Near-deterministic photon entanglement from a spin qudit in silicon using third quantisation

TL;DR

<3-5 sentence high-level summary>This paper proposes a near-term demonstration of Rudolph’s third quantization in silicon using a high-spin ${}^{123}$Sb donor to deterministically entangle photons spread across multiple time bins. By preparing two eight-mode single-photon states and distributing their modes among eight parties, the scheme yields a random multipartite Bell state with a theoretical upper bound efficiency of $0.875$ in the ideal limit, without non-deterministic gates. The work leverages time-bin multiplexing, strong Sb–cavity coupling (via EDSR), and qudit Hadamard decoupling to realize high-dimensional multipartite entanglement and points toward scalable pathways to universal quantum computation within silicon-based photonic architectures. It also analyzes practical error sources, photon loss, and the feasibility of extending to higher photon numbers and frequencies or cavities.

Abstract

Unlike other quantum hardware, photonic quantum architectures can produce millions of qubits from a single device. However, controlling photonic qubits remains challenging, even at small scales, due to their weak interactions, making non-deterministic gates in linear optics unavoidable. Nevertheless, a single photon can readily spread over multiple modes and create entanglement within the multiple modes deterministically. Rudolph's concept of third quantization leverages this feature by evolving multiple single-photons into multiple modes, distributing them uniformly and randomly to different parties, and creating multipartite entanglement without interactions between photons or non-deterministic gates. This method requires only classical communication and deterministic entanglement within multi-mode single-photon states and enables universal quantum computing. The multipartite entanglement generated within the third quantization framework is nearly deterministic, where ``deterministic'' is achieved in the asymptotic limit of a large system size. In this work, we propose a near-term experiment using antimony donor in a silicon chip to realize third quantization. Utilizing the eight energy levels of antimony, one can generate two eight-mode single-photon states independently and distribute them to parties. This enables a random multipartite Bell-state experiment, achieving a Bell state with an upper-bound efficiency of 87.5% among 56 random pairs without non-deterministic entangling gates. This approach opens alternative pathways for silicon-based photonic quantum computing.

Figures (6)

• Figure 1: Proposed Device for Third Quantisation Experiment and Antimony Donor a): Artist's impression of the random, multipartite Bell state device. The device has a microwave antenna to drive EDSR, ESR and NMR transitions. Additionally, a cavity, placed near the microwave antenna, is coupled with the electric dipole of the donor electron as shown with red arrows. The antimony donor is assumed positioned at an antinode of the cavity electric field, to achieve capacitive coupling to the donor's electric dipole and stimulate the EDSR transitions to emit photons coherently. (see details in \ref{['total_H']} for the total Hamiltonian of the system.) A microwave switch, represented by gray arrows, sends the emitted photon, the red circle, to one of the transmon qubits for readout. b) The energy spectrum of the neutral antimony donor. Antimony is a high-spin nuclei and it has 8 energy levels. When the antimony donor is in its neutral charge state, the spin of the electron couples with the spin of the antimony nucleus, resulting in a doubled Hilbert space dimension, which is a 16-level system. Curved arrows represent $\rm{NMR}_0^{\pm 1}$ transitions, while ESR is depicted using vertical solid arrows, and EDSR is indicated with dashed arrows. The $0$ subscript in NMR emphasise that the antimony donor is in its neutral charge state.
• Figure 2: Time-bin protocol for controlling multi-mode photonic states The EDSR frequency between the states $\ket{7/2}\ket{\downarrow} \leftrightarrow \ket{5/2}\ket{\uparrow}$ is chosen as fixed frequency for emitting photons. The purple rectangle represents the uniform superposition. The gray arrows represent the 'permutation operation' operation between nuclear spin states. $t_1$, $t_2$, $\cdots$, $t_8$ represent the time-bins into which the photon is emitted. (see details in \ref{['time-binsteps']}.)
• Figure 3: Effect of (non)uniform photon loss on success probability The total success rate is shown for two cases: when each mode has the exact same photon loss rate (blue stars), and when each mode has approximately the same loss rate, modeled as a normal distribution over the 16 modes with a standard deviation of 0.005 (orange dots).
• Figure 4: Effect of random photon loss on success probability a) The total success rate under random photon loss per mode. The x-axis indicates the interval within which each mode's loss rate is randomly selected. Compared to the previous plot, the photon loss is more randomized, but still follows a uniform distribution within the specified interval across the 16 modes. b)The distance from uniformity versus photon loss interval across modes is plotted. The distance from uniformity is calculated using the following formula:$\sum_i \left(q_i - \frac{1}{n_i}\right)^2$ where $i$ indexes the successful patterns (i.e., cases in which no party receives more than one photon), $n_i$ is the total number of successful patterns, and $q_i$ is the normalized success probability corresponding to pattern $i$.
• Figure A.1: Frequency multiplexing protocol The protocol uses as many microwave cavities as there are frequencies in the system (eight microwave cavities for ESR frequencies and seven microwave cavities for EDSR frequencies).