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Abstract
Multipartite entanglements are essential resources for proceeding tasks in quantum information science and technology. However, generating and verifying them present significant challenges, such as the stringent requirements for manipulations and the need for a huge number of building-blocks as the systems scale up. Here, we propose and experimentally demonstrate the heralded multipartite entanglements on a three-dimensional photonic chip. Integrated photonics provide a physically scalable way to achieve an extensive and adjustable architecture. Through sophisticated Hamiltonian engineering, we are able to control the coherent evolution of shared single photon in the multiple spatial modes, dynamically tuning the induced high-order W-states of different orders in a single photonic chip. Using an effective witness, we successfully observe and verify 61-partite quantum entanglements in a 121-site photonic lattice. Our results, together with the single-site-addressable platform, offer new insights into the accessible size of quantum entanglements and may facilitate the developments of large-scale quantum information processing applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Entanglements play a crucial role in quantum-based technologies and are vital resources for implementing many applications, including quantum teleportation, quantum key distributions and quantum metrology [1–5]. The ultimate goal is to establish scalable and controllable quantum systems that promote the development of quantum technologies [6]. As the number of particles involved increases, entanglements become more effective resources for practical quantum information processing and can also deepen our understanding of the quantum world [7]. Despite many prospective applications of entanglements, both the generation and detection of multipartite entangled states are still very challenging for both experimental and theoretical research [8–10].
Furthermore, entanglements are often fragile and sensitive to particle losses, requiring significant resources to build a reliable system. Among the various classes of multipartite entangled states, the W-states have received considerable attention due to their robustness against losses, even when tracing several modes [11,12]. The W-states have been used in quantum communications and other quantum tasks due to their easy of use and robustness to losses compared to other entangled states [3]. The W-states are obtained from a superposition of single qubits with equivalent amplitude distributions in $N$ spatial modes. After the introduction by a three-qubit archetype [13], there have been efforts to scale up the system to millions of qubits [9,11,12,14–17]. However, it remains challenging to address and control single qubits individually for practical applications.
Various methods have been developed to create single-addressed W-states, such as sharing a single photon coherently in a splitter-based one-dimensional network [17] or in multiplexed spatial modes of an atomic ensemble [9,18]. These methods allow the quantum superposition of single qubits, enabling the possibility of building complex quantum networks for quantum key distributions [3,19], quantum teleportation [20,21], stationary quantum interfaces for future quantum internet and testing the quantum non-locality of single-particle quantum states [9,22,23]. While generating single-particle W-states is comparatively easier than other multipartite quantum states, high-order W-states generation is still a challenging task for many platforms, particularly those with complex and demanding implementations [12,24]. Quantum walks, which are advanced tools in quantum information processing, can potentially operate quantum algorithms, realize quantum computing or simulation, and perform quantum state engineering [17,25–36]. Utilizing quantum walks on an integrated quantum optical platform has the potential to be a reliable and efficient way to achieve scalable photonic W-states, due to the adjustable and repeatable fabrication capabilities of integrated photonics platform [37–39].
Here, we present the generation and verification of two-dimensional (2D) high-order W-states on a three-dimensional (3D) integrated photonic chip in a scalable and repeatable fashion. The exploration of higher dimensions can lead to novel applications of quantum states and the investigation of complex physics on lower dimensions [30,35,39]. We experimentally demonstrate $3\times 3$, $4\times 4$, $5\times 5$ and $8\times 8$ modes W-states via femtosecond laser direct-writing technique [37]. The generated W-states present highly symmetrical distributions due to the central excitation in the designed square configurations, which satisfies the precondition of W-states on a 2D lattice. By utilizing an effective witness, we verify the multipartite entanglements of different orders, observing up to 61-partite W-states. In principle, our scheme can be extended to generate "classically-entangled" states by engineering multiple spatial modes of single photon [40–42], and can be scaled to any desired orders by sophisticatedly engineering the internal structures of the photonic chip [38,39,43–45].
