## Going quantum with metamaterials

Metasurfaces should allow wafer-thin surfaces to replace bulk optical components. Two reports now demonstrate that metasurfaces can be extended into the quantum optical regime. Wang *et al.* determined the quantum state of multiple photons by simply passing them through a dielectric metasurface, scattering them into single-photon detectors. Stav *et al.* used a dielectric metasurface to generate entanglement between spin and orbital angular momentum of single photons. The results should aid the development of integrated quantum optic circuits operating on a nanophotonic platform.

## Abstract

Metasurfaces based on resonant nanophotonic structures have enabled innovative types of flat-optics devices that often outperform the capabilities of bulk components, yet these advances remain largely unexplored for quantum applications. We show that nonclassical multiphoton interferences can be achieved at the subwavelength scale in all-dielectric metasurfaces. We simultaneously image multiple projections of quantum states with a single metasurface, enabling a robust reconstruction of amplitude, phase, coherence, and entanglement of multiphoton polarization-encoded states. One- and two-photon states are reconstructed through nonlocal photon correlation measurements with polarization-insensitive click detectors positioned after the metasurface, and the scalability to higher photon numbers is established theoretically. Our work illustrates the feasibility of ultrathin quantum metadevices for the manipulation and measurement of multiphoton quantum states, with applications in free-space quantum imaging and communications.

The field of nanostructured metasurfaces offers the possibility of replacing traditionally bulky imaging systems with flat optics devices (*1*), achieving high transmission based on all-dielectric platforms (*2*–*7*). The metasurfaces provide a freedom to tailor the light interference by coherently selecting and mixing different components on a subwavelength scale, enabling polarization-spatial conversion (*4*, *7*–*12*) and spin-orbital transformation (*13*). Such capabilities motivated multiple applications for the regime of classical light, yet the metasurfaces have the potential to emerge as essential components for quantum photonics (*14*–*17*).

The key manifestations of quantum light are associated with nonclassical multiphoton interference, which is an enabling phenomenon for the transformation and measurement of quantum states. Conventionally, manipulation of multiphoton states is performed through a sequence of beam-splitting optical elements, each realizing quantum interference (*18*–*20*). Recent advances in nanotechnology have enabled the integration of beam-splitters and couplers on tailored plasmonic structures (*21*, *22*); however, material losses and complex photon-plasmon coupling interfaces restrict the platform scalability. We realize several multiphoton interferences in a single flat all-dielectric metasurface. The parallel quantum state transformations are encoded in multiple interleaved metagratings, taking advantage of the transverse spatial coherence of the photon wave functions extending across the beam cross section. In the classical context, the interleaving approach was effectively used for polarization-sensitive beam splitting (*8*, *9*, *11*, *12*), yet it requires nontrivial development for the application to multiphoton states.

We formulate and realize an application of the metasurface-based interferences for multiphoton quantum state measurement and reconstruction. We develop a metasurface that incorporates a set of *M*/2 interleaved metagratings [see part 3 of (*23*)], where *M* is an even number of diffracted beams forming imaging spots. Each metasurface is composed of nanoresonators with specially varying dimensions and orientations, according to the principle of geometric phase (*8*), to split specific elliptical polarization states (*7*), which would not be possible with conventional gratings [see parts 1 and 2 of (*23*)]. This performs quantum projections in a multiphoton Hilbert space to *M* imaging spots, which can be considered as output ports. Each port corresponds to a different elliptical polarization state (Fig. 1A), which is essential to minimize the error amplification in quantum state reconstruction (*24*). Then, by directly measuring all possible *N*-photon correlations, where *N* is the number of photons, from the *M* output beams, it becomes possible to reconstruct the initial *N-*photon density matrix, providing full information on the multiphoton quantum entanglement. For example, in Fig. 1B, we show a sketch of three gratings (top) which realize an optimal set of projective bases shown as vectors on the Poincaré sphere (middle) for *M* = 6.

The photon correlations between *M* output ports can be obtained with simple polarization-insensitive single-photon click detectors. The metasurface can be potentially combined with single photon–sensitive electron-multiplying charge-coupled device (EMCCD) cameras (*25*, *26*) to determine the spatial correlations by processing multiple time-frame images of quantum states. We consider quantum states with a fixed photon number *N*, which is a widely used approach in photon detection (*27*–*30*). The *N*-fold correlation data, stored in an array with *N* dimensions, are obtained by averaging the coincidence events over multiple time frames. For example, in Fig. 1C, we sketch a case with *N* = 2 photons and *M* = 6. In each frame, two photons arrive at different combinations of spots. After summing up the coincidence events over multiple time frames, we obtain a correlation in two-dimensional space. Following the general measurement theory of (*30*), we establish that, for an indistinguishable detection of *N*-photon polarization states (i.e., the detectors cannot distinguish which is which of the *N* photons), the required number of output ports to perform the reconstruction scales linearly with the photon number as *M* ≥ *N* + 3 (see Fig. 1B, bottom). For instance, with *M* = 6, up to *N* = 3 photon states can be measured.

