Time-evolution simulation of a controlled-NOT gate with two coupled asymmetric quantum dots

We study a possible physical realization of a quantum controlled-NOT gate with the use of two weakly coupled asymmetric quantum dots. Solving the time-dependent Schrödinger equation for the model two-electron system, we simulate the infrared-radiation-induced quantum transitions that correspond to basic gate operations. We require the transition probabilities to be close to 1 and optimize the parameters of the nanostructure in order to make the gate operation time as short as possible. In the simulations, we have taken into account the entire energy spectrum, which can be populated by the absorption or emission of the infrared radiation. We discuss the consequences of the existence of many bound two-electron states on the probability of radiative transitions.

Figure 1

Confinement potential profile in the vertical direction for two coupled QDs. Shown is also the electron probability density for the one-electron ground (∣0⟩) and first excited (∣1⟩) states. $B$ is the barrier thickness, $D_L$ $\left(D_R\right)$ is the thickness of the deep potential-well region in the left (right) QD, and $S_L$ $\left(S_R\right)$ is the thickness of the shallow potential-well region in the left (right) QD.

Figure 2

Electron probability density for the one-electron ground and first excited states localized in (a) symmetric QD, (b) symmetric QD in an electric field, and (c) asymmetric QD. Arrows indicate the shift of the electron density in cases (b) and (c) with respect to case (a).

Figure 3

Contours of the probability density for the ten lowest-energy states of the two-electron system in the coupled QD nanostructure. Dashed lines separate different quadrants; $d$ denotes the diagonal.

Figure 4

Contour plots of the wave function (a) and the electron probability density (b) for the entangled state [Eq. (9)]. In (a) the values of the wave function are given in gray scale: the bright (dark) areas correspond to the positive (negative) wave-function values. A more detailed description is given in text.

Figure 5

One-electron probability density [Eq. (14)] for the computational-basis states for $t=0$. Also shown is the profile of the confinement potential.

Figure 6

Contour plots of the electron probability density as functions of time $t$ and spatial coordinate $z=z_1$ for the working cycle of the CNOT gate corresponding to the operation defined by Eq. (7a). (a,b) corresponds to the relaxation of the system to the ground state and simultaneously the preparation of the initial state. (c) corresponds to the CNOT gate operation defined by Eq. (7a).

Figure 7

Contour plots of the electron probability density as functions of time $t$ and spatial coordinate $z=z_1$ for the working cycle of the CNOT gate corresponding to the operation defined by Eq. (7b). (a) relaxation to the ground state, (b) preparation of initial state ∣01⟩, and (c) CNOT gate operation defined by Eq. (7b).

Figure 8

Contour plots of the electron probability density as functions of time $t$ and spatial coordinate $z=z_1$ for the working cycle of the CNOT gate corresponding to the operation defined by Eq. (7c). (a) Relaxation to the ground state, (b) preparation of initial state ∣10⟩, and (c) CNOT gate operation defined by Eq. (7c).

Figure 9

Contour plots of the electron probability density as functions of time $t$ and spatial coordinate $z=z_1$ for the working cycle of the CNOT gate corresponding to the operation defined by Eq. (7c). (a) Relaxation to the ground state, (b) two-step preparation of initial state ∣11⟩, and (c) CNOT gate operation defined by Eq. (7d).

Figure 10

Contour plots of the electron probability density as functions of time $t$ and spatial coordinate $z=z_1$ for the entanglement process defined by Eq. (8). (a) Relaxation to the ground state, (b) preparation of initial state $(\mid 00⟩+\mid 10⟩)\sqrt{2}$, and (c) entanglement production defined by Eq. (8).

Figure 11

Probability $P$ of transition from initial state ∣10⟩ to final state ∣11⟩ as a function of the number $N$ of gate operations for coherence time $T=1\mathrm{ns}$.