The Relation Between One—to—One Correspondent Orthonormal Eigenstates of H^0 and H（λ）=H^0＋λV

The relation between one-to-one correspondent orthonormal eigenstates of H0 and H(λ) ＝ H0 + λV is carefully studied with general perturbation theory. Attention is particularly paid to the analyticity and its local destruction due to nonlinear resonance. Numerical results are given to show such possibility with a special Jacobi diagonalization method. The conclusions show that for the system H(λ) belonging to the same class as H0, the relation between one-to-one correspondent orthonormal eigenstates |φi(λ)> and|φ0m(i)>can be expressed as an analytical unitary matrix which can be identified to the relevant quantum canonical transformation. But for the system H(λ) violated dynamical symmetry, the relation between one-to-one correspondent orthonormal eigenstates cannot be expressed as an analytical unitary matrix. Such a kind of unitary matrix cannot be taken as a quantum canonical transformation to define quantum mechanical quantities. This is a key point for studying the quantum chaos with the help of dynamical symmetry theory.

Numerical Investigation for the Properties of Nonintegrable Quantum SYstems Eigenspace in the Line of Quantum—Classical Correspondence

The properties of the eigenspace of nonintegrable quantum systems are explored in detail in the light of the viewpoint of quantum-classical completely correspondence proposed recently by Xu et al. The changes of the topological structure in the state space of autonomous quantum system due to the nonlinear resonance are displayed numerically with the uncertainty measure ofa special initial state ρα(λ) and the transformation matrix U ( λ + δλ, λ - δλ). The statistical behavior of the subspace occupied by the state in eigenspace of quantum nonintegrable system is discussed carefully with the help of a special renormalization method. The results show that the randomness of effective Hamiltonian matrix, the transition matrix and the nearest level spacings in this region can be described by random matrix theory. And the extent of agreement of our calculation with the prediction of GOE is in correspondence to the extent of the classical torus violation.

Anticontrol of Quantum Chaos in Hamiltonian Systems

We present a method of realizing anticontrol chaos in a quantum confined system consisting of N two-levelatoms within a cavity, using a time-delayed optic field. The delay time plays a construction and organization role forproducing temporal chaos, while the interaction between atoms and photons creates spatial chaos. The chaos is quitesensitive to small time delayed. The spectral decomposition of the Hamiltonian obtained by using projection methodologyreveals that evolution of the left eigenvectors shows quite complicated chaotic fashions. The method we proposed maybe easily tested in experiment, and provides a general method using a sort of driving optic field to achieve anticontrol ofchaos for quantum systems.

The Spatiotemporal Evolution of Wave Packets under Chaotic Condition

Using the minimum uncertainty state of quantum integrable system as initial state, the spatiotemporal evolution of the wave packet under the action of perturbed Hamiltonian is studied causally as in classical mechanics. Due to the existence of the avoided energy level crossing in the spectrum there exist nonlinear resonances between some pairs of neighboring components of the wave packet, the deterministic dynamical evolution becomes very complicated and appears to be chaotic. It is proposed to use expectation values for the whole set of basic dynamical variables and the corresponding spreading widths to describe the topological features concisely such that the quantum chaotic motion can be studied in contrast with the quantum regular motion and well characterized with the asymptotic behaviors. It has been demonstrated with numerical results that such a wave packet has indeed quantum behaviors of ergodicity as in corresponding classical case.