Phase-dependent ionization of Rydberg atoms in static fields

Using a laser intensity modulated synchronously with a 15.9 GHz microwave field we have examined laser excitation of Li atoms to the vicinity of the ionization limit in the presence of the microwave field and a parallel static field. The static field breaks the symmetry of ionization in the two microwave half cycles, and as the intensity modulated laser beam is delayed relative to the microwave field, maxima in the ionization are observed every microwave field cycle, not every half cycle, as in the absence of the static field. When the laser is tuned below the ionization limit there is an unexpected phase reversal of the modulation in the ionization, the origin of which is clarified by a classical model.

Figure 1

Photoelectrons produced by the 819-nm laser in the combined Coulomb and static fields. Both the laser and the static field are polarized in the $z$ direction. Photoelectrons leave the ion core moving both to the left and to the right, which we refer to as “uphill” or “downhill” electrons, respectively. The inset shows the $z$ polarized microwave field experienced by the electrons and the direction of the static field, both oriented along the $\widehat{z}$ axis.

Figure 2

Energy-level diagram for exciting a thermal beam of Li atoms from their ground state to high-lying Rydberg states or the continuum via the route $2s\stackrel{\text{671}\text{nm}}{\to }2p\stackrel{\text{610}\text{nm}}{\to }3d\stackrel{\text{819}\text{nm}}{\to }nf,\epsilon f$. The $2s\to 2p\to 3d$ excitation is driven by two simultaneous 20-ns dye laser pulses with 10-GHz linewidths. The final $3d\to nf,\epsilon f$ excitation occurs after the dye laser pulses, and is driven by a 20-ns pulse from an amplified, intensity modulated 819-nm laser. The 819- and 670-nm beams intersect the Li beam perpendicular to it, while the 610-nm beam propagates counter to the Li beam, creating a 1-${\mathrm{mm}}^3$ interaction region.

Figure 3

Temporal view of the microwave field (top) phase locked with the amplitude modulated laser intensity (bottom). Inside the interaction region, the peak laser intensity occurs at the phase $\omega t_0$ of the microwave field and can be adjusted by an optical delay line.

Figure 4

Schematic showing how the intensity modulated 819-nm laser is produced and locked to the microwave frequency, as well as how both the laser and microwave fields are delivered to the interaction region. The intensity modulated IR laser is generated by overlapping the DL-Pro and DL-100 laser beams on a beamsplitter (BS). One output is amplified and sent to the interaction region. The second is mixed with half the output of the microwave synthesizer to lock the intensity modulation to the microwaves using the heterodyne optical phase-locked loop (HOPLL). The other half of the synthesizer's output is formed into pulses by the PIN switch and amplified by the traveling wave tube amplifier (TWTA) before injection into the microwave cavity.

Figure 5

Rydberg state signal as the phase delay is scanned for various values of $E_s$. The central laser frequency is tuned to 2 GHz above the DIL. The vertical scale is normalized such that 1 is the total number of electrons excited by the 819-nm laser. When no static field is applied, there is no observed phase modulation. As the static field in increased, the phase dependence in the signal grows and then lessens as the total mean signal decreases. The experimental data are shown in grey, while the solid sinusoidal curves trace the best fit to Eq. (4). The traces at $+7.2$ (solid) and $-7.2$ (dashed) mV/cm show that reversing the static field inverts the amplitude of the phase modulation.

Figure 6

Rydberg signal obtained vs phase delay for laser tuning 14 GHz below the DIL in three different 14-mV/cm static fields ($E_s$) oriented parallel and perpendicular to the $z$ polarized microwave field. (a) $E_s$ in the $+\widehat{z}$ (vertical) direction, (b) $E_s$ in the $+\widehat{x}$ (horizontal) direction, and (c) $E_s$ in the $-\widehat{z}$ (vertical) direction. There is no dependence on the delay with the static field perpendicular to the microwave field, as expected from the symmetry of the problem.

Figure 7

(a) Peak to peak modulation $2S_m$, (b) mean signal $S_0$, and (c) phase ${\varphi }_S$ vs static field $E_s$ observed with the laser tuned 2 GHz above the DIL. These parameters are extracted by fitting delay scans such as those of Fig. 5 to Eq. (4) with the constraint $0\le {\varphi }_S\le \pi$. At zero static field, $E_s=0$, the mean signal is at a maximum, and there is no modulation. We observe that modulation emerges as the magnitude of $E_s$ is increased from zero, reaching a peak at $E_s=\pm 14$ mV/cm. For $|E_s|>14$ mV/cm the mean signal and modulation both decay toward zero. The phase is roughly identical for data collected at equal and opposite $E_s$, and there is a positive phase shift as the magnitude of $E_s$ is increased.

