Dynamics Of A Particle In Periodically Driven Systems

The periodically driving systems have been widely used in controlling quantum tunneling and manipulating quantum states.The dynamic of a single quantum par-ticle is the cut-in point for the study of quantum phenomena and plays an important role in the development of quantum mechanics.It has been a long history of the dynamics of a single quantum particle in the periodically driving systems in which the coherent destruction of tunneling(CDT)and the dynamical localization are two very important research findings.Recently,scientists have reported several impor-tant extensions of coherent destruction tunnelling such as nonlinear CDT,dark CDT and many-body CDT,etc.,which have greatly enriched our understanding of the fun-damental quantum phenomena.Along with the deepening of research,the external periodic driving field is introduced into various systems including spins,cold atoms,quantum dots and Josephson junctions,etc.and has become an effective way for us to explore more novel quantum physics.This paper will focus on the dynamics of a single quantum particle in a pe-riodically driving system and the contribution of the external fields on the system dynamics is investigated by applying an external periodically driving field to the har-monic trapping potential,one-dimension lattice array and coupled waveguide sys-tems.Firstly,we will discuss the standard CDT in a periodically driven double-well potential,and then the dark coherent destruction of tunneling in the driven three-well potential is introduced to compare the difference of standard CDT and dark CDT.Then on the base of these,we propose a driving scheme to realize the transition between standard CDT to dark CDT in the finite N-site driving system.More inter-estingly,an analog of photon assisted tunneling which has a new physical mechanism is found out in the driving system composed of odd lattice points.In the last chapter,the dynamics of Bose-Einstein condensate(BEC),which can be treated as a parti-cle in the mean-field approximation,trapped in the periodically driven lattice array is discussed and the research results may deepen our understanding about chaotic dynamics.Three main conclusions of this study are as follows:1.The transition between standard CDT and dark CDTIn the periodically driven N-site system,we try to realize the transition between standard CDT and dark CDT through an appropriate driving scheme.We have found that the dynamics of a quantum particle initially localized in the left boundary site under a fixed driving field can be manipulated by driving the right boundary site with a adjustable driving frequency or amplitude.The physic underlying such transition is to change the effective coupling strength between the right boundary site and its near-est neighbor by using the external periodically driving field,and eventually to achieve the purpose of increasing or decreasing a lattice site.Moreover,we can realize ei-ther localization to coherent tunneling transition or coherent tunneling to localization transition,which may provide theoretical support for the transport of atoms in the lattice array.Further more,this paper has revealed that our proposed transition is robust against the second-order coupling(SOC)between next-nearest-neighbor sites in a three-site system,whereas localization can be significantly enhanced by SOC in a four-site system.2.An analog of photon-assisted tunnelingWe theoretically report an analog of photon-assisted tunneling(PAT)in a peri-odically driven lattice array or a coupled waveguide system.Different from tradition PAT,this type of PAT proposed in this paper is due to dark Floquet state and the ini-tially static biased potential is non-essential here.In a non-high-frequency regime,the analog of PAT can be realized just by periodically modulating the top waveg-uide(site)and adjusting the distance between the bottom and its adjacent waveguide(site).The numerical results have shown that such PAT resonances also exist in the five-channel waveguide(five-site)system and probably exist in the waveguide(lattice)arrays with other odd numbers of waveguides(sites),but they will become weak as the number of waveguides(sites)increases.It is readily observable under currently accessible experimental conditions and we will also further demonstrate the feasibility of the experiment in this paper.3.Chaotic dynamics in the periodically driven systemWe have extended the dynamic characteristics of a single particle to the case of BEC in a periodically driving system.We report a type of chaos-related localization in a lattice array with an external periodically field acted on site 1 that allows us to realize the controllable chaotic dynamics with a tunable driving frequency.Through adjusting the driving frequency,we can make the system into or out the chaotic re-gions and realize the transition between the chaos assisted tunnelling and chaos re-lated localization.Interestingly,with the increase of nonlinearity,driving frequency or even second-order coupling,the relatively stable parameters windows for chaos-rclatcd localization with a sharp edge facilitate us to find the threshold of the switch between the chaos-assisted tunneling and chaos-related localization.In addition,the influence of the driving field phase is discussed.These results may deepen our un-derstanding of chaos-related localization in nonlinear driving system.In conclusion,this thesis discusses about the dynamics of a particle in the periodically driven system and proposes a scheme to realize the transition between standard CDT and dark CDT.Then,a new quantum phenomena analog to photon assisted tunneling is observed in the simple three-channel waveguide system.Finally,we observe the coexistence of chaos assisted tunneling and chaos related localization in analyzing the dynamics of BEC in the driven lattice array.So,in this paper,we have successfully extended our research object from a simple quantum particle to the Bose-Einstein condensate,the driving system from the simple two-state to finite N-state systems and the research method from the high frequency approximation to non-high-frequency approximation.The results may deepen our understanding to the quantum dynamics in periodically driving system and may provide new possibility to design quantum switch and manipulate atoms.