טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentAnat Klempner
SubjectThermalization in Open Quantum Systems and Density Matrix
in Steady States far from Equilibrium
DepartmentDepartment of Physics
Supervisor Professor Podolsky Daniel
Full Thesis textFull thesis text - English Version


Abstract

We study the statistical mechanics of open quantum systems coupled to external baths locally near the edges of the system. We consider the Lindblad formalism to treat the dynamics of such systems, and focus on a class of problems that allow for analytical solutions of the steady state and equilibration properties. In particular, we examine the quantum XY Model and the transverse field Ising model in 1D, with the outer spins coupled to external baths. We obtain the steady state properties of this system using Third Quantization, which provides an analytical method for solving the Lindblad equation. Numerical methods, Exact Diagonalization and Density Matrix Renormalization Group, are used as benchmarks for this method. We develop a method to extract the exact steady state density matrix, and examine the system for different sets of parameters and different couplings to the external baths. In the case of the XY model coupled to a bath on one side only, we found that the density matrix in steady-state factorizes into a product of density matrices in individual sites. For more general cases the structure of the density matrix is more complex. For instance, when the XY model is driven asymmetrically on both ends, we find correlations that are restricted to nearest neighbor sites. We provide a systematic study of current and its fluctuations in the driven XY model. Finally, we examine the relaxation time of the system to steady-state. We find that when both edges of the Ising model are coupled to baths the relaxation time exhibits L3 behavior, where L is the number of spins. However when only one edge is coupled to a bath, we find a transition between L3 and exponential behavior. This transition occurs at the critical field of the model in zero temperature. We provide a simple toy model to explain this result.