Motional-mode analysis of trapped ions

We present two methods for characterization of motional-mode configurations that are generally applicable to the weak- and strong-binding limit of single or multiple trapped atomic ions. Our methods are essential to realize control of the individual as well as the common motional degrees of freedom. In particular, when implementing scalable radio-frequency trap architectures with decreasing ion-electrode distances, local curvatures of electric potentials need to be measured and adjusted precisely, e.g., to tune phonon tunneling and control effective spin-spin interaction. We demonstrate both methods using single ${}^{25}\text{Mg}^{+}$ ions that are individually confined $40\mu \mathrm{m}$ above a surface-electrode trap array and prepared close to the ground state of motion in three dimensions.

Figure 1

Illustration of both mode-analysis methods applicable to the weak- and strong-binding limit. In our simplified example, the motional modes are rotated relative to the coordinate system by ${\varphi }_{\mathrm{z}}$. (a) In the weak-binding limit, the mode ${\mathbf{u}}_1$ is resonantly excited by an electric field ${\mathbf{E}}_{\text{exc}}$ oscillating at ${\omega }_{\mathrm{exc}}={\omega }_1$ to motional amplitude $A_1\propto \langle {\mathbf{u}}_1,{\mathbf{E}}_{\text{exc}}\rangle$ with $\langle n_1\rangle \gg 0$. Subsequently, a laser with wave vector ${\mathbf{k}}_{\mathrm{w}}$ along $x$ resonantly drives the transition $\left|g_{\mathrm{w}}\right.〉\leftrightarrow \left|e_{\mathrm{w}}\right.〉$. Blue bars illustrate the natural linewidth ${\mathrm{\Gamma }}_{\mathrm{w}}$. In turn, the mode orientation is derived considering the Doppler effect and the related decrease of the fluorescence rate ${\mathrm{\Omega }}_{\text{F}}$ as a function of $A_1\langle {\mathbf{u}}_1,{\mathbf{k}}_{\mathrm{w}}\rangle$. (b) In the strong-binding limit, a laser is tuned to ${\omega }_{\mathrm{s}}+{\omega }_1$ with wave vector ${\mathbf{k}}_{\mathrm{s}}$ along $x$, coherently coupling the electronic (gray bars) and motional states (colored bars), e.g., $|g_{\mathrm{s}},0,0,0\rangle$ to $|e_{\mathrm{s}},1,0,0\rangle$. Here, the coupling rate ${\mathrm{\Omega }}_{\mathrm{s}}\propto \langle e_{\mathrm{s}},1,0,0|e^{i\langle {\mathbf{k}}_{\mathrm{s}},{\mathbf{u}}_1\rangle }|g_{\mathrm{s}},0,0,0\rangle$ encodes the mode configuration.

Figure 2

Illustration of the setup showing a false colored scanning electron microscope image of our trap array. The array consists of two radio-frequency electrodes (light gray) and 30 control electrodes (dark gray), labeled $1\cdots 30$. Trapping sites are denoted ${\mathbf{T}}_j$ with $j\in \{0,1,2\}$ and are separated by $\approx 40\mu \mathrm{m}$; here, all experiments are carried out with single ${}^{25}\text{Mg}^{+}$ ions trapped near ${\mathbf{T}}_2$. Motional excitation fields ${\mathbf{E}}_{\mathrm{exc}}=U_{\mathrm{exc}}{\mathbf{E}}_l$ oscillating at ${\omega }_{\mathrm{exc}}$ are applied via the $l\mathrm{th}$ control electrode. A set of laser beams, propagating along ${\mathbf{k}}_{\mathrm{w}}$ and parallel to the magnetic field $|{\mathbf{B}}_0|\approx 4.65\mathrm{mT}$, is used for preparation and detection of the electronic degrees of freedom (blue arrow). Two more beams, labeled BR/RR (red arrows), with an effective wave vector $\mathrm{\Delta }\mathbf{k}\equiv {\mathbf{k}}_{\mathrm{s}}$, are implemented to coherently couple electronic and motional states via two-photon stimulated Raman transitions.

