Readout and dynamics of a qubit built on three quantum dots

We present a model of a qubit built of three coherently coupled quantum dots with three spins in a triangular geometry. The qubit states are encoded in the doublet subspace and they are controlled by a gate voltage, which breaks the triangular symmetry of the system. We show how to prepare the qubit and to perform one-qubit operations. A doublet blockade effect in the current will be discussed. The blockade is related to an asymmetry of transfer rates from the electrodes to different doublet states and is used to read out the dynamics of the qubit state. Our research also presents analysis of the Rabi oscillations, decoherence, and leakage processes in the doublets subspace.

Figure 1

The model of a triangular molecule placed in the effective electric field $\mathbf{E}$.

Figure 2

Density matrix elements ${\rho }_{D_1{D}_1}$ (solid black curve) and ${\rho }_{D_2{D}_2}$ (dashed red curve) vs $\theta$ for the ground state at a moderate value of the electric field $g_E=0.2,J=1$. The left and right inserts show direction of the electric field when the qubit is prepared in the states $|D_2\rangle$ and $|D_1\rangle$, respectively. These two opposite directions of the electric field correspond to a minimal (thin line) and maximal (thick line) bond polarization between dots 2 and 3.

Figure 3

Doublet blockade effect for the case with the two-electron ground state as the singlet state. (a) Current characteristic versus applied voltage. The vertical dashed lines show the positions of the state $|D_1\rangle$ and $|D_2\rangle$. (b) Probabilities of occupation of the states $|S\rangle$, black; $|D_1\rangle$, blue; and $|D_2\rangle$, red curve. The calculations were performed for the parameters $J=1$ (in the further calculations it is taken as unity), $\delta =-0.3,\gamma =0.08$ (which corresponds to $g_E=0.2$ and $\theta \approx 1.42\pi$), $t_0=-3,U_{i,i+1}=5$, for which the states are at $E_S=-1,E_{D_1}=13.35$, and $E_{D_2}=13.65$. The Fermi energy is taken at $E_F=14$, temperature $T=0.05$, and $\Gamma =0.05$.

Figure 4

Doublet blockade when the two-electron ground state is a triplet. (a) Current characteristic versus applied voltage. The vertical dashed lines show the positions of doublets $|D_1\rangle ,|D_2\rangle$ as well as quadruplets $|Q\rangle$. (b) Probabilities of occupation of the states $|T\rangle$, black; $|D_1\rangle$, blue; $|D_2\rangle$, red; and $|Q\rangle$, green curve. We use the parameters $J=1,\delta =0.3,\gamma =0.08,t_0=3,U_{ij}=5$, for which $E_T=-1,E_{D_1}=13.65,E_{D_2}=13.35$, and $E_Q=15$. The Fermi energy is $E_F=13.8$, temperature $T=0.05$, and $\Gamma =0.05$.

Figure 5

Coherent oscillations when the singlet ground state is engaged in the current flow. The plots are derived for the time-independent mixing $\gamma \left(t\right)={\gamma }_0$ of the doublet states $|D_1\rangle$ and $|D_2\rangle$. Panels (a) and (b) present the density matrix elements and the currents flowing through the junctions, respectively. The black-dashed line corresponds to the current in the stationary limit. Panel (c) shows the time evolution of the pseudospin on the Bloch sphere. The black arrow represents the initial state $|D_1\rangle$ and the green arrow the final state. The parameters taken in the calculations: $\delta =-0.3,{\gamma }_0=0.08,\Gamma =0.05,{\rho }_{D_1{D}_1}\left(0\right)=1$, and the other parameters as in Fig. 3. The relaxation rates calculated exact are $1/T_{\mathrm{leak}}=0.0306,1/T_1=0.0069$, and $1/T_2=0.0083$.

Figure 6

Rabi oscillations for the driven case $\gamma \left(t\right)={\gamma }_{0}exp(-i\omega t)$ at the resonance condition $\left(\omega =\delta \right)$. The notation and the parameters are the same as in Fig. 5. Notice that the results are presented in the rotating frame, whereas Fig. 5 presents the evolution in the laboratory frame. The relaxation rates are $1/T_{\mathrm{leak}}=0.0306$ and $1/T_2=0.0083$.

Figure 7

Coherent oscillations when the triplet ground state is engaged in the current flow. Figures are plotted for the time-independent mixing term with $\gamma \left(t\right)={\gamma }_0$ switched on at $t=0$. Panel (a) presents the time evolution of the density matrix elements, while (b) presents the current flowing through the left (blue curve) and right junction (black curve). The black-dashed line corresponds to the current in the stationary limit. Panel (c) shows the evolution of the pseudospin on the Bloch sphere. The initial state is $|D_2\rangle$ (the black arrow), and the final state is represented by the green arrow. The parameters taken in the calculations: $\delta =0.3,{\gamma }_0=0.08,\Gamma =0.05.$ The other parameters are the same as in Fig. 4. The relaxation rates for these parameters are $1/T_{\mathrm{leak}}=0.0405,1/T_{\mathrm{leak}}^{TQ}=0.0117,1/T_1=0.0033$, and $1/T_2=0.0055$. Notice that in the present case the currents are smaller than those in Fig. 5 because now the transfer rates are different.

Figure 8

Coherent oscillations including spin-flip processes when singlet in engaged in the current flow. Panel (a) presents the density matrix elements for all considered states; panel (b) shows the time dependence of the current flowing through the left junction with spin $+1/2$ (blue dotted curve), $-1/2$ (blue dashed curve) and right junction with spin $\pm 1/2$ (black dashed curve). Notice that $I_R^{\uparrow }\left(t\right)=I_R^{\downarrow }\left(t\right)$. The total currents $I_{L/R}\left(t\right)=I_{L/R}^{\uparrow }\left(t\right)+I_{L/R}^{\downarrow }\left(t\right)$ are plotted as a solid lines. The parameters taken in the calculations are the same as in Fig. 5. The relaxation rates are $1/T_{\mathrm{leak}}=0.05,1/T_1=0.0069,1/T_2=0.0083,1/T_{sf}^{+}=0.0163$, and $1/T_{sf}^{-}=0.0003$.