Sympathetic cooling of mixed-species two-ion crystals for precision spectroscopy

Sympathetic cooling of trapped ions has become an indispensable tool for quantum-information processing and precision spectroscopy. In the simplest situation a single Doppler-cooled ion sympathetically cools another ion which typically has a different mass. We analytically investigate the effect of the mass ratio of such an ion crystal on the achievable temperature limit in the presence of external heating. As an example, we show that cooling of a single Al${}^{+}$ with Be${}^{+}$, Mg${}^{+}$, and Ca${}^{+}$ ions provides similar results for heating rates typically observed in ion traps, whereas cooling ions with a larger mass perform worse. Furthermore, we present numerical simulation results of the rethermalization dynamics after a background gas collision for the Al${}^{+}$-Ca${}^{+}$ crystal for different cooling laser configurations.

Figure 1

Schematic of a linear ion trap. The trap consists of four blade-shaped electrodes of which two opposing ones are connected to an rf voltage while the other two are connected to ground. It also includes two endcap electrodes that are connected to a positive dc voltage. The line through the two endcaps defines the trap axis $z$ and the other two axes ($x$, $y$) are chosen such that two blades lie on each axis.

Figure 2

Normal mode frequencies and normalized amplitudes for a two-ion two-species crystal. The square of the normalized amplitude ($b_1^2$) and the in-phase (${\omega }_i$) and out-of-phase (${\omega }_o$) trap frequencies normalized to the axial trap frequency of a single ion of mass $m_1$ are shown for the axial and radial direction for different mass ratios $\mu =m_2/m_1$ and $\epsilon$ parameters.

Figure 3

Dependence of the cooling limits on the angle of the cooling laser. $k_{\parallel }/k_{\mathrm{total}}$ denotes the component of the laser radiation parallel to the investigated axis. The other two axes are assumed to be cooled equally.

Figure 4

Normalized axial clock ion energy plotted against the mass ratio $\mu$ of the ion crystal. The energy $E_c$ of the clock ion is the sum of the clock ion energy in both modes. The calculations were performed for varying electric field spectral densities $S_E$. The cooling laser intensity $I/I_0$ has been optimized for each value of $S_E$. The dashed lines show different logic ion species for an Al${}^{+}$ clock.

Figure 5

Normalized radial clock ion energy plotted against the mass ratio $\mu$ of the ion crystal for radial modes. The energy $E_c$ of the clock ion is the sum of the clock ion energy in both radial modes. The shaded region indicates parameters for which the ion crystal is no longer linear in axial direction. The cooling laser intensity $I/I_0$ has been optimized for each value of $S_E$.

Figure 6

Total clock ion energy (secular energy $+$ micromotion energy) of a clock ion in a Doppler-cooled crystal along one radial direction. Here the mass ratio $\mu$ and the $\epsilon$ parameters were varied and the energy was normalized to the Doppler-cooling energy $E_{\mathrm{D}}$. The absence of external heating was assumed. Due to large micromotion contributions, the total energy diverges at the points where the crystal becomes instable.

Figure 7

Comparison of secular motion-induced second-order Doppler shifts for Al${}^{+}$-$X^{+}$ crystals with different cooling ions $X^{+}=\{{\mathrm{Be}}^{+}$, Mg${}^{+}$, Ca${}^{+}$, Sr${}^{+}$, Yb${}^{+}\}$ in the presence of external heating with an electric field spectral density of $S_E$. The right figure shows the parameters $\epsilon$ and $I/I_0$ for best cooling performance for an Al${}^{+}$-Ca${}^{+}$ crystal. Those optimum parmaters for a spectral energy density of $S=0.005S_{E0}$ are given by $2.63,0.40$, $3.17,0.05$, $1.12,0.87$, and $1.04,1.61$ for the Be, Mg, Sr, and Yb crystals, respectively.

Figure 8

Time needed to cool an Al${}^{+}$-Ca${}^{+}$-ion pair from 19.3 K to 1000 Doppler limits ($\sim$800 mK). The $x$ axis shows the detuning of a second laser at saturation intensity that is cooling the ion crystal in addition to the standard $-\Gamma /2$-detuned beam.

Figure 9

Cooling evolution of an Al${}^{+}$-Ca${}^{+}$-ion pair after a collision event with an H${}_2$ molecule (blue line). The red line shows the cooling evolution of a single Ca${}^{+}$ ion in the same trap with the same initial energy. The horizontal dotted lines denote the Doppler-cooling limit and the inset shows the energies of the calcium and aluminum ion before and after crystallization occurred at around 7 ms.

Figure 10

Relative micromotion energy of the clock ion in the in-phase and out-of-phase radial modes. The energy is normalized to the secular energy of the clock ion in the respective mode. In this graph the absence of external heating was assumed. The dashed lines denote the asymptotic behavior for very large radial confinement ($\epsilon \to \infty$).