Controlled operations in a strongly correlated two-electron quantum ring

We have analyzed the electronic spectrum and wave-function characteristics of a strongly correlated two-electron quantum ring with model parameters close to those observed in experiments. The analysis is based on an exact diagonalization of the Hamiltonian in a large B-spline basis. We propose a qubit pair for storing quantum information, where one component is stored in the total electron spin and one multivalued “quMbit” is represented by the total angular momentum. In this scheme the controlled-NOT quantum gate is demonstrated with near 100% fidelity for a realistic far-infrared electromagnetic pulse.

Figure 1

(Color online) Energy levels as functions of the absolute value of the total angular momentum $\left|M_L\right|$. The full blue (dark gray) lines correspond to singlets and the dashed red (light gray) lines to triplets. The states used to realize the CNOT gate are labeled using the notation $|S{M}_L⟩$.

Figure 2

(Color online) Probability densities and currents of the two-electron ring plotted in the coordinates of the first electron [see Eqs. (4, 5)]. The left column depicts the lowest energy states (EXC$=0)$ for $M_L=0$, 1, and $-2$ counting downward. The right column depicts the first, second, and third excited states (EXC$=1$, 2, and 3) with $M_L=1$, also counting downward. Here the ring radius is set to $r_0=2\text{a}\text{.u}{\text{.}}^{\ast }\approx 19.6\text{nm}$.

Figure 3

(Color online) Relative probability densities and currents [see Eqs. (6, 7)] of the $M_L=2$ system. The left column depicts singlets and the right triplets, starting with the lowest lying singlet (triplet) at the top left (right) corner, continuing with the first-excited singlet (triplet) in the middle left (right) panel, and so on, compare Fig. 1. The label EXC$=0$ is used for ground states, EXC $=1$ for the first-excited state, etc. Here the ring radius is set to $r_0=2\text{a}\text{.u}{\text{.}}^{\ast }\approx 19.6\text{nm}$.

Figure 4

(Color online) The quotient between $\Delta E_{\text{triplet}}=E_{|10⟩}-E_{|11⟩}$ and $\Delta E_{\text{singlet}}=E_{|01⟩}-E_{|00⟩}$ (see Fig. 1) as a function of the ring radius $r_0$. The inset shows the absolute value of $\Delta E_{\text{triplet}}$ and $\Delta E_{\text{singlet}}$ as a function of the same. Here the effective Bohr $\text{radius}=1\text{a}{\text{.u}}^{\ast }\approx 9.8\text{nm}$.

Figure 5

(Color online) Upper panel: the time development of the populations of the different states when the central frequency, ${\omega }_L$, of the pulse corresponds to the energy shift between the two lowest states in the singlet system, $\Delta {\epsilon }_{|S{M}_L⟩}={\epsilon }_{|01⟩}-{\epsilon }_{|00⟩}\approx 3.8\text{meV}$ implying a laser frequency of 0.9 THz. The population of the state $|S{M}_L⟩=|00⟩$ is almost completely transferred to $|01⟩$, while the population of the $|10⟩$ is seen to be nearly constant. The inset shows the $x$ component of the electromagnetic pulse. Lower panel: the truth table shows just a small amount of unwanted population. The pulse length was chosen to be $T=500\text{a}{\text{.u}}^{\ast }\approx 28\text{ps}$.