Quantum entanglement [1] is not only a fundamental issue in physics but also plays a significant role in quantum information processing [2,3]. Recently, much research has been devoted to sources of multipartite entanglement, which is the key ingredient for quantum communication and computation.

Based on an optical system, schemes for generating multipartite entangled states are commonly achieved with a linear optical transformation of a beam-splitter network [3,4]. Recently, several impressive demonstrations of multi-partite entanglement have been shown, such as 8-photon cluster states [5] and an 8-beam state [6]. However, a practical source is not only scalable, but can also be compact, even when the number of modes is very large. Recently, larger continuous variable (CV) multipartite entanglement using an optical parametric oscillator (OPO) was demonstrated in the frequency domain [7,8] and ultra-larger-scale cluster states of more than 10,000 modes were reported in time domain [9].

Spatial freedom of light is an effective approach to scale the number of entangled states. Many great achievements, especially in the discrete variable (DV) domain, have promoted the development of quantum information [3]. Parallel to the DV domain, CV spatial mode entanglement has been a hot topic of investigation because of the applications in quantum information [10] and quantum metrology [11,12]. An eight-mode Greenberger-Horne-Zeilinger (GHZ) CV entangled state using a virtual beam-splitter network was obtained experimentally [13], but there are reports on entanglement experimentally generated by an OPO that are limited to two spatial modes [14,15]. Recently, Patera et al. [16] theoretically demonstrated that multimode OPOs can produce a great variety of multipartite CV entangled states through appropriate control of the parametric interaction.

Recently, we have simultaneously produced two pairs of entangled Hermite–Gauss modes with a type II OPO and demonstrated that the entanglement is a CV hyperentanglement with a type II OPO [17]. In this report, we used the ${\mathrm{HG}}_{02}^{45°}$ mode pump to control the parametric nonlinear interaction and directly produce spatial quadripartite CV entanglement among first-order Hermite–Gaussian modes in a type II OPO below the threshold. We also realized spatial nonlocal separation and measurement in four modes. To our knowledge, this is the first time that spatially separable CV quadripartite entanglement has been demonstrated with the same frequency by an OPO, which is beneficial for coherent manipulation. The experimental system is scalable to higher-dimensional spatial modes by changing the spatial profile of the pump beam. Entanglement will be a promising source for future spatial multichannel quantum information.

In principle, we used the characteristics of the spatial freedom of light to generate quadripartite entanglement. In our model, shown in Fig. 1, the spatially tailored pump beam with a frequency of $2\omega$ drives a type II crystal in an optical cavity and quadripartite entanglement with frequency $\omega$ is generated from the cavity. The entanglement is in first-order Hermite–Gaussian (HG) modes with the same “family index” ($f=1$) [18] (which are degenerate modes in the cavity): ${\mathrm{HG}}_{01}^i$, ${\mathrm{HG}}_{01}^s$, ${\mathrm{HG}}_{10}^i$, and ${\mathrm{HG}}_{10}^s$, where $i$ (idle) and $s$ (signal) are the polarizations of the beams. In the interaction picture, assuming perfect phase matching and exact resonance between the field and the cavity and assuming that the pump is not depleted, the interaction Hamiltonian of the system is

 

Fig. 1. Schematic illustration of the physical system. The spatially tailored pump beam ${ϵ}_p$ (green) drives an OPO with a type II nonlinear crystal (KTP) below threshold, and spatial quadripartite entanglement among four first-order HG modes (red) with different polarizations ($i$, $s$) is generated. (a) Relationship between the spatial mode of the pump field ${ϵ}_p$ (green) and the spatial modes of entangled down-conversion fields (red). EPR (red thick line) represents the entanglement between a pair of down-conversion fields with HG modes. (b) Spatial distribution for pump mode ${\mathrm{HG}}_{02}^{45°}$ used in the experiment, which is superposition of three types of HG modes. (c) Theoretical solution for VLF criteria below the OPO threshold.

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Here, ${\widehat{a}}_k^i$ and ${\widehat{a}}_j^s$ are annihilation operators describing the intracavity fundamental fields. $k$ and $j=01$, 10 denote the types of HG mode and $i$ and $s$ denote the polarizations of the HG mode. ${ϵ}_p$ is the average amplitude of the pump field. ${\kappa }_{kj}$ is the nonlinear coupling constant [18], which is dependent on the spatial overlap between the pump and down-conversion fields.

