Imaging electric fields in the vicinity of cryogenic surfaces using Rydberg atoms

The ability to characterize static and time-dependent electric fields in situ is an important prerequisite for quantum-optics experiments with atoms close to surfaces. Especially in experiments which aim at coupling Rydberg atoms to the near field of superconducting circuits, the identification and subsequent elimination of sources of stray fields are crucial. We present a technique that allows the determination of stray-electric-field distributions $[F_{\mathrm{x}}^{\mathrm{str}}\left(\vec{r}\right),F_{\mathrm{y}}^{\mathrm{str}}\left(\vec{r}\right),F_{\mathrm{z}}^{\mathrm{str}}\left(\vec{r}\right)]$ at distances of less than 2 mm from (cryogenic) surfaces using coherent Rydberg-Stark spectroscopy in a pulsed supersonic beam of metastable $1s^{1}2s^1{}^1{S}_0$ helium atoms. We demonstrate the capabilities of this technique by characterizing the electric stray field emanating from a structured superconducting surface. Exploiting coherent population transfer with microwave radiation from a coplanar waveguide, the same technique allows the characterization of the microwave-field distribution above the surface.

Figure 1

(a) Sketch of the experimental setup indicating the metastable-atom source, the skimmer, electrodes (orange), and the atom beam (light cyan). The laser beam and the microwave pulse for the two-photon excitation of the $34s$ Rydberg state are shown in blue and red, respectively. The electrodes of the einzel lens are indicated in gold, and the microchannel plate detector and the phosphor screen are indicated in dark gray and green, respectively. The sample (light gray) containing the superconducting coplanar waveguide (CPW, black) is also shown. (b) Cross section of the interaction region [black dashed line in (a)] above the CPW. Potentials $V_{\mathrm{c}},V_{\mathrm{r}},V_{\mathrm{l}}$, and $V_{\mathrm{s}}$ can be applied to the center conductor (green), the right (red) and the left (blue) ground planes of the CPW, and the metallic shield [brown, not shown in (a)]. The thickness of the superconducting layer of the CPW is shown not to scale. Charges $Q_{\mathrm{g}}$ and $Q_{\mathrm{s}}$ (light and dark yellow) may accumulate either in the insulating gaps of the waveguide or on an insulating layer on the superconductor. The dashed blue line indicates the cross section of the Rydberg beam limited by the collimating slit apertures.

Figure 2

Distribution of electric-field strengths for potentials $V_{\mathrm{c}}=4.00$ V, $V_{\mathrm{l}}=-0.02$ V, $V_{\mathrm{r}}=0.00$ V, and $V_{\mathrm{s}}=-0.11$ V. (a) Measured distribution of $34s$ Rydberg atoms after driving the $34s$ to $34p$ transition at a microwave detuning of 1.73 MHz. The spatial resolution of the images is $150\mu \mathrm{m}$ [see (c)–(f) for length scales]. The dots mark the positions where the field-ionization signal is minimal and represent the positions where the $34s$ to $34p$ transfer is maximal and hence the field strength (extracted from the Stark shift) is 55 mV/cm. (b) Experimental spectrum (dots) and Gaussian fit (solid line) of the $34s$ to $34p$ transition for He atoms located at the positions marked by the blue (upper trace) and red (lower trace) arrows in (a) after binning over $5\times 5$ pixels. (c) Spatial distribution of $34s$ to $34p$ transition frequencies (Stark shifts) extracted from the spectra measured at the different pixels. (d) Corresponding distribution of FWHM. (e) Electric-field distribution derived from the observed Stark shifts. (f) Simulated electric-field distribution using ${\sigma }_{\mathrm{g}}=-23.6\left(1\right)\times {10}^{-6}\mathrm{C}/{\mathrm{m}}^2$ and ${\sigma }_{\mathrm{s}}=-2.10\left(5\right)\times {10}^{-6}\mathrm{C}/{\mathrm{m}}^2$; see text.

Figure 3

(a)–(c) Distribution of stray-electric-field strengths extracted from Stark shifts measured for the potential configurations defined in each panel. The dotted line in (c) indicates a region of well-compensated stray fields (see text). (d) Electric-stray-field vectors extracted from measurements in (a)–(c). (f) Predicted electric-field distribution as measured in (e); see text. Except for (d), all color scales are identical to the color scale given in (a).

Figure 4

(a) Coherent population transfer between the $34s$ and $34p$ states as a function of CPW drive amplitude $\mathrm{\Theta }$ for two spatially selected atom positions marked by the blue and red arrows in (b). The black lines are fits to a simple model; see text. (b) Microwave electric field distribution for ${\mathrm{\Theta }}_{\max }$ extracted from the measured distribution of ${\mathrm{\Omega }}_{\mathrm{eff}}(x,y)$ using Eq. (2). The CPW is located at the origin $(0,0)$.