Observation of synchronization between two quantum van der Pol oscillators in trapped ions

TL;DR

This work addresses the challenge of observing synchronization between quantum limit-cycle oscillators by realizing two quantum van der Pol (vdP) oscillators in a mixed-isotope trapped-ion system with engineered collective dissipation. The two motional modes serve as the oscillators, whose relative phase defines the synchronization, observable only via joint measurements and tunable through the phase of the dissipative coupling. The authors demonstrate synchronization across near-classical and quantum regimes, quantify it with a phase-invariant mutual-information measure, and show robustness to detuning; they further show phase locking to an external drive, enabling sensing-like capabilities. The results establish a dissipation-based pathway to study complex quantum synchronization and pave the way for scalable networks and quantum metrology applications using trapped-ion platforms.

Abstract

Synchronization is a hallmark of collective behavior that emerges when nonlinear systems interact, spanning scales from mechanical oscillators to planetary orbits. As a universal phenomenon it underpins the study of complex systems and has far-reaching technological implications. While classical synchronization has a long and rich history, it has not been observed experimentally between multiple quantum limit-cycle oscillators despite a decade of theoretical investigations. We realize synchronization between two quantum van der Pol oscillators by engineering dissipation in a mixed-isotope trapped-ion quantum simulator. The synchronized state is encoded in a fixed relative phase between the oscillators that is inaccessible to local measurements and only revealed through joint readout of both oscillators, in stark contrast to the classical case where synchronization can be observed via individual phase measurements. We further show that the relative phase can be precisely controlled, and that the chain of two oscillators can synchronize to an external field, suggesting applications in sensing. Our results provide a promising pathway for studying more complex synchronized quantum dynamics beyond two oscillators, where a theoretical treatment becomes increasingly challenging, and it remains to be understood whether genuinely quantum features persist in such cases.

Figures (14)

• Figure 1: Synchronization of quantum van der Pol oscillators with trapped ions. (a) Schematics of experimental setup with a $\mathrm{^{40}Ca^+}-\mathrm{^{44}Ca^+}$ ion crystal trapped in a linear rf Paul trap. The two axial motional modes are used for the synchronization experiment, where the state of the $\mathrm{^{40}Ca^+}$-ion and the motional state of the ion crystal are controlled by a 729 nm laser light. (b) Illustration of the donut-shaped Wigner function of a quantum limit-cycle oscillator. Black dashed curves indicate classical trajectories attracted towards the classical limit cycle. (c) Experimental implementation of the corresponding dissipator to generate and synchronize two vdP oscillators. (1) Negative damping is realized by driving the blue sideband of $M_i$, followed by a qubit reset; (2) Nonlinear damping is realized by driving the second order red sideband of $M_i$, followed by a qubit reset; (3) Collective dissipation is generated by simultaneous driving the red sidebands of $M_1, M_2$ with a phase difference $\varphi$, followed by another qubit reset. (d) The Wigner function of each mode $M_1$ and $M_2$ individually does not show any phase preference. Instead, the synchronized dynamics only appears as stable relative phase relation between the two oscillators, which can only be observed by a joint measurement. (e) Joint probability distribution $P(x_1,x_2)$ for in-phase ($\varphi=0$) synchronized oscillators. Black dashed line shows the corresponding classical trajectory.
• Figure 2: The quantum circuit for synchronization between two quantum vdP oscillators and reconstruction of the motional states. (a) The circuit acts on one qubit and two motional modes, where the auxiliary qubit is to generate collective dissipation on the two vdP oscillators and to read out the two-mode state. Each cycle consists of a stroboscopic application of negative and nonlinear dampings on both modes [described in (c)(d)]. We repeat the sequence fifteen times to generate the two vdP oscillator states, with another ten cycles including the dissipative coupling [described in (e)] to synchronize them. For the motional reconstruction, the controlled displacement in the $\sigma_x$ basis (realized by SDF) maps the motional information onto the qubit. The imaginary part is read out with an additional $\pi/2$ qubit rotation along the $x$ direction prior to state detection. (b) The qubit reset is realized with 854 nm repumping laser light to reset it to the $\ket{\downarrow}_z$ state. (c)(d) The negative (nonlinear) damping on $M_i$ is realized by a coherent BSB (2RSB) drive followed by a qubit reset R. (e) Collective dissipation on the two motional modes is realized by simultaneously driving the RSB on $M_1$ and $M_2$ with a constant phase difference $\varphi$, followed by a qubit reset [see details in Appendix \ref{['app:Experimental_platform']}] .
• Figure 3: Crossover of the vdP oscillator from the near-classical to the deep quantum regime. (a) Top row shows various experimentally reconstructed Wigner function of vdP oscillator states with different mean phonon number when increasing $\kappa_-/\kappa_+$. The bottom row shows the corresponding numerical simulation. (b) The radius $\alpha_p$ is shown as a function of the ratio $\kappa_{-}/\kappa_{+}$ on a logarithmic scale. Red dots are the experimental data, blue solid curve is the numerical simulation using Lindblad equation and the orange dashed line represents the classical mean field theory. When $\kappa_-/\kappa_+$ increases, the oscillator becomes more quantum and deviates from the classical mean-field theory. During the experiment we keep $\kappa_+/2\pi=0.112$ kHz to be constant and change the strength of $\kappa_-$ by tunning the power of the BSB drive.
• Figure 4: Evolution towards the limit cycle of a single vdP oscillator. We create a cat state with $\alpha=3$ at $t=0$ ms and measure the time evolution to the vdP oscillator state with Wigner function reconstruction. Top row shows the experimental data and the bottom row shows the numerical simulation. The parameters used in experiment are $t_{\rm 2RSB}=48\,\rm \mu s, t_{\rm BSB}=24\,\rm \mu s$, $\Omega_{\rm BSB}/2\pi=0.06\rm$ MHz, $\Omega_{\rm 2RSB}/2\pi=0.1$ MHz.
• Figure 5: A single vdP oscillator synchronized to an external drive with different phases $\varphi_d = 0,\,\pi/2,\,\pi$. The dashed circles indicate the size (radius $\alpha_p$) of the vdP oscillator with no external drive.