Feasibility of coherent xuv spectroscopy on the $1S\text{−}2S$ transition in singly ionized helium

The $1S\text{−}2S$ two-photon transition in singly ionized helium is a highly interesting candidate for precision tests of bound-state quantum electrodynamics (QED). With the recent advent of extreme ultraviolet frequency combs, highly coherent quasi-continuous-wave light sources at 61 nm have become available, and precision spectroscopy of this transition now comes into reach for the first time. We discuss quantitatively the feasibility of such an experiment by analyzing excitation and ionization rates, propose an experimental scheme, and explore the potential for QED tests.

Figure 1

Relevant energy levels of ${{}^4\text{H}\text{e}}^{+}$. The inset shows the fine structure and Lamb shift of the $n=2$ levels. After two-photon excitation from the ground state, the third ionizing photon reaches 6.8 eV into the continuum.

Figure 2

Excited- (solid line) and ionized-state (dashed line) populations as functions of interaction time $t_0$ for an ${\text{He}}^{+}$ ion illuminated by an average intensity of $10\mu \text{W}/\mu {\text{m}}^2$ at 61 nm. We describe the $1S\text{−}2S$ two-photon transition including the ionization channel, as given by Eq. (2), and employ a three-level scheme consisting of the $1S$ ground state, the $2S$ excited state, and a continuum state.

Figure 3

Optimization of the interaction time $t_0$ with the exciting laser field to maximize either the excitation rate (solid line) or the resonant three-photon ionization rate (dashed line). The dead time is set to zero and the laser intensity is assumed to be $10\mu \text{W}/\mu {\text{m}}^2$.

Figure 4

Excitation (solid line) and resonant ionization rates (dashed line) for a single anticollinearly excited ${\text{He}}^{+}$ ion.

Figure 5

Numerically determined length $l_i$ of an ${\text{He}}^{+}$ ion chain with constant radial confinement of ${\omega }_r=2\pi \times 10\text{MHz}$ and variable axial confinement to maintain a linear string. The length is well approximated linearly by $l_i\sim N$ $4\mu \text{m}–5\mu \text{m}$.

Figure 6

Sideband spectrum of a collinearly excited ${\text{He}}^{+}$ ion with a secular frequency of ${\omega }_s=2\pi \times 10\text{MHz}$. The vertical line indicates the recoil frequency of 54 MHz. Circles and squares indicate whether the ion is assumed to be cooled to the ground state of motion or to 1 mK, respectively.

Figure 7

Relative magnitude of the strongest sideband as a function of secular frequency for ${\text{He}}^{+}$ ions cooled to 1 mK (squares) or to the ground state (circles).

Figure 8

Sideband spectrum of a string of two axially excited ${\text{He}}^{+}$ ions confined with a secular c.m. frequency of 10 MHz and cooled to 1 mK. The vertical line indicates the recoil frequency.

Figure 9

Calculated number of sidebands that exceed 1% of the strongest component as a function of the number of ions. The secular frequency is 10 MHz; the temperature is 1 mK. The dashed line shows for comparison a hypothetical linear increase in the number of sidebands.

Figure 10

Same as Fig. 8 except one ${\text{Mg}}^{+}$ cooling ion was added.

Figure 11

Maximum excitation (solid line) and ionization rates (dashed line) for one collinearly excited ${\text{He}}^{+}$ ion. The Rabi frequency has been reduced to 10%. Since the ionization cross section remains unchanged, the dynamics is slightly different from that in Fig. 4.