2. Experiment
As Fig. 1 shows, a single photon coherently excite an ensemble of $N$ spatial modes. The single photon undergoes a unitary operation to gradually spread into $N$ uniform modes, generating optical W-states $\vert W_{N}\rangle$. These spatial modes can be described in the form of bosonic creation operators $a_{n}^{\dagger}$ corresponding waveguide modes number $n=1,2,3,\ldots,N$. Consequently we get,
where $\phi _{n}$ and $\vert 0\rangle$ represent arbitrary relative phases and the vacuum state of the ensemble respectively.
Fig. 1. Diagram of multipartite entanglement generation and verification. A heraleded single photon is injected into the central port of the the lattice on the photonic chip. Harmiltonian egineering inside the chip realizes a unitary operation to spread the single photon into $N$ channels during the evolution process. The generated W-states are located on the output facets of the photonic chip, and thus detected and verified by the entanglement witness.
In our scheme, we prepare a series of $11\times 11$ square lattices which are made up of uniform waveguides. When we inject single photon into the central waveguide of the 2D lattice, single photon gradually spreads over the lattice [46] due to the quantum tunnelling between adjacent waveguides (see SMs Section 1 for details). The quantum tunnelling process can be regarded as single-photon propagating along longitudinal coordinate $z$ which represents the evolution length. Then we can obtain the evolution operator $U(z)=e^{-izH}$, where $H$ is the coupling matrix depand on the lattice structures. The evolution operator shares the same expression with continuous-time quantum walks [26,47]. Therefore, we can describe the final states $\vert \Psi (z)\rangle$ with the probability amplitudes $U_{c,n}(z)$ after propagating a fixed evolution length $z$: $\vert \Psi (z)\rangle = (\sum _{n=1}^{N}U_{c,n}(z)a_{n}^{\dagger})\vert 0\rangle$, where $c$ represents the central excitation. In addition, as Equation (1) shows, the photon number distributions of W-states require that all the channels have uniform intensity. In other words, all the excited channels in the generated W-states have almost the same photon number counts at the final evolution length $z_{f}$ in quantum regime.
To achieve the uniform distributions theoretically, we gradually increase the evolution lengths $z$ of the $11 \times 11$ square lattice in the quantum walks. We design a recursive algorithm to identify the matched evolution lengths for each channel and the initial W-states order is set to $3\times 3$. The algorithm records the evolution length when the number of channels with similar intensity distributions surpasses the previous record and continues searching for the next matched evolution lengths. When the photons reach the boundary of the lattice, the algorithm halts if the intensity of the boundary channels is higher than that of the adjacent channel inside. This approach ensures that the intensity distributions of each channel are nearly uniform, helping us to find multi-partite W-states with different orders in the $11\times 11$ square lattice.
We employ the above algorithm to simulate the photon number distributions with different evolution length $z$. Due to the highly symmetrical configuration of the 2D waveguide array, we theoretically find the 3$\times$3, 4$\times$4, 5$\times$5 and 8$\times$8 lattices shown in Fig. 2(a)-(d) exhibit nearly equal photon number distributions among excited channels. All the valid channels are labeled by dashed circles in Fig. 2 and the unlabeled channels can be considered as coupling noises along the evolution process. Then, the experimental lattices are fabricated by femtosecond laser direct writing technology with a fixed square lattice cross-section consisting of $11\times 11$ channels. By varying the final evolution lengths $z_{f}^{i}$, $i=1, 2, 3, 4$, we obtain different orders of W-states (see SMs Section 2 for details). As the single photon gradually spreads over the chip channels, we can observe non-trivial distributions, for example, 3$\times$3, 4$\times$4, 5$\times$5 and 8$\times$8 in Fig. 2(e)-(h). The straight waveguide structure of the chip maximizes the efficiency and integration, resulting in a smaller chip size. From direct observation, we can find that the intensity distributions of the excited channels are almost uniform, but excited channels are not always in adjacent waveguides.
