Synchronizing the Smallest Possible System

Alexandre Roulet and Christoph Bruder
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

The theory of synchronization [1] has been successfully applied to complex classical systems such as electronic and biological networks, and there is a growing interest in understanding how it applies to quantum systems. However, when moving from a classical to a quantum framework, one is inevitably confronted with the exponential growth of the state space as opposed to the linear classical scaling, which limits the tractable size of a network of quantum oscillators.

In this work [2], we approach this issue by addressing the fundamental question of the minimal dimension for a quantum system to be synchronized. We first show that the standard paradigm of synchronization does not apply to a two-level system due to the lack of a limit cycle. Surprisingly, we then find that a single spin 1 – a system with no classical analogue – can be phase-locked to a weak external signal of similar frequency and exhibits all the standard features of the theory of synchronization. Our results rely on a proposed phase portrait applicable to spin systems, in which we can identify the phase variable that lies at the core of the formalism.