Multipartite entanglement is an essential physical resource enabling more complex processing. With its use, the fundamental quantum properties of light have been examined and various quantum-information tasks, such as quantum computation ^[1,2], quantum error correction ^[3], and quantum communication ^[4,5], have been successfully implemented. In recent years, a special type of multipartite entangled state, the cluster state, has attracted attention. Being highly entangled, the cluster state maintains an entanglement structure that is robust to local measurements ^[6] and can be used for measurement-based quantum computation ^[1,2,7–11].

There have been numerous demonstrations of cluster states, including the generation of four-mode ^[12,13] and eight-mode ^[14] cluster states. However, in using many cavities and separate beams to build the quantum state, the schemes require complex layouts. Very recently, though, a novel and potentially scalable method was proposed to encode deterministically quantum modes propagating within one beam, enabling each quantum mode to be manipulated by the same optical components, and cluster states of an arbitrary size to be generated using a limited and fixed optical setup ^[15–27]. Previous attempts to create cluster states within one beam have been exploited with orthogonal quantum modes in the frequency domain ^[15–17] and in the time domain ^[18–21] by encoding each mode at different times. To date, an ultra-large-scale entangled state consisting of more than 10000 entangled wave packets of light has been experimentally demonstrated ^[20]. In the spatial domain, entanglement between co-propagating modes in one beam has been demonstrated previously ^[22], and a cluster state was also created by defining quantum modes to be combinations of different spatial regions of one beam ^[23]. Yet, to the best of our knowledge, there has been no report on cluster-state generation using different spatial modes on the basis of Hermite–Gaussian (HG) modes and Laguerre–Gaussian (LG) modes. Spatial multimode entanglement in a single multimode optical parametric amplifier (OPA) has been explored ^[24,25], with the generation of continuous-variable (CV) hyperentanglement ^[25,26] and spatial quadripartite Greenberger–Horne–Zeilinger (GHZ) entanglement ^[27] being accomplished.

In this paper, we describe our proposal and demonstration in experiments of spatial-mode quadripartite cluster-state generation in a single OPA operating in multimode. We further enhance the nonlinear efficiency to improve the entanglement by selecting a suitable pump-profile mode. These spatial domain explorations are expected to advance both theory and experimentation on spatial-mode one-way quantum computation.

To create the multipart cluster state, multimode entangled beams are prepared first. We employ the multimode OPA model of Refs. ^[25,26], which performs multiplexing in the spatial domain and generates hyperentanglement of the polarization and orbital angular momentum (OAM) modes. In the interaction picture, the interaction Hamiltonian of the system is given by ^[25,26] where a^p and a^lj denote the bosonic annihilation operators for the pump and fundamental fields, respectively, superscripts j(j=i,s) signify signal and idler fields, respectively, and subscripts l=±1 index the quantum number of the OAM modes LG01 and LG0−1; parameter χ denotes the nonlinear coupling strength, which depends on, e.g., the cavity parameters and spatial overlap between the pump and down-conversion fields ^[28,29]. From the interaction Hamiltonian, Eq. (1), the down-converted signal and idler photons are produced in two OAM modes, LG01 and LG0−1. By performing a simple-basis transformation from the LG modes to the HG modes, the generated down-converted signal and idler beams also produce four HG modes, HG01i, HG01s, HG10i, and HG10s. In the following, we use the HG mode basis and rewrite Hint as where subscript 01(10) labels the HG01(10) mode. The hyperentanglement criteria for the polarization and OAM modes can then also be restated as ^[25,26] where X^01/10=a^01/10+a^01/10† and Y^01/10=i(a^01/10−a^01/10†) are the amplitude and phase quadrature operators of the HG01/10 mode, respectively, and V(B)=⟨B2⟩−⟨B⟩2 denotes the variance of the observable B, which indicates whether both the HG01 and HG10 modes are entangled independently. After simple transformations, these two-mode entanglements are further transformed into a spatial cluster state [Fig. 1(a)]. Performing the transformation U introduces only a π2-phase shift in the two signal fields of the entangled output beam, resulting in a^01s→ia^01s and a^10s→ia^10s. Then, by coupling the two modes as a^135os=(ia^01s+ia^10s)/2 and a^45°s=(ia^01s−ia^10s)/2, the quadrature transformation relation becomes

