Structural Transitions of Ion Strings in Quantum Potentials

We analyze the stability and dynamics of an ion chain confined inside a high-finesse optical resonator. When the dipolar transition of the ions strongly couples to one cavity mode, the mechanical effects of light modify the chain properties close to a structural transition. We focus on the linear chain close to the zigzag instability and show that linear and zigzag arrays are bistable for certain strengths of the laser pumping the cavity. For these regimes the chain is cooled into one of the configurations by cavity-enhanced photon scattering. The excitations of these structures mix photonic and vibrational fluctuations, which can be entangled at steady state. These features are signaled by Fano-like resonances in the spectrum of light at the cavity output.

Figure 1

The dipolar transitions of ions forming a chain couple with one mode of a high-finesse optical cavity. The depth of the optical potential inside the resonator depends on the ions’ transverse positions and on the strength of the laser pumping the cavity. This property gives rise to hysteresis in the structural configuration and to quantum correlations between photonic and mechanical fluctuations. The scheme can be implemented in a setup like the one realized in Ref. [33, 34].

Figure 2

(a) Intensity $I_{\mathrm{out}}$ at the cavity output for varying pump strength $P=(\eta /{\eta }_0{)}^2$, with ${\eta }_0^2=\kappa {\omega }_a^2/(2{\omega }_R)$. $I_{\mathrm{out}}$ is in units of $I_1=I_{\mathrm{out}}(P=1)$ and is obtained for a configuration minimizing $V_{\mathrm{tot}}$, numerically found using linear and zigzag chains as initial guesses. (b) Potential $V_s$ as a function of the displacement $x$ of the central ion (in units of $1/k$) for $P=130$, 160, 190. The dashed lines indicate nodes at $kx=\pm \pi /2$. The plots correspond to a chain of 60 ${}^{40}\mathrm{Ca}^{+}$ ions with interparticle distance $4.3\mu \mathrm{m}$ in the central region, coupled to a cavity mode with wavelength 866 nm and transverse width $\sigma =14\mu \mathrm{m}$ ($N_{\mathrm{eff}}\simeq 5.7$). The other parameters are ${\omega }_a/2\pi =0.1\mathrm{MHz}$, ${\omega }_t/2\pi =2.26\mathrm{MHz}$ (the critical value is ${\omega }_{tc}/2\pi \simeq 2.216\mathrm{MHz}$), ${\Delta }_c=0$, ${\Delta }_0/2\pi =500\mathrm{MHz}$, $\kappa /2\pi =0.5\mathrm{MHz}$, $g_0/2\pi =9.4\mathrm{MHz}$, $\gamma /2\pi =10\mathrm{MHz}$, resulting in $\mathcal{C}=2$.

Figure 3

Spectrum $S(\nu )$ of the field at the cavity output (in units of $S_0=1/{\omega }_a$) for a zigzag chain of three ions when only one ion at the chain edge, the right one in (c), couples significantly to the cavity mode (with Lamb-Dicke parameter $\sim 0.1$) and the ions’ motion is cooled by the cavity field. The parameters are the same as in Fig. 2, except for ${\omega }_a=\kappa =2\pi \times 1\mathrm{MHz}$, ${\omega }_t=2\pi \times 1.57\mathrm{MHz}$, and (a) $\mathcal{C}=0.5$, $P=1$; (b) $\mathcal{C}=3$, $P=0.22$; the mode width $\sigma$ is 0.65 times the interparticle distance in the linear array, so $N_{\mathrm{eff}}\simeq 1.1$. The equilibrium positions are the same in (a) and (b). The motional modes contributing to the spectral peaks are sketched in panel (c) (not to scale; the resonance corresponding to mode 2 is not visible because this mode is too weakly coupled to the cavity). The Rayleigh peak is not shown.