Abstract
Metasurfaces have proven themselves an exotic ability to harness light at nanoscale, being important not only for classical but also for quantum optics. Dynamic manipulation of the quantum states is at the heart of quantum information processing; however, such function has been rarely realized with metasurfaces so far. Here, we report an alloptical dynamic modulation of the photonic quantum states using the nonlinear metasurface. The metasurface consists of a metallic nanostructure combined with a photoisomerizable azo layer. By tuning the plasmonic resonance through optically switching the azo molecules between their binary isomeric states, we have realized dynamic control of transmission efficiencies of orthogonally polarized photons and also the phase delay between them, thereby an entangled state was efficiently controlled. As an illustration, a quantum state distillation has been demonstrated to recover a Bell state from a nonmaximally entangled one to that with fidelities higher than 98%. Our work would enrich the functions of the metasurface in the quantum world, from static to dynamic modulation, making the quantum metasurface going practical.
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Introduction
Nonlinear metasurfaces have recently captured an intense interest. Compare to the passive ones showing fixed optical properties, the response of the nonlinear metasurfaces can be tuned by external optical stimuli in an active manner. Such a dynamic framework provides new degrees of freedom to manipulate light behaviors in the time domain, and facilitates a transition from the traditional static metadevices to their dynamic counterparts^{1,2}. In this context, various fascinating alloptical modulating functions have been demonstrated, including light phase modulation^{3}, nonlinear polarization conversion^{4}, spectral tuning^{5}, spatial light modulations^{6}, and so on^{7}. However, all these dynamic functions were mainly motivated by the applications for classical light. On the other hand, the interest in applying metasurfaces to quantum optics has also been growing rapidly since the first experimental verification that photon entanglement survives through plasmonic nanohole arrays^{8,9,10}. Various remarkable success has been demonstrated^{11}, including generation of entangled multiphotons^{12,13,14}, reconstructing entangled states^{15}, quantum edge detection using dielectric metasurfaces^{16}, coherent perfect absorptions of single photons^{17}, and so on. However, until now the above quantum metasurfaces are mostly static in nature. The dynamic functions of metasurfaces have rarely been used for manipulating photonic quantum states, which are beneficial to construct reconfigurable quantum information systems with a compact profile in the future.
Quantum information technique holds potential supremacy over classical technologies in computational efficiency and communication security. One of the most basic concepts and important resources for quantum optical applications is entangled photon pairs of the form^{18}
where \(\left V \right\rangle\) and \(\left H \right\rangle\) represent the vertical and horizontal polarization state of a photon, subscripts S and I represent the signal path and idle path, and α and β are complex amplitude of \(\left {V_S} \right\rangle \left {V_I} \right\rangle\) and \(\left {H_S} \right\rangle \left {H_I} \right\rangle\) components, respectively. The manipulations of such entangled states are fundamental in quantum algorithms for quantum computation and quantum communications^{19,20,21}. In general, maximally entangled quantum states (α/β = 1) are required. However, owing to decoherence and dissipation in the quantum system, practical states normally degraded to be less entangled (\(\alpha /\beta \ne 1\)). Thus, it is essential to recover the maximally entangled states from these imperfect ones. Quantum entanglement distillation is one important method to settle this problem, which transforms N copies of an arbitrary entangled state into some number of approximately pure Bell pairs using only local operations and classical communication^{22,23,24,25}. It has been reported that anisotropic structured metasurfaces can be used in quantum state distillation by employing their polarizationdependent transmission efficiencies. However, those metasurfaces were still static, and can only compensate for certain nonmaximally entangled quantum states without any tunable ranges, which is apparently inconvenient for practical use^{26}.
In this paper, we go beyond the above static metasurface scenario, and present the nonlinear metasurface for alloptical manipulation of the polarizationentangled photon states, and more specifically, the task of entanglement distillation, as shown by the schematic in Fig. 1(a). The metasurface consists of an anisotropic structured nanostructure layer covered by a photoisomerizable ethyl red film. By optically switching the transmission contrast and phase retardation between the orthogonally polarized photons through the metasurface, the states of polarizationentangled photon pairs were controlled efficiently. A nonmaximally entangled state was recovered to that with fidelities higher than 98%. Our results open up a new horizon towards reconfigurable quantum systems and have the potential for realizing integrated onchip quantum applications in the future.
