Spin-orbit splitting of ${\mathrm{Ar}}^{+}$, ${\mathrm{Kr}}^{+}$, and ${\mathrm{Kr}}^{2+}$ determined by strong-field ultrahigh-resolution Fourier-transform spectroscopy

By the Fourier transform of the yields of ${\mathrm{Ar}}^{2+}$ and ${\mathrm{Kr}}^{2+}$ obtained by the pump-probe measurements using intense near-infrared few-cycle laser pulses, we determined the spin-orbit splitting energies in the electronic ground states of ${\mathrm{Ar}}^{+}$, ${\mathrm{Kr}}^{+}$, and ${\mathrm{Kr}}^{2+}$ with high precision. For ${\mathrm{Ar}}^{+}$ and ${}^{84}\mathrm{Kr}^{+}$, the spin-orbit splitting energies for ${}^{2}P_{1/2}-{}^{2}P_{3/2}$ were determined to be 1431.583 33(12) and $5370.29642\left(50\right)\mathrm{c}{\mathrm{m}}^{\text{–}1}$ and the isotope effect was examined for ${}^{83}\mathrm{Kr}^{+}$, ${}^{84}\mathrm{Kr}^{+}$, and ${}^{86}\mathrm{Kr}^{+}$. For ${}^{84}\mathrm{Kr}^{2+}$, the spin-orbit splitting energies for ${}^{3}P_1-{}^{3}P_2$ and ${}^{3}P_0-{}^{3}P_2$ were determined to be 4548.2144(40) and $5312.8059\left(40\right)\mathrm{c}{\mathrm{m}}^{\text{–}1}$, respectively.

Figure 1

TOF spectra of Ar (a) and Kr (b) on a logarithmic scale. The inset in (b) shows the expanded TOF spectrum of ${\mathrm{Kr}}^{2+}$. The peaks with an asterisk in (a) are ringing signals of ${\mathrm{Ar}}^{+}$ and ${\mathrm{Ar}}^{2+}$ and those in (a) are ringing signals of ${}^{84}\mathrm{Kr}^{2+}$ and ${}^{86}\mathrm{Kr}^{2+}$ in (b), originating from the impedance mismatch in the electric circuit in the detection system.

Figure 2

(a) The time-delay dependent ion yield of ${\mathrm{Ar}}^{2+}$ (black curve) and ${}^{84}\mathrm{Kr}^{2+}$ (red curve) at $\mathrm{\Delta }t\sim 500$ ps. The ion yield of ${{\mathrm{D}}_2}^{+}$ (blue broken line) was used for the fine calibration of the Fourier transform spectra. Fourier transform spectra of the ion yields are shown for (b) ${\mathrm{Ar}}^{2+}$, (c) ${}^{84}\mathrm{Kr}^{2+}$, and (d) ${{\mathrm{D}}_2}^{+}$. An inset in each spectrum shows an expanded view of the peak profiles.

Figure 3

Energy diagram of a rare-gas atom and its cation. The pump pulse creates ${}^{2}P$ ($m_L=0$) state through a strong-field ionization of neutral atoms and the population of the ${}^{2}P$ ($m_L=0$) state oscillates by the spin-orbit interaction with the period corresponding to the spin-orbit energy separation of $E_{\mathrm{SO}}$.

Figure 4

The spin-orbit splitting energy, $E_{\mathrm{SO}}$, of (a) ${\mathrm{Ar}}^{+}$ and (b) ${}^{84}\mathrm{Kr}^{+}$. The gray area indicates the 1σ uncertainty of the averaged value.

Figure 5

(a) Fourier transform of the ion yields of ${}^{86}\mathrm{Kr}^{2+}$ (black curve) and ${}^{84}\mathrm{Kr}^{2+}$ (red broken curve) for a comparison. (b) Fourier transform of the ion yields of ${}^{83}\mathrm{Kr}^{2+}$. Black circles: The experimental results. Blue and red sticks: the line strengths of the hyperfine components, ($J=1/2$, $F^{\prime }=5$)–($J=3/2$, $F$) and ($J=1/2$, $F^{\prime }=4$)–($J=3/2$, $F$), calculated by using Eq. (16). Blue broken curve: theoretically obtained spectrum of ${}^{83}\mathrm{Kr}^{2+}$, which is a convolution of the theoretical line strengths and a sinc function. Purple curve: theoretical spectrum including the effect of the ringing signal of ${}^{82}\mathrm{Kr}^{2+}$. (c) The hyperfine structure of ${}^{83}\mathrm{Kr}^{+}$. The hyperfine splitting energies of $J=3/2$ and $J=1/2$ were taken from Refs. [9] and [12], respectively.

Figure 6

Energy diagram of ${\mathrm{Kr}}^{2+}$. The pump pulse creates ${}^{3}P$ ($m_L=\pm 1$) and the populations of ${}^{3}P$ ($m_L=\pm 1$) state and ${}^{3}P$ ($m_L=0$) state oscillate by spin-orbit interaction with the periods corresponding to $E_{\mathrm{SO}}^{++}$ and $E_{\mathrm{SO}}^{\prime ++}$.

Figure 7

The sum of the FT spectra obtained by the measurement sets of ${}^{84}\mathrm{Kr}$ from 5250 to $5450\mathrm{c}{\mathrm{m}}^{\text{–}1}$ (a) and from 4450 to $4650\mathrm{c}{\mathrm{m}}^{\text{–}1}$ (b).

Figure 8

Best-fit curves obtained by the least-squares fit to the peaks assigned to the spin-orbit splitting in the Fourier transform of the ion yields of (a) ${\mathrm{Ar}}^{2+}$, (b) ${}^{84}\mathrm{Kr}^{2+}$, and (c) ${}^{86}\mathrm{Kr}^{2+}$. The residuals of the least-squares fits of the peak profiles of (d) ${\mathrm{Ar}}^{2+}$, (e) ${}^{84}\mathrm{Kr}^{2+}$, and (f) ${}^{86}\mathrm{Kr}^{2+}$. The black and red circles in (a)–(c) represent the real and imaginary components of the FT spectra, respectively. The black and red curves represent the real and imaginary components of the fitting function given by Eq. (A3). The black and red circles in (d)–(f) represent the real and imaginary components of the residuals of the least-squares fits with their uncertainties (σ).