The dynamics of the intracavity fields are then given by the quantum Langevin equations [15]:

It follows from the definition of ${\kappa }_{kj}$ that the relationship between the spatial mode of the pump field (green) and the spatial modes of the entangled down-conversion fields (red) are shown in Fig. 1(a). To generate the entanglement (EPR) between the ${\mathrm{HG}}_{01}^i$ and ${\mathrm{HG}}_{01}^s$ down-conversion fields, the spatial profiles of the pump beam must be in ${\mathrm{HG}}_{00}$ or ${\mathrm{HG}}_{02}$ modes; for the entanglement between ${\mathrm{HG}}_{10}^i$ and ${\mathrm{HG}}_{10}^s$, the pump beam must be in ${\mathrm{HG}}_{00}$ or ${\mathrm{HG}}_{20}$ modes. Moreover, to generate entanglement between ${\mathrm{HG}}_{01}^i$ and ${\mathrm{HG}}_{10}^s$ or entanglement between ${\mathrm{HG}}_{10}^i$ and ${\mathrm{HG}}_{01}^s$, the spatial profile of the pump beam is only ${\mathrm{HG}}_{11}$ mode. Thus, to obtain spatial quadripartite CV entanglement among first-order Hermite–Gaussian (HG) modes (${\mathrm{HG}}_{01}^i$, ${\mathrm{HG}}_{01}^s$, ${\mathrm{HG}}_{10}^i$, and ${\mathrm{HG}}_{10}^s$), the effective spatial pump beam must be a coherent superposition of ${\mathrm{HG}}_{11}$, ${\mathrm{HG}}_{02}$, and ${\mathrm{HG}}_{20}$ or a superposition of ${\mathrm{HG}}_{11}$ and ${\mathrm{HG}}_{00}$. It is complex to experimentally generate the hybrid mode through coherent superposition of different spatial modes on beam splitters. Fortunately, ${\mathrm{HG}}_{02}^{45°}$, which is equal to the superposition of ${\mathrm{HG}}_{11}$, ${\mathrm{HG}}_{02}$, and ${\mathrm{HG}}_{20}$, is an effective and simple pump mode used to generate quadripartite entanglement among HG modes, as shown in Fig. 1(b).

With the ${\mathrm{HG}}_{02}^{45°}$ pump, ${\kappa }_{kj}=\frac{{\chi }_0}{2\sqrt{2}}$ is identical for all down-conversion fields. ${\chi }_0$ is the nonlinear coupling constant between the ${\mathrm{HG}}_{00}$ pump and the ${\mathrm{HG}}_{00}$ down-conversion fields. The Hamiltonian has the same form as quadripartite entanglement of the optical frequency comb in Ref. [8]. As shown in Fig. 1(c), below the OPO threshold, the van Loock-Furusawa (VLF) CV multipartite entanglement criteria [19] are violated, demonstrating quadripartite entanglement. The largest violation, and thus the maximum quadripartite entanglement, is near the threshold ${|{ϵ}_{\mathrm{th}}|}^2$, which is twice the threshold of a standard ${\mathrm{HG}}_{00}$ mode OPO.

The experimental setup is depicted in Fig. 2. We used a high-power CW solid-state laser as the laser source, which delivers light with wavelengths of 1080 and 540 nm. The 1080-nm light was transmitted through a three-mirror ring cavity (RC), which tailors the spatial profile of the beam into the ${\mathrm{HG}}_{10}$ mode, and was then injected into the OPO as the seed with a power of 3 mW and a polarization of 45°. The 540-nm beam was tailored by the RC into the ${\mathrm{HG}}_{20}$ mode and rotated to ${\mathrm{HG}}_{02}^{45°}$ by a Dove prism (MR), which then drives the OPO as the pump. The OPO cavity included two mirrors with a radius of curvature of 30 mm (M1 and M2 in Fig. 2). The piezo-actuated mirror, M1, was used as the input coupler (with a high reflectivity at 540 and 1080 nm), and mirror M2 was the output coupler of the entangled beams at 1080 nm (with a transmission of 4.1% at 1080 nm and an antireflective coating for 540 nm). A device for astigmatism compensation [17] was placed into the cavity to achieve simultaneous resonation of the four modes (${\mathrm{HG}}_{01}^i$, ${\mathrm{HG}}_{01}^s$, ${\mathrm{HG}}_{10}^i$ and ${\mathrm{HG}}_{10}^s$) in the cavity.

 

Fig. 2. Experimental setup. The quadripartite entanglement is generated by a type II OPO, and then it is separated into four nonlocal parts using two asymmetric Mach–Zehnder interferometers (MZ1 and MZ2) and verified by four sets of balanced homodyne detectors (BHD). RC, three-mirror-ring cavity for generation of HG mode; MR, mode rotator; DBS, dichroic beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; PZT, piezoelectric element for controlling phase; LO, local oscillator; MC, mode converter for generation of LO of the ${\mathrm{HG}}_{01}$ mode from the ${\mathrm{HG}}_{10}$ mode; D1 and D2, detectors; and SA, spectrum analyzer.

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In our setup, the OPO threshold of the ${\mathrm{HG}}_{00}$ mode is 400 mW. With the 540 mW ${\mathrm{HG}}_{02}^{45°}$ pump that is below the OPO threshold of the first-order HG mode, spatial quadripartite CV entanglement can be generated when the relative phase between the pump field and the injected signal is locked in a state of deamplification with PZT1. Using a dichroic beam splitter (DBS), entanglement was separated from the pump. The power of the entanglement was 25 μW.

