Quantum Dynamics Of Two-level System In Single Mode Cavity Or Dissipative Environments

Over the past several decades, quantum information science has emergedand many research groups around the world are working towards the highlyambitious goal of building a quantum computer, which would dramaticallyimprove computational power for particular tasks. In quantum computing,a quantum bit ( qubit ) is a unit of quantum information, and is basedon the coherence and entanglement of quantum instruments. As we know,any two-level system can be used as a qubit. And a number of physicalsystems are being developed for qubits, such as cavity quantum electrodynamics( QED ) qubits, trapped ion ( or atom ) qubits, nuclear magneticresonance ( NMR ) quibts, quantum-dot ( QD ) qubits, superconductorbasedqubits, etc. However, such a small quantum device of qubit is nota'closed box'. It must be greatly disturbed by the surrounding environment,which is always described by bosonic modes ( photons, phonons ).The effect of the dissipative environment leads to quantun decoherence.This may be the major challenge for the future of quantum informationscience. In this context, we study the quantum dynamics of two-levelsystem ( qubit ) coupled with various environments.People have exhibited various approximate analytical methods to dealwith the interaction of spin and bosons. The most popular example may be the rotating-wave approximation ( RWA ), which relies on the assumptionof near-resonance and weak-coupling. However, this condition may not besatisfied yet. The quibts in the physical devices above always suffer strongcouplingand far-off-resonance situation, which challenges the validity ofthe RWA. More and more results show that the counter-rotating termsplay an important role in quantum dynamics. In this work, a perturbativemethod based on a unitary transformation is developed, which avoids theRWA and is suitable in both weak and strong coupling regime, eitheron-resonance or off-resonance situation. By using this approach, severalmodels are studied. And the whole thesis is consisted of five chapters.In Chapter One, we first introduce the background of the quantuminformation and quantum computation. Then, the current experimentalresearches for qubits are given. Finally, we introduce the correspondingtheoretical description between qubits and two-level systems.In Chapter Two, we present an analytical approach based on a unitarytransformation for both the JC model and two-photon JC model. Sincethis approach takes into account the effect of counter-rotating terms butstill keeps the Hamiltonian with a simple mathematical structure of RWA,we usually call it transformed rotating-wave approximation ( TRWA ). Weinvestigate the energy levels of ground and lower-lying excited states, aswell as the time-dependent quantum dynamics. The rusults are comparedwith both the numerically exact ones and the RWA ones. We show that forstrong coupling and far-off-resonance our analytic calculations of TRWAare still quantitatively in good agreement with the exact ones, while theRWA may be invalid. Besides, our analytical approach for the two-photonJC model can naturally defind the bounding of the coupling strength, thatis g/ω< 1. The main purpose of Chapter Two is to check the validity andexpansibility of the TRWA method. In Chapter Three, we develop a modified Bloch-Redfield approach,which is based on the TRWA method, taking into account the effect ofthe counter-rotating terms of the system-environment interaction. Theapproach is used to study the interference between the driving and thedissipation of a dissipative two-level system in the Ohmic bath with Rabidriving. By calculating the nonequilibrium correlation P(t) for a finitedriving we show that the counter-rotating terms of the system-environmentinteraction play an important role in the dynamic evolution, and theirmainimpact, compared with the RWA, is twofold: One is to suppress the decayrate when driving is finite, and the other is to reduce the longtimeamplitude and to change the phase shift of the driven oscillation. Thismeans that the effect of counter-rotating terms helps keep the coherenceof system, and cannot be neglected even in the long-time behavior.In Chapter Four, the ground state and the spectral structure of lowerlyingexcited states of a dissipative two-level system coupled to a sub-Ohmic bath ( s = 1/2 ) with nonzero bias have been studied using theTRWA method. By calculating the ground-state entanglement entropy,the ground-state average of (σ_z)G, and the static susceptibility of the twolevelsystem, we explore the nature of the transition ( crossover ) betweenthe delocalized and localized state of the two-level system. Furthermore,we calculate the time-dependent expectation (σ_z(t)) and the time evolutionof the entanglement entropy to show that, when the system undergoesa transition ( crossover ) from the delocalized to the localized state, thetime evolution of the two-level system changes from coherent to decoherentdynamics.In Chapter Five, it is the main conclusions and the prospect.This work was supported by the National Natural Science Foundationof China under contract NO. 10734020 and NO. 90503007, as well as the National Minister of Education Program for ChangJiang Scholars andInnovative Research Team in University of IRT0524.