Ion chains in high-finesse cavities

We analyze the dynamics of a chain of singly charged ions confined in a linear Paul trap and which couple with the mode of a high-finesse optical resonator. In these settings the ions interact via the Coulomb repulsion and are subject to the mechanical forces due to scattering of cavity photons. We show that the interplay of these interactions can give rise to bistable equilibrium configurations, into which the chain can be cooled by cavity-enhanced photon scattering. We characterize the resulting equilibrium structures by determining the stationary state in the semiclassical limit for both cavity field and crystal motion. The mean occupation of the vibrational modes at steady state is evaluated, showing that the vibrational modes coupled to the cavity can be simultaneously cooled to low occupation numbers. It is also found that at steady state the vibrations are entangled with the cavity field fluctuations. The entanglement is quantified by means of the logarithmic negativity. The spectrum of the light at the cavity output is evaluated and the features signaling entanglement are identified.

Figure 1

The dipolar transitions of ions forming a chain strongly couple with one mode of a high-finesse optical cavity. The photon-mediated interaction between the ions gives rise to multistable structural configurations and to quantum correlations between photonic and mechanical fluctuations, which can be revealed by measuring the intensity and spectrum of the field at the cavity output.

Figure 2

Potential $V_s$ of the zigzag mode [Eq. (30)] as a function of the zigzag amplitude $b$ (in units of $1/k$) for different pumping powers $P$ (in units of $P_0=\kappa {\omega }_x^2/{\omega }_R$). The dashed lines indicate the nodes of the cavity mode at $kb=\pm \pi$. The cooperativity is (a) $\mathcal{C}=0.5$ and (b) $\mathcal{C}=3$. The coupling of the ions with the field is assumed to be homogeneous and $\vartheta =N{\omega }_s^2/{\omega }_x^2=0.22$.

Figure 3

(a) Same as Fig. 2 but for a chain in a linear Paul trap. A chain of 60 ${}^{40}$Ca${}^{+}$ ions is taken with interparticle distance 4.3 $\mu$m in the central region. The ions are coupled to a cavity mode with wavelength 866 nm and transverse width $w=14\mu$m ($N_{\mathrm{eff}}\simeq 5.7$). (b) Intensity $I_{\mathrm{out}}$ at the cavity output as a function of $P$. Here, the intensity is in units of $I_0=I_{\mathrm{out}}\left(P_0\right)$ and for a configuration minimizing $V_{\mathrm{tot}}$, numerically found using linear and zigzag chains as initial guesses. The other parameters are ${\omega }_y=2\pi \times 0.1$ MHz, ${\omega }_x=2\pi \times 2.26$ MHz (the critical value is ${\omega }_{\mathrm{crit}}\simeq 2\pi \times 2.216$ MHz), ${\Delta }_c=0$, ${\Delta }_0=2\pi \times 500$ MHz, $\kappa =2\pi \times 0.5$ MHz, $g_0=2\pi \times 9.4$ MHz, $\gamma =2\pi \times 10$ MHz, resulting in $\mathcal{C}=2$.

Figure 4

Equilibrium configurations in a chain of four ions as a function of the pump power $P$. The plots display the value of the $x$ coordinate (in units of $1/k$) of each of the ions as a function of $P$ (in units of $P_0=\kappa {\omega }_x^2/{\omega }_R$). The cooperativity is $\mathcal{C}=0.3$ (left panels) and $\mathcal{C}=3$ (right panels). The vertical dotted lines in the right panels indicate the limits of the bistability region. In (a) and (b) the chain is centered with respect to the cavity axis, while in (c) and (d) it is displaced by 1.1 $\mu$m in the $y$ direction with respect to the cavity center. The arrow in (b) indicates the value of $P$ for the eigenmodes shown in Fig. 5. The ions are labeled from $A$ to $D$, with $A$ and $D$ corresponding to the chain edges. The other parameters are ${\Delta }_c=0$, ${\Delta }_0=2\pi \times 500$ MHz, $\kappa =2\pi \times 0.5$ MHz, $\gamma =2\pi \times 10$ MHz, ${\omega }_y=\kappa =2\pi \times 1$ MHz, ${\omega }_x=2\pi \times 2.12$ MHz (the critical value is ${\omega }_{\mathrm{crit}}=2\pi \times 2.04$ MHz). The width $w$ of the Gaussian mode is equal to the distance between the central ions, which is 4.1 $\mu$m in the linear array.

Figure 5

Schematic representation of the motional modes (ordered by increasing frequency) for the parameters of Fig. 4b with $P=0.074P_0$ and in the zigzag configuration ($U_0$ is 1/5 of its value for the linear chain). The positions are not to scale. The modes that couple to the cavity mode are indicated by a double circle.

Figure 6

Mean excitation number $\langle b_n^{†}{b}_n\rangle$ for each of the motional modes in Fig. 5. The dotted line indicates the value $\langle b_n^{†}{b}_n\rangle =1$ (the vertical axis is in logarithmic scale).

Figure 7

Spectrum $S\left(\nu \right)$ of the field at the cavity output at steady-state for a zigzag of four ions (the Rayleigh peak is not shown). $S\left(\nu \right)$ is evaluated from Eq. (50) and reported in units of ${\omega }_y^{-1}$, while $\nu$ is in units of ${\omega }_y$. The top (bottom) panels refer to the symmetric (asymmetric) configuration, where in the asymmetric case the chain is displaced with respect to the cavity axis by 0.28 times the distance between the central ions. The parameters are the same as in Fig. 4 except for (a) $\mathcal{C}=0.3$ and $P=0.54$; (b) $\mathcal{C}=3$ and $P=0.074$; (c) $\mathcal{C}=0.3$ and $P=0.62$; (d) $\mathcal{C}=3$ and $P=0.084$ (the values of $P$ are such that $U_0$ is 1/5 of the value it takes when the ions form a linear array). The motion is assumed to couple to a thermal bath at temperature 1 mK and thermalization rate ${\Gamma }_n=100{\mathrm{s}}^{-1}$ for all modes.

Figure 8

Entanglement between field and motional fluctuations as a function of pumping power $P$. The entanglement is given by the logarithmic negativity. The continuous line shows the entanglement between the cavity fluctuations and the whole set of vibrational modes, the dashed line the entanglement between the cavity and the zigzag mode only. The parameters are the same as in Fig. 4 for (a)–(d), respectively, but the results here refer only to zigzag configurations and within the Lamb-Dicke regime. The dotted vertical lines indicate the values for which the spectra in Fig. 7 are evaluated.