Proposal of a full Bell state analyzer for spin qubits in a double quantum dot

We propose a scheme for full Bell state measurement of spin qubits in a double quantum dot. Our scheme consists of Pauli spin-blockade measurements and biaxial electron-spin resonance. In order to eliminate the average of the Zeeman fields, the double quantum dot is designed so that the Landé $g$-factors of first and second dots satisfy $g_1=-g_2$ with the use of $g$-factor engineering. Thus, we can swap one of the three spin-triplet states for the spin-singlet state without disturbing the other states. Our study shows that the sequential spin-to-charge conversions enable us to implement the full Bell state measurement of electron-spin qubits.

Figure 1

(Color online) Schematic view of (a) vertically and (b) laterally coupled double QDs. Both the QDs are located between the heterostructure barriers, and are then focused to small areas with the use of the surface metallic gates. We assume that the $g$-factors in the two QDs are designed to satisfy the condition $g_1=-g_2$. For the vertical double QD, such a condition can be achieved, e.g., by adjusting the thickness and the alloy composition of each quantum well. For the lateral QDs, we can achieve the condition by controlling the equilibrium electron positions in individual QDs when the top and bottom barrier layers have different $g$-factors. We take the $z$ axis as the direction of the static magnetic field, and apply oscillating magnetic fields in $x$ and $y$ directions common to the two QDs. The ac fields flip the electron spins confined in the QDs when the ESR condition is satisfied.

Figure 2

(Color online) The sequence of the Bell state measurement, we propose, is shown schematically. The solid lines indicate the energy costs to add an extra electron to the first or the second QD. The dotted line with a filled circle indicates the energy levels of the (0,1) charge state. (i) Initially the detuning energy is set to be a large negative, $ϵ={ϵ}_0⪡-\sqrt{2}T$. (ii) Applying gate voltage to the first QD (depicted in left), we apply the spin-blockade regime $\left(ϵ={ϵ}_1>\sqrt{2}T\right)$. The electron in the first QD can tunnel to reservoir when the two-electron state is in $|S⟩$. (iii) If no electron tunneling is detected, the system is brought back to the Coulomb-blockade regime $ϵ={ϵ}_0$. We apply the effective ac field $B_{\text{ac}}^x\left(t\right)$ for the time ${\tau }_{\text{ESR}}$ on both the QDs, and then we wait for the time $t_0$. This swaps $|T_{-}⟩$ for $|S⟩$. (iv) We apply the spin-blockade regime to the QDs, and determine whether the two-electron state was initially in $|T_{-}⟩$. If the electron tunneling is not detected again, (v) we flip $|T_{+}⟩$ into $|S⟩$ by $B_{\text{ac}}^y\left(t\right)$, and (vi) determine that the system was in $|T_{+}⟩$ or $|T_0⟩$.

Figure 3

(Color online) (a) Energy diagram for the relevant states. From initial point A with $S\left(0,2\right)$, we can reach point C by sweeping the detuning $ϵ$ adiabatically $\left(\text{A}\to \text{B}\right)$ and then nonadiabatically $\left(\text{B}\to \text{C}\right)$. (b) Enlarged view of the region indicated by square in panel (a). The hyperfine field allows for flip-flop process $\left(|T_0⟩\to |\uparrow \uparrow ⟩\right)$ when $h_n^{\parallel }>0$. Then driving the detuning up adiabatically from C can reduce the total hyperfine field $h_n^{\parallel }$. For $h_n^{\parallel }<0$, flip-flop process $\left(|T_0⟩\to |\downarrow \downarrow ⟩\right)$ can reduce $\left|h_n^{\parallel }\right|$ in a similar way.