Interplay between thermal Rydberg gases and plasmas

We investigate the phenomenon of bistability in a thermal gas of cesium atoms excited to Rydberg states. We present both measurements and a numerical model of the phenomena based on collisions. By directly measuring the plasma frequency, we show that the origin of the bistable behavior lies in the creation of a plasma formed by ionized Rydberg atoms. Recombination of ions and electrons manifests as fluorescence which allows us to characterize the plasma properties and study the transient dynamics of the hysteresis that occurs. We determine scaling parameters for the point of plasma formation, and verify our numerical model by comparing measured and simulated spectra. These measurements yield a detailed microscopic picture of ionization and avalanche processes occurring in thermal Rydberg gases. From this set of measurements, we conclude that plasma formation is a fundamental ingredient in the optical bistability taking place in thermal Rydberg gases and imposes a limit on usable Rydberg densities for many applications.

Figure 1

Experimental setup. Two counterpropagation laser beams excite the Cs atoms in a vapor cell to a Rydberg state. The change in the radio-frequency transmission through the atomic sample is measured with the dipole antennas shown in the figure. The atomic fluorescence is captured with a fiber, bandpass-filtered, and detected on a PMT.

Figure 2

(a) Three-level diagram for optical excitation with ionization to an additional plasma state. The plasma causes a shift and broadening of the Rydberg level, and recombination results in fluorescence. (b) and (c) Illustration of the processes leading to plasma formation and bistability, as described in Sec. 3. (b) Rydberg-ground-state collisions ionize some atoms leading to production of fast electrons. (c) Avalanche ionization due to fast electrons and broadening due to steep electric field gradients can facilitate higher Rydberg population. (d) Fluorescence due to recombination in the plasma.

Figure 3

(a) Side view of the cell. The lasers propagate between the dotted lines; 455-nm laser from left to right, 1064-nm laser from right to left. (b)–(d) Fluorescence image for wavelengths between 500 and $800\mathrm{nm}$ with background subtracted. Scanning the Rydberg laser across resonance creates a lightsaber like fluorescing ray. When appearing, the glowing beam extends across the whole cell at once (b), and then gradually retracts from right to left as the detuning is scanned (c) and (d), before smoothly vanishing completely. The 455-nm laser intensity decreases from left to right due to absorption. At the transition between light and dark side, the conditions for plasma formation are not fulfilled any further along the beam. The defects on the left- and right-hand side are due to quartz sublimate on the surface of the cell. The position $z$ along the cell is indicated on the bottom axis.

Figure 4

Fluorescence vs radio-frequency (top) and radio-frequency transmission change (bottom) for the $30D_{5/2}$ Rydberg state for the same experimental conditions, with both lasers locked. Experimental parameters are $P_{\mathrm{RF}}=-15\mathrm{dBm},{\mathrm{\Omega }}_{\mathrm{B}}/2\pi =3.6\mathrm{MHz}$, and ${\mathcal{N}}_{\mathrm{g}}=3\times {10}^{12}{\mathrm{cm}}^{-3}$. The dashed lines are Gaussian fits.

Figure 5

Plasma resonance peaks in the fluorescence signal. Experimental parameters are as follows: radio-frequency power $P_{\mathrm{RF}}=-16\mathrm{dBm}$, density ${\mathcal{N}}_{\mathrm{g}}=3\times {10}^{12}{\mathrm{cm}}^{-3},{\mathrm{\Omega }}_{\mathrm{B}}/2\pi =3.6\mathrm{MHz}$. The $30D_{5/2}$ Rydberg state was used for the measurements.

Figure 6

Fluorescence signal collected from different spatial positions along the vapor cell, with increasing distance from the entrance window of the blue laser beam. (a) Scan from red to blue detuning (i.e., left to right). (b) Scan from blue to red detuning. All traces in (b) feature the same sharp edge at approximately $500\mathrm{MHz}$. The dashed line shows the mean of the fluorescence signal. The mean is similar to the integrated signal one would obtain by measuring the probe transmission, e.g., as in [21, 26]. Experimental parameters are ${\mathcal{N}}_{\mathrm{g}}=3\times {10}^{12}{\mathrm{cm}}^{-3},{\mathrm{\Omega }}_{\mathrm{B}}/2\pi =2.5\mathrm{MHz}$, and ${\mathrm{\Omega }}_{\mathrm{R}}/2\pi =18\mathrm{MHz}$. Data is shown for the $42D_{5/2}$ Rydberg state.

Figure 7

Map of the bistability frequency edge ${\mathrm{\Delta }}_{\mathrm{E}}$ as a function of the scaling parameters $S$ and $S^{★}$, according to Eq. (5). (a) Red to blue detuning. (b) Blue to red detuning. For comparison, the larger spread of the distributions with the respective raw and absorption adjusted Rabi frequencies are shown.

Figure 8

Comparison between fluorescence obtained as a function of 1064-nm laser detuning and calculations. Each row shows the variation of only one experimental parameter: ground-state atomic density (a), ${\mathrm{\Omega }}_{\mathrm{B}}$ (b), and ${\mathrm{\Omega }}_{\mathrm{R}}$ (c). The measurement is shown in orange dots (scan from red towards blue wavelengths) and purple diamonds (scan from blue towards red wavelengths). For the simulation we use orange solid and dashed purple lines, respectively. Experimental settings are used as input parameter for the simulation. Only the fluorescence signal amplitude is adjusted to the data. Experimental settings are (a) ${\mathrm{\Omega }}_{\mathrm{B}}/2\pi =2.6\mathrm{MHz},{\mathrm{\Omega }}_{\mathrm{R}}/2\pi =15.4\mathrm{MHz}$. (b) ${\mathrm{\Omega }}_{\mathrm{R}}/2\pi =15.4\mathrm{MHz},{\mathcal{N}}_{\mathrm{g}}=1.3\times {10}^{12}{\mathrm{cm}}^{-3}$. (c) ${\mathrm{\Omega }}_{\mathrm{B}}/2\pi =5.3\mathrm{MHz}$, ${\mathcal{N}}_{\mathrm{g}}=0.2\times {10}^{12}{\mathrm{cm}}^{-3}$.

Figure 9

Stark map for the $42{\mathrm{D}}_{5/2}$ state with all three $m_{\mathrm{J}}$ sublevels, and approximately $1\mathrm{GHz}$ below the $42{\mathrm{D}}_{3/2}$ state. The overlap to the $42{\mathrm{D}}_{5/2},m_{\mathrm{J}}=1/2$ is indicated by the color code for electric field in parallel to the laser polarization. To get an effective mapping from electric field strength to a certain Stark shift, we integrate over all angles for the electric field direction, and weight with the respective overlap to the state the lasers couple to. The dotted line indicates the mapping we used for the Stark shift in our model.

Figure 10

Flow chart of the algorithm. At each iteration, the current charge density determines an electric field distribution and hence the broadening and shift of the Rydberg energy level. The charge density also influences the ionization rate. With these values, the ensemble of steady-state solutions of the master equation is calculated. When averaged over all velocity and electric field components, an improved estimation of the charge density at each laser detuning is achieved. The procedure is repeated until the system converges to a steady-state solution.