Mechanically mediated optical-microwave quantum state transfer by feedback

TL;DR

This work establishes a broadband optical-to-microwave quantum transduction protocol using measurement-based feedback in a sideband-unresolved optical cavity, followed by a mechanical-to-microwave conversion, achieving high fidelity and quantum-compatible noise with realistic parameters. It introduces a single-valued quantum transfer witness, W_T, which captures both non-unity-gain performance and the ability to preserve Gaussian entanglement, and demonstrates that W_T<1 identifies regimes outperforming classical LOCC and preserving input entanglement. The study shows optical coupling inefficiency as the primary bottleneck, yet indicates current platforms can reach near-unit transfer with added noise below vacuum, and extends to bidirectional transfer via coherent optical feedback. Together, these results broaden the transducer-design landscape, enabling robust quantum links between distant nodes in a quantum network.

Abstract

State transfer between light and microwaves is a key challenge in quantum networks. Promising transducers use a mechanical intermediary that couples to both fields via radiation pressure. Such electro-optomechanical devices have achieved high efficiencies, yet require resolved-sideband cavities, and generally compromise in scalability and noise performance. Here, we relax this constraint by extending the protocol of Navarathna et al. that transfers optical quantum information onto a mechanical resonator using a broadband, sideband-unresolved cavity and feedback. Combining this with parametric mechanical-to-microwave conversion, we show that continuous optical-to-microwave quantum state transfer is possible using measurement-based feedback, while all-optical coherent feedback enables bidirectional transfer. To assess the transfer, we introduce the quantum transfer witness $\mathcal{W}_T$, which -- though similar to the input-referred added noise -- also identifies whether a channel is capable of both preserving Gaussian entanglement and outperforming classical transduction schemes. Finally, we show that quantum-compatible noise performance is within reach of current experimental capabilities. Our results unlock a new design space for electro-optomechanical transducers and strengthens their candidacy as scalable quantum links between distant nodes.

Figures (8)

