Self-stabilized high-dimensional quantum key distribution on a metropolitan free-space link

Abstract

Quantum communication technologies capable of operating reliably across heterogeneous optical channels are essential for scalable metropolitan quantum networks. Here we demonstrate high-dimensional time-bin-encoded quantum key distribution over a hybrid metropolitan link comprising 1.7 km free-space transmission and 685 m of optical fiber. Operating at a clock rate of 500 MHz in the C-band, we implement both 2- and 4-dimensional protocols, and obtain estimated secure finite-key rates of (95 +- 28) kbit/s for 4D at (25.0 +- 2.0) dB loss and (59 +- 27) kbit/s for 2D at (23.5 +- 2.3) dB loss. Crucially, we achieve continuous operation over 48 h in a fully self-referenced architecture: initial synchronization, interferometric phase stabilization, and long-term drift compensation are performed exclusively using the detected quantum signals, without auxiliary optical reference channels. Our results thus establish a practical and versatile platform for hybrid free-space-to-fiber quantum communication and show that the encoding dimensionality can be adapted to the optimal operating regime of realistic metropolitan channels, providing a pathway toward efficient, autonomous and deployable quantum network nodes.

Figures (8)

• Figure 1: High-dimensional quantum key distribution on the campus of Fraunhofer IOF and Abbe Center of Photonics. The weak coherent source is placed in a mobile container on the rooftop of a local provider of energy and water (Stadtwerke Jena Gruppe). 2D and 4D time-bin states are transferred over a 1.7 km free-space link to another mobile container (QuBUS) located at Fraunhofer IOF, and guided to the nearby Abbe Center of Photonics via fiber and measured with interferometers and nanowire single-photon detectors.
• Figure 2: Experimental setup for higher dimensional encoding.a, The setup consists of a distributed feedback laser (DFB) and electro-optical modulators to prepare the time-bin states. b, The state sequence is loaded and output by a FPGA that is locked to an external rubidium clock. The photons are sent through telescopes to the remote receiver. The free-space link infrastructure includes tip-tilt wavefront correction through beacon lasers. Noise is filtered by coupling the 1550 nm light into a single-mode fiber that acts as a spatial filter. The receiver incorporates two interferometers that are phase-locked using the quantum bit error rate obtained in the parameter estimation step as the error signal. c, The arrival time is measured through nanowire single-photon detectors (SNSPD) and time-to-digital converters (TDC). d, Probability of detection when each input state is detected in each interferometer, as characterized in laboratory conditions without the free-space channel.
• Figure 3: Results with 2-dimensional and 4-dimensional encoding on the 1.7 km free-space link.a, Piezo control voltage that is sent to the interferometer for stabilization. b, Difference in the relative frequency of the rubidium oscillator at sender and quartz oscillator at the receiver. c, Timing stability before and after quantum time transfer. The noise figures improve from flicker phase modulation noise ($\tau^{-1}$) and flicker frequency modulation noise ($\tau^{0}$) to only white phase modulation noise ($\tau^{-3/2}$). d, The timing offset varies by up to 1 ms before stabilization. e, After stabilization, the fluctuation of the timing offset reduce substantially with standard deviation of 58 ps. f, Distribution of mean photon number over the full measurement time. g, Sifted key rate extracted from both Alice and Bob choosing the Z basis. h, State error rate in Z basis estimated from X basis $\Phi_Z$ and quantum bit error rate (QBER) estimated from a 10 % subsample of states in Z basis. i, The finite secure-key rate. Data over 10 seconds is acquired and considered with finite-size key effects. Every data point represents the mean key rate over 30 min. All information in X basis is published and used for estimating the error $\Phi_Z$, as well as to stabilize the X-basis interferometer. In the Z basis, 10 % of the detections are used to estimate the QBER for error correction and to stabilize the Z-basis interferometer. j, End-to-end estimation of loss on the link, based on the single-photon count rate. The subplots a,b,c,d,e refer to the 4-dimensional protocol. All data was recorded on 8$^{\text{th}}$-12$^{\text{th}}$ February 2024 (see Methods \ref{['sec:Environmental and turbulence measurements']}, Fig. \ref{['fig:fig_8_envirnmental_measurements']} for weather and turbulence data).
• Figure 4: Variation of the 4D-QKD secure key rate under atmospheric turbulence. Atmospheric turbulence leads to fluctuations in the channel transmission, enabling the extraction of secure key rates over a broad range of attenuation values. Each recorded data point, acquired within 100 ms integration time, is sorted into 1 dB-wide attenuation bins according to the instantaneous link transmission, which follows the probability distribution indicated in the bar chart. A finite key analysis is performed for each attenuation bin, introducing a block size of $n_Z = 10^7$. Note that we only consider data points acquired during night-time operation. During daytime operation, fluctuations in background-light-induced noise counts introduced significant bias in channel-loss estimates derived from detected count rates. Simulation parameters: probability of an error = 1%, probability of a dead count = 4e-07, dead time of the detectors = 5e-08, probability for both Alice and Bob to choose the Z basis = 90%, probability to choose the signal state = 76%, signal state mean photon number = 0.37, decoy state mean photon number = 0.13, $\epsilon_{cor} = 10^{-9}$, $\epsilon_{sec} = 10^{-15}$, generation rate 500 MHz, fidelity of the states $c = 0.297$, error correction inefficiency $f_e$ = 1.08.
• Figure 5: Environmental and turbulence data during the experiment.a, The temperature (Temp.) with mean value and standard deviation during the 2D and 4D QKD experiments. b, Rain data with maximum values. c, Wind speed with mean value and standard deviation during the 2D and 4D QKD experiments. d, Surface Shortwave Downward Radiation (SWDR) with maximum values. e, Turbulence parameter $C^2_n$ with mean value and standard deviation during 2D and 4D QKD experiments. f, Fried parameter $r_0$. For reference, our beam diameter is approximately 8 cm during free-space transmission. The weather data (a-d) was sourced from the weather station Beutenberg, Max-Planck-Insitut für Biogeochemie, Jena. The turbulence data (a-d) has been measured with a scintillometer. All data was recorded on 8$^{\text{th}}$-12$^{\text{th}}$ February 2024.