Quantum Dynamics Of Dissipative Two-state System With Nite Bias

The emerging science of quantum information theory, which includesthree parts: quantum control, quantum computing and quantum com-munication, has broad application prospects and great values of scienti?cresearch. Currently, the chip technology is close to the case of quantumlimit and new technologies have to be developed to break through thisbottleneck. In addition, the natural parallel quantum computation andthe quantum no-cloning theorem attract a large number of researchers tojoin in the quantum information ?eld. Due to the stability of solid-statequantum devices and their easiest large-scale assemblable, the researchesof solid-state quantum devices are of great interest in quantum informationtheory, especially in the ?eld of quantum computing. However, these smallquantum devices are greatly a?ected by their surrounding environment.Particularly the decoherence e?ect, caused by the dissipative environment,is fatal. Therefore, the quantum dynamics of the open quantum system,which is coupled to the environment, are very important research subjects,both experimentally and theoretically. Our work is mainly theoretical. Inthe ?eld of the dissipative quantum systems, the present theoretical meth-ods have made many positive developments, however, there are still manyshortcomings: analytical work is mainly based on the rotating wave ap-proximation (RWA) and/or the Markov approximation, which works well in the weak coupling regime. However, the coupling between the solid-statedevices and their surrounding environment are usually stronger, which isbeyond the validity of the above approximations. Therefore, we proposean analytical method, which is suitable in the strong coupling regime,as well as weak coupling regime, beyond the RWA and without Markovapproximation. Our approach is a perturbation method based on uni-tary transformations. To solve the non-Markovian quantum master equa-tion (QME) and to reach the analytical expression of the non-Markoviandensity operator for any ?nite biased quantum system, we can calculateout the non-Markovian dynamics and other interested physical quantities.Since the qubit in quantum computation is a two level quantum system(TLS), our main interest is the dissipative TLS. The whole thesis is con-sisted of six chapters.In Chapter One, it introduces the background and the current exper-imental researches for the dissipative TLS model, as well as a quantummodel—the spin-boson model (SBM). In addition, it also describes thetypical Markov approximation method.In Chapter Two, it studies the model of a TLS coupled to a single-mode boson to check the validity of our method. The model, which can'tbe solved analytically, is equivalent to many other widely used single-bosonmodels, e.g. Jaynes-Cummings model. We have applied two analyticmethods to reach the analytical expressions for the ground state energyfor arbitrary ?nite bias. For zero bias, we also have reached the analyticalexpressions for the low-lying excited states, and the analytical dynamicswith arbitrary initial state; our results have been checked by the exactnumerical results. For nonzero bias, we have o?ered analytical expressionsfor the expectation value of the matrix elementσx, which has also beenchecked by the exact numerical results. Our approach has shown the mathematical simple form as RWA, and the good consistency with theexact numerical results, which is suitable for the stronger coupling regimeand beyond the weak coupling regime of RWA.The work in Chapter Two can be easily extended to the biased SBM,which are studied in Chapter Three, Four and Five, with the sub-Ohmicbath, super-Ohmic bath and Lorentzian bath, respectively, and in whichthe sum rule and the Shiba's relation are exactly satis?ed via numericalcalculation in the coherent regime.In Chapter Three, it studies the quantum dynamics of the dissipa-tive TLS with nonzero bias and sub-ohmic bath. Our approach has beendiscussed here in detail. It provides an analytical expression for the non-equilibrium correlation. By comparing with the Markov approximation,we ?nd that non-Markov is an essential to investigate the short-time evo-lution of the system for the sub-Ohmic bath. In addition, we have alsoreached an analytical expression for the susceptibilityχ′′(ω), and deter-mined the little studied coherent-incoherent transition pointαc. Mean-while, it shows that non-zero bias can help to enhance the short-timecoherence.In Chapter Four, we have applied two ways to solve the SBM with thesuper-Ohmic bath: hamiltonian diagonalization approach and the QMEapproach. By each of them, it presents analytical expressions for the dy-namics. It shows that for su?cient strong coupling, Markov approximationis invalid for short-time evolution. In addition, when s→1 and ? ?ωcfor zero bias, it shows the existence of coherent-incoherent transition pointαc, while it is usually in the coherent regime.In Chapter Five, it studies the biased SBM with a Lorentzian spec-tral density, which is equivalent to: a biased qubit coupled through a harmonic oscillator to an ohmic environment. We have applied similarprocedures in Chapter Three to calculate out analytical expressions forthe non-Markovian dynamics P (t) and the corresponding spectrum func-tion S (ω), and to explain the physical nature of the spectrum functions inthree ways. As a proof, our results have been checked by literature resultsas well as the numerical QUAPI methods, and we ?nd that QUAPI is notapplicable for the nonzero bias case. Meanwhile, we have provided thelocalize-delocalized transition pointαL and the coherent-incoherent tran-sition pointαc, which have not been provided in the literature (exceptαcfor zero bias). In addition, we ?nd a new peak in the spectrum which isclearly shown.In Chapter Six, it is the main conclusions and the prospect.This work was supported by the National Natural Science Founda-tion of China under contract NO. 10734020 and NO. 90503007, as well asthe National Minister of Education Program for ChangJiang Scholars andInnovative Research Team in University of IRT0524.