Theoretical Study And Application Of Quantum Lyapunov Control

As a burgeoning interdiscipline, quantum control is a fundamental theory to realize quantum computer and quantum communication. The development of quantum control can facilitate the fusion of physics, chemistry and biology, and promote the progress of quantum technology. The research of quantum control mainly contains a set of essential topics such as the manipulation and preparation of quantum state, and the construction of decoherence-free space. The research of quantum control will also contribute to raise the status of quantum information in the future communication. The thesis is mainly contributed to several appli-cations of Lyapunov control in different quantum systems, including the spin-1/2 chains, the ID Kitaev's chain of spinless fermions, double-well 1D optical lattice, the hybrid systems of quantum dots coupled with superconductor and the 1D Fermi gas trapped in optical lattices. These applications will definitely stimulate the development of future quantum information processing. The main research work starts from Chapter 3.In Chapters 1 and 2, the background of this thesis is introduced, and the Lyapunov stability theorem and the invariant set theorem are given as well. Those theorems provide theoretical foundation for our work. Besides, a briefly introduction of several methods of constructing Lyapunov functions is presented.In Chapter 3, a scheme to realize quantum state transfer with high fidelity in spin-1/2 chain is presented by only modulating the interaction between boundary spins and neighbour spins or the Larmor frequency of boundary spins. Compared to the conventional transmission protocols, the present scheme possesses the following advantages:the final state is a steady state; it does not require precise manipulations of the control time; it is robust against the fluctuations in the control fields. The scheme can also works for variable spin-1/2 chains with different periodie structures.In Chapter 4, a scheme to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian is presented. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the uoninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The possibility to replace the continuous control field with square wave pulses is finally discussed.In Chapter 5. with the assistance of a pair of Majorana fermions, four different schemes are considered to entangle two quantum dots, i.e., the teleportation scheme, the crossed Andreev reflection scheme, the intradot spin flip scheme and the scheme beyond the intradot spinflip. We demonstrate that the entanglement can be generated by Lyapunov control and adiabatic passage. In contrast to adiabatic passage, the Lyapunov control manifests two advantages at flexibility designing control fields and accelerating control time.In Chapter 6, a proposal to prepare edge states in the Aubry-Andre-Harper model is presented. The advantage is that we only control the energies of boundary sites to realize edge states. The we adopt the deformation Lyapunov function to design control fields to achieve boundary-boundary entangled state, that is, the maximal entanglement between the two edge states. The method offers a new way to the manipulation of edge state.The conclusion and prospection are given in Chapter 7.