High-purity entanglement mediated by magnons despite weak coupling

Technical Summary: High-purity entanglement mediated by magnons despite weak coupling

Problem Statement
Entangling distant qubits via a shared magnonic bus typically encounters a fundamental tradeoff: increasing the spin-magnon coupling strength improves entanglement fidelity but simultaneously accelerates the decay of the spins into the magnon bath. Existing proposals often require the coupling to exceed the magnon dissipation rate to generate significant entanglement, a condition that is experimentally challenging. Furthermore, detuning spins to reduce decay often weakens the coupling, seemingly making the tradeoff unavoidable. The authors aim to break this tradeoff to enable high-fidelity entanglement even under weak coupling conditions.

Methodology
The paper proposes a probabilistic, heralded protocol to entangle two nitrogen-vacancy (NV) centers situated near a magnetic wire. The approach adapts the Barrett-Kok (BK) protocol from quantum optics to a magnonic system, utilizing collective decoherence as a resource.

• System Setup: Two NV centers are placed near a thin magnetic wire (modeled as a cylinder) with a static magnetic field $H$ defining the quantization axis. The NVs couple to magnons via dipolar interactions.
• Computational Basis: The protocol utilizes the $|0\rangle$ and $|\bar{1}\rangle$ states of the NV spin-1 system as the computational basis. Crucially, the magnons are coupled only to the $|0\rangle \leftrightarrow |1\rangle$ transition, which lies outside the computational basis. This ensures that the computational states $|0\rangle$ and $|\bar{1}\rangle$ do not directly decay into the magnon bath.
• Protocol Steps:
  1. Initialization: Prepare each NV in a superposition of $|\bar{1}\rangle$ and $|1\rangle$.
  2. First Wait ($t_1$): Wait for a fixed time. If a magnon is detected (or lost to the environment, acting as a measurement), the system undergoes a quantum jump. If no magnon is detected, the run is discarded.
  3. Cyclic Gate ($G$): Apply a gate $G \equiv |0\rangle \to |\bar{1}\rangle \to |1\rangle \to |0\rangle$ to both NVs. This swaps the roles of the bright (coupled) and dark (decoupled) states.
  4. Second Wait ($t_2$): Wait for a second magnon detection. A detection here heralds the successful creation of the entangled state.
• Theoretical Framework: The dynamics are analyzed using the Monte-Carlo wavefunction approach, a stochastic unraveling of the Lindblad master equation. This tracks individual quantum trajectories involving coherent exchange, magnon loss, magnon measurement, and NV dephasing.
• Signal Amplification: To address the experimental challenge of detecting single magnons at non-zero wavevectors, the authors propose a parametric amplification scheme using three-magnon splitting within the same nanowire. A pump mode splits into the NV-coupled mode and an idler mode, amplifying the signal for detection via Brillouin light scattering (BLS) or another NV.

Key Contributions

• Breaking the Tradeoff: The primary contribution is a protocol where weak coupling reduces the success probability but does not degrade the fidelity of the entangled state. By coupling magnons only to a transition outside the computational basis, the computational states are protected from direct decay.
• Unit Fidelity in Ideal Conditions: Theoretical analysis shows that in the absence of NV dephasing, the protocol can reach unit fidelity with a maximally entangled Bell state ($|0\bar{1}\rangle + |\bar{1}0\rangle$) for arbitrarily weak coupling, provided the waiting times are sufficiently long to allow bright components to decay while dark components survive.
• Phase Control: The protocol addresses the phase randomness inherent in the entangled state by imposing a geometric condition ($2k_{mag}x = 2n\pi$) on the NV positions and frequencies, ensuring all successful runs converge to the same Bell state.
• Experimental Feasibility: The paper provides a concrete simulation setup using Yttrium Iron Garnet (YIG) and NV centers, demonstrating that high fidelity is achievable even when the coupling strength ($g$) is significantly smaller than the magnon linewidth ($\kappa$).

Results
Numerical simulations using the Monte-Carlo wavefunction approach were performed for two NV dephasing scenarios:

1. Low Dephasing ($\kappa_{NV}^{low} \approx 2\pi \times 1$ Hz): With a coherence time of 160 ms, the protocol achieves a fidelity $F > 0.99$ for waiting times $t_1 \ge 11$ $\mu$s. The success probability saturates at approximately 0.6%.
2. Moderate Dephasing ($\kappa_{NV}^{mod} \approx 2\pi \times 1$ kHz): With a coherence time of 160 $\mu$s, the fidelity peaks at $F = 0.91$ at optimal waiting times ($t_1 = 11$ $\mu$s, $t_2 = 8$ $\mu$s). The success probability remains around 0.57%.

• Coupling Regime: The simulations assume a coupling strength $g = 2\pi \times 30$ kHz, which is one-third of the assumed magnon linewidth ($\kappa = 2\pi \times 0.1$ MHz).
• Entanglement Rate: At the optimal point for moderate dephasing, the rate of generating Bell pairs is calculated to be 0.5 kHz.

Significance and Claims
The paper claims to offer a scalable solution for on-chip entanglement of NV centers without requiring individual microwave addressing, relying instead on global control. The significance lies in demonstrating that high-purity entanglement is possible even when the spin-magnon coupling is weak and the coupling is an order of magnitude below the dissipation rate.

The authors explicitly note that the most experimentally challenging component is the detection of single magnons at non-zero wavevectors. They propose the three-magnon splitting mechanism as a theoretical solution to amplify these signals, though they acknowledge that a full analysis of the time-dependent measurement rate in this amplification regime is beyond the scope of the current work. The protocol is presented as a step toward modular quantum processors and quantum networks, leveraging the long coherence times of NV centers and the scalability of ferromagnetic magnonic buses.