Fluxon Time-Delay Readout of a Superconducting Qubit Protected by a Spectral Gap in a Josephson Transmission Line

Abstract

We theoretically investigate a readout scheme of the quantum state of a superconducting qubit based on time delay of a single flux quantum (SFQ), also known as a fluxon, propagating in a Josephson transmission line (JTL). We concretely study the time-delay readout based on capacitive coupling between a transmon qubit and a JTL, and we evaluate the time delay depending on the qubit state. We also reveal a feature of the absence of fluxon pinning and exponential suppression of nonadiabatic transitions caused by the propagating fluxon, which is advantageous for the time-delay readout. We extend the analysis to a multi-level transmon as well. Owing to the spectral gap in the JTL, the radiative decay of the qubit mediated by the JTL is exponentially suppressed, and thus the transmission line itself also serves as a filter protecting the qubit. The readout scheme requires neither complicated wiring to low-temperature stages nor bulky microwave components, which are bottlenecks for integration of a large-scale superconducting quantum computer.

Figures (5)

• Figure 1: Schematic diagram of a discrete JTL, also called a lumped-element JTL, which is made of JJs connected each other via linear inductors. Each cell of length $\Delta x$ is characterized by the junction capacitance $C_{\text{J}}$, the junction critical current $I_{\text{c}}$, and the inductance $L$ of the upper electrode. Each junction is subject to a bias current $I_{\text{b}}$, and a resistor $R_{\text{J}} = 1/G_{\text{J}}$ describes normal current flowing across the junction. The discrete JTL gives a circuit model of a continuous JTL, which is also known as a long Josephson junction (LJJ).
• Figure 2: Schematic diagram of a transmon capacitively coupled to a JTL. The transmon is characterized by the total capacitance $C_{\Sigma} \coloneqq C_{\text{S}} + C_{\text{J,tr}}$ ---the summation of the shunt capacitance $C_{\text{S}}$ and the transmon junction capacitance $C_{\text{J,tr}}$ ---and the critical current $I_{\text{c,tr}}$ of the junction of the transmon. The transmon charging and Josephson energies $E_{\text{C,tr}} \coloneqq e^2 / (2 C_{\Sigma})$ and $E_{\text{J,tr}} \coloneqq \varphi_0 I_{\text{c,tr}}$ determine the properties of the transmon. The transmon is coupled to the $m$-th junction of the JTL via the coupling capacitor $C_{\text{c}}$.
• Figure 3: (circles) Numerical results of the time delay $T_{\text{d}} = (\tau_{\ket{\downarrow}} - \tau_{\ket{\uparrow}}) \omega_{\text{p}}^{-1}$ calculated by numerically solving Eqs. (\ref{['eq:ODE_u_qubit']}) and (\ref{['eq:ODE_Xi_qubit']}) for several values of the damping coefficient $\alpha$ and (line) analitycal result in Eq. (\ref{['eq:Td_qubit_ana01']}). We adopt the values of circuit parameters in Table \ref{['tab:circuit_params']}.
• Figure 4: Numerically obtained snapshots of accelerated fluxons as dimensionless voltage pulses $V (\xi, \tau) / (\varphi_0 \omega_{\text{p}}) = - \partial_{\tau} \phi (\xi, \tau)$ for two different initial positions of centroid $\xi_0 = 5.75$ and $5.85$. The grey area satisfying $0 \le \xi < 15$ represents the nonbiased region, while the white area satisfying $\xi \ge 15$ represents the biased region by the dimensionless bias $\gamma = -0.1$. The initial velocity is $u_0 = 0.01$ for both cases, and the damping coefficient is $\alpha = 1.0 \times 10^{-6}$. We assume the closed boundary conditions at the ends of the JTL, i.e. $\phi (0,\tau) = 0$ and $\phi (100, \tau) = 2\pi$ for an arbitrary time $\tau$. When we assume the values of circuit parameters in Table \ref{['tab:circuit_params']}, the unities of time $\tau$, coordinate $\xi$, and voltage are $2.8 \,$ps, $12 \, \mu$m, and $0.12 \,$mV, respectively.
• Figure 5: (a) A transmon capacitively coupled to a JTL at one end via a coupling capacitor $C_{\text{c}}$, and the other end is terminated by an external source made of a resistor $R_{\text{in}}$ and an inductor $L_{\text{in}}$. The red region represents the JTL in which a fluxon propagates. (b) A lumped-element circuit diagram of the system. The junctions in the JTL are approximated by inductors $L_{\text{J}}$, where the Josephson energy $E_{\text{J}}$ and the corresponding inductance $L_{\text{J}}$ are associated with each other, as $L_{\text{J}} \coloneqq \varphi_0^2/E_{\text{J}} = \varphi_0 / I_{\text{c}}$ per unit cell of the JTL. The approximated transmission line is called a KGTL. The small square resperents a subgap resistor $R_{\text{J}}$ per unit cell. Other parameters are the same as the previous ones.