Results
Our metasurface consists of an array of asymmetric split ring (ASR) slits, which were manufactured by focused ion beam (FIB) milling through a 45 nm thick gold film rested on a fused silica substrate. The total array occupies a footprint of 21 × 21 μm^{2}. Schematic of one unit cell and scanning electron microscope (SEM) image are illustrated in Fig. 1(b). The length and the width of the slits are \(l = 250\,{{{\mathrm{nm}}}}\) and \(w = 45\,{{{\mathrm{nm}}}}\), respectively, and the gap d between “C” and “I” slits is 95 nm. The lattice constant of the array \(P = 300\,{{{\mathrm{nm}}}}\) is much smaller than the wavelength of the photon quanta used in our experiments (808 nm), so that photons can transmit through the metasurface without diffraction. Furthermore, the polymer film (mixture of ethyl red and polymethylmethacrylate, see method) was spincoated on top of the sample, which was used as the nonlinear optical switch layer. Upon external control laser excitation (532 nm in our experiment), the molecular chains of the ethyl red would fold, leading to a structural transformation from trans state to cis state. By removing the optical stimulus, the ethyl red molecules return to the initial trans state through the relaxation process. Such molecular folding causes a decrease in the refractive index of the polymer film, thereby changing the optical response of the metasurface in a reversible manner.
The ASR is intrinsically anisotropic in structure, thus the metasurface would respond differently to the photons with orthogonal polarizations, leading to different transmission spectra T. We first performed numerical studies of the optical properties of the sample using a finite element method (COMSOL Multiphysics). The refractive indices of the quartz substrate, gold, and ethyl red used in simulations were adopted from ellipsometric measurements (Fig. S3 in the Supplementary information). The transmission spectra for vertically (V) and horizontally (H) polarized incidence without the control light are given by solid lines in Fig. 2(a). The metasurface shows a transmission valley around 800 nm and a peak near 850 nm for Hpolarized incidence, while the Vpolarized spectrum presents swapped wavelength positions for valley and peak. The transmission ratio (\(T_V/T_H\)) and relative phase delay ϕ between V and Hpolarizations are given in the right two panels of Fig. 2(a). The role of the nonlinear excitation by the control light was considered in the simulation by reducing the refractive index of the ethyl red film by 0.05^{4}. The results are indicated by dashed lines in Fig. 2(a), and it is obvious that the spectra T, along with the transmission ratio \(T_V/T_H\) and the phase delay ϕ, show distinct blueshifts by control light excitation, leading to an efficient modulation to the quantum state \(\left {{{\mathrm{{\Psi}}}}} \right\rangle\) described in Eq. (1).
To experimentally characterize the properties of the metasurface, we built a spectrometer system to measure the transmittance T and the relative phase ϕ between V and Hpolarizations (Section I of the Supplementary information). A halogen lamp was used as the light source, whose output was collected by an objective (20×, 0.4 N.A.) to illuminate the sample. Another objective (20×, 0.45 N.A.) was used to collect the transmitted light, which was finally analyzed by a spectrometer (HR4000, Ocean Optics). A polarizer was inserted before the sample to polarize the incident light along a specific direction. To measure the transmittance of the Vpolarization, the first polarizer was set vertically, which was again aligned horizontally for measuring the Hpolarization. By further setting the incident polarization along +45° and analyzing the spectra for the ±45°, oppositelyhandedcircular polarization components in the transmission, the relative phase ϕ between V and Hpolarizations was obtained. To stimulate the nonlinear response of the sample, a 532 nm continuouswave (CW) laser (MGLIII532, CNI Corp.) was used as the control light. The green light had a power P_{C} of 20 mW, and was focused down to a diameter of about 13 μm, leading to the light intensity of about 15.1 kW cm^{−2}. The green light was combined with the halogen light using a polarizationindependent beam splitter (BS). The measured spectra are given in Fig. 2(b), which match reasonably well with the simulated results in Fig. 2(a). However, the experimental results are flatter than simulations, which can be mainly contributed to the imperfect fabrications of the sample (Section II of the Supplementary information). Under the stimulus of the green light, the spectra of T and ϕ all experience blueshifts as expected. Such spectral shifts lead to modification to both the value of \(T_V/T_H\) and ϕ at a certain wavelength, such as 808 nm as indicated by vertical black dashed lines, resulting in a nonlinearly tunable α/β. Consequently, the quantum states in Eq. (1) could be controlled.
