and the waist of signal beam is about 300 $\mu$m. Combined by a Glan-Laser polarizer (GL), the two parallel pump beams focused by lens (L) and the signal beam are crossed in the center of the $^{85}Rb$ vapor cell. The signal beam is symmetrically crossed with the two pump beams. The angle between the signal beam and plane of the two pump beams is about 4 mrad, making the multiple output beams spatially separated. The two residual pump beams after the CFWM process are eliminated by a Glan-Thompson polarizer (GT) with an extinction ratio of $10^5$:1. Under the same physical mechanism with our previous works [31,32], the signal beam can interact with each pump beam respectively through the normal single-pump FWM process [21]. Moreover, with the satisfaction of the energy and momentum conservation, the signal beam can also interact with both of the two pump beams simultaneously. In this sense, we name such FWM process as dual-pump FWM interaction. During such dual-pump FWM interaction, each pump beam annihilates one photon, the signal beam generates one photon, and the idler beam generates one photon. The generated beams can also interact with each pump beam or with both of the two pump beams as long as the energy and momentum conservation is satisfied. As a result, through all these possible single-pump and dual-pump interactions, the multiple quantum correlated beams are generated. For instance, all possible single-pump (indicated as black straight lines) and dual-pump (indicated as red straight lines) FWM interactions for generating the 14 quantum correlated beams are shown in Fig. 1(c). The output signal beams are sent to one photodetector (${D}_{1}$), while the output idler beams are sent to the other photodetector (${D}_{2}$). The photodetectors’ transimpedance gain is $10^5$ V/A and the quantum efficiency is about 96%. The radio-frequency (rf) components of ${D}_{1}$ and ${D}_{2}$ are subtracted from each other by using rf subtractor S. Then, this obtained photocurrent (${i}$) is analyzed by a spectrum analyzer (SA). The SA is set to a 30 kHz resolution bandwidth (RBW) and a 300 Hz video bandwidth (VBW).

 

Fig. 1. The detailed experimental layout for two-beam pumped CFWM scheme (beam diameter not to scale). (a) The experimental scheme of the CFWM process. HWP, half wave plate; PBS, polarization beam-splitter; AOM, acousto-optic modulator; SLM, spatial light modulator; L, lens; GL, Glan-Laser polarizer; GT, Glan-Thompson polarizer; ${D}_{1}$-${D}_{2}$, photodetector; rf, radio-frequency; S, subtractor; SA, spectrum analyzer; Yellow, signal beams; Blue, idler beams. Red, pump beams. ${i}$, photocurrent. (b) Double-$\Lambda$ energy level diagram of $^{85}Rb$ D1 line for CFWM process. $\Delta$, one-photon detuning; $\delta$, two-photon detuning. (c) The interaction structure for generating the 14 quantum correlated beams for the CFWM. The black straight lines indicate the possible single-pump FWM processes, while the red straight lines indicate the possible dual-pump FWM processes.

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3. Results

In order to explore how to maximize both the number and the quantum correlation of the multiple correlated beams from this two-beam pumped CFWM system, we firstly investigate the relationship between the number of multiple quantum correlated beams and the angle between the two pump beams because the multiple-beam quantum correlations of the two-beam pumped CFWM process are only due to the introduction of an additional pump beam compared with the normal single-pump FWM process. We set one-photon detuning $\Delta$=0.9 GHz (blue detuned 0.9 GHz), two-photon detuning $\delta$=4 MHz (red detuned 4 MHz) and scan the angle between the two pump beams from about 2 mrad to 8.6 mrad. The changing of the angle between the two pump beams is realized by changing the separation between two parallel pump beams controlled by SLM. For each angle, we measure the intensity-difference noise power spectrum and its corresponding shot noise limit (SNL), which gives us the degree of squeezing [32] shown as the blue dot curve in Fig. 2. The green dashed line at 0 dB is the normalized SNL. At the same time, the transverse pattern of the output beams in the far field from the $^{85}Rb$ vapor cell is captured by the CCD camera for each angle point as shown in Fig. 2. As we can see, we have observed the notable transition of the number of the output beams (from 6 beams to 12 beams) with the decreasing of the angle between the two pump beams from 8.6 mrad to 3.6 mrad. However, it is difficult to further increase the number of the output beams by simply keeping decreasing the angle between the two pump beams as shown in Fig. 2. When the angle decreases from about 3.6 mrad to 2 mrad, the adjacent beams at the output begin to overlap with each other. The intensity-difference squeezing in our CFWM scheme has almost the same trend with the number of the output beams. The degree of the intensity-difference squeezing in our CFWM scheme increases as the decreasing of the angle between the two pump beams from 8.6 mrad to 3.6 mrad. It reaches the value of −6.56 $\pm$ 0.20 dB at the sideband of 1.6 MHz when the angle is 3.6 mrad. If we further decrease the angle between the two pump beams from 3.6 mrad to 2 mrad, the intensity-difference squeezing decreases.

