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arXiv:1005.2446 · Physical Review A

Lyapunov control on quantum open systems in decoherence-free subspaces

Wei Wang, L. C. Wang, X. X. Yi

Published Sep 30, 2010• 51 citations
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Abstract

A scheme to drive and manipulate a finite-dimensional quantum system in decoherence-free subspaces (DFSs) by Lyapunov control is proposed. Control fields are established by the Lyapunov function. This proposal can drive the open quantum system into DFSs and manipulate it to any desired eigenstate of the free Hamiltonian. An example which consists of a four-level system with three long-lived states driven by two lasers is presented to demonstrate the scheme. We have performed numerical simulations of the dynamics of the four-level system, which show that the scheme works well.

DOI
10.1103/physreva.82.034308

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A scheme to drive and manipulate a finite-dimensional quantum system in decoherence-free subspaces (DFSs) by Lyapunov control is proposed. Control fields are established by the Lyapunov function. This proposal can drive the open quantum system into DFSs and manipulate it to any desired eigenstate of the free Hamiltonian. An example which consists of a four-level system with three long-lived states driven by two lasers is presented to demonstrate the scheme. We have performed numerical simulations of the dynamics of the four-level system, which show that the scheme works well.
We present a scheme to drive a finite-dimensional quantum system into the decoherence-free subspaces (DFS) by Lyapunov control. Control fields are established by Lyapunov function. This proposal works well for both closed and open quantum systems, with replacing the DFS by desired subspaces for closed systems. An example which consists of a four-level system with three degenerate states driven by three lasers is presented to gain further insight on the scheme. Numerical simulations for the dynamics of the system are performed and the results are good.
The problem of quantum state control in open quantum systems based on the Lyapunov method is studied. The control laws are designed to drive the system to a pure target state in decoherence-free subspace (DFS) in the interaction picture, where the average of an observable operator P is chosen as a Lyapunov function. By analyzing the positive limit set via Barbalat’s lemma, we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state. Under this condition, it is proved that the control laws can drive the open system to any pure target state in DFS in the interaction picture. Also, we provide a method to construct the observable operator via Schmidt orthogonalization. Numerical simulations on a three-level system validate the effectiveness of the proposed control strategy.
We review the scheme of quantum Lyapunov control and its applications into quantum systems. After a brief review on the general method of quantum Lyapunov control in closed and open quantum systems, we apply it into controlling quantum states and quantum operations. The control of a spin-1/2 quantum system, driving an open quantum system into its decoherence free subspace (DFS), constructing single qubit and two-qubit logic gates are taken to illustrate the scheme. The optimalization of the Lyapunov control is also reviewed in this article.
A convergent state control strategy of open quantum system based on Lyapunov method is proposed for the target state being a pure state in decoherence-free subspaces. Based on the assumptions of all the transition frequencies being different and all the levels being directly coupled,a sufficient condition is proposed for the observation operator to make that the largest invariant set only contains target state.On the basis of choosing the average of observable operators as a Lyapunov function,the control laws are designed in interaction picture and the largest invariant set of the controlled system is analyzed via Barbalat's Lemma. It is shown that any pure target state in decoherence-free subspace is globally asymptotic stable under the sufficient condition given,and the system controlled could be transferred to target state from arbitrary initial state.Also,a method is provided to construct the observation operator via Schmidt orthogonalization.Numerical experiments on a three-level system verify the validity of the proposed control strategy.
To maintain state coherence in an open quantum systems, the decoherence-free subspace (DFS) is an effective method. The target state is usually needed to be in DFS. In the case the target state is not in DFS, we first design external control field I to construct a DFS included target state. Then another control field II is used to transfer the state of the system to be controlled to the expected target state. As is well known, invariant set contains all stable states of the system. Control Hamiltonian is designed to shrink invariant set and increase the transition probability from the system state to target state. In such a way, the state transfer and coherence maintain are completed. Numerical simulation is given for a four-level energy open quantum system to demonstrate the effectiveness of the control strategy proposed.
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.