To illustrate the nonlinear control over the photonic quantum states by the metasurface, we replaced the above halogen lamp with a quantum light source. As depicted in Fig. 3(a), a 404 nm CW laser (MLLIII404, CNI Tech., 200 mW) was used to pump a pair of typeI phasematched βbarium borate crystals (BBO, CASTECH Inc.), which were stacked together with their optical axes along H and Vdirections respectively. VVpolarized (HHpolarized) photon pairs at the wavelength of 808 nm were generated on the first (second) BBO crystal via a process of spontaneous parametric downconversion (SPDC). Subsequently, the entangled state of \(\left {{{\mathrm{{\Psi}}}}} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {e^{i\varphi }\left {V_S} \right\rangle \left {V_I} \right\rangle + \left {H_S} \right\rangle \left {H_I} \right\rangle } \right)\) obtains. φ is determined by the phase matching condition and the BBO crystal thickness, and can be compensated to 0 by a birefringent quartz crystal (QC). Thus, the generated entangled state by typeI BBO crystals shows \(\left {{{{\mathrm{{\Psi}}}}}_0} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left {V_S} \right\rangle \left {V_I} \right\rangle + \left {H_S} \right\rangle \left {H_I} \right\rangle } \right)\)^{27}. The twin photons traveled along two paths separately [labeled as signal and idle paths in Fig. 3(a)]. The metasurface was inserted between the two objectives in the signal path. Before coupling into singlemode fibers (SMFs) for spatial mode selection, the photons passed through a series of quarterwaveplate (QWP), halfwaveplate (HWP), polarizing beam splitter (PBS), and interference bandpass filters (IFs) placed in respective paths. The photon pairs were then detected by avalanche photodiodes (APDs, EXCELITAS Tech.). The output signals from the two detectors are combined electronically in a timecorrelated singlephoton counting system (PicoHarp 300, PicoQuant GmbH), whose integration time was set as 10 s, and the coincidence window was 4 ns. The postadjusted electronic time delay was applied to compensate for the nonequivalent optical path length between the two paths. The losses of the metasurface will cause the drop of photon counts in the signal path, thus decreasing the coincident counts of photon pairs proportionally. Because of the nonlocal property of the quantum entangled state, any local operation (i.e., operation on one photon) is equivalent to operating the whole entangled state^{28}. We characterized the generated quantum states following standard quantum state tomography (QST)^{29}, in which the measurement basis states \(\left H \right\rangle\), \(\left V \right\rangle\), \(\left D \right\rangle = \left( {\left H \right\rangle + \left V \right\rangle } \right)/\sqrt 2\), and \(\left R \right. = \left( {\left H \right. + i\left V \right.} \right)/\sqrt 2\) were chosen by selective combinations of the aforementioned QWP, HWP, and PBS. Hence the density matrix ρ, concurrence, and fidelity of the quantum states can be derived (Section III of the Supplementary information).
To demonstrate the entanglement distillation by the nonlinear metasurface, the initial photon state (including the effect of the sample) was deliberately generated as a nonmaximally entangled one \(\left {{{\mathrm{{\Psi}}}}} \right\rangle = \frac{1}{{\sqrt {\left \alpha \right^2 + \left \beta \right^2} }}\left( {\alpha \left {V_S} \right\rangle \left {V_I} \right\rangle + \beta \left {H_S} \right\rangle \left {H_I} \right\rangle } \right)\) (\(\alpha /\beta \ne 1\)) by adjusting the 404 nm HWP and QC compensators. The measured density matrix ρ of this initial state is shown in Fig. 3(b), in which the left panel is the real part [Re(ρ)], and the right one is the imaginary part [Im(ρ)]. Four corner components show unbalanced values. Furthermore, the concurrence and fidelity (with respect to the Bell state) were calculated as 88.9 ± 2.4% and 89.8 ± 1.7%, respectively [as shown by the data points of P_{C} = 0 in Fig. 3(d)]. Then the alloptical nonlinear entanglement distillation process was performed by irradiating the sample with the 532 nm laser. As a result of the nonlinear change of the transmission ratio \(T_V/T_H\) and relative phase ϕ between H and Vphotons, the quantum state is changed accordingly. The density matrix ρ of the distilled quantum state is shown in Fig. 3(c). Compared with the initial state (redrawn here by dashed lines for a clear comparison), the four corner elements in the Re(ρ) matrix tend to be balanced, and two antidiagonal elements of Im(ρ) approach zero, confirming the applicability of nonlinear metasurface arrays for the state distillation and the recovery to the Bell state \(\frac{1}{{\sqrt 2 }}\left( {\left {V_S} \right\rangle \left {V_I} \right\rangle + \left {H_S} \right\rangle \left {H_I} \right\rangle } \right)\) (for \(\alpha /\beta = 1\)). As shown in Fig. 3(d), the concurrence, fidelity, and certain characteristic corner elements of the ρ matrix all vary monotonically when the power of the control light P_{C} was gradually increased to 20 mW. The concurrence and fidelity rise to 99.1 ± 0.4% and 98.7 ± 0.5% for \(P_{{{\mathrm{C}}}} = 14\,{{{\mathrm{mW}}}}\), and become saturated for larger P_{C}. The ρ matrix element \(\left\langle {H_S} \right\left\langle {H_I} \right{{{\mathrm{Re}}}}\left( \rho \right)\left {H_S} \right\rangle \left {H_I} \right\rangle\) becomes equal to \(\left\langle {V_S} \right\left\langle {V_I} \right{{{\mathrm{Re}}}}\left( \rho \right)\left {V_S} \right\rangle \left {V_I} \right\rangle\) at \(P_{{{\mathrm{C}}}} = 14\,{{{\mathrm{mW}}}}\); however, they deviate from each other for larger P_{C} due to overmodulation. On the other hand, the \(\left\langle {H_S} \right\left\langle {H_I} \right{{{\mathrm{Im}}}}\left( \rho \right)\left {V_S} \right\rangle \left {V_I} \right\rangle\) approaches zero monotonically as P_{C} increases.
