Parity readout in Majorana box qubits from the dispersive to the resonant regime

Abstract

We study theoretical models for charge reflectometry and capacitive readout of the Majorana parity degree of freedom in Majorana box qubits, taking into account decoherence channels within the framework of the Lindblad master equation. Noting that a parity-dependent dynamical susceptibility $χ_z(ω)$ governs both readout schemes, we provide a general expression for $χ_z(ω)$ which covers the full crossover from the resonant regime to the off-resonant dispersive regime. In addition, we re-examine previous results which were obtained under a semiclassical factorization assumption. Using three different error measures, we show that this approximation is quantitatively justified in the dispersive regime. In the resonant regime, however, we find deviations from exact reference data, obtained by numerical solution for the steady state of the full Lindblad equation. These deviations are typically of the order of a few percent in the considered error measures.

Figures (4)

• Figure 1: Schematic readout schemes for the Majorana qubit parity $\hat{z}=i\gamma_2\gamma_3$ with eigenvalues $z=\pm 1$, see Eq. \ref{['MBQpauli']}. The Majorana states $\gamma_{1,2,3,4}$ are boundary states of a pair of topological superconductor wires connected by a superconducting bridge ($S$) Plugge_2017Karzig2017. The floating Majorana box is operated under Coulomb valley conditions. A pair of small quantum dots (1 and 2, shown as green squares), with the double dot in the single-electron occupancy regime, is tunnel-coupled to the wire ends harboring $\gamma_2$ and $\gamma_3$, respectively. In addition to elastic cotunneling through the MBQ, coherent tunneling between dots 1 and 2 can proceed through a tunnel-coupled reference arm ($R$). The phase difference $\varphi$ between both interfering tunneling paths can be tuned by gate voltages or by a weak perpendicular magnetic field. The fluctuating charge on dot 2 is monitored in the frequency domain by capacitively coupling dot 2 to a microwave cavity (photon resonator) which in turn is coupled to the readout circuit. (a) Charge reflectometry setup, where the outgoing intensity $I_{\rm out}(\omega)$ for given intensity $I_{\rm in}(\omega)$ in the input line is measured in a microwave circuit. (b) Quantum capacitance measurement, where the cavity is used for gate sensing and capacitance readout Aghaee2025.
• Figure 2: Relative deviation $\varepsilon$ in Eq. \ref{['vareps']} vs normalized detuning $\varrho$ in Eq. \ref{['detuning']}, comparing the full susceptibility expression \ref{['suscgen']} to approximate expressions. Note the logarithmic scales for $\varepsilon$. Here $\varrho$ is changed by ramping up $\omega_0-2\omega_z$ at given coupling strength $g$. Since $\varepsilon(-\varrho)\simeq \varepsilon(\varrho)$, we depict only $\varepsilon(\varrho\ge 0)$. Solid curves show $\varepsilon$ using the strong-coupling approximation for $\chi_z$ in Eq. \ref{['susc']}, and dashed curves refer to the weak-coupling approximation in Eq. \ref{['susc2']}. The horizontal dotted line indicates a $2\%$ error threshold. (a) $\varepsilon$ vs $\varrho$ for several values of $g$ in the range $4~{\rm MHz}\le g\le 40$ MHz with fixed total decoherence rate $\Gamma_{\rm tot}=9$ MHz. (b) Same as (a) but for several values of $\Gamma_{\mathrm{tot}}$ in the range $9~{\rm MHz}\le \Gamma_{\rm tot}\le 60$ MHz with fixed coupling strength $g=20$ MHz.
• Figure 3: Comparison of semiclassical and exact results in the strong- and weak-coupling limits. We use the dimensionless numbers $\{ \varepsilon_{a\tau_z}, \varepsilon_A, \varepsilon_{\phi}\}$ in Eqs. \ref{['var1']}, \ref{['var2']}, and \ref{['var3']}, which are shown on logarithmic scales. Employing the scale $\lambda_0 = 0.2~\mathrm{GHz}$, we assume the tunnel couplings $|t_0| = 5 \lambda_0$ and $|t_1| = \lambda_0$. For $z=+1$ and $\varphi=0$, we have $|t_z| = |t_0 + t_1|=6\lambda_0$. The cavity linewidth is $\kappa = 0.04 \lambda_0$. Results are shown at temperature $T=30$ mK. Panels (a)--(d):Strong-coupling limit. Unless noted otherwise, we choose the quantum-dot detuning $\varepsilon=0$, see Eq. \ref{['hamiltonian']}, with $\omega_z$ resulting from Eq. \ref{['omegaz']}, the bare cavity frequency $\omega_0 = 11.99 \lambda_0$, the rates $\gamma = 0.003 \lambda_0$ and $\Gamma_\phi = 0.1 \lambda_0$, the drive frequency $\omega=\omega_0$, and the cavity-dot coupling $\lambda = 0.1 \lambda_0$, with $g$ obtained from Eq. \ref{['gdef']}. Panel (a) shows the phase of $A_\omega$ vs $\omega$. The dotted vertical line indicates the resonance at $\omega=\omega_0$. Panels (b), (c), and (d) show the error measures as function of $g$, of the detuning $\Delta_z=2\omega_z-\omega$, and of $\varepsilon$, respectively. Dottel horizontal lines mark an error level of $1\%$. Panels (e)--(h): Same as (a)--(d) but for the weak-coupling limit. Unless noted otherwise, we here put $\varepsilon = 6 \lambda_0$, $\lambda = 0.25 \lambda_0$, $\gamma = 0.015 \lambda_0$, $\Gamma_\phi = 0.05 \lambda_0$, $\omega_0 = 16 \lambda_0$, and $\omega=\omega_0$. In that case, $|\omega_0 - 2\omega_z|\sim {\cal O}(\lambda)$.
• Figure 4: Dependence of the error measures $\varepsilon_{a\tau_z}$, $\varepsilon_{\mathrm A}$, and $\varepsilon_\phi$ on the relaxation rate $\gamma$ of the double dot in the weak-coupling limit. Semiclassical results are compared to the exact analytical expressions in Sec. \ref{['sec2b']}, see, in particular, Eq. \ref{['weakcouplingexactA']}. The dashed horizontal line indicates the $1\%$ level. All energies are expressed in units of $\lambda_0 = 0.2~\mathrm{GHz}$.