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International Journal of Modern Physics B

Lyapunov-based state transfer and maintenance for non-Markovian quantum system

Juju Hu, Qiang Ke, Yinghua Ji

Published Sep 5, 2016• 23 citations
Publisher

Abstract

One main problem of quantum control in quantum systems is the manipulation of states including state transfer and state tracking. We design control fields to realize arbitrary state transfer for open quantum system by Lyapunov stability theory, and investigate the state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under Lyapunov control function by an external steady control field of proper amplitude, with a success rate of more than 98%. When the right control field and function are implemented, not only the decoherence is compensated completely but also the purity of quantum states is maintained in the process of state transfer.

DOI
10.1142/s0217979216501770

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One main problem of quantum control in quantum systems is the manipulation of states including state transfer and state tracking. We design control fields to realize arbitrary state transfer for open quantum system by Lyapunov stability theory, and investigate the state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under Lyapunov control function by an external steady control field of proper amplitude, with a success rate of more than 98%. When the right control field and function are implemented, not only the decoherence is compensated completely but also the purity of quantum states is maintained in the process of state transfer.
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In quantum state transfer and tracking, control of quantum states are effective against decoherence caused by external environment and information processing. In this paper, utilizing model reference adaptive control theory and Lyapunov stability theorem, we derive the adaptive law for the model reference adaptive system. Then we design the Lyapunov control law by double control functions and we investigate the orbit tracking of quantum state for Markovian quantum system. Meanwhile, through counteracting the shift term by simplization, the control law avoids oscillating state trajectory in a two-level Markovian quantum system. Finally, numerical simulations for a two-level open quantum system validate the effectiveness of the method.
The weak-coupled two-level open quantum system described by non-Markovian Time-convolution-less master equation is investigated in this paper. The cut-off frequency, coupling constant and transition frequency, which impact on the system's decay rate, coherence factor and purity, are investigated. The appropriate parameters used in system simulation experiments are determined by comparing analysis results of different values of parameters for the effects of system performance. The control laws used to transfer the system states are designed on the basis of the Lyapunov stability theorem. Numerical simulation experiments are implemented under the MATLAB environment. The features of the free evolution trajectory of the non-Markovian systems and the states transfer from a pure state to a desired pure state under the action of the proposed control laws are studied, respectively. By comparing the experimental results, the effectiveness of the proposed quantum Lyapunov control method applied to the state transfer in non-Markovian open quantum systems is verified. Meanwhile, the influences of different control parameters and cut-off frequencies on the system performance are analyzed.
In the case of fixed system frequency and ambient temperature, we design a Lyapunov-based controller to transfer and maintain states for the Time-convolution-less (TCL) master equation of two-level non-Markovian open quantum systems. Basis on the state transferring from the superposition state to superposition state, we maintain the state in the superposition state. The system simulation experiments are implemented under the MATLAB environment. The comparison experiments of the fault-tolerant performance before and after maintaining state demonstrate the superiority of the designed controller. At the same time, the influence of control parameters to length of time maintaining the state of the system are analyzed, and the best control parameters are given.
A control method for trajectory tracking in non-Markovian open quantum systems is proposed in this paper. By inducing the error between the controlled state and the target state as the error state, the trajectory tracking problem of the quantum system is divided into two steps, which the regulation control of the error state and the keeping control of the error state. With the help of Lyapunov stability theorem, the tracking control law is designed to enable the non-Markovian open quantum system to track the evolutionary target quantum system over time. The control parameters are adjusted by simulation experiments, and the numerical simulation experiments are carried out. The tracking control of the quantum system when the initial states of the controlled system and the initial states of the target system are superposition states is implemented in simulation experiments. The experimental results demonstrate the feasibility of the proposed control method and the analysis is given.
We investigate in depth the Lyapunov-based quantum control methods including the relation between the Lyapunov-based control and the optimal control,the Lyapunov function and the performance index,the range of application and the existing problems.Combining our results with intrinsic properties of quantum system,we develop several control schemes with improvements for different objectives such as the regulation of eigenstates,superposition state or mixed state.The design ideas,the physical significance and the range of suitable applications are analyzed.The designs method employing Lyapunov stability theorem in controlling quantum system is systemized.