Discussion
In conclusion, we have realized the dynamic alloptical control of the photonic quantum states using the nonlinear metasurface. Specifically, the function of entanglement distillation was illustrated, which is essential to settle the problem of quantum state degradation by decoherence and dissipation in the practical quantum system. A nonmaximally entangled state has been successfully recovered to the ones with concurrence and fidelity higher than 99% and 98%, respectively. Compared with the mechanical method^{30}, and other nonlinearities such as Kerr effect^{31,32} or thermooptical effect^{33,34,35}, the photoisomerization process demonstrated here has the advantages of low pumping power (only needs milliwatts CW pump power) and fast response time (on the order of milliseconds, Section V of the Supplementary information)^{4}. Thus, utilizing a feedback loop to control the pump light power of our metasurface, a novel adaptive quantum optical source with the optimum degree of entanglement may be built. Furthermore, apart from entanglement distillation, we believe our nonlinear metasurface framework can be extended to accomplish more complex quantum state engineering tasks, such as logic gates in quantum information processing and quantum computations. The nanoscale sizes of metasurfaces also make them appealing to construct integrated quantum systems.
Materials and methods
Fabrication of the metasurface
A 45 nm Au layer was sputtered on a fused silica substrate. The nanostructures were patterned by FIB milling under 30 kV Gallium beam and the beam current was 1.8 pA. The overall area of the sample is about 21 × 21 μm^{2}. The ethyl red solution was made by dissolving ethyl red powder in ethanol with the concentration of about 18 wt%. Then the solution was mixed with polymethylmethacrylate (PMMA) at the volume ratio of 1:1. The mixture was spincoated on top of the metasurface at the speed of 1500 rpm, which formed a polymer film with the thickness of about 300 nm.
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Acknowledgements
This work was supported by Guangdong Major Project of Basic and Applied Basic Research (2020B0301030009); National Key R&D Program of China (2017YFA0305100, 2017YFA0303800, 2019YFA0705000); National Natural Science Foundation of China (92050114, 12174202, 11774333, 91750204, 62061160487, 12004373, 61775106, 11904182, 12074200, 11774185); The Hong Kong Research Grant Council (AoE/P502/20, 17309021); Anhui Initiative in Quantum Information Technologies (AHY130300); The Strategic Priority Research Program of the Chinese Academy of Sciences (XDB24030601); 111 Project (B07013); PCSIRT (IRT0149); Open Research Program of Key Laboratory of 3D Micro/NanoFabrication and Characterization of Zhejiang Province; Fundamental Research Funds for the Central Universities (01063201003, 01063201008, 01063201009, 01063211001); Tianjin Youth Talent Support Program. We thank the Nanofabrication Platform at Nankai University and USTC Center for sample fabrication.
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D.Z., M.X.R., and X.F.R. conceived and performed the design. D.Z., Y.C., and S.C.G. performed the optical measurement. D.Z., M.X.R., and X.F.R. analyzed data. W.W. and D.Z. fabricated the samples. M.X.R., X.F.R., and J.J.X. organized and led the project. All authors discussed the results and prepared the manuscript.
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Zhang, D., Chen, Y., Gong, S. et al. Alloptical modulation of quantum states by nonlinear metasurface. Light Sci Appl 11, 58 (2022). https://doi.org/10.1038/s41377022007445
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DOI: https://doi.org/10.1038/s41377022007445
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