The output of the entangled beams was divided by a polarizing beam splitter (PBS) into two parts ($i$ and $s$) with orthogonal polarization. Each part includes both ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ modes. The two arms of the asymmetric Mach–Zehnder interferometers (MZ1 and MZ2) [20] have odd and even numbers of reflectors, respectively. Because of the different interference conditions for the ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ modes through asymmetric Mach–Zehnder interferometers, ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ modes were separated and independently output from the two ports. In the experiment, both MZ1 and MZ2 have transmissions of 98%. Detectors D1 and D2 provide a control signal for locking PZT2 and PZT3 of the Mach–Zehnder. As the result, the quadripartite entanglement is separated into four nonlocal parts: ${\mathrm{HG}}_{01}^i$ (mode 1), ${\mathrm{HG}}_{10}^i$ (mode 2), ${\mathrm{HG}}_{01}^s$ (mode 3), and ${\mathrm{HG}}_{10}^s$ (mode 4). Then, the quadripartite entanglement was verified by four sets of balanced homodyne detectors (BHD1–BHD4) with spatially tailored local oscillators (LOs) using a mode converter (MC). The LO beam was either in the ${\mathrm{HG}}_{10}$ or ${\mathrm{HG}}_{01}$ mode, depending on which spatial mode was measured. The measured correlation variances were then analyzed with a spectrum analyzer (SA). The measured efficiency of the spatial overlaps between the signal and the LO in the homodyne detector for balanced homodyne detectors (BHD) 1, 2, 3, and 4 were 92%, 93%, 92%, and 93%, respectively, and the measured photodiode (FND-500) efficiency was 92%.

The correlation variances were obtained by joint BHD according to the VLF criteria at an analysis frequency of 6 MHz. As shown in Fig. 3, the trace (1) is the shot-noise limit (SNL) and the trace (2) is the correlation variance normalized to SNL. $⟨{(\delta P_{01}^i-\delta P_{01}^s)}^2⟩$, $⟨{(\delta P_{01}^s-\delta P_{10}^i)}^2⟩$, $⟨{(\delta P_{10}^i-\delta P_{10}^s)}^2⟩$, $⟨{(\delta P_{10}^s-\delta P_{01}^i)}^2⟩$, and $⟨{(\delta X_{01}^i+\delta X_{01}^s+\delta X_{10}^i+\delta X_{10}^s)}^2⟩$ are $1.02\pm 0.18\mathrm{dB}$, $0.89\pm 0.21\mathrm{dB}$, $1.12\pm 0.21\mathrm{dB}$, $0.68\pm 0.20\mathrm{dB}$, and $1.44\pm 0.20\mathrm{dB}$ below the SNL, respectively. Here, $X=(\widehat{a}+{\widehat{a}}^{†})/2$ is the amplitude quadrature and $P=(\widehat{a}-{\widehat{a}}^{†})/2i$ is the phase quadrature. According to the VLF criteria for quadripartite GHZ CV entanglement,

 

Fig. 3. Results of spatial quadripartite GHZ entanglement. (a) $⟨{(\delta P_{01}^i-\delta P_{01}^s)}^2⟩$, (b) $⟨{(\delta P_{01}^s-\delta P_{10}^i)}^2⟩$, (c) $⟨{(\delta P_{10}^i-\delta P_{10}^s)}^2⟩$, (d) $⟨{(\delta P_{10}^s-\delta P_{01}^i)}^2⟩$, and (e) $⟨{(\delta X_{01}^i+\delta X_{01}^s+\delta X_{10}^i+\delta X_{10}^i)}^2⟩$. Trace (1) is the SNL and trace (2) is the correlation variance normalized to SNL. The measured parameters of SA are a resolution bandwidth of 1 MHz and a video bandwidth of 100 Hz.

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All of the inequalities in Eq. (3) are satisfied, and the quadripartite GHZ entanglement among the four spatial modes was thus demonstrated experimentally. $g_i(i=1,2,\dots ,8)$ are arbitrary real parameters that were used to optimize the violation of these inequalities. In this experiment, $g_i=1$, which is not optimal [19] but is sufficient to verify the existence of quadripartite GHZ entanglement.

In the experiment, there are factors that degrade the correlation. One major factor is the various loss of our setup, including the cavity loss and a total measurement loss owing to propagation loss and BHD and Mach–Zehnder interferometer inefficiencies. Another main factor is the intensity imbalance between ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ modes, because only an ${\mathrm{HG}}_{10}$ seed affects the correlation between ${\mathrm{HG}}_{01}$ and ${\mathrm{HG}}_{10}$ modes. Also, the optimum pump power was not reached in the experiment and the purity of the pump spatial profile needs to be further improved. The entanglement can be improved by optimizing the experimental setup to meet the needs of particular applications.

We experimentally demonstrated spatial quadripartite GHZ CV entanglement. The presented experimental system can also be used to generate spatial cluster entanglement and be expanded to higher-dimensional entanglement. The presented experimental system and scheme pave an efficient way to generate spatial multimode CV entanglement, which would improve the quality of an image in quantum imaging [21] and has advantages in terms of the complexity of protocols [10,22] to expand quantum channels. Another promising application is to build an interface between light and atoms and to hopefully generate multi-system CV entanglement of atoms [23–25].