Steady-state spin synchronization through the collective motion of trapped ions

Ultranarrow-linewidth atoms coupled to a lossy optical cavity mode synchronize, i.e., develop correlations, and exhibit steady-state superradiance when continuously repumped. This type of system displays rich collective physics and promises metrological applications. These features inspire us to investigate if analogous spin synchronization is possible in a different platform that is one of the most robust and controllable experimental testbeds currently available: ion-trap systems. We design a system with a primary and secondary species of ions that share a common set of normal modes of vibration. In analogy to the lossy optical mode, we propose to use a lossy normal mode, obtained by sympathetic cooling with the secondary species of ions, to mediate spin synchronization in the primary species of ions. Our numerical study shows that spin-spin correlations develop, leading to a macroscopic collective spin in steady state. We propose an experimental method based on Ramsey interferometry to detect signatures of this spin synchronization; we predict that correlations prolong the visibility of Ramsey fringes, and that population statistics at the end of the Ramsey sequence can be used to directly infer spin-spin correlations.

Figure 1

Mapping cavity steady-state superradiance onto an ion-trap system. In the left panel, we show the model for cavity steady-state superradiance, where the cavity mode serves as a mediator for collective decay of the spins formed by the $\sigma$ atoms. In the right panel, we show the ion trap system where a normal mode of vibration serves as a mediator for collective decay of the spins formed by the $\sigma$ ions. The figure illustrates the model with a two-dimensional crystal. More generally our model can also be applied to 1D crystals of ions.

Figure 2

Level diagram of a $\tau$ ion. The $\tau$ ions are driven using a cooling laser that is red detuned from the dipole allowed $|e\rangle \leftrightarrow |g\rangle$ transition. This results in cooling of the normal modes of vibration of the ion-trap system.

Figure 3

Level diagram of a $\sigma$ ion. The three level configuration $\{|1\rangle ,|2\rangle ,|3\rangle \}$ is used to drive stimulated Raman transitions in a far detuned regime, giving rise to an effective two-level system in the $\{|1\rangle ,|3\rangle \}$ manifold. The detuning $\delta$ can be adjusted to drive red sideband transitions coupling the electronic dynamics with an external normal mode of vibration. Incoherent pumping through an excited state $|a\rangle$ replenishes energy lost via Doppler cooling (not shown here) of the normal mode.

Figure 4

Level structure of ${}^{24}\mathrm{Mg}^{+}$ and ${}^{25}\mathrm{Mg}^{+}$ ions in high field [29, 30]. The hyperfine shifts between the two species are not shown here. The laser configurations to be used are also indicated. The repump laser drives the $3s{}^2{S}_{1/2}(m_J=-1/2)\leftrightarrow 3p{}^2{P}_{3/2}(m_J=+1/2)$ transition in ${}^{25}\mathrm{Mg}^{+}$, and the upper state rapidly decays to $3s{}^2{S}_{1/2}(m_J=-1/2)$ and $3s{}^2{S}_{1/2}(m_J=+1/2)$ with branching ratios of $1/3$ and $2/3$, respectively.

Figure 5

Steady-state (a) inversion and (b) spin-spin correlation as a function of repump strength for three different system sizes (SS) ($N_{\sigma }, N$). The corresponding values for a minimal cavity model with $N_{\sigma }=40$ atoms are also plotted. As the system size increases, the inversion and correlation in the ion trap case become similar to the cavity case.

Figure 6

(a) Ramsey pulse sequence to probe the spin synchronization. During the interrogation time, the $\sigma$ ions interact with a heavily damped normal mode while being continuously repumped. (b) Decay of the Ramsey fringe envelope for uncorrelated ions and a system of correlated ions with SS (124, 217) and $w=N_{\sigma }{\mathrm{\Gamma }}_c/2$. Once the $\sigma$ spins have phase locked, the Ramsey fringe decays at a slower rate than when the spins are uncorrelated. (Inset) Fringe decay as a function of time for three different system sizes for $w=N_{\sigma }{\mathrm{\Gamma }}_c/2$.

Figure 7

(a) Decay rate of the Ramsey fringe envelope (dots) as a function of repump strength for SS (124, 217). The solid line shows the decay rate if only repumping is present. (b) Decay rate of the Ramsey fringe envelope (dots) as a function of the relative branching ratio for SS (124, 217) and $w=N_{\sigma }{\mathrm{\Gamma }}_c/2$. The solid line shows the decay rate if only repumping (with branching) is present. The repumping scheme proposed in this paper (see Fig. 4) with the ${}^{25}\mathrm{Mg}^{+}$ ions has a relative branching ratio of 0.5, and is indicated by a green triangle.

Figure 8

Decay rate of the Ramsey fringe envelope (dots) as a function of repump strength for three different system sizes. The fringes persist for longer with increasing $N$.

Figure 9

Steady-state variance of inversion as a function of repump strength $w$ for SS (124, 217) at the end of the Ramsey pulse sequence, normalized to the projection noise for uncorrelated ions ($N_{\sigma }/4$). (Inset) Normalized steady-state variance of inversion as a function of relative branching ratio $\chi$ for SS (124, 217) and $w=N_{\sigma }{\mathrm{\Gamma }}_c/2$. The repumping scheme proposed in this paper (see Fig. 4) with the ${}^{25}\mathrm{Mg}^{+}$ ions has a relative branching ratio of 0.5, and is indicated by a green triangle.

Figure 10

(a) Decay rate of the Ramsey fringe envelope as a function of the mean occupation ${\overline{n}}_{\text{COM}}$ of the center-of-mass (COM) mode for $N_{\sigma }=124$ ions. The rates shown here are calculated using the SU(4) method for a minimal model of a single mode (COM) interacting with the $\sigma$ ions [Eq. (23)]. The Doppler cooling scheme proposed is shown by a green triangle. (b) Decay rate of the Ramsey fringe envelope as a function of repump strength $w$ for $N_{\sigma }=124$ ions. The minimal model of Eq. (23) is used with ${\overline{n}}_{\text{COM}}=0$, but with additional spontaneous emission and dephasing processes for the individual ions (${\mathrm{\Gamma }}_{\text{sp}}={\mathrm{\Gamma }}_{\text{d}}=5{\mathrm{\Gamma }}_c$). The decay rate for uncorrelated ions is shown by the horizontal line ($\lambda =5{\mathrm{\Gamma }}_c$). Synchronization can prolong visibility of Ramsey fringes.