GoKawiilThe modern world has been shaped by semiconductor technologies, grounded in the intrinsic band structures of materials and in our ability to engineer those structures with precision6,7. Similarly, the emergence of photonic crystals four decades ago not only transformed the field of optics but also laid the groundwork for compact quantum technologies8,9. The possibility of replacing bulky optical set-ups with two-dimensional nanostructures, typically referred to as metasurfaces, has stimulated widespread interest in exploring their potential for the preparation, manipulation and detection of quantum light fields3,10. However, most implementations so far have focused on integrating single-photon emitters with metasurfaces and manipulating several degrees of freedom, such as frequency, polarization and orbital angular momentum2,11,12,13,14,15. Similar efforts have also been reported for entangled photon pairs14,16,17. Yet, given the enormous implications that controlling larger multiparticle systems on metasurfaces would have for scalable quantum technologies, numerous continuing efforts aim to demonstrate this capability2,3,18. Nevertheless, this goal has remained elusive so far.
Interest in multiphoton quantum systems originates from the complex interference phenomena they can host19,20,21,22, which are particularly valuable for quantum information technologies14,23,24,25. The nature of these interference processes depends on the quantum coherence properties of the multiphoton system, which are, in turn, determined by the quantum statistical characteristics of the corresponding light fields19,21,26,27. These fundamental properties define different kinds of light, such as single photon sources, coherent light and thermal light21,28,29. Unlike other degrees of freedom, such as polarization or frequency, which can be investigated and filtered using photonic metasurfaces2,3, the statistical properties of multiphoton systems cannot be directly accessed. So far, their identification has required characterizing the collective behaviour of the entire multiphoton system21,25,29. Consequently, no material has yet been shown to exhibit sensitivity to the statistical fluctuations or coherence properties of multiphoton systems. As a result, the implementation of operations based on the quantum coherence of multiphoton systems has remained unattainable so far.
Here we introduce, to our knowledge, the first class of room-temperature quantum materials that are intrinsically sensitive to the quantum statistical properties defining all forms of light. In close analogy with the emergence of allowed and forbidden bands in semiconductors and photonic crystals, the meta-atoms composing quantum statistical plasmonic metacrystals result in quantum statistical bands that enable selective transmission of light according to its quantum coherence28. We show that the response of these plasmonic metacrystals is governed by the geometry of the constituent meta-atoms and by their collective arrangement within the crystal lattice30. As a result, many-particle interactions mediated by the plasmonic metacrystal suppress forbidden quantum statistical fluctuations, which cannot propagate through the metasurface, whereas multiphoton fields supported by allowed statistical bands propagate robustly and without distortion. These statistical bands therefore enable the controlled transport of otherwise fragile multiphoton quantum states. The demonstration of the first room-temperature quantum material intrinsically sensitive to the quantum coherence of many-body systems has direct implications for improving the efficiency of energy-harvesting processes, which are fundamentally influenced by the coherence properties of light31,32,33,34. The ability to control these properties using a coherence-sensitive materials platform operating under ambient conditions opens transformative opportunities for solar energy conversion and the development of next-generation optoelectronic devices5,31,32. More broadly, our approach lays the groundwork for robust many-body quantum technologies operating beyond cryogenic environments1,3,5,18,22,23,30,35.