Fig. 2. Photon number distributions of 2D W-states. (a), (b), (c), (d) are simulated output photon number distributions of photonic W-states, 3$\times$3, 4$\times$4, 5$\times$5 and 8$\times$8 modes from left to right respectively, by tuning evolution lengths. Effective channels in the generated W-states are labeled by dashed circles. (e), (f), (g), (h) are corresponding experimental results of photonic W-states accumulated by IsCMOS camera, in which one of the correlated photons is injected to the photonic chip while the other one acts as a trigger to herald the IsCMOS camera with a time window of 3 ns.
The single-photon distributions have been shown in Fig. 2. The first row (a) $\sim$(d) displays the theoretical simulations of different evolution lengths $z_{f}^{i}$, corresponding to the experimental results (e) $\sim$ (h) shown in the second row. Coherence reveals the superposition of single-photon W-state. Classical random walks will result in a uniform distribution among all the channels after a long evolution, without quantum superposition. However, as Fig. 2 shows, quantum walks with different evolution lengths correspond to different probability distributions, which are not uniform among all the channels. Therefore, the W-states produced in our work are indeed coherent superposition of single photon. To estimate the deviation between experiment and theory, we introduce the similarity $S_{i} = (\sum _{j}\sqrt {p_{i,j}^{exp}\cdot p_{i,j}^{th}})^{2}/(\sum _{j}p_{i,j}^{exp}\sum _{j}p_{i,j}^{th})$, where the $p_{i,j}$ is the probability of detecting one photon at $j$ when excited $i$. The similarity results are 95.5 $\%\pm$0.6$\%$, 96.3 $\%\pm$0.5$\%$, 97.2 $\%\pm$0.5$\%$ and 96.8 $\%\pm$0.4$\%$ when the pumping power is 100 $mW$.
As we mentioned above, the W-states are robust to particle losses [48]. For the $N$-partite W-state $\vert W_{N} \rangle$, we can define the state $\left | W_N \right \rangle \equiv (1/\sqrt {N})\left | N-1,1 \right \rangle$, where $\left | N-1,1 \right \rangle$ represents $N-1$ zeros and $1$ ones are included in the totally symmetric state [13]. If any $N-2$ parties lose the information, by symmetry of the state $\vert W_{N} \rangle$, the reduced density operators $\rho _{k,u}$ of $\vert W_{N} \rangle$ can be described as:
The detailed experimental setup is illustrated in Fig. 3. It consists of three main parts: source generation, chip function and detection/verification implementations. First, we prepare a heralded single-photon source to excite the lattice. We focus 780$nm$ femtosecond laser pulses on an $LiB_{3}O_{5}$ (LBO) crystal to generate ultraviolet pulses with a wavelength of 390 $nm$. The ultraviolet pulses then pump a $\beta -BaB_{2}O_{4}$ (BBO) crystal that satisfies type-II phase-matching, generating correlated photons via spontaneous parametric downconversion in a Beam-like scheme [49]. One of the correlated photons is collected by a polarization manipulation fiber and coupled into the central channel of the photonic chip, which is labeled by red color waveguide in the window view shown in Fig. 3. Under the action of the evolution operators $U$, the heralded single photon dynamically propagates in the photonic lattices, generating W-states located on the output facets of the corresponding lattices that undergo different evolution lengths. Meanwhile, the other photon serves as a trigger to herald the generated W-states. To measure the single-photon number distribution, we accumulate the single-photon counts with an image-sensor complementary metal oxide semiconductor (IsCMOS) camera triggered by the corresponding correlated photon, as shown in Fig. 3. We set the coincident time window to 3$ns$ to reduce the environment noises.
Fig. 3. Sketch of the experimental setup. A 780$nm$ fs laser pumps the $LiB_{3}O_{5}$ (LBO) crystal to generate ultra-violet pulses at the wavelength of 390 $nm$. Then the ultra-violet pulses are focused into a $\beta -BaB_{2}O_{4}$ (BBO) crystal to generate the correlated photon pairs in a Beam-like scheme satisfying type-II phase-matching. To measure single photon number distribution, we accumulate single photon counts by IsCMOS camera which is triggered by the corresponding correlated photon. The generated W-states are located on the output facets of corresponding lattices which undergo different evolution lengths. We utilize a balanced beam splitter (BS) to separate the output photons to measure higher-order excitations, second order in our experiment. We use multi-mode fibers attached to two single photon detectors to measure the $g_{2}(0)$ correlations and calculate the second order excitations.
