Andreev spin qubit protected by Franck-Condon blockade

Abstract

Andreev levels localized in a weak link between two superconductors can trap a superconducting quasiparticle. If there is a spin-orbit coupling in the link, the spin of the quasiparticle couples to the Josephson current. This effect can be leveraged to control and readout the spin of the quasiparticle thus using it as a qubit. One of the factors limiting the performance of such an Andreev spin qubit is spin relaxation. Here, we theoretically demonstrate that the relaxation lifetime can be enhanced by utilizing the coupling between the Andreev spin and the supercurrent in a transmon circuit. The coupling ensures that the flip of the quasiparticle spin can only happen if it is accompanied by the excitation of multiple plasmons, as dictated by the Franck-Condon principle. This blocks spin relaxation at temperatures small compared to plasmon energy.

Figures (3)

• Figure 1: Protected Andreev spin qubit. Left: A spin-$1/2$ quasiparticle (green) trapped inside a Josephson junction forming a resonant transmon circuit. The quasiparticle spin couples to the supercurrent flowing in the circuit via the spin-orbit coupling, leading to a spin-dependent Josephson potential $U_\pm(\phi)$. The noise in the magnetic environment of the spin (dark yellow spins) causes it to relax. The quasiparticle spin can also be externally driven (blue arrow), e.g. via a local gate. Right: The potential energy of the circuit splits in two branches (dashed lines), one for spin up (blue) and one for spin down (red). The transmon ground state wave functions for opposite spins are disjoint, which protects the spin from relaxation. The protection can be compromised by a finite temperature, which activates transitions where circuit plasmons are excited in addition to the spin flip (vertical arrow). By the Franck-Condon principle, the matrix elements for such transitions are larger than those without plasmons (horizontal arrow).
• Figure 2: (a): Temperature dependence of the spin-flip rate at $J_z=0$. We assume that the system is initialized in the plasmon ground state $k=0$ for one of the two spins. The solid line is Eq. \ref{['eq:rate_sum']}, the purple dashed line is the approximate expression \ref{['eq:gamma_vs_T']}, and the horizontal dashed line is the $T=0$ result \ref{['eq:simple_protection']}. The parameters are $E_c/E_\textrm{so}=0.035$ and $E_0/E_\textrm{so}=1$, yielding $\omega_p/E_\textrm{so}\approx 0.63$ and, using Eq. \ref{['eq:phi0']} and \ref{['eq:phic']}, $\xi_0\approx 2.8$. (b)-(c): Dependence of the spin-flip rate on temperature and magnetic field parallel to the spin-orbit axis, starting from either spin down (b) or spin up (c). The rates display step-like features as a function of the magnetic field, which are washed out by increasing temperature. The vertical and horizontal lines correspond to the line cuts shown in panels (a) and (d) respectively. (d): Magnetic field dependence of the spin-flip rate for initial states with opposite spin (initialized in a state with $k=0$). The solid curves are computed at $T/\omega_p = 0.1$, while the dashed ones at $T/\omega_p = 1$. In the low temperature curves, the steps occur at half-integer values of $J_z/\omega_p$, and have heights given by the coefficients $w_k$. The results presented in this figure are derived for a simple bath of two-level systems, with a constant density of states and energy-independent coupling to the Andreev spin.
• Figure 3: Spin-flip rate in response to an external drive at frequency $\omega$ as a function of the coupling $E_\textrm{so}$, using Eq. \ref{['eq:drive_response']}. We set $T=0$, $J_z/2\pi=0.2$ GHz, $E_0/2\pi=1$ GHz, $E_c/2\pi=35$ MHz, $\Gamma/2\pi=50$ MHz. The dark lines visible in the figure correspond to distinct spin-flip transitions, each accompanied by the simultaneous excitations of $k$ plasmons, with increasing values $k=0,1,2,3,\dots$ as indicated. The visibility of each transition depends on $E_\textrm{so}$ via corresponding Franck-Condon factors $w_k$, as in Eq. \ref{['eq:drive_response']}. The results illustrate that in the presence of spin-orbit coupling, the spin-flip transition fans out into a set of plasmonic transitions with spacing $\omega_p$.