Time-resolved certification of frequency-bin entanglement over multi-mode channels

Abstract

Frequency-bin entangled photons can be efficiently produced on-chip which offers a scalable, robust and low-footprint platform for quantum communication, particularly well-suited for resource-constrained settings such as mobile or satellite-based systems. However, analyzing such entangled states typically requires active and lossy components, limiting scalability and multi-mode compatibility. We demonstrate a novel technique for processing frequency-encoded photons using linear interferometry and time-resolved detection. Our approach is fully passive and compatible with spatially multi-mode light, making it suitable for free-space and satellite to ground applications. As a proof-of-concept, we utilize frequency-bin entangled photons generated from a high-brightness multi-resonator source integrated on-chip to show the ability to perform arbitrary projective measurements over both single- and multi-mode channels. We report the first measurement of the joint temporal intensity between frequency-bin entangled photons, which allows us to certify entanglement by violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, with a measured value of $|S|=2.32\pm0.05$ over multi-mode fiber. By combining time-resolved detection with energy-correlation measurements, we perform full quantum state tomography, yielding a state fidelity of up to $91\%$. We further assess our ability to produce non-classical states via a violation of time-energy entropic uncertainty relations and investigate the feasibility of a quantum key distribution protocol. Our work establishes a resource-efficient and scalable approach toward the deployment of robust frequency-bin entanglement over free-space and satellite-based links.

Figures (10)

• Figure 1: . State generation and time-resolved detection.(a) Experimental layout for state generation and equatorial basis measurement. (b) Schematic of the mechanism for the generation of the entangled state: the idler (blue) and signal (red) frequency bins are generated from the bichromatic pump (green) via spontaneous four-wave mixing (SFWM). All the resonances (as well as the pump fields) are spaced by $\Delta\omega/2\pi=820MHz$. (c) Experimental transmission spectra and Lorentzian fit of the device at the signal, pump and idler frequencies. Pump spectra show split resonances due to Rayleigh backscattering (see Methods) and are fitted using a two-Lorentzian model. RF: radio-frequency; BPF: band-pass filter; SMF: single-mode fiber; MMF: multi-mode fiber; SNSPD: superconducting nanowire single photon detector; APD: avalanche photodiode; SFWM: spontaneous four-wave mixing.
• Figure 2: . Equatorial basis measurements.(a) Theoretical JTI for ringdown time $1/\gamma=0.90$$\rm{ns}$, the counts were scaled to match the experimental data. (b) Measured JTI over single-mode fiber, with a fitted ringdown time $1/\gamma_\mathrm{fit}=0.90(4)$$\rm{ns}$. (c) Visual representation of the equatorial basis projection for signal and idler qubits, $\sigma_X\sigma_X$ projections on the Bloch spheres are pictorially represented as colored points on the JTI. (d) Diagonal profile of the measured JTI, a biphoton temporal beating can be observed in the coincidences whereas the singles exhibit no oscillation highlighting the nonlocal nature of the correlations. The single-mode and multi-mode signal propagation scenarios are denoted respectively as SM and MM. (e) Antidiagonal profile of measured JTI, the cavity lifetime leads to a double exponential decay proportional to $\gamma$. (f) CHSH expectation values $\langle A_\mathrm{i} B_j \rangle,$$i,j \in \{0,1\}$ as a function of the absolute time $(t_\mathrm{s}=t_\mathrm{i}) \mod T_\mathrm{b}$, and resulting (g) CHSH $S$ parameter (see Eq. \ref{['eq:CHSH']}). The time axis spans 0 to 610 ps, corresponding to one beat period $T_\mathrm{b}=(\pi/\Delta\omega)$ and $i,j$ denote different azimuthal angles on the equator of the Bloch sphere. The integration time for the equatorial basis measurement was 600 seconds.
• Figure 3: . Interferometric detection scheme.(a) Experimental layout for the $\sigma_Z\sigma_Z$ measurement. (b) A detailed schematic of the field-widened interferometer. A 10cm N-BK7 glass rod in the long arm of the interferometer is used to compensate for spatial distortions of the optical mode of the frequency-bin qubit. The basis choice is implemented via an optical shutter in the short arm of the interferometer. Flip-mirrors at the interferometer's outputs enable coupling into single- and multi-mode fibers. (c) Conceptual diagram of the interferometric frequency demultiplexing. The signal (idler) resonances are spaced by half of the interferometer free spectral range (FSR), which acts as a narrowband filter. (d) Visual representation of the projection for the closed (equatorial basis) and (e) open (computational basis) shutter configurations. (f) Joint temporal intensity (JTI) for the four physical measurement settings required for quantum state tomography: (i) both photons in the equatorial basis, (ii) signal in the equatorial basis and idler in the $\sigma_Z$ basis, (iii) signal in the $\sigma_Z$ basis and idler in the equatorial basis, and (iv) both photons in the $\sigma_Z$ basis. Quantum state tomography: reconstructed real (g) and imaginary (h) part of the density matrix of the state measured over multi-mode fiber.
• Figure 4: . Quantum key distribution.(a) Conceptual deployed scenario in a dual-downlink configuration, the entangled photon source (EPS) is onboard the satellite. (b)$C$-parameter ($C$) and normalized asymptotic key rate ($K/R$), calculated according to Eqs. \ref{['eq:C']}-\ref{['eq:SKR']}.
• Figure 5: . Point spread function.(a) PSF, imaged with a beam-profiling camera (New Imaging Technologies WiDy SenS 640), of the multi-mode beam in the analyzer, after propagation through a 5-meter long graded-index (GRIN) multi-mode fiber (Thorlabs GIF625). (b) Reconstruction of the PSF from modal decomposition using the first 30 Hermite-Gaussian modes.