Manipulation Of Many-body Quantum State For Hermitian And Non-Hermitian Bose-Hubbard Dimer

The physical properties and quantum control of single-particle systems always are investigative focus of physicist, which has acquired great advances. In recent years, the development of atom coolling and trapping technology has provided ex-perimental condition for the investigation of interacting many-body system. The realization of Bose-Einstein condensate (BEC) of ultra-cold atoms in optical lat-tices and the creation of atom chips are two typical examples of many-body system. They not only have potential applications in quantum information processing and other advanced technology but also provide an ideal system for the investigation of atom interferometers and lasers, atom diodes and transistors. The precise tailor-ing and manipulation of optical lattices, on the other hand, enable us to investigate complex solidstate phenomena, such as the transition from Mott-insulator to su-perfiuid, Josephson effect, Anderson localization, and the Bose-glass transitions. Especially, it is envisioned that the emerging field of atomtronics, i.e. the atom analog of electronics materials and circuits, will be able to provide nanoscale de-vices of unprecedented quality.Among all the exciting issues raised in the framework of interacting Bose sys-tems, manipulation of quantum state is a basic issue, which not only is the heart of quantum control, but also is one of major tasks in quantum engineering and quantum information processing. Because BEC is a many-body system charac-terized by convenient manipulation, manipulating its many-body quantum state by adjusting external laser field can become typical example of quantum control. Therefore, this thesis focuses on studying manipulation of many-body quantum state via utilizing external field for a Bose-Hubbard dimer that models a BEC in a double well potential or a two-component BEC. The works are structured as follows: In the first chapter, we introduce Bose-Einstein condensate theory and two mode Bose-Hubbard model.In Chapter 2, we initiate the study of exact coherent control to time-dependent driving two mode Bose-Hubbard model. By employing mean-field approximation and balance conditions we construct new exact Floquet solutions with the de-generate Floquet energy, and give the balance region on the parameter space. It is revealed that for some different parameters in the balance region the Floquet solutions exactly describe the macroscopic quantum self-trapping, nonlinear coher-ent construction and destruction of quantum tunneling, respectively. Therefore, we can accurately perform the coherent control by suitably adjusting the system parameters.In Chapter 3, we show manipulation of many-body quantum states via single-photon resonance for a Bose-Hubbard dimer. By treating the periodic driving in time-dependent driving two mode Bose-Hubbard Hamiltonian as a weak pertur-bation, the transition probabilities up to second-order approximation are given as functions of the driving parameters, which are considerable only for the single-photon resonance case. Due to some transition matrix elements vanishing, the first-order quantum transition obeys a selection rule. The non-forbidden transi-tions involve states of different entanglement entropies and the forbidden transi-tions relate to the entropy balances between two states. The results provide a new route for manipulating many-body quantum states and entanglement entropies, and controlling the atomic tunnelings of the Bose-Hubbard dimer.In Chapter 4, We investigate incoherent control in a non-Hermitian Bose-Hubbard dimer without PT symmetry. It is analytically and numerically found that a real energy appears uniquely when a balance between the interaction strength and non-Hermiticity is established. The corresponding quantum state is a station-ary state which does not decay in time evolution. Any one of the balance conditions leads to a unique real energy and a nondegenerate stationary state. Changing the parameter values to fit a new balance condition can produce a new real energy and a new stationary state. This provides a route for enhancing survival probability of an open many-body system and for incoherently controlling quantum transition between the stationary states by modulating the interaction strength to fit the different balance conditions.In Chapter 5, we study the dynamics and population switch of an open two-component Bose-Einstein condensate in the Bloch representation. We find that different dissipation forms lead to different steady states. Specially, one can obtain the switching between the self-trapping states in the two hyperfine levels or induce the switching to the macroscopic quantum tunneling regime by making use of the combined effect between nonlinearity and dissipation.In the last part of this paper we give a simple summary and discussion to the above-mentioned works and present some expectations in this fieldThroughout this thesis we will approach the manipulation of many-body quan-tum state, dynamics and dissipation phenomenon in a Bose-Hubbard dimer. Our main works are involved in chapters two, three, four and five.