Bell Inequality Violation with Vacuum-One-Photon Number Superposition States

Abstract

Entanglement is a central resource in quantum technologies, and the realization of photonic entanglement necessarily relies on interaction with matter. Resonance fluorescence (RF), originating from the coherent interaction between a driving field and a two-level system, plays a pivotal role in quantum optics. Here, we demonstrate a novel route to entanglement generation based on RF from a single quantum dot. Rather than relying on generation of multiphoton states, our approach directly exploits vacuum-one-photon number superposition states created under resonant excitation. By delocalizing this superposition via a beam splitter, we realize time-bin entanglement and observe a clear violation of the Clauser-Horn-Shimony-Holt Bell inequality using Franson-type interferometry. Our scheme removes the need for multiphoton generation, simplifies the experimental requirements, and establishes a scalable pathway toward solid-state entangled photon sources.

Figures (4)

• Figure 1: Experimental setup.(a), The setup is composed by a quantum light source (light pink area) with a QD-micropillar device, resonantly driven by a CW laser, same resonant scheme as in Ref. wu2023mollow. The RF signal is split by a fiber beam splitter (FBS) and directed into two independent time-bin analyzers consisting of two AMZIs located at A and B sites(light blue areas). Coincidence counts are recorded by single-photon detectors (see detectors $A_1$, $A_2$ and $B_1$, $B_2$, respectively). Independent $\phi_A$ and $\phi_B$ phases of the two AMZIs are stabilized to desired values using a PID feedback loop based on the intensity of the phase locking laser. (b), High-resolution RF spectra measured with a Fabry-Pèrot interferometer (FPI). (c), Auto-correlation function $g^{(2)}(\tau)$ measured at an average photon number $\bar{n} = 0.01$ using a Hanbury Brown-Twiss (HBT) setup. FC, fiber collimator; QWP, quarter-wave plate; HWP, half-wave plate; MR, mirror; PZT, piezoelectric transducer; BS, beam splitter; PBS, polarizing beam splitter. $A_1$, $A_2$, $B_1$, and $B_2$ are the labels for the superconducting nanowire single photon detectors (SNSPDs).
• Figure 2: Mappings of the second-order correlation function $g^{(2)}(\Delta t)$. The laser driving power was set at $\bar{n} = 0.01$. (a-d), Normalized second-order correlation measurements of resonance fluorescence after the Franson interferometer between detectors $A_1$ and $B_1$, with $\phi_B$ fixed at 0, $\pi/2$, $\pi$, and $3\pi/2$, respectively, each $g^{(2)}(\Delta t)$ trace acquired with an integration time of four minutes. (e-h), Theoretical simulations corresponding to panels (a-d), respectively. The color intensity in the figure indicates the coincidence count rate at different phases, with all counts normalized to the maximum value. The color scale thus provides a direct visual representation of the variation in interference visibility.
• Figure 3: Franson interference fringes and observation of a CHSH inequality violation.(a), Correlation function $N$ for simultaneous coincidences in $A_1$ and $B_1$ as a function of $\phi_A$ for different values of $\phi_B$ = 0, $\pi$/2, $\pi$, 3$\pi$/2. Each point indicates the coincidence counts accumulated over a four-minute interval for a given phase setting. Dashed lines are fits using $N(\phi_A,\phi_B)=N_1\left[1+\cos(\phi_A-\phi_B)\right]+N_2$, with $N_1$, $N_2$ being fitting parameters. The average visibility of four correlation lines is 92.8 $\pm$ 2.6$\%$, which indicates the presence of time-entanglement. (b), Pump power dependence of the S parameter and $g^{(2)}(0)$, with the dashed line denoting the theoretical prediction. The inset shows the measured correlation functions at an average input photon number $\bar{n} = 0.01$.
• Figure 4: Effect of the first-order interference.(a), Baseline coincidence counts as a function of $\phi_A$ for different $\phi_B$ values, measured at the laser driving power of $\bar{n} = 0.01$. (b), Experimental second-order correlation measured under the phase configuration $(\phi_A,\phi_B) = (\pi, \pi)$. In both panels, the dots represent the experimental data, while the dashed lines show simulations obtained using the same calculation method as in Fig. \ref{['fig2']}(b).