3. Results
In addition to the coupling noises (unlabeled channels), the higher-order excitations of the photons also disturb the quality of W-states generation from the ideal. In our experiment, we can estimate the generated states by the following expression:
where $p_{0}, p_{1}$ and $p_{2}$ represent the population probability of different order excitations, and meet the realationship $p_{0}+p_{1}+p_{2}=1$ and $p_{2}=p_{1}^{2}g_{2}(0)/2$ (see SMs Section 3 for details). $\rho _{0}$, $\rho _{1}$ and $\rho _{2}$ are corresponding density matrix of zero, one and double excitations. We only consider second order excitation because the higher ones are too small in contrast with one excitation. The state fidelity $F$ can be defined as $F = \langle W_{N}\vert \rho \vert W_{N}\rangle = p_{1}\langle W_{N} \vert \rho _{1} \vert W_{N} \rangle$.To measure $g_{2}(0)$, we use a balanced beam splitter to separate the generated states and evaluate the influence of higher-order excitations as shown in Fig. 3. We collect the divided two patterns into two multi-mode fibers through the lens and then the fibers are attached to two single-photon detectors (avalanche photodiode). The other photon of correlated photon pairs is directly connected to another single-photon detector to trigger the two-photon events. The multi-channel coincidence module (MCCM) records all the coincidence clicks from single-photon detectors. The heralded $g_{2}(0)$ correlation is defined by $g_{2}(0)=\frac {N_{T}N_{T,2,3}}{N_{T,2}N_{T,3}}$, where $N_{T}$ is the number of trigger photon, $N_{T,2(3)}$ is the coincidence number of trigger photon and detector 2 (3), and $N_{T,2,3}$ represents the total coincidence counts of all three single-photon detectors.
Figure 4 shows the heralded $g_2(0)$ correlations at different pumping powers ranging from 100$mW$ to 500$mW$. There is an apparent tendency that $g_{2}(0)$ increases with the growing of pumping power. The increase in $g_{2}(0)$ is due to the increase in multi-photon events, which reduce the entanglement depth and the state fidelity. We estimate the error deviations based on the assumption that the photons follow Poisson distributions. Since the pumping power can affect the brightness and single-photon properties of the source, one of the key points of the experiment is to balance the brightness and the value of $g_{2}(0)$ required for the experiment. Fig. 4(b) shows that the generated W-states of different orders have similar correlation $g_{2}(0)$ on the same pumping level, which originate from the photon sources. Therefore, we can select a lower pumping power that is bright enough for the experiment but has a lower $g_2(0)$ value, which guarantee the single-photon property and state fidelity. The $g_{2}(0)$ is approximately 0.03 at 100$mW$, demonstrating a low probability of second-order excitation. Higher excitations (third order and above) are in the level of square or cube compared with the second-order excitation, which can be ignored.
Fig. 4. Measurement of the single-photon heralded $g_2(0)$ correlations. (a) We measure the $g_{2}(0)$ correlations at different pumping powers from 100 to 500 $mW$ with an interval of 100 $mW$. The two-particle excitation events can be obtained from this correlations. It shows an obvious increasing tendency from lower power to a high power pump conditions. The photon number distribution of the photon pairs generated by spontaneous parametric down conversion which can be estimated as Poisson distribution. The error bars of $N$ photons can be estimated by $\sqrt {N}$. Then the error deviations of $g_2(0)$ are calculated by error propagation formula. The results of 3$\times$3, 4$\times$4, 5$\times$5 and 8$\times$8 lattices are laeled by different colors. (b) All the $g_{2}(0)$ correlations are arranged in one graph. They show the same tendency during the increasing pumping power. For simplicity, a linear fitted gray line is used to present the generality and tendency of these results.
