All-optical modulation of quantum states by nonlinear metasurface

Metasurfaces have proven themselves an exotic ability to harness light at nano-scale, 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 all-optical 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 non-maximally 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.


I. SPECTRAL MEASUREMENTS OF SAMPLE
FIG. S1. Experimental setup to measure classical optical spectra. Transmission spectra T of metasurface for V -and H-polarized incidence were measured with halogen light illumination and were recorded by a multi-mode fiber coupled spectrometer. To determine the relative phase ϕ between the transmitted V -and H-polarizations, the incident light was further set along +45 • direction. And using a combination of QWP, HWP and GT, the transmitted light was decomposed into the ±45 • , left-and right-handed circular polarization components, whose spectra were measured successively. To introduce nonlinear optical stimulation, a 532 nm laser was combined with the halogen light by a polarization-independent BS. GT: Glan-Taylor polarizer; BS: beam splitter; QWP: quarter-waveplate; HWP: half-waveplate; LF: long-pass filter; FC: fiber coupler.
We measured transmission spectra of sample under V -and H-linear polarizations using a home-made spectrometer system, as shown in Fig. S1. A halogen lamp was used as light source, and two objectives (20×, 0.40 N.A. and 20×, 0.45 N.A.) were used to illuminate the sample and collect the transmitted signal, respectively. The transmitted light was analyzed by a multi-mode fiber coupled spectrometer (HR4000, Ocean Optics). The relative phase ϕ between the transmitted V -and H-polarizations was measured by further setting the incidence along +45 • direction. And the transmitted light was decomposed into ±45 • (labeled as T ±45 ), left-and right-handed circular (T L and T R ) polarization components successively by different orientation combinations of QWP, HWP and GT on the transmitted side (as detailed in Table S1), and the corresponding spectra were measured separately. Thus, ϕ can be calculated via ϕ = arctan T L −T R T +45 −T −45 . A 532 nm laser was aligned with the halogen light by a polarization-independent beam splitter (BS), which stimulated the sample to nonlinearly control the transmission spectra. The results of the nonlinear controlled spectra for T and ϕ are shown in Fig. 2 in the main text. The linear absorption spectra are shown in Fig. S2.

II. SIMULATION OF THE SPECTRA OF METASURFACE
The refractive indices of Au, SiO 2 , and ethyl red used in the simulation are shown in The spectra of the metasurface between the simulation and experiment show similar

III. CHARACTERIZATION OF ENTANGLED PHOTON PAIRS
The experimental setup is shown in Fig. 3(a) of the main text. The measured coincident photon counts in each polarization base are shown in Table S2. We performed quantum state tomography (QST) to characterize the entanglement properties of photon pairs. We used standard QST followed by maximum likelihood estimation (MLE) to reconstruct the density matrix ρ of the state, which was proposed by D. F. V. James, et al. in Ref. [3] . The density matrix of two-photon state can be expressed through 2-photon Stokes parameters S i1,i2 and Pauli operators σ i : where the single-qubit measurement operators µ i relate with Pauli matrices by where And the photon counts n i1,i2 in various polarization-projection measurements are related with S i1,i2 by where Y −1 i,j are elements of By substituting the selected measurement bases and photon counts in the above formulas, the density matrix ρ QST can be reconstructed. This will always work when assuming perfect experimental conditions. However, in reality, it may produce results that violate basic properties, such as positivity. The MLE is then used to avoid this problem.
The MLE generally contains the following steps.
(1) Generating a 'physical' density matrix ρ p = (T † T )/Tr(T † T) that matches the condition of normalization, Hermiticity, and positivity. Where T shows: in which ρ ij is the element in the density matrix ρ QST reconstructed in the QST process; ij is the determinant of the 3 × 3 matrix formed by deleting the ith row and jth column of ρ QST ; M (2) ij,kl is the determinant of the 2 × 2 matrix formed by deleting the ith and kth rows and jth and lth columns of ρ QST ; and ∆ = Det(ρ QST ).
(2) Introducing a 'likelihood function': in which |ψ ν ⟩ are states of the measurement bases. The task is then an optimization problem to find the minimum of the L function, which represents the 'likelihood' that the matrix ρ p could produce the measured data n ν .
(3) Numerical optimization. We set ρ QST to determine the initial set of values for t 1 , t 2 , ..., t 16 , and used standard numerical optimization technique to optimize the function L to find a minimum value.
Finally, we can find the optimized density matrix ρ p as the measured density matrix ρ.

IV. TUNING BEHAVIORS OF NONLINEAR METASURFACE
The numerically simulated potential tuning behaviors are shown in Fig. S5. The black curves exemplify how the quantum state evolves when the refractive index change of the nonlinear polymer film varies from ∆n ER = 0 to −0.5. The concurrence and fidelity of the initial quantum state (point of ∆n ER = 0) are much lower than 100%, and then they gradually increase as the refractive index decreases. When ∆n ER = −0.05, the quantum state recovers, and the concurrence and fidelity are about 100%, which corresponds to the expected distilled state. As the refractive index ∆n ER further decreases, the entangled state is over-tuned, and the concurrence and fidelity drop.

V. TEMPORAL RESPONSE OF NONLINEAR METASURFACE
To characterize the temporal response of the metasurface, the control light was mechanically chopped, thereby inducing periodic modulation over the signal light. The QWP, HWP, and GT served as the polarization analyzer, and were properly oriented to eliminate the transmitted signal light at the initial state. When the control light illuminated the metasurface, the polarization of the transmitted signal light was changed due to the different variations in transmittance between the horizontal and the vertical polarization components, which subsequently induced modulation in the leaked light intensity, which was monitored by an oscilloscope. Figure S7 shows the result of the dynamic response of our metasurface.
Using the biexponential fitting, 4 the rise and decay rates of the fast (and slow) components are obtained as 0.61 (and 13.80) and 0.24 (and 27.83) ms, respectively.

MEASUREMENTS
We fabricated another metasurface array and reperformed the nonlinear spectral measurements, as shown in Fig. S8. The results are reasonably agreed with Fig. 2(b) in the main text, confirming the well reproducibility of the experiments.