• Figure 1: Optical-to-microwave quantum transducer.(a) Transduction from an optical input channel $a_\text{in} = (X_\text{L,in} + \rmi Y_\text{L,in})/\sqrt{2}$ to a microwave output channel $c_\text{out} = (X_\text{M,out} + \rmi Y_\text{M,out})/\sqrt{2}$ is facilitated by cascading an optical ($a$, input coupling rate $\eta_\text{L}\kappa_\text{L}$), a mechanical ($b$) and a microwave ($c$, output coupling rate $\eta_\text{M}\kappa_\text{M}$) resonance. Modes interact with optomechanical and electromechanical coupling rates $g_\text{L}$ and $g_\text{M}$, respectively. The mechanical resonator couples to the thermal environment with rate $\Gamma$. Optical output quadrature $Y_\text{L,out}$ is detected and fed back onto $b$. (b) Schematic opto-electromechanical transfer system. A resonant optical pump is coupled into an optomechanical cavity, imprinting the amplitude quadrature $X_\text{L,in}$ onto $b$ through radiation pressure. Phase quadrature $Y_\text{L,in}$ contributes to output quadrature $Y_\text{L,out}$, which is measured through homodyne detection, delayed by time $\tau$ and fed back onto the momentum of $b$. Mechanical position $Q$ and momentum $P$ are transferred to $c$ by a microwave pump tuned to the red electromechanical sideband. Finally, microwave output quadratures $X_\text{M,out}$ and $Y_\text{M,out}$ carry the transferred quantum state out. (c) Lab-frame frequency diagram of the transfer process. The optical resonance at $\omega_\text{L}$ has linewidth $\kappa_\text{L}$ larger than the mechanical frequency $\Omega$. The upper optomechanical sideband (grey) is transferred to the mechanical resonator by a resonant pump and feedback. The microwave resonance at $\omega_\text{M}$ is sideband-resolved ($\kappa_\text{M} \ll \Omega$), so that a red-detuned microwave pump (at $\omega_\text{M} - \Omega$) transfers the mechanical state to the microwave resonator.
• Figure 2: Bandwidth of the transfer process. Transmission $T_{\infty}(\omega)$ as a function of signal detuning $\omega$ from the upper mechanical sideband for different linewidth ratios $\beta = \Gamma'/\kappa_\text{M}$ and electromechanical cooperativities $C_\text{M}' = 4g_\text{M}^2/\Gamma'\kappa_\text{M}$. For $C_\text{M}' \leq 1$, the transmission exhibits a single peak at $\omega=0$. For $C_\text{M}' > 1$, when the microwave cavity and the feedback-broadened resonator are strongly coupled, the transmission spectrum splits into two peaks corresponding to hybridised modes. For any value of $\beta$, transfer can be achieved with unity efficiency by setting $C_\text{M}' = 1$. Moreover, for matched linewidths $\beta=1$, this can be achieved for any $C_\text{M}' \geq 1$.
• Figure 3: Transducer performance with optomechanical cooperativity. (a) Contributions to the noise variance in Eq. (\ref{['eq:V_contributions']}) added to the output microwave field amplitude as a function of $C_\text{L}/\bar{n}$. Here, $(\eta_L, \eta_M, \eta_d) = (0.95,0.98, 0.85)$ and $\bar{n} = 10^3$. (b) Transfer fidelity $\mathcal{F}$ (Eq. (\ref{['eq:fidelity_def']})) for various pure input states and unit efficiencies. (c) Total negativity $\mathcal{N}$ (Eq. (\ref{['eq:negativity_def']})) of the output Wigner function relative to that of the pure input state, $\mathcal{N}_\text{pure}$. Solid curves indicate unit efficiencies ($\eta_L, \, \eta_M, \, \eta_d = 1$), while dotted curves indicate the efficiency choices in (a). Insets: output Wigner function $W_\text{out}(\mathbf{r})$ (Eq. (\ref{['eq:def_W_out']})) in phase space $\mathbf{r}=(X,Y)^T$ generated at the cooperativities identified by grey vertical lines (left-to-right: $C_\text{L}/\bar{n}=0.1,\, 1, \, 10$). The pure input state is identified with a solid colour border. The transferred state at $C_\text{L}/\bar{n} = 10$ for the efficiencies in (a) is identified with a dotted border.
• Figure 4: Quantum transfer witness $\mathcal{W}_T$ as a figure of merit for non-unity gain transfer. (a) The signal transmission coefficient $T_{ac}$ (Eq. (\ref{['eq:Tac_max']})) and added noise variance $V_X^\text{add}$ (Eq. (\ref{['eq:V_contributions']})) traced as a function of each loss parameter $\eta_L, \, \eta_M, \, \eta_d$ (black lines) from an initial starting point where $\eta_L, \, \eta_M, \, \eta_d = 1$ and $C_\text{L}/\bar{n}=10$ (white circle). Black ticks along the parametric curves indicate $10\%$ loss increments. In the blue region where $\mathcal{W}_T<1$ (see Eq. (\ref{['eq:def_quantum_transfer_witness']})), performance exceeds that of an LOCC transducer and the inseparability of a perfectly entangled Gaussian input is preserved. The grey forbidden region is identified from Heisenberg uncertainty requirements (SI Section \ref{['sec:apx:witness_beyond_LOCC']}). Gold square: $(\eta_L, \eta_M, \eta_d, C_\text{L}/\bar{n}) = (0.95, 0.98, 0.85, 10)$. Gold star: $(\eta_L, \eta_M, \eta_d, C_\text{L}/\bar{n}) = (0.98,0.98,0.96, 100)$. (b) The transducer's capacity to preserve entanglement ($\mathcal{W}_T<1$) is insensitive to microwave cavity external coupling losses $\eta_M$, and relatively insensitive to that of the optical cavity $\eta_L$. Here, $C_L/n_\text{th}=10$ and $\eta_d = 0.85$.
• Figure 5: Information content of the homodyne photocurrent. Transmission coefficients $T_{ja}(\Omega + \omega)$ of the optical (blue), mechanical (red) and microwave (green) input fields to the optical output around the upper mechanical sideband. Here, $C_\text{M}' = 1$ and $\beta = 1$. For finite cooperativity $C_\text{L}$ and limited transfer efficiency $\eta_\text{OM} = 0.7$ (left), the optical output is contains contributions from each channel. In the limit $C_\text{L} \to \infty$ where $\eta_\text{OM} \to 1$ (right), optical and mechanical fluctuations are fully suppressed at $\omega = \Omega$, and replaced by noise from the microwave input channel. In all cases, $T_{aa}(\omega) + T_{ba}(\omega) + T_{ca}(\omega) = 1$.