Highly charged ions: Optical clocks and applications in fundamental physics

Recent developments in frequency metrology and optical clocks have been based on electronic transitions in atoms and singly charged ions as references. The control over all relevant degrees of freedom in these atoms has enabled relative frequency uncertainties at a level of ${10}^{-18}$. This accomplishment not only allows for extremely accurate time and frequency measurements, but also to probe our understanding of fundamental physics, such as a possible variation of fundamental constants, a violation of the local Lorentz invariance, and the existence of forces beyond the standard model of physics. In addition, novel clocks are driving the development of sophisticated technical applications. Crucial for applications of clocks in fundamental physics are a high sensitivity to effects beyond the standard model and a small frequency uncertainty of the clock. Highly charged ions offer both. They possess optical transitions which can be extremely narrow and less sensitive to external perturbations compared to current atomic clock species. The large selection of highly charged ions offers narrow transitions that are among the most sensitive ones for the “new physics” effects. Recent experimental advances in trapping and sympathetic cooling of highly charged ions will in the future enable advanced quantum logic techniques for controlling motional and internal degrees of freedom and thus enable high-accuracy optical spectroscopy. Theoretical progress in calculating the properties of selected highly charged ions has allowed the evaluation of systematic shifts and the prediction of the sensitivity to the physics beyond the standard model. New theoretical challenges and opportunities emerge from relativistic, quantum electrodynamics, and nuclear-size contributions that become comparable with interelectronic correlations. This article reviews the current status of the field, addresses specific electronic configurations and systems which show the most promising properties for research, their potential limitations, and the techniques for their study.

Figure 1

Schematic of the shell order in neutral atoms (left) and in hydrogenlike ions (right) ([52]). One can see that the “diving” of $4d$ and $4f$ shells results in level crossings in the areas marked by circles.

Figure 2

Energy levels and radiative lifetimes of low-lying levels of Ag-like ${\mathrm{Nd}}^{13+}$. Vertical intervals are not to scale. The lowest state lifetime includes $E3$ and $M2$ transitions. Inclusion of the hyperfine-induced $E1$ transition for ${}^{143}{\mathrm{Nd}}^{13+}$ decreases the lifetime to 12.5–14 days, depending on the hyperfine component of the transition ([149]). From [371].

Figure 3

Energy levels and radiative lifetimes of low-lying levels of In-like ${\mathrm{Pr}}^{10+}$. From [372].

Figure 4

Low-lying energy levels of ${\mathrm{Cf}}^{15+}$. Leading configurations are shown on the top. $E2$ clock transitions are between the ground and first excited states.

Figure 5

Low-lying energy levels of ${\mathrm{Es}}^{16+}$. Leading configurations are shown on the top.

Figure 6

Energy levels and radiative lifetimes of low-lying levels of Sn-like ${\mathrm{Pr}}^{9+}$. From [371].

Figure 7

Schematic depiction of an EBIT ([280], [281]). A zone of high electron density (with values up to ${10}^{13}{e}^{-}/{\mathrm{cm}}^3$ and a diameter of less than 0.1 mm) is formed in the central region of the ultrahigh vacuum device by the interaction of a magnetic field of typically a few T with an electron beam (up to 1 A) emitted from a cathode of a few mm diameter. The negative space-charge potential of this charge distribution is capable of radially trapping positive ions. Axial confinement along the beam propagation direction is achieved by an ensemble of adequately biased, hollow cylindrical electrodes (drift tubes) arranged along the beam axis. Production of positive ions is driven by electron-impact ionization resulting from the beam interaction with injected atomic species (e.g., an atomic beam). A rather narrow distribution of charge states can be reached of which the highest value depends on the electron-beam energy. Because of their high ionization rate, strong electrostatic confinement as well as am efficiently suppressed ion-recombination rate, EBITs can quickly generate and trap ions in charge states up to ${\mathrm{U}}^{92+}$ ([296]) and retain them for periods of time on the order of hours.

Figure 8

Line profile of an optical transition within the $4f^{13}5s$ configuration of ${\mathrm{Ir}}^{17+}$ ion obtained at the Heidelberg EBIT. The Zeeman splitting due to the 8-T magnetic field of the trap is clearly visible. From [46].

