Synchronization in quantum systems

The phenomenon of synchronization occurs in many different situations and has been extensively studied for many years. If an autonomous oscillating system is coupled to another such system or to an external driving force it can synchronize its frequency and phase to the external system. Examples are coupled pendulums, circadian rhythms in living systems or  synchronization of fireflies flashing. Common to these systems are the fact that they need to have a stable limit cycle, which means they must be dissipative, so that they can return to the stable cycle after a perturbation, and contain an energy source, so that they can sustain oscillations indefinitely in the presence of dissipation. For all about synchronization in classical systems, see the book Arkady Pikovsky, J Kurths, and Michael Rosenblum. Synchronization: A Universal Concept in Nonlinear Sciences , Cambridge University Press, 2001.

 

Recently, the concept of synchronization has been applied to quantum systems, which can bee seen to be equivalent to classical systems with noise. This noise is usually interpreted as quantum noise, although it is not clear if this is really the correct interpretation, or that it can in some cases be considered as thermal noise. In classical systems, thermal noise can under certain conditions be reduced, and as part of this project we can try to do the same for quantum systems.

We have previously studied the simplest example, a two-level systems synchronized to an external signal

(Álvaro Parra-López and Joakim Bergli,: Synchronization in two-level quantum systems arXiv:1904.11763). We could demonstrate that the phase of the two-level system synchronizes to the external signal, but as mentioned before, not perfectly since there is some random noise that perturbs the phase. In classical synchronization one can study frequency locking in addition to phase locking. That is, that the frequency of the oscillator becomes identical to the driving frequency (for noise-free systems) or that the average frequency comes closer to the driving frequency (for systems with noise). In the quantum case, we could not determine the average frequency, as we only worked with the Lindblad equation for the density matrix. This gives information about the average state of the system, and the frequency of the motion of the average. But the frequency of the average is not the same as the average frequency. To determine the average frequency, we will apply the quantum trajectories theory, which separates the dynamics of the individual pure states that make up the mixed average state.