The validation of whether an prepared state is entangled or not is a challenging task in quantum information experiments. To verify the multipartite quantum entanglements in our experiment [50], we adopt a loose entanglement witness to lower bound the high-order W-states (see SMs Section 4 for details), whose form can be expressed as [9],
To bound the entanglement depth $k$, we experimentally measure the population $p_{n}(n=0,1,2)$ and the fidelity $F$ for different pumping powers and lattice orders (see SMs Section 5 for details). To obtain the population $p_1=\eta _c\eta _h$, with the setup shown in Fig. 3, we measure the chip efficiency $\eta _c$ and the heralding efficiecy $\eta _h$ of the quantum source. We also calculate $p_2=p_{1}^{2}g_{2}(0)/2$ by measuring the heralded $g_{2}(0)$ as shown in Fig. 4 and then $p_0=1-p_1-p_2$. The fidelity $F=p_{1}p_v$, where $p_v$ represents the probability of the valid points.
By utilizing the entanglement witness, we verify the entanglement depth $k$ of our chip-based high-order multipartite entanglement. We use a genetic algorithm to optimize all the parameters (see SMs Section 6 for details) and scan through the parameter space. The value of $\zeta$ simultaneously changes along with the change of the parameters $\alpha _{k}$, $\beta _{k}$ $\gamma _{k}$ and $k$. We first set a parameter range of $\alpha _{k}$, $\beta _{k}$ $\gamma _{k}$, if there are no $\zeta$ values smaller than 0 for a long search in such parameter range, we reduce the entanglement depth $k$ to lower the witness requirement. Table 1 displays a set of optimal parameter settings with 100 $mW$ and different lattice orders, where we detect 61-partite W-states at most. Fig. 5 shows the entanglement depth $k$ of generated W-states for different pumping powers and orders. We observe the entanglement depth decreases with the pumping power increases. This is because higher $g_{2}(0)$ correlations lead to higher probability of producing multi-photon events, which destroys the fidelity of W-states. The experimental results demonstrate the successful preparation of high-order W-states.
Fig. 5. Multipartite entanglement verification. The final results of entanglement depth are shown in histograms. As the pumping power increases, that is the $g_{2}(0)$ correlations increases, the entanglement depth decreasess. By using a effective witness, 61-partite quantum entanglement is observed in a 121-site photonic lattice.
Table 1. Parameter settings for witness.a
4. Conclusions and discussions
In summary, we demonstrate the heralded multipartite entanglements on a three-dimensional photonic chip. We successfully observe and verify up to 61-partite quantum entanglement in a 121-site photonic lattice by utilizing an effective witness. We obtain W-states of different orders by only changing the evolution length of the waveguide lattice, which is very efficient and scalable. Our scheme provides a stable and scalable paradigm for large-scale quantum information processing tasks, and also provides the methods for both classical and quantum optical connection and state engineering. The photonic W-states chip can meet the requirements of on-chip quantum random number generation relying on the tunable distributions of 2D detection array [17,52,53]. Through sophisticated Hamiltonian engineering of 2D topology lattice, we may also design non-regular lattices and introduce new degree of freedom to tune the novel structures for state generation or manipulation. Our results pave the way towards robust and versatile sources used for the wide-scale deployment of quantum protocols, as well as new applications of quantum information processing tasks in 2D lattices.
Funding
National Key Research and Development Program of China (2017YFA0303700, 2019YFA0308700, 2019YFA0706302); National Natural Science Foundation of China (11690033, 11761141014, 11904229, 61734005, 62235012); Shanghai Municipal Education Commission (2019SHZDZX01, 20JC1416300, 2017-01-07-00-02-E00049); China Postdoctoral Science Foundation (2020M671091).
Acknowledgments
X.-M.J. acknowledges additional support from a Shanghai talent program and support from Zhiyuan Innovative Research Center of Shanghai Jiao Tong University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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