Hence, the entanglement criteria, Eq. (3), read which satisfy the definition of a CV cluster-state given by ^[13,30] where A is the adjacency matrix, representing the graph of a given CV state. Therefore, the spatial multimode OPA can be used to generate a spatial-mode cluster state. We give the graph-state picture representation [Fig. 1(b)] describing the “square” correlation among the four modes. Each cluster node, corresponding to a spatial mode, is represented by a circle. Neighboring nodes are connected by lines and represent the bipartite entanglement between two spatial modes. Using the quadrature transformation relations, Eq. (4), the entanglement criteria, Eq. (5), can be restated as phase quadrature variances of the HG modes, HG01, HG10, HG45° and HG135°:

Clearly, the spatial-mode cluster state, Eq. (5), can be characterized using phase quadrature correlations for all four modes and is easy to measure in experiments.

Essentially, the spatial-state generation in optical parametric oscillator (OPO) depends closely on the transverse profile matching between pump mode and generated modes ^[26,27]. Different pump-mode profiles result in different nonlinear efficiencies and different pump thresholds, thus leading to different degrees of entanglement. More importantly, maximum entanglement can be obtained only with an optimal pump profile.

For the spatial cluster state, which generates HG01 and HG10 down-conversion modes simultaneously, the transverse distribution of the pump mode is expressed as νp(r→)=Σn=0∞cn[νn0(r→)+ν0n(r→)] ^[28,29], where νn0 and ν0n denote the transverse profiles of the HGn0 and HG0n modes, respectively, and cn is the corresponding coefficient. We can calculate the pump profile based on the definition of the coupling coefficient, which describes the spatial overlap between the pump and the two down-conversion modes in the transverse plane, and is defined by where μs(r→) and μi(r→) are the transverse distributions of the signal and idler modes, respectively, and α, the normalization coefficient of the squared signal, is defined as α2=∫−∞+∞μ4(r→)dr→. For the HG01/10 down-conversion mode, only two nonzero overlapping coefficients result from Eq. (8): for which Γ002+Γ02/202=1 holds, and subscripts 00 and 02/20 of Γ indicate the transverse distribution of the pump mode HG00 and HG02/20. Hence, the pump-mode profile for HG01/10 down-conversion modes can be HG00 and HG02/20 modes, and the optimal pump mode is the superposition of the HG00 and HG02/20 modes, of which the HG02/20 accounts for two thirds. The only pump-mode profiles for the spatial cluster state are therefore the HG00 and the superposition of HG02 and HG20 modes, where the superposition mode is equal to LG10 mode. The optimal pump profile is the superposition of the HG00 and LG10 modes, of which LG10 accounts for two thirds, and we have Γopt=1. When pumping in the HG00, LG10, and the optimal profile, the system increases its threshold inversely proportional to the square of the overlap coefficient. In 2017, Pereira et al. also found similar phenomena in type II second-harmonic generation, which is the reverse process of OPA ^[31]. By mixing opposite topological charges +1 and −1, they obtained the superposition between HG00 and LG10 modes. Moreover, the generation of higher radial orders was demonstrated when higher opposite charges were mixed in the crystal. This can motivate future investigations with higher radial orders and higher topological charges in the OPO as well.

Although the spatial profile of an ideal pump is the superposition of HG00 and LG10 modes, the efficiency of the LG10 mode pump is two times that of HG00. We can almost achieve optimal entanglement by simply using only the LG10 pump mode.

The experimental setup (Fig. 2) entails entanglement source generation, LG pump generation, and balanced homodyne detection. Our OPA cavity is formed by the rear surface of a type-II phase-matched potassium titanyl phosphate (KTP) crystal (KTP1) and an external mirror (radius of curvature is 50 mm, and R=95% at 1080 nm). An additional KTP crystal (KTP2) of identical size is placed with the z axis orthogonal to KTP1, to compensate the Gouy phase difference induced by astigmatism ^[32], and a 1080-nm half-wave plate oriented at 22.5° is inserted between the two crystals, so that the HG10 and HG01 modes of different polarizations resonate simultaneously in the OPA. This degeneracy is further optimized by individually controlling the temperature of the two KTP crystals. The length of the cavity is approximately 57.8 mm, resulting in a free spectral range of about 2.6 GHz and a finesse of 91 at 1080 nm. The cavity has a waist of 40 μm at 1080 nm and 28.5 μm at 540 nm. The cavity loss is approximately 1.44%.

