Probing Time Dilation in Coulomb Crystals in a High-Precision Ion Trap

Trapped-ion optical clocks are capable of achieving systematic fractional frequency uncertainties of ${10}^{-18}$ and possibly below. However, the stability of current ion clocks is fundamentally limited by the weak signal of single-ion interrogation. We present an operational, scalable platform for extending clock spectroscopy to arrays of Coulomb crystals consisting of several tens of ions while allowing systematic shifts as low as ${10}^{-19}$. We observe three-dimensional excess micromotion amplitudes inside a Coulomb crystal with atomic spatial resolution and subnanometer amplitude uncertainties, and show that in ion Coulomb crystals of length $400\mu \text{m}$ and 2 mm, time-dilation shifts of ${\mathrm{In}}^{+}$ ions due to micromotion can be close to $1\times {10}^{-19}$ and below ${10}^{-18}$, respectively. In previous ion traps, excess micromotion would have dominated the uncertainty budget for spectroscopy of even a few ions. By minimizing its contribution and providing a means to quantify it, we open up a path to precision spectroscopy in many-body ion systems, enabling entanglement-enhanced ion clocks and providing a well-controlled, strongly coupled quantum system.

Figure 1

Precision scalable ion-trap array. (a) Electrode geometry: the trap consists of one 2-mm-long segment and seven 1-mm-long segments, ‘comp’ refers to stray-field compensation voltages. (b) Trap assembly from a stack of four wafers. (c) Photograph of the trap, showing the onboard low-pass filters and one thermistor. (d) Large Coulomb crystal (here ${\mathrm{Yb}}^{+}$ ions), which can serve as a high-stability frequency reference. (e) Linear ${\mathrm{In}}^{+}$ (dark)/${\mathrm{Yb}}^{+}$ (fluorescing) crystals. The bottom configuration is chosen for high-accuracy clock operation. Its axial length is $219\mu \text{m}$ ($61\mu \text{m}$) for ${\nu }_{\mathrm{ax}}=30\text{kHz}$ (${\nu }_{\mathrm{ax}}=205\text{kHz}$), the parameters used in Ref. [33].

Figure 2

Single-ion measurements of the axial rf-field component in two trap segments of length 1 and 2 mm, respectively, at an rf amplitude of $U_{\mathrm{rf}}\approx 800\text{V}$. The shaded areas correspond to time-dilation shifts below ${10}^{-18}$ (${10}^{-19}$) for ${}^{115}{\mathrm{In}}^{+}$.

Figure 3

(a) Setup for stroboscopic micromotion measurements across Coulomb crystals with atomic resolution. (b) Stroboscopic image of a 14-ion ${}^{172}{\mathrm{Yb}}^{+}$ crystal for one rf-phase and laser direction setting with a total integration time of 2700 s. FPGA, field-programmable gate array; HV, high voltage; MCP, microchannel plate.

Figure 4

Measured excess micromotion in Coulomb crystals. (a) Measured 3D micromotion amplitudes of individual ions inside a Coulomb crystal in the 2-mm segment. The small amplitudes require the use of two different length scales: ion positions refer to the black micrometer scale, while micromotion amplitudes are depicted on a nanometer scale (white bar). (b) Time-dilation shifts for ${}^{115}{\mathrm{In}}^{+}$ ions due to $E_{\mathrm{rf}}$ at the respective positions. (c) Comparison of the axial rf fields determined in single-ion PMT measurements and by stroboscopic imaging of whole Coulomb crystals. The integration times were 2700 and 900 s per phase and direction for crystals 1 and 2, respectively. Data from the same measurement (see Supplemental Material [42] for values and uncertainties) are used in (a),(b) and in the part for crystal 1 in (c).