Longitudinal coupling between electrically driven spin qubits and a resonator

At the core of the success of semiconducting spin qubits is the ability to manipulate them electrically, enabled by the spin-orbit interactions. However, most implementations require external magnetic fields to define the spin qubit, which in turn activate various charge-noise mechanisms. Here we study spin qubits confined in quantum dots at zero magnetic fields that are driven periodically by electrical fields and are coupled to a microwave resonator. Using Floquet theory, we identify a well-defined Floquet spin-qubit originating from the lowest degenerate spin states in the absence of driving. We find both transverse and longitudinal couplings between the Floquet spin qubit and the resonator, which can be selectively activated by modifying the driving frequency. We show how these couplings can facilitate fast qubit readout and the implementation of a two-qubit CPHASE gate. Finally, we use adiabatic perturbation theory to demonstrate that the spin-photon couplings originate from the non-Abelian geometry of states endowed by the spin-orbit interactions, rendering these findings general and applicable to a wide range of solid-state spin qubits.

Figure 1

Schematics of the setup. Several single electron or hole spins confined in QDs with level spacing ${\omega }_0$ and coupled to the electric field (green arrows) of a microwave resonator. Applying time-dependent electrical fields $\mathit{E}\left(t\right)$ oscillating at a frequency $\mathrm{\Omega }\ll {\omega }_0$, displace the QDs on closed trajectories in the semiconductor, inducing in turn rotations of the lowest degenerate spinors on the Bloch sphere. The Berry phase accumulated during the rotation, ${\gamma }_B$, defines an effective Floquet spin-qubit Hamiltonian with a splitting ${\epsilon }_q\sim {\gamma }_B\mathrm{\Omega }$. The QD dynamics triggers both transverse ($\propto {\epsilon }_q{\widehat{\mathit{E}}}_c{\tau }_{+}^F$) and longitudinal ($\propto \mathrm{\Omega }{\widehat{\mathit{E}}}_c{\tau }_z^F$) spin-photon interactions, which are activated for ${\omega }_c\sim {\epsilon }_q$ and ${\omega }_c\sim \mathrm{\Omega }$, respectively, allowing us to detect the Floquet spin-qubit state and entangle remote spins interacting with a common resonator.

Figure 2

Variation of the longitudinal, ${\tilde{g}}_z^{x,y}(k=1)\equiv m_z^{x,y}(k=1)$, and transverse, ${\tilde{g}}_{+}^{x,y}(k=0)={\gamma }_B{m}_{+}^{x,y}(k=0)$, spin-photon coupling strengths with the ellipticity $R_{0,x}/R_{0,y}$ for $R_{0,y}/{\lambda }_{\text{SO}}=0.1$ (left) and $R_{0,y}/{\lambda }_{\text{SO}}=0.5$ (right).

Figure 3

The relaxation (left) and the pure dephasing (right) rates as a function of the driving frequency $\mathrm{\Omega }$ for holes and electron spins confined in various semiconducting QDs. We have used ${\lambda }_0/{\lambda }_{\text{SO}}=0.2$ for holes (black, red, and blue), while ${\lambda }_0/{\lambda }_{\text{SO}}=0.1$ for electrons (green). For all plots, we assumed $R_0/{\lambda }_{\text{SO}}=0.25$, and $T=40$ mK.

Figure 4

The overlap $|\langle {\mathrm{\Psi }}_i^F\left(t\right)|{\mathrm{\Psi }}_1^{\mathrm{inst}}\left(t\right)\rangle {|}^2$ of the fixed instantaneous state $|{\mathrm{\Psi }}_1^{\mathrm{inst}}\left(t\right)\rangle$ with the Floquet states $\left\{\right|{\mathrm{\Psi }}_i^F\left(t\right)\rangle {\}}_{i=1,\cdots ,i_{\text{max}}}$, evaluated numerically, with $i_{\text{max}}=12$. We observe that the Floquet state with index $i=12, |{\mathrm{\Psi }}_{12}^F\left(t\right)\rangle$, exhibits nearly perfect overlap with $|{\mathrm{\Psi }}_1^{\mathrm{inst}}\left(t\right)\rangle$, and hence the $|{\mathrm{\Psi }}_{12}^F\left(t\right)\rangle$ state is selected as the first Floquet qubit state. We assume a driven harmonic potential with parameters $E_0=0.1, \mathrm{\Omega }=0.3$, and ${\omega }_0=1$, which pertain to the adiabatic regime defined in the main text.

Figure 5

The overlap $|\langle {\mathrm{\Psi }}_i^F\left(t\right)|{\mathrm{\Psi }}_2^{\mathrm{inst}}\left(t\right)\rangle {|}^2$ of the fixed instantaneous state $|{\mathrm{\Psi }}_2^{\mathrm{inst}}\left(t\right)\rangle$ with the Floquet states $\left\{\right|{\mathrm{\Psi }}_i^F\left(t\right)\rangle {\}}_{i=1,\cdots ,i_{\text{max}}}$, evaluated numerically, with $i_{\text{max}}=12$. We observe that the Floquet state with index $i=12, |{\mathrm{\Psi }}_{11}^F\left(t\right)\rangle$, exhibits a nearly perfect overlap with $|{\mathrm{\Psi }}_1^{\mathrm{inst}}\left(t\right)\rangle$, and hence the $|{\mathrm{\Psi }}_{11}^F\left(t\right)\rangle$ state is selected as the second Floquet qubit state. All other parameters are the same as in Fig. 4.