Coherent control of localization, entanglement, and state superpositions in a double quantum dot with two electrons

We have recently proposed a quantum control method based on the knowledge of the energy spectrum as a function of an external control parameter [G. E. Murgida, D. A. Wisniacki, and P. I. Tamborenea, Phys. Rev. Lett. 99, 036806 (2007)]. So far, our method has been applied to connect the ground state to target states that were in all cases energy eigenstates. In this paper we extend that method in order to obtain more general target states, working, for concreteness, with a system of two interacting electrons confined in semiconductor double quantum wells. Namely, we have shown that the same basic method can be employed to obtain localization, entanglement, and general superpositions of eigenstates of the system.

Figure 1

Schematic plot of the building block of our control method. An avoided crossing and a schematic drawing of the diabatic (fast), adiabatic (slow), and intermediate ways to cross it. (a) Long (short) arrows represent fast (slow) variations in the control parameter; (b) medium arrows represent the intermediate velocity.

Figure 2

Confining double-well potential in the longitudinal direction of the coupled quantum-dot structure. The external electric field is $E=0$ (solid lines) and $E=12\text{kV}/\text{cm}$ (dashed lines).

Figure 3

Main panel: The energy spectrum of the two interacting electrons confined in a quasi-one-dimensional double-well semiconductor nanostructure as a function of an external uniform electric field. The first 31 energy levels are shown. Side panels: Spatial wave functions ${\varphi }_i\left(E,z_1,z_2\right)$ corresponding to labels (a) to (h) of the main panel. States (a) and (b) have one electron in each well, wave functions (c) and (d) are localized in the left well, and wave functions (e) and (f) have both electrons in the right well. States (a)–(f) are far from avoided crossings and therefore have well-defined localization properties. This is not the case for eigenstates (g) and (h), which are at avoided crossings.

Figure 4

Probability $P_{LL}$ as a function of the time-dependent electric field $E\left(t\right)$. The velocities of the electric field are as follows: (i) 4, (ii) 1, (iii) 0.4, (iv) 0.2, and (v) 0.04 (kV/cm)/ps. With circles we represent $P_{LL}$ of the adiabatic ground state as a function of the electric field (see text for details.) Inset: closer view of the first avoided crossing involved in these processes. The arrows indicate the adiabatic path.

Figure 5

Localization with the sudden-switch method. While the electric field stays at the avoided-crossing value $\left(E=4.81\text{kV}/\text{cm}\right)$ the probability $P_{LL}$ oscillates with the frequency corresponding to the energy gap (dashed lines). The method consists of increasing suddenly the electric field when $P_{LL}$ is maximal, freezing the localization on its highest value (full lines). Upper panel: probability that both electrons are in the left dot. Lower panel: step-wise constant electric field.

Figure 6

(Color online) Control paths for two electrons leading to target states which are superpositions of states with $RR$ and $LL$ localization. The lengths of the arrows indicate the velocity of the transitions. The target states are: (a) $\left(|RR⟩+|LL⟩\right)/\sqrt{2}$; (b) $\left(|RR⟩-|LL⟩\right)/\sqrt{2}$; (c) state given in Eq. (3).

Figure 7

Numerical simulation of the path of Fig. 6a in order to reach the Bell-type state $\left(|RR⟩+|LL⟩\right)/\sqrt{2}$. The electric field used (a) and the probabilities obtained (b) are shown as functions of time in semi-log plots.

Figure 8

(Color online) Schematic diagram of the path to obtain the target which is a superposition of three adiabatic states with different localization [Eq. (4)]. For clarity, we use different colors (online version) when the electric field is increasing (blue and red) or decreasing (green). Our initial state, the ground state without electric field, is indicated by a filled square. The desired target state is a combination of the three adiabatic states indicated by $\oplus$.

Figure 9

Upper panel: electric field as a function of time used to carry out the control strategy schematically shown in Fig. 8. Lower panel: overlap of the evolving state with the first six adiabatic states. We can clearly see that the final state approximates well an even linear combination of the adiabatic states 1, 5, and 6 at the corresponding final electric field.

Figure 10

Evolving wave function at various times during the time evolution schematically shown in Fig. 8 and described quantitatively in Fig. 9. See text for details.