Sharing similarities with the formation of allowed and forbidden bands in semiconductors and photonic crystals, the repeating arrangement of meta-atoms in our plasmonic metacrystal results in multiparticle interference processes that are sensitive to the statistical fluctuations defining different kinds of light22,26,27. As illustrated in Fig. 1a, these processes establish allowed and forbidden quantum statistical bands whose emergence depends on the geometry of the plasmonic metacrystal. This response enables the first kind of optical materials that are sensitive to the quantum statistical properties of light. We characterize the quantum statistical fluctuations of multiphoton fields using the degree of second-order coherence, \({g}^{(2)}(0)=1+(\langle {(\Delta \hat{n})}^{2}\rangle -\langle \hat{n}\rangle )/{\langle \hat{n}\rangle }^{2}\), in which \(\hat{n}\) is the photon-number operator and \(\Delta \hat{n}=\hat{n}-\langle \hat{n}\rangle \) denotes the photon-number fluctuation operator21,28,29. Notably, our plasmonic metacrystal transmits multiphoton fields whose degrees of coherence fall within the allowed statistical bands, whereas fields lying in forbidden bands are filtered and thermalized until their statistics converge to the nearest allowed band. In general, this transmitted multiphoton field can be described as an average over transverse spatial configurations Σ as
$${\hat{\rho }}_{{\rm{out}}}=\int {\rm{d}}{\varSigma \bigotimes }_{i,j}|{\alpha }_{0}\rangle {\langle {\alpha }_{0}|}_{{\theta }_{{ij}},{\varSigma }_{{ij}}}.$$ (1)
Here \({|{\alpha }_{0}\rangle }_{{\theta }_{{ij}},{\varSigma }_{{ij}}}\) denotes the coherent state of amplitude α 0 associated with the meta-atom at position (i, j), with its linear polarization specified by the angle θ ij (refs. 36,37). In particular, θ ij = 0 corresponds to vertical polarization and θ ij = π/2 corresponds to horizontal polarization. The transverse spatial distribution of these photons is given by \({\varSigma }_{{ij}}({\bf{x}})=\sin ({\theta }_{{ij}}){S}_{{ij}}({\bf{x}})\varSigma ({\bf{x}})\), in which x denotes the transverse position and S ij (x) describes the masking function of the meta-atoms. The factor sin(θ ij ) accounts for the coupling efficiency of that meta-atom to the horizontal polarization component of the input field. Further details on the functional integral ∫dΣ and the form of S ij (x) are provided in the Supplementary Information. The description of the plasmonic metacrystal response presented below applies to all forms of input light fields29,31,32,33,34,38,39. Specifically, taking Σ(x) to be complex corresponds to sub-thermal input fields, with degree of second-order coherence 1 < g(2)(0) < 2, whereas restricting Σ(x) to be real yields superthermal multiphoton fields, with degree of second-order coherence 2 < g(2)(0) < 3.
Fig. 1: Quantum statistical plasmonic metacrystals. Full size image a, Operation of a quantum statistical plasmonic metacrystal composed of 100 nanoantennas that act as meta-atoms. The plasmonic field propagating along the gold surface of the structure mediates coupling between neighbouring meta-atoms, resulting in multimodal quantum multiparticle interference. These interactions lead to the formation of allowed and forbidden statistical bands that respectively transport or filter multiphoton fields according to their quantum statistics. b, Experimental verification of the phenomenon, in which multiphoton fields with varying degrees of second-order coherence are prepared to illuminate coupling gratings. These gratings generate propagating surface plasmons, which subsequently excite the meta-atoms of the plasmonic metacrystal. The transmitted multiphoton field, propagating perpendicular to the metacrystal surface, is collected by a microscope objective and imaged using a tunable telescope, enabling the examination of different propagation planes within the paraxial near-field region of the metacrystal, in which the formation of quantum statistical bands is confined. We refer to this region as the crystal depth (Supplementary Information). The selected plane is directed through a beam splitter and analysed using two PNR detectors. c, The plasmonic metacrystal consists of coupling input gratings and 100 nanoantennas measuring 200 × 400 nm with varying orientations, patterned on a 110-nm-thick gold film deposited on a 175-μm-thick glass substrate, with adjacent nanoantennas separated by 1 μm. The roughness of the gold film is measured to be approximately 0.5 nm. The red spots in this figure depict the surface plasmon mode and the yellow arrow marks the propagation direction towards the plasmonic metacrystal. Further information about the coupling gratings and plasmonic metacrystal is provided in Methods and the Supplementary Information. Scale bar, 10 μm.
