Quantum Walks Of Few Particles In One-dimensional Optical Lattices

Quantum walk is a natural and straightforward generalization of the classical random walk to the quantum world.Its main task is to study the random dynamics of quantum particles on a discrete lattice.It is well known that the investigation on classical random walk has realized a lot of important applications in physical experiments,algorithm designing,and even stock marketing.Having witnessed the huge success of classical random walks,people believe that quantum walks may have widespread applications in quantum algorithms,quantum computing,quantum information,quantum simulation and so on.And compared to the classical random walks,dramatic different behavior is shown in quantum walks.For example,a quantum walker can propagate linearly with respect to the expansion time,which is much faster than its classical counterpart.In recent years,ultracold atom physics has developed rapidly.Ultracold atom systems have become a versatile platform for simulating intriguing condensed-matter physics,especially the ultracold atom systems in ideal optical lattices.In this thesis,we mainly investigate physical properties of the few-particle quantum walks on one-dimensional optical lattices,including,spin-flipping accompanied quantum walks in one-dimensional optical lattices and quantum walks in the commensurate off-diagonal AAH model.For the spin-flipping accompanied quantum walks,we have also observed the linear expansion behavior.Moreover,another interesting phenomenon is shown.As the strength of the spin-flipping term grows,we find an interesting spin-flipping-induced localization in the time evolution of the density distribution.The interesting point is that one part of the density is localized but the other part of the density is expanding as normal.This may be called two-speed quantum walk.The intrinsic origin of this exotic phenomenon is attributed to the emergence of a flat band in the single-particle spectrum of the system.When the flat band disappears,the localization phenomenon will not appear.Therefore,this interesting behavior of the quantum walker can be used as detection of flat band.For the quantum walks in the commensurate off-diagonal AAH model,the investigation is carried out mainly on the period-2 case.Due to the topological nature of the AAH model,edge state has been found in the system.In this situation,quantum walker set out from the edge sites will be localized as long as the strength of the kinetic modulation is strong enough.We find that localization properties of the quantum walker are mainly determined by the IPRs of the edge states.More interestingly,the edge states are shown to have an evident repulsion effect on quantum walker initiated from lattice sites inside the bulk.This may have potential applications in the futures' s quantum information techniques.