Synchronizing the Smallest Possible System

We investigate the minimal Hilbert-space dimension for a system to be synchronized. We first show that qubits cannot be synchronized due to the lack of a limit cycle. Moving to larger spin values, we demonstrate that a single spin 1 can be phase locked to a weak external signal of similar frequency and exhibits all the standard features of the theory of synchronization. Our findings rely on the Husimi $Q$ representation based on spin coherent states which we propose as a tool to obtain a phase portrait.

Figure 1

The limit cycle. (a) The $Q$ function of the target state $|1,0⟩$ represented using the Winkel tripel earth projection. The north and south poles correspond respectively to $\theta =0$ and $\theta =\pi$ while $\varphi$ specifies the longitude. The $\varphi$-symmetric distribution centered around the equator is reminiscent of the ring stabilized in the Wigner representation of a Van der Pol limit cycle [6]. (b) The spin-1 ladder, with the eigenstates separated by the natural frequency ${\omega }_0$. The action of the dissipators is to transfer the populations from the extremal states towards the target state.

Figure 2

Phase locking to a resonant ($\mathrm{\Delta }=0$) external signal represented by the steady-state $Q$ function. The signal strength is chosen as $ϵ=0.1{\gamma }_{\min }$, where ${\gamma }_{\min }=\min ({\gamma }_g,{\gamma }_d)$, ensuring that we are in the weakly perturbed regime. (a) ${\gamma }_g/{\gamma }_d=0.1$, ${\gamma }_g=10\epsilon$, the distribution is localized around $\varphi =0$ while remaining approximately on the equator. (b) ${\gamma }_g/{\gamma }_d=1$, ${\gamma }_g=10\epsilon$, the limit cycle is slightly distorted with no visible signature of phase locking. (c) ${\gamma }_g/{\gamma }_d=10$, ${\gamma }_d=10\epsilon$, same as (a) but the distribution is now localized around $\varphi =\pi$. Thus, phase locking is possible when one of the rates dominates.

Figure 3

The distribution of the phase $S(\varphi )$ as a measure of phase locking for ${\gamma }_g=0.1{\gamma }_d$. (a) The phase locks to different values when the detuning $\mathrm{\Delta }$ of the signal is varied. This is indicated by the dashed line that follows the peak $\max S(\varphi )$ of the distribution. (b) Increasing the signal strength $ϵ$ leads to a more pronounced phase locking. (c) The Arnold tongue. Significant phase locking is achieved over a broader range of detuning $\mathrm{\Delta }$ by increasing the signal strength $ϵ$. The tongue does not touch the bottom axis due to the presence of noise.

Figure 4

Influence of the signal strength $ϵ$ onto the limit cycle for zero detuning, $\mathrm{\Delta }=0$. In the gray-shaded area, the effect of the signal is not limited to a perturbation of the free phase $\varphi$ but instead substantially deforms the limit cycle by increasing or decreasing the energy of the spin, depending on the ratio ${\gamma }_g/{\gamma }_d$. The corresponding $Q$ functions are shown for $ϵ={\gamma }_{\min }$.