As a highly entangled quantum network, the cluster state has the potential for greater information capacity and use in measurement-based quantum computation. Here, we report generating a continuous-variable quadripartite “square” cluster state of multiplexing orthogonal spatial modes in a single optical parametric amplifier (OPA), and further improve the quality of entanglement by optimizing the pump profile. We produce multimode entanglement of two first-order Hermite–Gauss modes within one beam in a single multimode OPA and transform it into a cluster state by phase correction. Furthermore, the pump-profile dependence of the entanglement of this state is explored. Compared with fundamental mode pumping, an enhancement of approximately 33% is achieved using the suitable pump-profile mode. Our approach is potentially scalable to multimode entanglement in the spatial domain. Such spatial cluster states may contribute to future schemes in spatial quantum information processing.
© 2018 Chinese Laser Press
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 , 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  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  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 . In the spatial domain, entanglement between co-propagating modes in one beam has been demonstrated previously , and a cluster state was also created by defining quantum modes to be combinations of different spatial regions of one beam . 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  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.
2. THEORETICAL DESCRIPTION
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]28,29]. From the interaction Hamiltonian, Eq. (1), the down-converted signal and idler photons are produced in two OAM modes, and . 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, , , , and . In the following, we use the HG mode basis and rewrite as 25,26] 1(a)]. Performing the transformation introduces only a -phase shift in the two signal fields of the entangled output beam, resulting in and . Then, by coupling the two modes as and , the quadrature transformation relation becomes
Hence, the entanglement criteria, Eq. (3), read13,30] 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, , , and :
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 and down-conversion modes simultaneously, the transverse distribution of the pump mode is expressed as [28,29], where and denote the transverse profiles of the and modes, respectively, and 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 by8): 31]. By mixing opposite topological charges and , they obtained the superposition between and 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 and modes, the efficiency of the mode pump is two times that of . We can almost achieve optimal entanglement by simply using only the pump mode.
3. EXPERIMENTAL SETUP AND RESULTS
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 at 1080 nm). An additional KTP crystal (KTP2) of identical size is placed with the axis orthogonal to KTP1, to compensate the Gouy phase difference induced by astigmatism , and a 1080-nm half-wave plate oriented at 22.5° is inserted between the two crystals, so that the and 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 weak field at 1080 nm is injected into the OPA cavity as a seed beam, and a fundamental mode or light beam operating at 540 nm is used as the pump field to drive the OPA. Here, we achieve the -mode by converting the -mode with a mode converter , where the -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 beam and a vacuum-entangled 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 , , , and modes, respectively. We spatially tailor the measurement basis using a Mach–Zehnder interferometer with a Dove prism in one arm to convert a mode into 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 , 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 - and -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 , , , and modes. When pumping with the -mode beam (Fig. 3), we obtained the correlation variances with mean values of , , , and , 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 -mode as a pump provides better efficiency. All four correlation variances show improvements (Fig. 4); mean values of , , , and were obtained. Compared with 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 , where , , and are respective of the measured propagation efficiency, photodiode (ETX500) efficiency, and spatial overlap efficiency in the homodyne detector. Thus, the inferred inseparability values are: , , , and , and the inferred correlation variances (Fig. 4) are: , , , and .
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)  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 pump profile. The approach described can be extended to other more complex graph states if a suitable pump profile is used [35,36].
National Natural Science Foundation of China (NSFC) (91536222, 11674205); Ministry of Science and Technology of the People’s Republic of China (MOST) (2016YFA0301404); Program for the Outstanding Innovative Team of Higher Learning Institution of Shanxi and Shanxi “1331 Project.”
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