The parallel realization of multiphoton interferences with a single metasurface offers practical advantages for quantum state measurements. Conventional quantum state tomography (*27*) methods based on reconfigurable setups can require extra time and potentially suffer from errors associated with the movement of bulk optical components (*27*) or tuning of optical interference elements (*31*). Moreover, the conventionally used sequential implementations of projective measurements present a fundamental limit for miniaturization while being inherently sensitive to fluctuations or misalignment between different elements, especially for higher–photon number states. The emerging methods based on static transformations implemented with bulk optical components (*19*) or integrated waveguides (*28*–*30*) still require multiple stages of interferences. By contrast, our quantum metasurface provides an ultimately robust and compact solution, the speed of which is limited only by the detectors.

We fabricate silicon-on-glass metasurfaces designed for *M* = 6 and 8 imaging spots using standard semiconductor fabrication technology [see parts 4 and 7 of (*23*) for details]. The experimentally determined polarization projective bases obtained through classical characterization are plotted on the Poincaré sphere in Fig. 2A for a metasurface with *M* = 6 that is used later for quantum experiments. The transfer matrix measurements confirm that the polarization projective bases are close to the optimal frame. The condition number, a measure of error amplification in the reconstruction [see part 1 of (*23*)], is 2.08, close to the fundamental theoretical minimum of . The reconstruction is immune to fabrication imperfections because their effect is fully taken into consideration by performing an experimental metasurface characterization with classical light after the fabrication [see parts 6 and 10 of (*23*)].

First, we show that our metasurface enables accurate reconstruction of the quantum-polarization state of single photons. A heralded photon source is used at a wavelength of 1570.6 nm on the basis of spontaneous parametric down conversion (SPDC) in a nonlinear waveguide [see parts 5, 8, 9, and 11 of (*23*) for details]. The heralded single photons are initially linearly polarized. They are prepared in different polarization states by varying the angle of a quarter-wave plate (QWP), then sent to the metasurface, and each diffracted photon beam is collected by a fiber-coupled interface to the single-photon detectors. By measuring the correlations with the master detector, we reconstruct the quantum-polarization state from the photon counts at the six ports. The results are shown in Fig. 2B, where the curves are theoretical predictions and dots are experimental measurements. We observe that the measurement errors are dominated by the single-photon detection shot noise, which is proportional to the square root of the photon counts, as indicated by the error bars. We use the measured photon counts to reconstruct the input single-photon states by performing a maximum-likelihood estimation (*27*) and plot them on a Poincaré sphere in Fig. 2C. The reconstructed states present a high average fidelity of 99.35% with respect to the prepared states.

Next, we realize two-photon interference, the setup of which is conceptually sketched in Fig. 3A. The SPDC source generates a photon pair with horizontal (H) and vertical (V) polarizations, with the path length difference between polarization components controllable by a delay line [see part 12 of (*23*) for details]. We measure the effect of delay on the two-photon interference, analogous to the Hong-Ou-Mandel (HOM) experiment (*32*). In such a nontrivially generalized two-photon interference, we expect a dip or peak depending on the two-by-two transfer matrix from the two-dimensional polarization state vector to a chosen pair of ports, where † denotes transpose conjugate and **u*** _{a}* and

**u**

*are the projective bases of ports*

_{b}*a*and

*b*, respectively. We note that

**T**

*corresponds to an effective Hermitian Hamiltonian only if*

_{a,b}**u**

*and*

_{a}**u**

*are orthogonal, resulting in a conventional HOM dip, whereas otherwise, a HOM peak can appear analogous to a lossy beam splitter (*

_{b}*22*). Here we set the angle of the QWP at θ = 0°, which means that the photon pairs are in a state ρ(θ = 0°), where one photon is H polarized and another is V polarized. As reflected in the Poincaré sphere of Fig. 3B—where the red arrows denote projective bases of the two ports (

**u**

*and*

_{a}**u**

*) and the blue arrows represent the polarization of the photon pairs, one photon in H and the other in V polarization—we see that the state vector*

_{b}**u**

_{1}points in the opposite direction of

**u**

_{6}. We find that, in this case, photons with cross-polarized entanglement in H-V basis will give rise to a dip in the interference pattern with the variation of path-length difference (see Fig. 3B, left). Such a behavior is directly caused by the coalescence nature of bosons. The situation is quite different if we measure such an interference between ports

