Spin Quantum Bit with Ferromagnetic Contacts for Circuit QED

We theoretically propose a scheme for a spin quantum bit based on a double quantum dot contacted to ferromagnetic elements. Interface exchange effects enable an all electric manipulation of the spin and a switchable strong coupling to a superconducting coplanar waveguide cavity. Our setup does not rely on any specific band structure and can in principle be realized with many different types of nanoconductors. This allows us to envision on-chip single spin manipulation and readout using cavity QED techniques.

Figure 1

(a) The two quantum dots $L$ and $R$ (transparent spheres) are connected together through a tunnel junction with hoping constant $t$ (small green disk). Each dot is connected to a grounded normal reservoir (yellow rectangle) through a ferromagnetic insulator (FI) (large green disks). The magnetizations ${\overrightarrow{m}}_L$ and ${\overrightarrow{m}}_R$ of the two FIs (red arrows) form an angle $\theta$. The double dot circuit is placed between the center and ground conductors of a superconducting coplanar waveguide (green tapes), which conveys a voltage $V_{\mathrm{ac}}$. The dot $L(R)$ is capacitively coupled to a gate with voltage $V_g^{L(R)}$. We tune $V_g^L$ and $V_g^R$ such that there is a single active electron on the double dot system, mainly localized on dot $L$ (magenta cloud). (b) Top view of the same device on a wider scale. The double dot is placed at an antinode of the cavity voltage standing wave. For clarity we have not respected the relative scales.

Figure 2

(a) Stability diagram indicating the double dot most stable state for $t=0$, versus the reduced gate voltages $n_g^{L(R)}=C_g^{L(R)}{V}_g^{L(R)}/e$. We have used the capacitance values indicated in the supplementary material [21], and $V_{\mathrm{ac}}=0$. The circuit can be operated along the dashed line $n_g^R=0.5+[(C_{\Sigma }^R+C_m)/(C_{\Sigma }^L+C_m)](0.5-n_g^L)$ to maximize the energy separation between the singly occupied levels $|\varnothing ,{\uparrow }_R⟩$,$|\varnothing ,{\downarrow }_R⟩$,$|{\uparrow }_L,\varnothing ⟩$ and $|{\downarrow }_L,\varnothing ⟩$ and other energy levels corresponding to a different double dot occupation. This yields a detuning $D=e^2(2n_g^L-1)/(C_{\Sigma }^L+C_m)$ between the states $|{\uparrow }_L,0⟩$ and $|0,{\uparrow }_R⟩$. (b) Eigenenergies of the double dot versus $D$ for $\delta =16\mu \mathrm{eV}$, $\theta =\pi /6$, $t=0$ (dashed lines), and $t=2\delta /3$ (solid lines). When $t=0$, the levels $|{\downarrow }_L,0⟩$ and $|0,{\uparrow }_R⟩$ cross at $D=2\delta$. This crossing is replaced by an anticrossing for $t\ne 0$ and $\theta \ne 0[2\pi ]$. At $t=2\delta /3$, $D=2.8\delta$ and $\theta =\pi /6$ (on point), we obtain transition frequencies ${\nu }_{01}=7.68\mathrm{GHz}$ and ${\nu }_{12}=4.41\mathrm{GHz}$.

Figure 3

(a) Spin–electric-field coupling $g$ of the spin qubit, in terms of vacuum Rabi frequency $g/\pi$. This quantity decreases with $D$, vanishes for $\theta =0[\pi ]$, and reaches a maximum for values $\theta =\pm {\theta }_{\max }(D)$ close to $\pm \pi /2$. At the on point ($D=2.8\delta$, $\theta =\pi /6$), we obtain $g/\pi =5.6\mathrm{MHz}$. At the off point ($D=20\delta$, $\theta =\pi /6$), which is indicated in Fig. 2a, we obtain $g=13\mathrm{kHz}$. (b) Derivative of the transition frequency ${\nu }_{01}$ with respect to $D$. This quantity mainly determines the sensitivity of the qubit to charge noise mediated by fluctuations of $D$.

Figure 4

Inset: Implementation of our setup with a SWNT. The dots $L$ and $R$ are delimited by the gates enclosing the insulating layer $Is$ and the ferromagnetic insulators FI1 and FI2. The outer normal metal contacts $N1$ and $N2$ are used to control the dots’ occupation numbers. Main frame: Half-effective Zeeman splitting ${\delta }^L$ calculated for dot $L$ in the framework of Eq. (2), with $|{\uparrow }_L,\varnothing ⟩$ and $|{\downarrow }_L,\varnothing ⟩$ corresponding to the 29th lowest pair of spin-dependent levels in the dot conduction band. This quantity depends on the chemical potential shift $E_b-E_g^L$ between dot $L$ and the neighboring SWNT sections. We obtain a sweet spot $\partial {\delta }^L/\partial E_g^L=0$ (black point) which can be used to reduce charge noise-induced dephasing mediated by fluctuations of ${\delta }^L$. In this Figure, we use $E_b=E_{Is}$, $R=1\mathrm{nm}$, $\lambda =100\mathrm{nm}$, and $E_{\mathrm{ex}}=3.7\mathrm{meV}$.