Preparing Quantum State By Lyapunov Control

The emergence of quantum information stems from the effective combination of quantum mechanics and information science.Since the sensitivity of the quantum system,how to prepare quantum state in a noise environment has became the main problem of quantum information processing.With the rapid development of the researching in quantum information technology during the past years,people are increasingly focusing on the control of quantum state.The various applications of quantum information require that people have the ability to manipulate and regulate the quantum systems.In classical control,in order to control the whole evolution process via this ways,people usually determines the control field by the condition of system.This method is simple and effective in the classical field,but it is difficult to complete in the quantum system since the state of quantum system must be destroyed during the measurement.Fortunately,in the last decades people found an effective method by continuous researching to obtain the open-loop control law which simulating the dynamics equation of closed-loop control,namely Lyapunov control.The advantage of Lyapunov control is that it can provide a reasonable control function to ensure the quantum system evolve to a stable target state rapidly.It has been widely adopted in the quatun state control and preparation in recent years,such as preparing quantum state from pure states to entangled states? the realization of quantum gates and the cooling of mechanical oscillators.Meanwhile,the researching of other control methods has been emerged during the past days.Nowadays,the development of control theory has gone into every field of human life deeply,and promoted the fusion between the quantum informatics and natural subjects such as Biology and Chemistry.The rapid development of quantum control can offer a greatly assisted for understanding the basic physics problems?operating small nanoscale device and solving the problem of algorithm,it also promotes the progress in natural science researching.As we all know,entanglement has been an important index to describe the correlation of systems,which is at the core of quantum mechanics category.How to realize the preparation and conversion of entanglement effectively have became the focus of people's research.The Graph state and Dicke state are considered the most typical entanglement state at present,and many scheme were proposed to achieve these goals.The quantum Zeno effect opens up a new way to study such questions.It can obtains an effective Hamiltonian which has the same effect as that of non-unitary dynamics,by using the continuous coupling between the interacting system and the observing instrument.Inspired by this method,we propose a scheme to implement 4 atoms W state by Lyapunov control which combination of Zeno dynamics in the third part.According to the effective transformation law of Zeno dynamics,we construct an effective Hamiltonian using the eigenstates of Zeno invariant subspace.We can obtain a perfect W state by approximate selecting the parameters of system,and the fidelity can reach a stable value.Considering that spontaneous emission and cavity damping of atoms are inevitable in the process of interaction,we also calculate the effects of these dissipations of the system,and the numerical simulation also be discussed.Our scheme has certain robustness and insensitive with regard to these effects in simulation results.At the end of this section,we apply the scheme to implement the 1?M economical phase covariant cloning,and we implement the preparation of target state successfully.Quantum Hall effect is described as a specially phenomenon of two-dimensional electron gas in usual,Laughlin realizes the physical interpretation of the procedure by variational method.The approximate wave functions which proposed by Laughlin are agreement with the experimental results commendably.Such as incompressibility?arbitrary excitation with fractional statistical significance and some characteristics are also obtained by this process.In the fourth part,we mainly discuss the implementation of Fractional Quantum Hall(FQH)states by Lyapunov control.In this content,we adopt the general control method,which not only describes the selection of parameters in FQH state? but also the form of control expression and control process of Lyapunov control function.We also applies it into a specific optical lattice system,according to the different input states,we choose the homologous control field to achieve control effectively.Then we also use the pulse control and optimization control fields to optimize the procedure,which are better than conventional method.Finally,the robustness of the method are discussed in this section,and the results indicate that it match with the expectations.A summary for the whole work was illustrated in the final part,and we also figured out the advantage and deficiency of our scheme.The anticipated of future work are also be illustrated in it.