To characterize the second-order coherence at the output of the metacrystal, we first evaluate the corresponding first-order and second-order intensity moments \({G}_{{\rm{out}}}^{(1)}(0)=\langle \hat{n}\rangle \) and \({G}_{{\rm{out}}}^{(2)}(0)=\langle :{\hat{n}}^{2}:\rangle \) (ref. 37). The notation :⋅: is used to indicate normal ordering. Moreover, the photon number operator \(\hat{n}\) is given by \(\hat{n}=\int \frac{{{\rm{d}}}^{2}x}{{(2\pi )}^{2}}[{\hat{a}}_{{\rm{H}}}^{\dagger }({\bf{x}}){\hat{a}}_{{\rm{H}}}({\bf{x}})+{\hat{a}}_{{\rm{V}}}^{\dagger }({\bf{x}}){\hat{a}}_{{\rm{V}}}({\bf{x}})]\). Here \({\hat{a}}_{s}({\bf{x}})\) annihilates photon density at position x, with s = H, V denoting horizontal and vertical polarizations. The coherent states \({\otimes }_{{ij}}{|{\alpha }_{0}\rangle }_{{\theta }_{{ij}},{\varSigma }_{{ij}}}\) from equation (1) are eigenstates of these annihilation operators (Supplementary Information). In particular, the eigenvalue of \({\hat{a}}_{{\rm{H}}}({\bf{x}})\) is given by \({\sum }_{{ij}}{\alpha }_{0}(2\pi )\sin ({\theta }_{{ij}})\sin ({\theta }_{{ij}}){S}_{{ij}}({\bf{x}})\varSigma ({\bf{x}})\), whereas that associated with \({\hat{a}}_{{\rm{V}}}({\bf{x}})\) becomes \({\sum }_{{ij}}{\alpha }_{0}(2\pi )\sin ({\theta }_{{ij}})\cos ({\theta }_{{ij}}){S}_{{ij}}({\bf{x}})\varSigma ({\bf{x}})\). These eigenvalue relations directly yield \({g}_{{\rm{out}}}^{(2)}(0)={G}_{{\rm{out}}}^{(2)}(0)/{[{G}_{{\rm{out}}}^{(1)}(0)]}^{2}\) with
$$\begin{array}{c}{G}_{{\rm{out}}}^{(1)}(0)=|{\alpha }_{0}{|}^{2}\int {\rm{d}}\varSigma \int {{\rm{d}}}^{2}x{\varSigma }^{\ast }({\bf{x}})\varSigma ({\bf{x}})\sum _{i,j}{\sin }^{2}({\theta }_{{ij}}){S}_{{ij}}({\bf{x}}),\\ {G}_{{\rm{out}}}^{(2)}(0)=|{\alpha }_{0}{|}^{4}\int {\rm{d}}\varSigma \int {{\rm{d}}}^{2}{x}_{1}{{\rm{d}}}^{2}{x}_{2}{\varSigma }^{\ast }({{\bf{x}}}_{1}){\varSigma }^{\ast }({{\bf{x}}}_{2})\varSigma ({{\bf{x}}}_{2})\varSigma ({{\bf{x}}}_{1})\\ \,\times \sum _{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}{\sin }^{2}({\theta }_{{i}_{1}{j}_{1}}){\sin }^{2}({\theta }_{{i}_{2}{j}_{2}}){S}_{{i}_{1}{j}_{1}}({{\bf{x}}}_{1}){S}_{{i}_{2}{j}_{2}}({{\bf{x}}}_{2}).\end{array}$$ (2)
The response of each individual meta-atom is captured by S ij (x). By contrast, the array that forms the plasmonic metacrystal is specified by the set of polarization angles θ ij , which encodes the collective multipolar response arising from the arrangement of meta-atoms. Numerical evaluation of the multiphoton field transmitted through a statistical plasmonic metacrystal reveals clear design principles for quantum statistical control. We report numerical calculations in the Supplementary Information, showing that the size of each meta-atom sets the allowed values of the second-order coherence, whereas the number of meta-atoms and their relative orientations fine-tune the statistical bandwidth of the crystal. As a result, quantum statistical bands do not necessarily arise in arbitrary plasmonic structures2,3,17,19,30. When a plasmonic aperture becomes sub-wavelength, higher-order multipolar oscillations are suppressed40, producing a localized oscillating meta-atom with a uniform phase and enabling the selection of specific values of the second-order coherence. As described by equation (2), near-field coupling between meta-atoms further induces distinguishable and indistinguishable multiparticle interactions that govern the width of the statistical bands36,37. Meta-atoms aligned along the same direction lead to indistinguishable multiparticle interference, whereas differently oriented meta-atoms produce distinguishable multipolar effects.
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