Multiprobe Time Reversal for High-Fidelity Vortex-Mode-Division Multiplexing Over a Turbulent Free-Space Link

The orbital angular momentum (OAM) of photons presents a degree of freedom for enhancing the secure key rate of free-space quantum key distribution (QKD) through mode-division multiplexing (MDM). However, atmospheric turbulence can lead to substantial modal crosstalk, which is a long-standing challenge to MDM for free-space QKD. Here, we show that the digital generation of time-reversed wavefronts for multiple probe beams is an effective method for mitigating atmospheric turbulence. We experimentally characterize seven OAM modes after propagation through a 340-m outdoor free-space link and observe an average modal crosstalk as low as 13.2% by implementing real-time time reversal. The crosstalk can be further reduced to 3.4% when adopting a mode spacing $\mathrm{\Delta }\ell$ of 2. We implement a classical MDM system as a proof-of-principle demonstration, and the bit error rate is reduced from $3.6\times {10}^{-3}$ to be less than $1.3\times {10}^{-7}$ through the use of time reversal. We also propose a practical and scalable scheme for high-speed, mode-multiplexed QKD through a turbulent link. The modal crosstalk can be further reduced by using faster equipment. Our method can be useful to various free-space applications that require crosstalk suppression.

Figure 1

The previously reported mode fidelity measured in 150- [16], 340- [17], and 1600-m [23] links when using an OAM mode spacing $\mathrm{\Delta }\ell =1$. The dashed line represents the fidelity threshold of 76.3% for secure QKD with seven-dimensional encoding.

Figure 2

Illustration of time reversal to transmit high-fidelity modes of $\ell =-1,0,1$. Alice uses a time-reversing mirror to generate the phase conjugates of the probe beams transmitted by Bob and sends it to Bob as signal beams. Information can be transferred from Alice to Bob by using a high-speed optical modulator to modulate the time-reversed beams at Alice’s side. The arrows under the spatial modes indicate the propagation direction.

Figure 3

Schematic of the experiment. The upper-left image shows an aerial view of the free-space link and is courtesy of Map data © 2020 Google. The retroreflector is installed on a building rooftop. Digital time reversal is realized by displaying a computer-generated diffractive hologram on SLM 2 based on the feedback from a camera (cam 1). AWG is the arbitrary waveform generator. BS is the beam splitter. PBS is the polarizing beam splitter. FC is the fiber coupler. Ch. 1 is channel 1. Ch. 2 is channel 2. Additional experimental details are presented in Sec. I of the Supplemental Material [24].

Figure 4

(a) Typical examples of the spatial modes received by Alice (top row) and Bob (bottom row). Scale bar is 0.5 mm. (b) Experimentally measured crosstalk matrix without time reversal. The average crosstalk is 37.0%. (c) Crosstalk matrix with time reversal. The average crosstalk is 13.2%. (d) Crosstalk matrix using a mode spacing $\mathrm{\Delta }\ell$ of 2 without time reversal. The average crosstalk becomes 10.0%. (e) Crosstalk matrix using a mode spacing $\mathrm{\Delta }\ell$ of 2 with time reversal. The average crosstalk is reduced to 3.4%.

Figure 5

(a) Eye diagrams of different realizations when transmitting the $\ell =2$ and $\ell =3$ modes and receiving the $\ell =2$ mode. The diagrams with time reversal are displayed in the top row, and those without time reversal are shown in the bottom row. (b) Proposed free-space QKD system with $N$-channel OAM multiplexing using time reversal for modal crosstalk suppression. The quantum signal streams are represented by binary on-off keying signals for simplicity. DM is the dichroic mirror. WFS is the wavefront sensor.