Figure 3

Determination of mode configurations, i.e., motional frequencies and mode orientations, in the weak-binding limit. (a) Experimental sequences (not to scale; durations between subsequent operations remain $<500\mathrm{ns}$) performed with a single ion trapped near ${\mathbf{T}}_2$, beginning with Doppler cooling and preparation of $|g\rangle$, followed by a motional excitation pulse of duration $t_{\text{exc}}=10\mu \mathrm{s}$, and finalized by the measurement of ${\mathrm{\Omega }}_{\text{F}}$, revealing the normalized fluorescence $\mathcal{F}$. (b) Excitation pulses are applied using ${\mathbf{E}}_{\mathrm{exc}}\propto {\mathbf{E}}_l$, representatively shown for $l\in \{22,26,28\}$ (from top to bottom), with corresponding $\mathcal{F}$ as a function of the excitation frequency ${\omega }_{\text{exc}}$. Data points are taken in random order and averaged over 200 repetitions of sequences with fixed parameter settings, where error bars denote the statistic uncertainties. A combined model fit (solid lines) considering ten different electrodes $l\in \{21\cdots 30\}$ yields mode configurations: $\{{\varphi }_x,{\varphi }_y,{\varphi }_z\}={\{-4{\left(1\right)}_{\text{stat}}{\left(2\right)}_{\text{sys}},-36{\left(1\right)}_{\text{stat}}{\left(4\right)}_{\text{sys}},-3{\left(1\right)}_{\text{stat}}{\left(1\right)}_{\text{sys}}\}}^{\circ }$ and $\mathbf{\omega }/\left(2\pi \right)=\{3.584{\left(2\right)}_{\mathrm{stat}},4.833{\left(3\right)}_{\mathrm{stat}},5.878{\left(4\right)}_{\mathrm{stat}}\}$ MHz. Residuals are shown below each spectrum.

Figure 4

Determination of mode orientations in the strong-binding limit. (a) After Doppler cooling and preparation of $|g\rangle$ and $\langle n_i\rangle \le 1$ via resolved sideband cooling, we apply a coherent coupling of electronic and motional states with lasers BR and RR tuned to ${\omega }_{\mathrm{R}}$ for duration $t_{\mathrm{pulse}}$ and derive $P_{|g\rangle }$, the probability of detecting the ion in $|g\rangle$. Here, $\mathbf{\omega }$ is determined by calibration measurements (not to scale; durations between subsequent operations remain $<500\mathrm{ns}$). In subsequent measurements, we record $P_{|g\rangle }$ as a function of $t_{\mathrm{pulse}}$ for different couplings, where ${\omega }_{\mathrm{R}}\approx {\omega }_{\mathrm{s}}$ [see (b)] and ${\omega }_{\mathrm{R}}\approx \{{\omega }_{\mathrm{s}}+{\omega }_1,{\omega }_{\mathrm{s}}+{\omega }_2,{\omega }_{\mathrm{s}}+{\omega }_3\}$ [see (c)]. A combined model fit to the data (solid lines) and results from complementary measurements yield $\{{\varphi }_x,{\varphi }_y,{\varphi }_z\}={\{-9{\left(2\right)}_{\text{stat}}{\left(5\right)}_{\text{sys}},-51{\left(2\right)}_{\text{stat}}{\left(5\right)}_{\text{sys}},-15{\left(1\right)}_{\text{stat}}{\left(5\right)}_{\text{sys}}\}}^{\circ }$, $\{\langle n_1\rangle ,\langle n_2\rangle ,\langle n_3\rangle \}=\{0.5{\left(1\right)}_{\mathrm{stat}},1.0{\left(1\right)}_{\mathrm{stat}},0.44{\left(5\right)}_{\mathrm{stat}}\}$. Corresponding calculated thermal distributions ${\mathcal{P}}_i\left(n_i\right)$ of each mode are shown in bar charts on the right.

Figure 5

Evolution of statistical uncertainties in dependence of the number of electrodes ${\mathcal{N}}_{\mathrm{el}}$. (a) Number of fits ${\mathcal{N}}_{\mathrm{fit}}$ that have to be performed for all possible combinations of electrodes. (b) Mean angles $\langle {\varphi }_{\alpha }\rangle$ averaged over all combinations, where error bars are given by ${\sigma }_{\alpha }$. Horizontal lines represent the result obtained for ${\mathcal{N}}_{\mathrm{el}}=10$. (c) Evolution of $S\left({\mathcal{N}}_{\mathrm{el}}\right)$, where $S\left(5\right)\approx 4^{\circ }$ yields values comparable to the maximum estimated systematic uncertainty for this dataset.