Figure 8

(a) Peak to peak modulation $2S_m$, (b) mean signal $S_0$, and (c) phase ${\varphi }_S$ vs static field $E_s$ observed with the laser tuned 14 GHz below the DIL. These parameters are extracted by fitting delay scans such as those of Fig. 5 to Eq. (4) with the constraint [phi 0] $0\le {\varphi }_S\le \pi$. At zero static field, we again observe the maximum mean signal, and there is no modulation. For small static fields, $|E_s|<27$ mV/cm, the modulation has the opposite sign to that shown in Fig. 7 and reaches its first peak at $E_s=\pm 14$ mV/cm. As the magnitude of the static field is increased modulation decreases and crosses zero at $E_s=\pm 27$ mV/cm. For $|E_s|>27\mathrm{mV}$/cm the modulation has the same sign as in Fig. 7, reaching its maximum at $E_s=\pm 60$ mV/cm, and decreasing for higher magnitudes of the static field. A slight positive shift of the phase is observed as $|E_s|$ is increased.

Figure 9

(a) Peak to peak modulation $2S_m$, (b) mean signal $S_0$, and (c) phase ${\varphi }_S$ vs static field $E_s$ observed with the laser tuned 30 GHz below the DIL. These parameters are extracted by fitting delay scans such as those of Fig. 5 to Eq. (4) with the constraint $0\le {\varphi }_S\le \pi$. As in Figs. 7 and 8, there is a maximum in the mean signal at $E_s=0$. Unlike Figs. 7 and 8, modulation is not immediately observed for $E_s\ne 0$. Modulation is not observed until $E_s=\pm 30$ mV/cm, at which point it has the same sign as in Fig. 8. It reaches its first maximum at $E_s=50$ mV/cm, crosses zero at $E_s=\pm 65$ mV/cm, reaches its second maximum at $E_s=\pm 144$ mV/cm, and decays to zero as $|E_s|$ is further increased. The phase appears to be constant.

Figure 10

Schematic showing the beginning of (A) “uphill” and (B) “downhill” electron orbits in the two-dimensional simulation, with the ion core at $(x,z)=(0,0)$. Given initial radius and velocity $r_0$ and $v_0$, downhill electrons are launched from $(0,-r_0)$, with velocity in the $-\widehat{x}$ direction. Uphill electrons are launched from $(0,+r_0)$ with velocity along $+\widehat{x}$. Both orbits shown have angular momentum aligned in the $+\widehat{y}$ direction. In the absence of a static field, bound electrons make highly elliptical orbits with the semimajor axis along the $z$ axis. The field $E_s$ applies a force in the $+\widehat{z}$ direction on electrons during their orbits.

Figure 11

Calculated observed signal from our two-dimensional model, with initial energy $W_0=0$ GHz, and $E_s=0$ (a), 36 (b), and 100 mV/cm (c). The contributions from downhill and uphill electrons are in orange (dotted) and blue (dashed), respectively, with the total expected signal in green (solid). In (a) with $E_s=0$ mV/cm, uphill and downhill signals have opposite phase dependence, resulting in a flat total signal. As the static field $E_s$ increases in (b) and (c), ionization of downhill electrons depresses their mean signal and modulation, and the total signal becomes dominated by contributions from uphill electrons.

Figure 12

Calculated observed signal from our two-dimensional model, with initial energy $W_0=-20$ GHz, and $E_s=0$ (a), 7.2 (b), and 100 mV/cm (c). Downhill and uphill signals are in orange (dotted) and blue (dashed), respectively, with total signal in green (solid). As in Fig. 11, in (a) with $E_s=0$ mV/cm, combining the equal and opposite uphill and downhill signals results in a flat total signal. In (b) with $E_s=7.2$ mV/cm, the downhill signal is diminished, but its phase dependence increases. The total signal shows a phase dependence shifted by $\pi$ from the $E_s=36$ mV/cm signal for $W_0=0$ GHz in Fig. 11. In (c) with $E_s=100$ mV/cm, downhill electrons almost all ionize, and the uphill signal dominates the total signal.