High-Probability Heralded Entanglement via Repeated Spin-Photon Phase Encoding with Moderate Cooperativity

TL;DR

This work tackles the challenge of generating high-fidelity remote entanglement between moderate-cooperativity spin-cavity registers by introducing a repeated, phase-encoding protocol that recycles a single photon for multiple spin-cavity interactions. Operating in a far-detuned regime, the protocol accumulates a spin-dependent phase of $\pi$ after $N$ rounds, enabling heralded entanglement with high fidelity even when $C\sim1$, and it gains higher phase-encoding efficiency via width-scaling pulses. The authors develop a comprehensive input-output framework, optimize encoding duration, and analyze realistic imperfections including parameter variations, phase stability, losses, and mode mismatch, offering strategies such as mode compensation and dynamical decoupling for robustness. Extensions to a three-level register and a transmission-based variant widen the applicability to solid-state platforms (e.g., quantum dots, NV centers) and point toward photon-loss-tolerant distributed quantum computing with moderate cooperativity.

Abstract

We propose a heralded high-probability scheme to generate remote entanglement between moderate-cooperativity spin-cavity registers with high fidelity. In conventional single-shot interfaces, limited cooperativity restricts the spin-conditional optical response and thus strongly suppresses the success probability. Our proposal instead recycles a single incident photon for repeated interactions with the spin-cavity register, such that a small spin-conditional phase shift acquired on each round trip accumulates coherently to enable remote entanglement. Moreover, the repeated scheme enables higher spin-photon encoding efficiency by using a spectral-width-scaling photon pulse with a shorter duration. We show that, for realistic imperfections and losses, this repeated phase-encoding approach produces high-fidelity entangled states with an appreciable success probability even at cooperativity $C\sim1$. Our protocol is particularly well suited to weakly coupled, cavity-based solid-state spin platforms and provides a route toward hybrid, photon-loss-tolerant distributed quantum computing.

Figures (15)

• Figure 1: The heralded entangling based on repeated spin-photon phase encoding. (a) The schematic view of the entangling circuit, in which two registers $R_A$ and $R_B$ are connected by a Mach-Zehnder interferometer. Here, the upper side is labeled $A$. A single photon is sent from port $A$ and splits at a $50/50$ beam splitter (BS), and the conditional phase encoding is implemented with the corresponding register. After the encoding, the reflected photons are routed back and combined at BS. The remote entanglement is generated when one photon reaches the port $A$ or $B$ and the detector clicks. (b) A local register exhibiting a four-level system and an optical cavity and an additional optical cycle (green curve). (c) A single cavity mode couples with two optical transitions at a strength $g$.
• Figure 2: Simulation of the entangling proposal. (a) The input Gaussian spectral envelope $\tilde{u}(\omega)$ is centered in the entangling window (gray region) around $\omega/\gamma=0$, where $|r_+(\omega)|^2\approx0$ (purple line) and $N(\theta_0-\theta_1)\approx\pi$ (green line). Parameters: $\Delta/\gamma=6.2826$, $N=5$, spectral width $\sigma_w =\gamma/5$ and cooperativity $C=1$. (b) Simulated infidelities for the monochromatic pulse, $1-F_A^{(\delta)}$ (purple line), and for a Gaussian pulse, $1-\tilde{F}_A$ (blue line), as functions of the detuning $\Delta/\gamma$ with $C=1$. The blue squares denote the optimal detuning values $\Delta$ to maximize the fidelity. (c) The required repetition number $N$ as a function of $\Delta$ for $C=0.5,1,2$, shown as red circles, blue squares, and green triangles, respectively, which lie very close to the monochromatic result $N=\pi\Delta/(2C\gamma)$ (solid lines). (d,e) The simulated infidelities $1-\tilde{F}_A$, total success probabilities $P_t$ as a function of the repetition $N$ for varying $C$ of a Gaussian pulse. The solid lines in (e) show the corresponding monochromatic probabilities. Parameters: $\kappa/\gamma=200$ throughout this paper.
• Figure 3: (a) The simulated infidelity with different spectral width $\sigma_\omega = \Delta/N_\omega$ at each $N$. (b) Maximal encoding rates for fidelity thresholds $F_A^{(\mathrm{t})}=0.99$, $0.995$, and $0.999$ are shown by red diamonds, blue circles, and green triangles, respectively, where each point uses the optimal $N_\omega$ for a given $N$. The width-fixed result $\sigma_\omega=\gamma/5$ is indicated by the purple $N_\omega$-independent surface. Parameters: $N_t=10$, $C=2$.
• Figure 4: (a) The simulated infidelities for four transitions with cooperativity values $C=1.5,\,2,\,2.5,\,3$. The infidelities $1-F_{A(B)}^{(c)}$, denoted by left(right)-pointing green(blue) triangles, are improved compared to the uncorrected infidelity $1-F_{A(B)}$ denoted by red squares(orange circles), approaching the infidelity with four identical transitions with $C=2$ indicated by the purple line. (b) The success probability $P_t$ versus cooperativity $C$ and repetition $N$ with the blank region having $F_A<0.99$. The efficiencies are $\eta_r,\eta_i=0.9886,\,0.99$. (c) The probability $P_t$ achieved with the correction $U_{\pi}$ for different overlaps $\eta_m$ with the same efficiencies in (b) at $C=2$.
• Figure 5: (a) Three-level system. (b) Dynamical-decoupling-based phase encoding. The $\pi$ pulses (green blocks) periodically flip $|0\rangle$ and $|1\rangle$ with the separation $\tau_{\mathrm{DD}}=N_t\sigma_t$. The stepwise (purple) and sinusoidal (blue) detunings $\Delta(t)$ are synchronized with the DD sequence. (c) Simulated infidelity $1-F_A^{(\sin)}$ for sinusoidal detuning with $N_t=12,16,20$ (blue diamonds, red circles, and yellow triangles). The purple curve shows $1-F_A^{(\mathrm{step})}$ for stepwise detuning with $N_t=10$. Parameters: $C=2$.