Simultaneous High-Fidelity Readout and Strong Coupling for a Donor-Based Spin Qubit

TL;DR

This paper addresses the challenge of achieving strong spin–photon coupling while maintaining high-fidelity dispersive readout for a donor-based flip-flop qubit in silicon. By modeling the qubit–resonator system with two Schrieffer–Wolff transformations, it derives a dispersive Hamiltonian characterized by a state-dependent resonator shift χ_z and analyzes decoherence, relaxation, and critical photon-number constraints. The authors map regions in parameter space where high readout fidelity (SNR^2 ≥ 282) and strong coupling (g_s ≥ κ) coexist, highlighting that intermediate tunnel couplings maximize the viable window, and showing that squeezing input fields can significantly expand it. The results indicate that modest squeezing can enable simultaneous high-fidelity readout and strong coupling in realistic devices, providing a general framework applicable to quantum-dot and donor-based qubits for scalable quantum information processing in cQED platforms.

Abstract

Superconducting resonators coupled to solid-state qubits offer a scalable architecture for long-range entangling operations and fast, high-fidelity readout. Realizing this requires low photon-loss rates and qubits with tunable electric dipole moments that couple strongly to the resonator's electric field while maintaining long coherence times. For spin qubits, spin-photon coupling is typically achieved via spin-charge hybridization. However, this introduces a fundamental trade-off: a large spin-charge admixture enhances the coupling strength, which boosts readout and resonator-mediated gate speeds, but exposes the qubit to increased decoherence, thereby increasing the threshold required for strong coupling and limiting the time available for accurate state measurement. This makes it essential to identify optimal operating points for each qubit platform. We address this for the donor-based flip-flop qubit, whose microwave-controllable electron-nuclear spin states make it suitable for coupling to microwave resonators. We demonstrate that, by choosing intermediate tunnel couplings that balance strong interaction with long qubit lifetimes, high-fidelity readout and strong coupling are simultaneously achievable. We also map out the respective charge-photon couplings and photon-loss rates required. Furthermore, we show that experimental constraints on charge-photon coupling and photon loss can be mitigated using squeezed input fields. As similar trade-offs appear in quantum-dot-based qubits, our methods and insights extend naturally to these platforms, offering a potential route toward scalable architectures.

Figures (8)

• Figure 1: (a) Schematic of the flip-flop qubit. The electron wavefunction (grey) can be delocalized from the donor state towards the silicon-insulator interface with an electric field $E_z$ from a top gate. The donor and interface states are coupled by $V_t$. (b) Energy level diagram of the flip-flop qubit. The electron-nuclear spin states $|\uparrow\Downarrow\rangle,|\downarrow\Uparrow\rangle$ are separated approximately by the Zeeman splitting $\omega_B$, while the orbital splitting $\omega_0$ separates the ground (red) and excited (blue) charge manifold. The flip-flop qubit is encoded in the spin states of the ground manifold (red). (c) Schematic of dispersive readout. The flip-flop qubit is coupled to the resonator of frequency $\omega_r$ with charge-photon coupling strength $g_c$. The qubit experiences an effective relaxation rate $\gamma$ and the resonator photon-loss rate is $\kappa$. $b_{\text{in}}$ and $b_{\text{out}}$ are the known input and measured output drives that are used to read the qubit state.
• Figure 2: Readout efficiency function $D$ for different $|\chi_z|/\kappa$ regimes. For $|\chi_z|/\kappa \leq 0.5$, the optimal driving frequency is on resonance ($\delta_d = 0$, $\omega_d = \tilde{\omega_r}$), though $D<1$ for $|\chi_z|/\kappa < 0.5$. Conversely, when $|\chi_z|/\kappa > 0.5$, $D$ reaches unity at $\delta_d = \pm \sqrt{\chi_z^2 - \left(0.5\kappa\right)^2}$.
• Figure 3: Using the parameters $g_c = 2\pi\times 50\text{ MHz}$, $\kappa = 2\pi\times 1.8\text{ MHz}$: (a) Full $|\chi_z|$ landscape over tunnel coupling $0\le V_t \le 2\pi\times 20\text{ GHz}$ and detuning $|\varepsilon| \le 2\pi\times 20 \text{ GHz}$. (b) Analytical approximation $|\chi_z-\chi_{\text{cor}}|$. (c) Magnitude of correction $|\chi_{\text{cor}}|$. Region A (white semi-ellipse) violate energy scales of Eq. \ref{['eq:escale']}, \ref{['eq:spinlike']} or causes the dispersive approximation to break down ($\braket{n}<1$, Eq. \ref{['eq:meanphoton']}) from $n_{c,1}\rightarrow 0$. Region B (white arcs) violate the unwanted transition bound of Eq. \ref{['eq:bound']}. Comparison of (c) with (a,b) confirms that $\chi_{\text{cor}}$ is negligible outside the regime where $\omega_0 \approx \omega_r + \omega_B$.
• Figure 4: Landscape of $\text{SNR}^2$, $\Gamma_m$ and $\gamma$ for $\omega_r = 2\pi \times 6.5 \text{ GHz}$ and $\omega_B = 2\pi \times 6.6 \text{ GHz}$. (a-c) Good parameters ($Q = 10^4$). (d-f) Sub-optimal parameters ($Q \approx 3.6\times 10^3$). We observe that high SNR$^2$ is concentrated at $\varepsilon = 0$ and high $V_t$. This is driven by the exponential suppression of the relaxation rate $\gamma$ from its maximum value of 0.012 MHz, which compensates for the reduction in measurement rate $\Gamma_m$. Conversely, $\Gamma_m$ peaks at lower $V_t$. The region circumscribed by the red contour indicates strong coupling ($g_{s}/\kappa \ge 1$), but this is only the small accessible part of the regime due to the invalid region, with the area being much smaller with sub-optimal parameters in (d-f). High $g_{s}/\kappa$ and $\Gamma_m$ can be simultaneously achieved, but the overlap between high SNR$^2$ and strong coupling is minimal. Maximizing SNR$^2$ typically requires operating in the weak coupling regime ($g_{s}/\kappa < 1$).
• Figure 5: Regimes of simultaneous high fidelity and strong coupling. (a-c) Map of different $\text{SNR}^2$ (black contours: $F \ge 90\%$ ($\text{SNR}^2 \ge 13$), $F \ge 95\%$ ($\text{SNR}^2 \ge 36$), and $F \ge 99\%$ ($\text{SNR}^2 \ge 282$)) and $g_{s}/\kappa$ (red contours) thresholds for fixed tunnel couplings $V_t$. The minimum photon-loss rate of the plot is $\kappa = 2\pi \times6.5\ \text{kHz}$, and corresponding quality factors are given. Striped areas highlight the target regime where strong coupling ($g_{s}/\kappa \ge 1$) and high fidelity ($F \ge 99\%$) coexist. This area is maximized at intermediate tunnel couplings (comparing (b) to (a)). The yellow contour ($|\chi_z|/\kappa = 0.5$) demarcates the saturation of readout efficiency $D$. To the left of this line ($D=1$), increasing $g_c/\kappa$ yields no efficiency gain and instead degrades $\text{SNR}^2$ via increased Purcell decay and lowered mean photon number $\braket{n}$, as observed in (c). Numbered points 1–6 correspond to the specific parameter sets from Fig. \ref{['fig:fig4']}.