Quantum synchronisation and the validity of its derivation

Abstract

In this bachelor thesis, the paper ”Classical synchronization indicates persistent entanglement in isolated quantum systems” by Dirk Witthaut et al. is looked into and discussed. First, synchronization in a classic sense is explained. The Kuramoto model is introduced, and a few properties of this model are defined. In the next part of the theoretical background, the creation and annihilation operators are defined. An example is given on how to derive these operators, and how to write the Hamiltonian in the form of creation and annihilation operators. It becomes clear that the Hamiltonian of a vector potential in vacuum does not have any coupling terms. This is why no synchronization will occur here. Assuming the Hamiltonian has a different form with two-body interactions and a coupling factor, then it will have some sort of interaction between modes. Dirk Witthaut published in his paper a way to derive the Kuramoto equation from this Hamiltonian. First, the time derivative of the expecta- tion value of the â n operator is evaluated by using the Ehrenfest theorem. This returns multiple three point functions. These can also be evaluated by the Ehrenfest theorem, but that will only result in more coupled equations and five point functions. To solve it, a first order mean field approximation is used. According to Witthaut, this results in a series of coupled complex differential equations which can be rewritten into the Kuramoto equations. Witthaut then further elaborates this result in the remainder of his paper. My calculations point towards a different conclusion. Witthaut made a mistake when calculating the commutation relations that were used in the Ehrenfest theorem. This resulted in a different system of coupled equations, which I couldn’t elaborate into the Kuramoto model. This is the conclusion of this bachelor thesis.