Controlling systematic frequency uncertainties at the ${10}^{-19}$ level in linear Coulomb crystals

Trapped ions are ideally suited for precision spectroscopy, as is evident from the remarkably low systematic uncertainties of single-ion clocks. The major weakness of these clocks is the long averaging time, necessitated by the low signal of a single atom. An increased number of ions can overcome this limitation and allow for the implementation of novel clock schemes. However, this presents the challenge to maintain the excellent control over systematic shifts of a single particle in spatially extended and strongly coupled many-body systems. We measure and deduce systematic frequency uncertainties related to spectroscopy with ion chains in an rf trap array designed for precision spectroscopy on simultaneously trapped ion ensembles. For the example of an ${\mathrm{In}}^{+}$ clock, sympathetically cooled with ${\mathrm{Yb}}^{+}$ ions, we show in our system that the expected systematic frequency uncertainties related to multi-ion operation can be below $1\times {10}^{-19}$. Our results pave the way to advanced spectroscopy schemes such as entangled clock spectroscopy and cascaded clock operation.

Figure 1

Scalable precision ion trap array. (a) Trap assembly from a stack of four wafers (taken from [30]). (b) Photograph of the trap, showing the onboard low-pass filters and one thermistor. (c) Close-up of the trapping region. (d) Electrode geometry. The trap consists of one 2-mm-long and seven 1-mm-long segments. The rf electrodes are shown in red, and static voltages applied to the blue rf ground electrodes are used to confine the ions axially, compensate for stray electric fields, and control the radial mode frequency splitting and orientations.

Figure 2

Coulomb crystals. (a) Large Coulomb crystal (here fluorescing ${\mathrm{Yb}}^{+}$ ions), which can serve as a high-stability frequency reference. (b) Crystal configuration considered for spectroscopy with high accuracy (see Table 1). The sketch corresponds to the lower of the three displayed crystal examples of linear ${\mathrm{In}}^{+}$ - ${\mathrm{Yb}}^{+}$ crystals. The axial extension is $219\mu \text{m}$ ($61\mu \text{m}$) for ${\nu }_{\mathrm{ax}}=30\text{kHz}$ (${\nu }_{\mathrm{ax}}=205\text{kHz}$) (figure taken from [30]). (c) Coulomb crystal with a symmetric topological defect (as used in [32]).

Figure 3

Electric-field gradients in a 13-ion chain as a function of the axial confinement. The key refers to the crystal configuration shown in the inset. The right-hand scale shows the corresponding quadrupole shift for the transition from ${}^1{S}_0$, $m_F=\pm 9/2$ to ${}^3{P}_0$, $m_{F^{\prime }}=\pm 7/2$ in ${}^{115}{\mathrm{In}}^{+}$. The bottom graph shows the respective total chain length. Vertical bars indicate the parameters for indium cooled and sympathetically cooled configurations A and B, respectively.

Figure 4

Time dilation shift due to excess micromotion (from [30]). A 14-ion ${}^{172}{\mathrm{Yb}}^{+}$ crystal was used to probe rf electric-field amplitudes along the line of minimal potential in the 2-mm segment. The graph shows the corresponding time dilation shifts for ${}^{115}{\mathrm{In}}^{+}$ ions.

Figure 5

Frequency dependence of the radial heating rates, determined using a single ${}^{172}{\mathrm{Yb}}^{+}$ ion cooled to the ground state. The fitted line corresponds to $\dot{\overline{n}}=2.8\left(2\right)\times {10}^{11}{\text{s}}^{-1}{\text{Hz}}^2/{\nu }_{\mathrm{radial}}^2$.

Figure 6

Fractional frequency shifts of each clock ion [cf. Fig. 2] due to thermal motion. Closed symbols depict the results for configuration A (cooling on the narrow intercombination line in ${\mathrm{In}}^{+}$); open symbols represent configuration B (sympathetic cooling on ${\mathrm{Yb}}^{+}$ only). (a) Time dilation shift due to secular motion and intrinsic micromotion for all $3N$ modes. (b) ac Stark shift due to intrinsic micromotion.

Figure 7

Debye-Waller effect in clock interrogation. (a) Mean Rabi frequencies and rms fluctuations, normalized to the Rabi frequency of a free atom, for configurations A (closed circles) and B (open circles). (b) Effective broadening of the interrogation laser due the Rabi frequency fluctuations between interrogations in Ramsey spectroscopy with $T_{\mathrm{int}}=195\text{ms}$, as predicted by Eq. (29). Closed and open symbols identify the respective impact of the rms spreads shown in (a). The Bloch spheres in the inset illustrate the effect of a Rabi frequency mismatch on the error signal: On the left, the same Rabi frequency is assumed for both interrogations, while on the right, the Rabi frequency of the red (front) trace is increased by $20\%$, leading to a higher difference in observed populations.

Figure 8

Trap temperature increase at the two integrated Pt100 sensors as a function of the rf voltage amplitude. Here thermistor 1 refers to the sensor on the rf wafer closest to the carrier board and thermistor 2 is on the other rf wafer. The mean temperature increase of the trap for an rf voltage amplitude of $750\text{V}$ ($1.5\text{kV}$) at ${\mathrm{\Omega }}_{\mathrm{rf}}=2\pi \times 24.4\text{MHz}$ is $0.6\text{K}$ ($2.2\text{K}$). The voltage uncertainty of each data point is less than $10\text{V}$ and temperature rise uncertainties are around $15\text{mK}$ and originate from fit errors of the experimental data. The lines are quadratic fits to the data points.