Certifying non-classicality and non-Gaussianity through optical parametric amplification

TL;DR

The paper demonstrates theoretically and experimentally that optical parametric amplification (OPA) combined with conventional intensity detectors can certify non-Gaussianity (NG) and non-classicality (NC) without photon-number-resolving detectors. By measuring the mean photon number relative to the amplified vacuum, $\mu_{\mathrm{rel}}$, and the post-amplification second-order coherence, $g^{(2)}$, after amplification, NG/NC witnesses become robust to gain for $G>3$, enabling verification across bright and faint regimes. Using a heralded quasi-single-photon state seeded into an OPA, the authors show NG up to low brightness and track its transition to NC and then classical behavior as brightness increases, with results consistent with coincidence-based benchmarks and theoretical bounds. The approach supports broadband, multimode certification and paves the way for high-dimensional quantum technologies that rely on non-Gaussian resources, while avoiding the need for photon-number-resolving detection. Corrections for losses and mode-matching are shown to be feasible, making the method practical for realistic quantum optics experiments.

Abstract

Non-Gaussian states of light are essential for numerous quantum information protocols; thus, certifying non-Gaussianity is crucial. Full quantum state tomography, commonly used for this purpose, is a complicated procedure and yields inconclusive results for strongly mixed states. Certifying non-Gaussianity through directly measurable parameters is a simpler alternative, typically achieved by measuring photon-number probabilities - either directly, using photon-number resolving detectors, or through Hanbury Brown--Twiss type measurements with single-photon detectors. Here, we demonstrate theoretically and experimentally that optical parametric amplification combined with conventional intensity detectors can effectively replace this approach without the need for photon-number resolution. In our method, we measure the mean photon number and the second-order correlation function for the amplified state. Using it, we successfully certify the non-Gaussianity of a heralded quasi-single-photon state. Since optical parametric amplification is a broadband and multimode process, our method provides a foundation for developing high-dimensional quantum technologies utilizing broadband multimode non-Gaussian states.

Figures (8)

• Figure 1: (a) Non-Gaussianity (NG) and non-classicality (NC) witnesses in terms of vacuum ($p_0$) and single-photon ($p_1$) probabilities Radim2011_NG. The area above the dotted line (blue and green parts) is not accessible by classical states, while the green area above the blue line is not accessible by Gaussian states. The cross-hatched area is non-physical. Inset: the same criteria in terms of single-photon ($p_1$) and higher photon-number ($p_{2+}$) probabilities. (b) The recently proposed Racz2025b NG witness in terms of photon-number mean $m$ and variance $s^2$ (black line). The blue line shows the photon-number probability-based NG witness converted to $m,s^2$ variables. The dotted line is the boundary of the anti-bunching non-classicality witness $g^{(2)} < 1$. (c) NG and NC witnesses in terms of the post-amplification second-order correlation function $g^{(2)}$ and mean photon number relative to the one of the amplified vacuum $\mu_{\mathrm{rel}}$, applicable to phase-independent states. The dashed area is where phase-independent states cannot reach. The dashed line for $\mu_{\mathrm{rel}}\in (1, 3)$ corresponds to different input mixtures of a vacuum and a single photon and for $\mu_{\mathrm{rel}}\in (3, 5)$, to input mixtures of single-photon and two-photon states. White areas in all panels correspond to points that classical states can reach. The black triangular, circular, and square markers correspond to Fock states $|0\rangle, |1\rangle,$ and $|2\rangle$, respectively.
• Figure 2: Experimental scheme. Quantum states are prepared (green shaded box) through heralding on the idler photons of type-II SPDC using a single-photon detector (SPD1) after filtering with a band-pass filter (BP1) and a single-mode fibre (SMF1). For certification (yellow shaded box), heralded signal photons are fed into optical parametric amplifier OPA, whose output, after similar filtering with BP2 and SMF2, is registered by photodetector PD (a camera). The efficiency $\eta$ includes both the detection and coupling into the OPA. The dashed box shows an HBT setup with single-photon detectors SPD2,3 used for the characterization of the heralded state. Inset (i): the histogram of coincidences between SPD1 and SPD2/SPD3 for the brightness of the source, measured by SPD1, set at 0.02 photons per pulse at the source output. Inset (ii): single- and multiphoton probabilities obtained for heralded states of varying brightness, quantified through the mean number of photons per pulse at the source output. The magenta line shows the analytic curve for the fixed heralding efficiency of 51%. Inset (iii): photon numbers for a sequence of $35000$ pulses in the cases of vacuum (left) and heralded photon (right) states at the input.
• Figure 3: Photon-number distributions of the amplified vacuum (black solid line) and amplified heralded state with a brightness of 0.1 photon per pulse (red solid line), measured over 35000 pulses. The dotted/dashed black line shows the theoretical photon-number distribution of amplified vacuum/ideally amplified single photon. The blue and green lines show the theoretical distributions for input states with $p_0=0.67,p_1=0.33, p_{2+}=0$ (non-Gaussian) and $p_0=0.715,p_1=0.239,p_{2+}=0.046$ (non-classical), respectively. Inset: points corresponding to these states (blue and green crosses, respectively) in the ($p_0, p_1$) diagram.
• Figure 4: Experimental post-amplification $g^{(2)}$ and relative mean photon numbers $\mu_{\mathrm{rel}}$. Blue circles correspond to heralded states for different source brightnesses, quantified through mean number of photons per pulse at the output. The blue and green areas correspond to the NC and NG witnesses from Fig. \ref{['fig:NG-original']}(c). The magenta line shows the dependence calculated analytically for a fixed effective transmittance of $26\%$ while increasing the brightness of the source (from left to right). Red points show the results for a thermal state obtained from the same source without heralding. The measurement shows $g^{(2)}$ close to the theoretical value 3 (black dash-dotted line) and no nonclassicality.
• Figure 5: Estimated photon-number mean and variance values from the coincidence measurement (blue triangles) and the amplified measurement (orange disks). The numeric values next to the symbols correspond to the mean number of photons per pulse at the source output. The solid black line shows the boundary of the non-Gaussianity witness, the dotted black line is the boundary of the non-classicality (anti-bunching) witness, $s^2 = m$.