Synchronisation along quantum trajectories of three coupled VdP oscillators

Abstract

Synchronisation is the remarkable phenomenon of in-phase movement of coupled oscillators during a prolonged period of time, which even occurs when these have different natural frequencies. This property is used in many fields like physics, biology and chemistry, to model various system behaviours, for instance: circadian rhythms in the chemistry of the eyes \cite{Rompala2007}.
\newline\noindent This thesis focuses on synchronisation and entanglement in systems consisting of quantum Van der Pol oscillators. After looking at the exemplary properties of the classical VdP oscillator, it follows the methods of \cite{EshaqiSani2020} to explore the behaviour of two coupled QVdPOs, quantum Van der Pol oscillators, using Monte Carlo simulations for trajectories of this system. The system is then expanded to three-oscillator systems with all-to-all and chain coupling. The validity of the extension of the properties found in two QVdPOs in \cite{EshaqiSani2020} with regard to synchronisation and entanglement is tested. Does an Arnold tongue, the region of parameters for which a system shows synchronisation, still exist and if so has its shape changed? Do synchronisation and strong entanglement still show a positive correlation? \newline\noindent Simulations of the 2-oscillator system gave results which were just slightly different from those obtained in\cite{EshaqiSani2020}, validating the occurrence of synchronisation within the Arnold tongue and showing a positive correlation between synchronisation and entanglement of the system. Both the 3-oscillator systems showed synchronisation as well in their respective Arnold tongues, which were increasingly smaller for the all-to-all coupled and the chain coupled system when compared to the 2-oscillator system. Overall, the amount of synchronisation, shown in three oscillators was very close to the amount shown in two oscillators when strongly coupled and just a little less for weak coupling. The correlation between synchronisation and strong entanglement was also found for three oscillators, though for weaker coupling the chain coupled system showed a little more entanglement than before.