 

Fig. 2. The effect of the angle between the two pump beams on the number of multiple quantum correlated beams generated by two-beam pumped CFWM process. The blue dot curve is the intensity-difference squeezing of the multiple beams produced by the CFWM scheme versus the angle between the two pump beams. The transverse patterns of the output beams for our CFWM scheme in the far field captured by CCD camera are shown near the corresponding blue dots. The green dashed line at 0 dB is the normalized SNL. The error bars are obtained from the standard deviations of multiple repeated measurements.

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Secondly, to study the effect of one-photon detuning on the number of multiple quantum correlated beams generated by our scheme, we set the angle between the two pump beams at about 3.6 mrad and $\delta$=4 MHz. As shown in Fig. 3, we make a series of measurements similar to Fig. 2 with changing one-photon detunings from 0.6 GHz to 1.2 GHz. The blue dot curve in Fig. 3 is the normalized intensity-difference squeezing of the quantum correlated multiple beams from the CFWM scheme. The output beams in the far field are also captured by the CCD camera for each point. The green dashed line at 0 dB is the normalized SNL. We can see that the number of multiple quantum correlated beams increases with the decreasing of one-photon detuning. The number of the multiple quantum correlated beams obtains its maximum value (14 beams) when 0.6 GHz $\leq \Delta \leq$ 0.8 GHz. The optimal intensity-difference squeezing with the 14 quantum correlated beams is −6.29 $\pm$ 0.20 dB at $\Delta$=0.8 GHz [−7.93 $\pm$ 0.64 dB after taking into account the optical propagation loss (about 5$\%$) and detection loss (about 4$\%$)].

 

Fig. 3. The effect of one-photon detuning on the number of multiple quantum correlated beams generated by two-beam pumped CFWM process. The blue dot curve is the intensity-difference squeezing of the multiple beams produced by the CFWM scheme versus one-photon detuning. The transverse patterns of the output beams for our CFWM scheme in the far field captured by CCD camera are shown near the corresponding blue dots. The green dashed line at 0 dB is the normalized SNL. The error bars are obtained from the standard deviations of multiple repeated measurements.

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Fig. 4. The effect of two-photon detuning on the number of multiple quantum correlated beams generated by two-beam pumped CFWM process. The blue dot curve is the intensity-difference squeezing of the multiple beams produced by the CFWM scheme versus two-photon detuning. The transverse patterns of the output beams for our CFWM scheme in the far field captured by CCD camera are shown near the corresponding blue dots. The green dashed line at 0 dB is the normalized SNL. The error bars are obtained from the standard deviations of multiple repeated measurements.

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Thirdly, to investigate the effect of two-photon detuning on the number of multiple quantum correlated beams generated by our scheme, we set the angle between the two pump beams at about 3.6 mrad, $\Delta$=0.9 GHz and scan two-photon detuning from −26 MHz (blue detuned 26 MHz) to 22 MHz (red detuned 22 MHz), as shown in Fig. 4. It can be found that the number of multiple quantum correlated beams is almost the same (12 beams) in the quantum region (−2 MHz $\leq \delta \leq$ 22 MHz). The maximum squeezing of −6.61 $\pm$ 0.18 dB is obtained with $\delta$=16 MHz. The number of the output beams becomes 15 when $\delta$=−14 MHz. However, the intensity-difference squeezing disappears in this situation, showing that these multiple beams are not quantum correlated any more. Therefore, the maximum number of quantum correlated beams from our system is still 14.

4. Conclusions

We have experimentally maximized the number of the quantum correlated beams emitted from the two-beam pumped CFWM process by varying the system parameters. With the optimal system parameters (the angle between the two pump beams 3.6 mrad, one-photon detuning 0.8 GHz and two-photon detuning 4 MHz), 14 quantum correlated beams with the intensity-difference squeezing of −6.29 $\pm$ 0.20 dB (−7.93 $\pm$ 0.64 dB after accounting for losses) have been experimentally achieved. Further increasing the number of quantum correlated beams could be implemented by shining more pump beams to induce other CFWM in different directions. Due to the fact that these quantum correlated beams are spatially separated, one could send different beams to different distant locations to build a multi-user quantum network. One could also imprint different physical parameters onto different beams, then use the whole quantum correlation to reduce the measurement noise floor and thus improve the corresponding signal-to-noise ratio. In other words, such CFWM system could be used to realize multi-parameter quantum metrology.