Towards experimental realization of synchronization of quantum VdP oscillators with trapped calcium ions

Synchronization is a fundamental phenomenon in nonlinear dynamical systems, where interacting systems adjust their dynamics to achieve a unified rhythm. While well understood in classical systems, synchronization in the quantum regime presents new challenges due to the intrinsic effects of quantum fluctuations and the absence of well-defined phase-space trajectories imposed by Heisenberg's uncertainty principle. Here, we report our experimental progress toward realizing quantum synchronization of two coupled van der Pol oscillators—an archetypal model of nonlinear oscillators exhibiting limit-cycle behavior. By encoding the two oscillators in the axial motions of a 40Ca+ 44Ca+ ion chain, we demonstrate the generation of quantum van der Pol oscillators and perform Wigner function tomography to characterize their steady state dynamics. To explore synchronization, we aim to establish phase locking between the two oscillators and to systematically map the phase diagram of synchronization as a function of interaction strength and frequency detuning. Our exploration of synchronization in the quantum regime has potential applications in quantum sensing, metrology, and information processing.