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Quantum Information and Computation

Generation of quantum correlation in quantum dots by Lyapunov control methods

Da-Wei Luo, Jing‐Bo Xu

Published Nov 1, 2014• 1 citation
Publisher

Abstract

We present a scheme to generate steady-state quantum correlations in quantum dots. The shape of the control field is obtained by using the Lyapunov control method, and the controlled state is found to approach the target state monotonically. We also explore the possibility of replacing the continuous control field with a train of discreet rectangular pulses, which is much easier to implement experimentally. The discretized control field is found to be still able to drive the initial state to the target state with very small errors, which suggests that the Lyapunov based control is still effective under practical limitations.

DOI
10.26421/qic14.15-16-6

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We present a scheme to generate steady-state quantum correlations in quantum dots. The shape of the control field is obtained by using the Lyapunov control method, and the controlled state is found to approach the target state monotonically. We also explore the possibility of replacing the continuous control field with a train of discreet rectangular pulses, which is much easier to implement experimentally. The discretized control field is found to be still able to drive the initial state to the target state with very small errors, which suggests that the Lyapunov based control is still effective under practical limitations.
The Lyapunov control theory was used in the manipulation of a single qubit in the two-level double quantum-dot (DQD) system. The proposed control process is composed of three parts: firstly a slope pulse takes the system from a positive detuning adiabatically to the anti-crossing point, which corresponds to the resonance state of the system; then a Lyapunov-based control pulse drives the charge qubit transfer non-adiabatically; finally another slope pulse takes the system away from the anti-crossing point to keep the system stable. The charge state probability P|L〉and the curve of Lyapunov-based control pulse were studied under different control parameters. Simulation results showed that: the designed Lyapunov-based control pulse has a rise time ~100ps, which is in the scope of the Aglient 81134A pulse generator for the implement, and it results in a significant probability 86% for transition from the initial charge state |R〉 to the desired target charge state |L〉. Although our control scheme is designed specific for the DQD system, the technique is independent of the physical architecture and has potential for wider applications.
Lyapunov-based quantum control has the advantage that it is free from the measurement-induced decoherence and it includes the instantaneous information of the system in the control. The Lyapunov control is often confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time delay on the Lyapunov control and explore the possibility of replacing the control field with a pulse train or a bang–bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the system. These results suggest that the Lyapunov control is robust against time delay, easy to realize and effective for high-dimensional quantum systems.
For a double quantum dot (DQD) system, we propose a Lyapunov control method to transfer the state from initial state to the target state on the picosecond scale, which open a significant new area of research for the system control theories. The control laws are composed of two-direction components, one is used to eliminate the dissipation in the system, another is used to transfer the state. The Lyapunov stability theorem ensures the system can be transferred to the target state in high probability, and the coefficients in control laws leads very fast convergence. The role of eliminating the dissipation plays the suppression of decoherence effect. Numerical simulation results show that the transfer probability and fidelity can be increased up to 98.79% and 98.97%, respectively. This is the first result directly applicable to a DQD system's state transferring using the Lyapunov control method. We also give specific experimental realization scheme.
Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.
For a two-level double quantum-dot (DQD) system, we proposed an alternative ultrafast manipulation approach, Lyapunov-based control method, to transfer the charge qubit from the initial charge state |R) to the target charge state |L) on the picosecond scale. Better control performance is obtained compared with that via Landau-Zener-Stu̅ckberge-Majorana interference, and the designed control fields can be implemented in the actual experiment facility. The control process is composed of three parts: first, a slope pulse takes the system from a positive detuning adiabatically to the anticrossing point, which corresponds to the resonance state of the system; then, a Lyapunov-based control pulse drives the charge qubit transfer nonadiabatically; finally, another slope pulse takes the system away from the anticrossing point to keep the system stable. The charge qubit transfer performance and the function of Lyapunov-based control pulse were studied under different control parameters. Simulation results showed that the designed Lyapunov-based control pulse has a rise time ~20 ps, which is in the scope of the Tektronix AWG70000A series arbitrary waveform generator and results in a significant probability ~96% for the transition from the initial charge state |R) to the target charge state |L). The fidelity of the control process designed can achieve ~94%. This is the first result directly applicable for a DQD charge qubit transfer using the Lyapunov-based control method.