A 45°-polarized HG01 weak field at 1080 nm is injected into the OPA cavity as a seed beam, and a fundamental mode or LG10 light beam operating at 540 nm is used as the pump field to drive the OPA. Here, we achieve the LG10-mode by converting the HG11-mode with a HG11−LG10 mode converter ^[33], where the HG11-mode is achieved by tailoring the fundamental mode with a four-quadrant phase mask and a filtering cavity. When operated in the de-amplification regime, the OPA produces simultaneously a bright entangled HG01 beam and a vacuum-entangled HG10 beam in the cavity. The output multimode entangled beam then passes through a quarter-wave plate with its fast axis aligned with the s-polarization axis of beam splitter output to induce a phase shift in the s-polarization with respect to p-polarization. Then the beam is analyzed using the balanced homodyne detection systems with the local oscillator operating in the HG01, HG10, HG45°, and HG135° modes, respectively. We spatially tailor the measurement basis using a Mach–Zehnder interferometer with a Dove prism in one arm to convert a HG01 mode into HG10 mode. This arrangement can be used to generate an arbitrary first-order mode when both the beam splitting ratio and the relative phase between two arms can be controlled. Measurements are conducted with the relative phase between the local oscillator and the signal beam locked to π2, so that the phase quadrature variances can be obtained.

We show the measured correlation variance spectra (Figs. 3 and 4) for the range of 1–6 MHz obtained using Eq. (7) with the HG00- and LG10-mode pump beams, respectively. Traces (2) correspond to the shot noise limit (SNL), which is obtained by blocking the signal beam, and traces (1) and (3) are the correlation variance and anti-correlation variance, respectively. In both figures, (a)–(d) correspond to the phase quadrature correlations for the HG01, HG10, HG45°, and HG135° modes. When pumping with the HG00-mode beam (Fig. 3), we obtained the correlation variances with mean values of 2.20±0.12dB, 2.30±0.12dB, 1.93±0.13dB, and 2.15±0.14dB, respectively, over the entire frequency range. As the analyzing frequency increases, the traces of the anti-correlation variance diminish because of the limited bandwidth of the electron gain of the detectors. The parameter settings of the spectrum analyzer are 300 kHz and 390 Hz for the resolution and video bandwidths, respectively. From the well-established inseparability criteria inequalities for the cluster-state entanglement, Eq. (7), we obtain

The LG10-mode as a pump provides better efficiency. All four correlation variances show improvements (Fig. 4); mean values of 3.15±0.11dB, 3.63±0.12dB, 3.20±0.12dB, and 3.32±0.14dB were obtained. Compared with HG00 pumping, an enhancement of approximately 33% is achieved. Their corresponding values for non-separability (Fig. 5) are also improved:

The measured inseparabilities and correlation are affected by various inefficiencies in the experiment. We estimate the total detection efficiency to be ηtotal=ηpropηphotηhd, where ηprop=0.96±0.02, ηphot=0.92±0.02, and ηhd=0.96±0.02 are respective of the measured propagation efficiency, photodiode (ETX500) efficiency, and spatial overlap efficiency in the homodyne detector. Thus, the inferred inseparability values are: V(Y^01i−Y^01s)=0.36±0.02, V(Y^10i−Y^10s)=0.30±0.02, V(Y^45°i−Y^45°s)=0.36±0.02, and V(Y^135°i−Y^135°s)=0.35±0.02, and the inferred correlation variances (Fig. 4) are: −4.44±0.11dB, −5.23±0.12dB, −4.44±0.12dB, and −4.56±0.14dB.

In addition to the inefficiencies in measurement discussed above, cavity loss resulting from the non-optimal dielectric coatings on optical elements is also a main factor that limits the quality of entanglement. Also, extra noise in the laser, imperfect phase locking, and so on, will also degrade the correlation. The entanglement can be improved by optimizing the experimental setup for future applications.

We have experimentally demonstrated the CV quadripartite spatial-mode “square” cluster-state generation by multiplexing orthogonal spatial modes within one beam in a single multimode OPA. The degree of freedom is extended in the spatial mode domain, and the generation setup is much simpler with only one OPA cavity. Realizing a compact Gaussian quantum computation using a multi-pixel detector or charge coupled device (CCD) ^[34] looks promising. Using this infinite spatial basis within a single beam, and the possible manipulation of the modes, makes it a practical contender for spatial multimode quantum communication systems. Furthermore, we explored the dependence of entanglement of the spatial cluster state on the pump profile, and demonstrated the enhancement for the spatial cluster state with a LG10 pump profile. The approach described can be extended to other more complex graph states if a suitable pump profile is used ^[35,36].