Study Of Properties Of Eigenfunction Of Quantum Chaotic System And Its Appliciation In Quantum Thermalization

With the improvement of experimental techniques, the topic of properties of mesoscale and nano-scale system have recieved more and more attention in recent years. It was found that many of the concepts of thermodynamics make good sense in systems of scale. However, with the decreasing of the system size, it is not clear that to extent to which these concepts still make sense. At the same time the quantum fluctuation effect of the system will be nonegligible if the system size is sufficiently small. Then the pro-plem quantum thermalization of small systems and gets more and more attention. The study of this problem, will enhance the understanding of quantum heat engines, thermo-electric effects, quantum batteries, and many of the problems in the field of biophysics.Quantum chaos plays an important role in the study of quantum thermalization of small systems. It has been found that the emerge of statistical behaviors in quantum systems is often related to quantum chaos. The solution of many important problem in the field of quantum thermalization need a better understanding of properteis of eigenfunctions of quantum chaos systems.In our work, first we use the semi-perturbative method to study the properties of the eigenfunction of quantum chaotic systems. It is predicted in the theory that eigen-function can be divided into two parts: the perturbative region and the nonperturbative region. With the semi-classical theory, we study the expansion of the eigenfunctions of the chaotic system on the basis of engenstate of nonperturbative Hamiltonian. We find that the nonperturbative region has a close relationship with the classical allowed region. We also study the statistical properties of the energy eigenfunctions in the non-perturbative region of the quantum chaotic system using Berry's conjecture. It is found that the statistical distribution of the components is consistent with the prediction of random matrix theory after a certain degree of rescale method. Moreover, in the study of the process from near integrable to chaos, the deviation between the statistical distri-bution of components of eigenfunction in nonperturbative region and the prediction of random matrix in theory can be used to characterize the chaotic properties of the system,and can even be used as a judgement of chaos.We apply the above results to the study of the thermalization of small quantum sys-tems, especially to the important problem of the internal temperature of small quantum systems. First, we use a small quantum probe (single quantum bit) to study whether we can assign a internal temperature to a small quantum chaotic system. The diffi-culty of the problem is that the measurement results often depend on the interaction between the probe and the system, the couple strengh, the position of the contact, and the Hamiltonian and initial states of the probe, making it difficult to determine whether the results of measurement can reflect the properties of the system itself. We have stud-ied this difficult problem by consideration of dynamics evolution and found that under certain conditions, we can get a result which is insensitive to none of the factors above.Thus, for small quantum chaotic systems, we get an operational definition of its internal temperature. we further find that the temperature has a Boltzmann temperature form.This temperature measurement method is also experimentally feasible. Then, we study the thermalization of two quantum chaotic systems with weak coupling, especially the equilibrium temperature. Specifically, we find that after the relaxation time, the state of the whole systemis similar to the typical state, and the two subsystems have the same temperature.