Quantum Synchronization of Quantum van der Pol Oscillators with Trapped Ions

The van der Pol oscillator is the prototypical self-sustained oscillator and has been used to model nonlinear behavior in biological and other classical processes. We investigate how quantum fluctuations affect phase locking of one or many van der Pol oscillators. We find that phase locking is much more robust in the quantum model than in the equivalent classical model. Trapped-ion experiments are ideally suited to simulate van der Pol oscillators in the quantum regime via sideband heating and cooling of motional modes. We provide realistic experimental parameters for ${}^{171}\mathrm{Yb}^{+}$ achievable with current technology.

Figure 1

Wigner function for an oscillator without external drive. (a)–(c) Classical limit with ${\kappa }_2=0.05{\kappa }_1$: (a) $W_c$, (b) $W_q$, and (c) both $W_c$ (black, dashed line) and $W_q$ (red, solid line). (d)–(f) Same, but for the quantum limit with ${\kappa }_2=20{\kappa }_1$.

Figure 2

Wigner function for an oscillator with external drive $E={\kappa }_1$. (a)–(c) Classical limit with ${\kappa }_2=0.05{\kappa }_1$: (a) $W_c$, (b) $W_q$, and (c) both $W_c$ (black, dashed line) and $W_q$ (red, solid line) after integrating out $|\alpha |$. (d)–(f) Same, but for the quantum limit with ${\kappa }_2=20{\kappa }_1$.

Figure 3

Wigner function for two coupled oscillators with $V=3{\kappa }_1$, showing $W_c$ (black, dashed line) and $W_q$ (red, solid line) as a function of the phase difference. (a) Classical limit with ${\kappa }_2=0.05{\kappa }_1$. (b) Quantum limit with ${\kappa }_2=10{\kappa }_1$.

Figure 4

Numerical phase diagrams for globally coupled vdP oscillators, comparing the classical model with $N=3000$ (black triangles) with the quantum model (red circles). (a) Steady states in classical limit with ${\kappa }_2=0.005{\kappa }_1$. (b) Boundary between synchronized and unsynchronized phases.

Figure 5

Level scheme for an ion with trap frequency ${\omega }_o$. (a) Negative damping comes from exciting the blue sideband of $|g⟩\to |e⟩$ (blue arrow). Nonlinear damping comes from exciting the double red sideband of $|g⟩\to |e^{\prime }⟩$ (red arrow). (b) Two modes can be coupled by off-resonantly exciting their blue sidebands of $|g⟩\to |e^{\prime \prime }⟩$.