Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers

We investigate quantum synchronization theoretically in a system consisting of two cold ions in microtraps. The ions' motion is damped by a standing-wave laser while also being driven by a blue-detuned laser which results in self-oscillation. Working in a nonclassical regime, where these oscillations contain only a few phonons and have a sub-Poissonian number variance, we explore how synchronization occurs when the two ions are weakly coupled using a probability distribution for the relative phase. We show that strong correlations arise between the spin and vibrational degrees of freedom within each ion and find that when two ions synchronize their spin degrees of freedom in turn become correlated. This allows one to indirectly infer the presence of synchronization by measuring the ions' internal state.

Figure 1

(a) Trapped-ion setup. Each ion is damped at a rate $\mathrm{\Gamma }$ by a standing-wave laser and driven by a blue-detuned laser of strength ${\mathrm{\Omega }}_{j=1,2}$. The phonons have a dipole interaction of strength $J$ and the trap frequencies are ${\omega }_1$ and ${\omega }_2$. (b) Internal electronic states of each ion. The “spin” states are pumped by a laser blue detuned by frequency ${\omega }_{j=1,2}$ and undergo spontaneous emission at a rate $\gamma$. The damping is achieved using a red-detuned drive on a different electronic transition (not shown) and is eliminated adiabatically.

Figure 2

(a) Average phonon number $\langle n\rangle$ (blue hexagons) and most likely phonon number $n_{\mathrm{m}}$ (green octagons) of the phonon distribution plotted against damping strength $\gamma /\mathrm{\Gamma }$; we also plot the Mandel-Q parameter (red stars) which becomes negative around the lasing transition. (b) Wigner distribution of the phonons for $\mathrm{\Gamma }/\gamma =1/3$. (c) Wigner distribution of the phonons after projection onto different spin eigenstates ${\rho }_{\mathrm{ss}}^{j,\pm }$ $\left(\mathrm{\Gamma }/\gamma =1/3\right)$. ${\sigma }^{x/y}$ operators are correlated with phase and ${\sigma }^z$ operators are correlated with number, which is confirmed by calculating the correlations, given beneath their corresponding Wigner distributions. Here $\mathrm{\Omega }/\gamma =1$, and $\mathrm{\Delta }=0$ throughout.

Figure 3

Phase distribution $P\left(\varphi \right)$ (a) when the system is perfectly symmetric $\left({\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=1,\mathrm{\Delta }=0\right)$; (b) while the oscillators are detuned $\mathrm{\Delta }/\gamma$ $\left({\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=1\right)$; and (c) for different relative pumping strengths ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2$ $\left(\mathrm{\Delta }=0\right)$. In (d) and (e) the synchronization measure $S$ (black solid) is plotted against the same parameters as (b) and (c), respectively. Here ${\mathrm{\Omega }}_2/\gamma =1,\mathrm{\Gamma }/\gamma =1/3$, and $J/\gamma =1/10$ throughout.

Figure 4

(a) and (c) $C({\sigma }_1^z,{\sigma }_2^z)$ (green circles), $C({\sigma }_1^x,{\sigma }_2^x)$ (blue up triangles), $C({\sigma }_1^x,{\sigma }_2^y)$ (red pentagons), ${\mathrm{\Phi }}_c$ (cyan diamonds), and ${\mathrm{\Phi }}_s$ (magenta squares) are plotted against $\mathrm{\Delta }$. In (a) the driving is balanced $\left({\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2=1\right)$ while in (c) ${\mathrm{\Omega }}_1$ is stronger $({\mathrm{\Omega }}_1/\gamma =5/4$ and ${\mathrm{\Omega }}_2/\gamma =1)$. (b) and (d) $S$ (black solid) is plotted for comparison with the same parameters as (a) and (c), respectively. Here ${\mathrm{\Omega }}_1/\gamma =1,\mathrm{\Gamma }/\gamma =1/3$, and $J/\gamma =1/10$. In (c), the red down-triangles, purple stars, and yellow hexagons come from calculations of $C({\sigma }_1^x,{\sigma }_2^x),C({\sigma }_1^x,{\sigma }_2^y)$, and $C({\sigma }_1^z,{\sigma }_2^z)$, respectively, using a version of Eq. (1) in which we neglect only terms $O\left({\eta }^4\right)$ and higher. They agree well with the results obtained using Eq. (2).