Synchronisation in complex networks

Abstract

The main investigation in this thesis is the research on a future quantum internet. The focus is laid on
the network structure of this quantum network. The network is rst examined by the investigation of the
Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all
nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to
verify why and when synchronisation takes place. After modelling the calculated equations of motion, an
interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but
by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion
leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee
transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters
used for the oscillators.
The analysis is then continued in the quantum domain where optomechanical systems are introduced. At
rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a
Dung circuit. The main investigation is again into the synchronisation, which implies that the operators
belonging to both systems behave the same as a function of time. Then several synchronisation measures
are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise
in terms of the operators for each system. Then a quantum network is introduced, where a complex yet
ecient network is created. This quantum network is a small world network, where a transmitter node is
able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place
at the same time and links which are not connected do not synchronise with the transmitter node. After
modelling this network, the results are in compliance with the theory available.
Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical
systems outside laboratory circumstances. Also, the network should be tested for practical use and
expanded in a much larger size both in theoretical analysis as well as in experiments.