*a*= 1 and

*b*= 5, because

**u**

_{1}and

**u**

_{5}are far from being orthogonal. This can be seen from the red arrows in the Poincaré sphere of Fig. 3C, where the angle between the two vectors representing

**u**

_{1}and

**u**

_{5}is much smaller than π. For entangled photons with H and V polarization in a pair, interference under the transfer matrix

**T**

_{1,5}leads to a peak instead of a dip when varying the path difference in the delay line. Indeed, in Fig. 3C (left) we observe a peak, which is related to the anticoalescence of bosons in transformations induced by non-Hermitian Hamiltonians, a nontrivial generalization of the HOM interference analogous to (

*22*). For details of the theoretical predictions and experimental methods, see part 4 of (

*23*).

As a following step, we measure all 15 twofold nonlocal correlations between the *M* = 6 outputs from the metasurface for a given input state where the time delay is fixed to zero. This provides us full information to accurately reconstruct the input two-photon density matrix. We use two single-photon detectors to map out all possible output combinations, although this could potentially be accomplished even more easily with an EMCCD camera. We show representative results for two different states ρ(θ = 0°) and ρ(θ = 37.5°) in Fig. 3, D and E, and Fig. 3, F and G, respectively. Note that ρ(θ = 0°) is a state in which photon pairs have cross-polarized entanglement beyond the classical limit, yet it is not fully pure [see part 4 of (*23*)], providing a suitable test case for reconstruction of general mixed states. In Fig. 3D, we show the measured twofold correlations for the input state ρ(θ = 0°), and the reconstructed density matrix is shown in Fig. 3E. That only the bunched four central elements are nonzero confirms the cross-polarized property of our photon pairs in H-V basis. Moreover, the nonzero element implies the presence of two-photon entanglement. It is smaller compared to the diagonal element , indicating that the polarization state is not fully pure. Although ρ(θ = 0°) only has nonzero elements in the real part of the density matrix, we also show the measurement and reconstruction of ρ(θ = 37.5°), which contains nontrivial imaginary elements, in Fig. 3, F and G. In both cases, we achieve a very good agreement between the predicted and reconstructed density matrices, as evidenced by high fidelity exceeding 95%. The correlation counts are obtained by a Gaussian fitting to the correlation histogram to remove the background, which is less than 10% of the signal for all measurements shown in Fig. 3F; see details in part 12 of (*23*).

Our results illustrate the manifestation of multiphoton quantum interference on metasurfaces. We formulate a concept of parallel quantum state transformation with metasurfaces, enabling single- and multiphoton state measurements solely based on the interaction of light with subwavelength thin nanostructures and nonlocal correlation measurements without a requirement of photon number–resolvable detectors. This provides ultimate miniaturization and stability combined with high accuracy and robustness, as we demonstrate experimentally via reconstruction of one- and two-photon quantum-polarization states, including the amplitude, phase, coherence, and quantum entanglement. In general, our approach is particularly suitable for imaging-based measurements of multiphoton polarization states, where the metasurface can act as a quantum lens to transform the photons to a suitable format for the camera to recognize and retrieve more information. Furthermore, there is the potential to capture other degrees of freedom associated with spatially varying polarization states for the manipulation and measurement of high-dimensional quantum states of light, with applications including free-space communications and quantum imaging.

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We gratefully thank H. Bachor, M. Scully, I. Walmsley, and F. Setzpfandt for fruitful discussions; R. Schiek and Y. Zarate for help in developing ovens for waveguide temperature control; and M. Liu for advice on numerical simulations.

**Funding:**This work was supported by the Australian Research Council (including projects DP160100619, DP150103733, and DE180100070) and the Ministry of Science and Technology (MOST), Taiwan, under contract 106-2221-E-008-068-MY3. A portion of this research was conducted at the Center for Nanophase Materials Sciences, which is a U.S. DOE Office of Science User Facility.

**Author contributions:**K.W., D.N.N., and A.A.S. conceived and designed the research; K.W. and L.X. performed numerical modeling of metasurface design; S.S.K. and I.I.K. fabricated the dielectric metasurfaces; H.-P.C. and Y.-H.C. fabricated nonlinear waveguides; K.W., J.G.T., H.-P.C., M.P., and A.S.S. performed optical experimental measurements and data analysis; A.A.S., D.N.N., and Y.S.K. supervised the work; K.W., A.A.S., D.N.N., and Y.S.K. prepared the manuscript and supplementary materials in coordination with all authors.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in this study are presented in the paper or in the supplementary materials.