Metrology of Rydberg states of the hydrogen atom

We present a method to precisely measure the frequencies of transitions to high-$n$ Rydberg states of the hydrogen atom which are not subject to uncontrolled systematic shifts caused by stray electric fields. The method consists in recording Stark spectra of the field-insensitive $k=0$ Stark states and the field-sensitive $k=\pm 2$ Stark states, which are used to calibrate the electric field strength. We illustrate this method with measurements of transitions from the $2s(f=0\text{and}1)$ hyperfine levels in the presence of intentionally applied electric fields with strengths in the range between 0.4 and $1.6\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$. The slightly field-dependent $k=0$ level energies are corrected with a precisely calculated shift to obtain the corresponding Bohr energies $(-c{R}_{\mathrm{H}}/n^2)$. The energy difference between $n=20$ and $n=24$ obtained with our method agrees with Bohr's formula within the 10-kHz experimental uncertainty. We also determined the hyperfine splitting of the $2s$ state by taking the difference between transition frequencies from the $2s(f=0\text{and}1)$ levels to the $n=20,k=0$ Stark states. Our results demonstrate the possibility of carrying out precision measurements in high-$n$ hydrogenic quantum states.

Figure 1

Schematic representation of the experimental setup. Upper part: laser system and geometry of the photoexcitation from the metastable $2s$ state of H to Rydberg states. Lower part: vacuum chambers in which the supersonic beam of H atoms is generated, these atoms are photoexcited to Rydberg states and the Rydberg states are detected by pulsed field ionization. Top right inset: Configuration of laser and supersonic beams used for the determination of Doppler-free frequencies. See text for details.

Figure 2

Schematic electric-circuit diagram of the laser-stabilization electronics (see text for details). Color-shaded inset: Spectral density (SD) of the in-loop beat note ${\nu }_b$ recorded with a bandwidth of $3\mathrm{k}\mathrm{Hz}$ with (black) and without (gray) active stabilization using the intracavity EOM.

Figure 3

Stark effect in the $n=20,m_f=0$ manifold of the H atom. (a) Field dependence of the $k=0$ state revealing a quadratic shift below $50\mathrm{m}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$ caused by the intramanifold mixing of different orbital-angular-momentum components, and a smaller quadratic shift at larger fields arising from the interaction between different $n$ manifolds. (b) Overview of the field dependence of all $m_{\ell }=0$ Stark states, which is essentially linear. (c) Calculated spectra for different electric-field strengths and electric-field vectors $\vec{\mathcal{F}}$ pointing parallel or perpendicular to the laser polarization ${\vec{\epsilon }}_{\mathrm{p}}$.

Figure 4

Expansion coefficients of the $k=0, |m_f|=0,1$, and 2 Rydberg-Stark wave functions in the $|ljf{m}_f\rangle$ angular-momentum basis as labeled in the figure. Only basis states with odd orbital-angular-momentum quantum numbers make significant contributions.

Figure 5

Energy level structure of the eight $n=20,k=0$ Stark states with $m_{\ell }=1$ character, calculated at an electric field strength $\mathcal{F}=0.8\mathrm{V}{\mathrm{cm}}^{-1}$. These states split into two groups of four states each, separated by $\sim 600\mathrm{k}\mathrm{Hz}$. The Zeeman effect induced by a magnetic field pointing along the quantization axis is schematically illustrated on the right side and lifts all remaining degeneracies.

Figure 6

Stark shifts of the metastable $2s$ levels of the H atom calculated for electric fields in the range between 0 and $2\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$.

Figure 7

PFI spectra of the $n=20$ Rydberg-Stark states of H recorded from the $2s(f=1$) hyperfine component in an electric field ${\mathcal{F}}_{\mathrm{dc}}\approx 200\mathrm{m}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$. The direction of the strong pulsed electric-field (${\mathcal{F}}_{\text{PFI}}=5.7\mathrm{k}\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$) used for ionization was set parallel to ${\mathcal{F}}_{\mathrm{dc}}^{\uparrow \uparrow }$ to record the upper spectrum and antiparallel ${\mathcal{F}}_{\mathrm{dc}}^{\uparrow \downarrow }$ to record the lower, inverted spectrum. The red and blue stick spectra represent the calculated intensity distributions. Inset: The alignment of the two fields leads to ionization without change of the field polarity (red) or to ionization after a diabatic state inversion upon reversal of the field polarity.

Figure 8

(a) Typical experimental (dots) and fitted (blue) spectra of the three ($k=0,\pm 2$) Rydberg-Stark states near the center of the $n=20$ Stark manifold of H, each exhibiting two Doppler components. (b) Weighted residuals in the spectral regions where the Poisson noise dominates.

Figure 9

Spectra of the $n=20,k=0,\pm 2,|m_{\ell }|=1\leftarrow 2s(f=1)$ transitions measured when applying nominal electric fields of 0.4, 0.8, 1.2 and $1.6\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$, respectively. Each spectrum represents the sum of three independent scans as described in Section 2. Right: Relative positions of the line center ${\nu }_0$ with respect to the line center measured at a nominal field strength of $0.4\mathrm{V}\mathrm{c}{\mathrm{m}}^{-1}$. The error bars represent $1\sigma$ uncertainties.

Figure 10

Spectra of the $n=20,k=0,\pm 2,|m_{\ell }|=1\leftarrow 2s(f=1)$ (blue) and $n=20,k=0,\pm 2,|m_{\ell }|=1\leftarrow 2s(f=0)$ (red). The difference of the two central frequencies ${\nu }_0$ corresponds to the hyperfine interval of the $2s$ state.

Figure 11

Spectra of the $n=20,k=0,\pm 2,|m_{\ell }|=1\leftarrow 2\text{s}(f=1)$ (top) and $n=24,k=0,\pm 2,|m_{\ell }|=1\leftarrow 2\text{s}(f=1)$ (bottom). The energy scale has its origin at the $n=20$ Bohr energy which is located $\approx 2.2\mathrm{M}\mathrm{Hz}$ above the $n=20,k=0$ Stark states (see Fig. 3). The Bohr energies are indicated by the dashed lines in the two panels.