The Quantum Coherent Control Of Bose-Einstein Condensates

In this article,using the well-known Landau-Zener model and Rosen-Zener model, we demonstrate how does the nonlinear interaction between BEC atoms brings new physical effects for the issue of quantum coherent manipulation & control.In the 1^st chapter,we firstly introduce the experimental backgroud of Bose-Einstein Condensation and quantum coherent manipulation & control.One of the most important character of BEC is the nonlinear many-body interaction between atoms. Such interaction has constantly brought out new phenomena never seen in linear quantum systems,e.g.,self-trapping,adiabatic tunneling,superfluidity and instability,to name only a few.We thus expect that it will certainly offer new challenges to quantum coherent manipulation & control.For example,some references has exposed that the Stimulated Adiabatic Rapid Passage(STIRAP),one of the most important technique in the area of coherent manipulation & control,is no longer valid for certain nonlinearity. How does the nonlinearity affects the old techniques,and how to design new techniques are two key topics of our article.In this chapter,we also gives the theoretical frame to describe BEC dynamics.Starting withe the second quantized Hamiltonian we have work out the Gross-Pitaevskii equation under the mean field approximation.We also rewrite this equation into the discrete matrix form,and introduce a method to construct the equivalent classical Hamiltonian for this quantum system.In the 2^nd chapter,we comprehensively analyze the Landau-Zener tunneling in a nonlinear three level system.For weak nonlinearity,we find that the level structure for the nonlinear system is similar to its linear counterpart,all three levels are quite smooth.However,the quantum state starts form the mid-level can't adiabatically evolve to the other side of this level because the mid-level is unstable.This is quite different from the two level case,where the breakdown of adiabaticity is certainly accompanied by the deformation of energy levels.For strong nonlinearity,a double-loop structure emerges in the mid-level and ain the upper level a butterfly structure appears.The deformation of energy levels would certainly lead to the break down of adiabaticity. As the external field sweeps,the system evolve to the tip of the butterfly structure, the energy level disappear suddenly,thus the quantum state must jump to other levels, leading to nonzero tunneling probability.Is is even more interesting that the tunneling probability is sensitive to the sweeping rate.A slight change of the sweeping rate would lead to very different tunneling probability.We attribute this phenomenon to the appearance of chaos,as exposed in the Poincare section.Such phenomenon is also never seen in a nonlinear two-level system.For the sudden limit,we derive an analytical expression for the tunneling probability with stationary phase approximation and show that the tunneling probability increase suddenly when the nonlinear"internal field" resonates with the external field.We also discuss the asymmetry of the tunneling probability induced by the nonlinearity.Our predictions can is realizable in a tripe-well trapped BEC.In the 3^rd chapter,we study the Rosen-Zener transition in a nonlinear two level system.We first consider the degenerate case.Under the adiabatic limit,for moderate nonlinearity,the transition probability varies rectangularly with the scanning period, and the oscillating period increase with the enhancement of nonlinearity.This kind of pattern is of practice interest,because it means that in wide range of parameters people can realize the 100%transformation of the population between two levels.The effect of the external field is just like a switch.For some certain parameters,the chunnel between the two level is thoroughly "open",the population can be completely transformed to the other level.For other parameters,the population is totally restricted in the initial level. For strong nonlinearity,the quantum transition is completely blocked.We also extend our discussion to the nondegenerate case,and find that the transition probability also oscillates rectangularly with the scanning period.But the amplitude of the oscillation decreases monotonously with the increase of energy bias.It suggests that the probability of the population tranferred to the other level can be designed at will by tuning the energy bias,thus is benefit for the preparation of coherent state.Nonlinear Rosen-Zener transition is observable in two mode BEC systems,and has a lot of prospective applications in various branches of the physics such as the the solid-state theory and quantum optics.