Math

This is a summary of the computational method I used for my master’s thesis [1]. I will focus on the Hartree-Fock approximation. Model The problem we were trying to attack was the correct modeling of the Majorana zero modes in a superconducting nanowire with a quantum dot attached at one of the ends of the wire. The Hamiltonian describing the system is $$ \begin{align*} H &= \sum_{j=1}^{N-1} \left( -t c_{j+1}^\dagger c_j + \Delta c_{j+1}^\dagger c_j^\dagger + \text{H.c.} \right) - \mu \sum_{j=1}^{N} c_j^\dagger c_j \\ &+ \epsilon_d d^\dagger d - t' \left( d^\dagger c_1 + \text{H.c.} \right) \\ &+ V \left( n_d - \frac{1}{2} \right) \left( n_1 - \frac{1}{2} \right) \end{align*} $$ The line is the Hamiltonian for the Kitaev chain, the second for the quantum dot and its hopping with the first site of the nanowire, and the third one accounts for the Coulomb repulsion between the quantum dot and the first site. ...

Final term paper for the course of Open Quantum Systems. Introduction Master equations govern the time evolution of a quantum system interacting with an environment and can be written in a variety of forms. Time-independent or memoryless master equations can be expressed in the well-known Lindblad form. In fact, any master equation local in time, Markovian or non-Markovian, can also be written in a Lindblad-like form. A diagonalization procedure results in a unique, canonical representation of the equation, which can be used to fully characterize the non-Markovianity of the time evolution. Several different measures of non-Markovianity have recently been presented that reflect, to different degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose to use the negative decoherence rates themselves as they appear in the canonical form of the master equation to fully characterize non-Markovianity. ...