Figure 9

Design of the compact EBIT operating at PTB for the dedicated production of HCI in an optical clock experiment. The inset shows light emission induced by electron-impact excitation of the trapped ions. From [302].

Figure 10

Schematic representation of an experiment using sympathetic cooling of HCI for optical clock applications ([383]; [384]). HCI are produced in an electron-beam ion trap (left), extracted from it, decelerated and bunched, and implanted into a Coulomb crystal of singly carged ions at mK temperatures. The implanted HCI strongly repel their fluorescing, laser-cooled ${\mathrm{Be}}^{+}$ neighbors. This provides information about the location and charge of the implanted ions.

Figure 11

Scheme of the experiment in which ${\mathrm{Ar}}^{13+}$ ions were sympathetically cooled by laser-cooled ${\mathrm{Be}}^{+}$ ions using a laser at $\lambda =313\mathrm{nm}$. From [384].

Figure 12

Doppler cooling of ${\mathrm{Be}}^{+}$ ions as applied by [383], [384], and [385]. The cooling radiation at $\lambda =313\mathrm{nm}$ and the repumper are produced using the technique proposed by [469].

Figure 13

Simulation of a Coulomb crystal containing 300 laser-cooled ${\mathrm{Be}}^{+}$ ions cooling six ${\mathrm{Ar}}^{5+}$ with a very close charge-to-mass ratio $Q/m$ equal to 0.111 and 0.125, respectively. The thermalization in this case is very good, and the radial and transversal temperatures of all components are about 4 mK. However, the crystalline order is less apparent due to the close similarity of the $Q/m$ for the two ion species. This leads to very close trap frequencies ($2\pi {\omega }_{xy}=273$ vs 309 kHz, $2\pi {\omega }_{xy}=139$ vs 147 kHz) in both the axial and the radial direction, and to intermixing as well as easy interchangeability of the positions for the ${\mathrm{Be}}^{+}$ and the ${\mathrm{Ar}}^{5+}$ ions. For larger differences in $Q/m$, the different species arrange themselves in a well-separated manner. From [333].

Figure 14

Images of ${\mathrm{Be}}^{+}$ Coulomb crystals with implanted HCI. (a) Ellipsoidal crystal containing five ${\mathrm{Ar}}^{13+}$ ions. Unbalanced light pressure due to one-sided laser cooling caused the cocrystalized HCI to preferentially accumulate in one side. HCIs close to the crystal edge occupied nonequidistant equilibrium positions. (b) Single ${\mathrm{Ar}}^{13+}$ ion cooled by a single ${\mathrm{Be}}^{+}$ ion ([384]) under modified trapping parameters (note the change of scale in the images). This latter configuration allows for the best sympathetic cooling conditions and eventually for ground-state cooling, which enables quantum logic detection ([381]).

Figure 15

Stability parameter for a HCI in a linear Paul trap. The stability parameter $q$ is plotted for the radial modes of a two-ion crystal consisting of a singly charged ${}^9{\mathrm{Be}}^{+}$ and an ${{}^{40}\mathrm{Ar}}^{Q+}$ ion as a function of the Ar-ion’s charge $Q$ for a single ${\mathrm{Be}}^{+}$ radial trapping frequency of ${\omega }_r=2\pi \times 2.2\mathrm{MHz}$ and ${\mathrm{\Omega }}_{\mathrm{rf}}=2\pi \times 50\mathrm{MHz}$. All charge states satisfy the stability criterion $q<0.908$.

Figure 16

Coupled axial mode parameters for an ${\mathrm{Ar}}^{Q+}/{\mathrm{Be}}^{+}$ two-ion crystal in a linear Paul trap as a function of the HCI charge $Q$. Top: mode frequencies; bottom: mode amplitudes. The position of the mode amplitude crossing point is determined. For details see text.

Figure 17

Coupled radial mode parameters for an ${\mathrm{Ar}}^{Q+}/{\mathrm{Be}}^{+}$ two-ion crystal in a linear Paul trap as a function of the HCI charge $Q$. Top: mode frequencies; bottom: mode amplitudes. The position of the mode amplitude crossing point is determined. For details see text.