Line shapes and time dynamics of the Förster resonances between two Rydberg atoms in a time-varying electric field

The observation of the Stark-tuned Förster resonances between Rydberg atoms excited by narrowband cw laser radiation requires usage of a Stark-switching technique in order to excite the atoms first in a fixed electric field and then to induce the interactions in a varied electric field, which is scanned across the Förster resonance. In our experiments with a few cold Rb Rydberg atoms, we have found that the transients at the edges of the electric pulses strongly affect the line shapes of the Förster resonances, since the population transfer at the resonances occurs on a time scale of $\sim 100$ ns, which is comparable with the duration of the transients. For example, a short-term ringing at a certain frequency causes additional radio-frequency-assisted Förster resonances, while nonsharp edges lead to asymmetry. The intentional application of the radio-frequency field induces transitions between collective states, whose line shape depends on the interaction strengths and time. Spatial averaging over the atom positions in a single interaction volume yields a cusped line shape of the Förster resonance. We present a detailed experimental and theoretical analysis of the line shape and time dynamics of the Stark-tuned Förster resonances $\mathrm{Rb}\left(n{P}_{3/2}\right)+\mathrm{Rb}\left(n{P}_{3/2}\right)\to \mathrm{Rb}\left(n{S}_{1/2}\right)+\mathrm{Rb}\left([n+1]S_{1/2}\right)$ for two Rb Rydberg atoms interacting in a time-varying electric field.

Figure 1

(a) Förster resonance $\mathrm{Rb}\left(nP\right)+\mathrm{Rb}\left(nP\right)\to \mathrm{Rb}\left(nS\right)+\mathrm{Rb}([n+1]S)$ for two Rb Rydberg atoms. Dipole-dipole interaction induces transitions from the initial $n{P}_{3/2}\left(\right|M_J|=1/2)$ state to the final $n{S}_{1/2}\left(\right|M_J|=1/2)$ and $(n+1)S_{1/2}\left(\right|M_J|=1/2)$ states in both atoms. (b) Timing diagram of the laser and electric-field pulses. (c) Collective states of the Förster resonance $\mathrm{Rb}\left(37P\right)+\mathrm{Rb}\left(37P\right)\to \mathrm{Rb}\left(37S\right)+\mathrm{Rb}\left(38S\right)$ in the dc electric field. Intersection at 1.79 V/cm corresponds to a Stark-tuned Förster resonance. (d) Collective states of the Förster resonance $\mathrm{Rb}\left(39P\right)+\mathrm{Rb}\left(39P\right)\to \mathrm{Rb}\left(39S\right)+\mathrm{Rb}\left(40S\right)$ in the dc electric field. A dc Stark-tuned Förster resonance is impossible, but it can be induced by adding a $\sim 100$ MHz radio-frequency field (green arrow) that binds the collective states [10].

Figure 2

(a) Three-photon laser excitation of $\mathrm{Rb}\left(nP\right)$ Rydberg atoms. The first step $5S\to 5P$ is detuned by ${\delta }_1=+92\mathrm{MHz}$, while the second step $6S\to nP$ is tuned on-resonance ${\delta }_2=0$. The solid arrows indicate coherent three-photon excitation and the dashed arrows indicate incoherent three-step excitation, as discussed in Ref. [9]. (b) Spectrum of the three-photon laser excitation of the $37P_{3/2}$ state in a zero electric field. The Rabi frequencies of the intermediate transitions are ${\mathrm{\Omega }}_1=10$, ${\mathrm{\Omega }}_2=26$, and ${\mathrm{\Omega }}_3=2\mathrm{MHz}$. The blue (thin) line is a theoretical calculation in a four-level model taking into account the laser linewidths according to Ref. [9]. The small peak at ${\delta }_3=0$ is a reference due to the added cooling laser radiation on the first step. (c) Stark effect in the laser excitation spectrum of the $37P_{3/2}$ state in a weak dc electric field.

Figure 3

Line shapes of the Förster resonance $\mathrm{Rb}\left(37P\right)+\mathrm{Rb}\left(37P\right)\to \mathrm{Rb}\left(37S\right)+\mathrm{Rb}\left(38S\right)$ recorded for $N=2–5$ detected Rydberg atoms at various time dependences of the controlling electric-field pulse shown on the right-hand panels (the dashed line indicates the resonant electric field). (a) The pulse has a short-term ringing at 20 MHz on the leading edge. (b) The ringing is smoothed using a capacitor. (c) The pulse has a nearly square shape with short edges. (d) A radio-frequency (rf) pulse is intentionally added to induce additional rf-assisted Förster resonances.

Figure 4

(a) Rydberg states related to the Förster resonance $\mathrm{Rb}\left(nP\right)+\mathrm{Rb}\left(nP\right)\to \mathrm{Rb}\left(nS\right)+\mathrm{Rb}\left[(n+1)S\right]$. (b) Collective states of two Rydberg atoms interacting via Förster resonance with dipole-dipole matrix element $V$. The energy defect of the Förster resonance $\mathrm{\Delta }$ is controlled by the electric field. (c) Effective two-level system can replace the three-level system in the theoretical calculations, as collective states |$13\rangle$ and |$31\rangle$ are equivalent and behave identically. The reduced dipole-dipole matrix element is now $\sqrt{2}V$.

Figure 5

(a)–(h) Comparison between experimental and theoretical line shapes of the Förster resonance $\mathrm{Rb}\left(37P\right)+\mathrm{Rb}\left(37P\right)\to \mathrm{Rb}\left(37S\right)+\mathrm{Rb}\left(38S\right)$ for two Rydberg atoms at various interaction times. (i) Comparison between experiment and theory for the resonance amplitude at various interaction times. (k) Comparison between experiment and theory for the resonance width at various interaction times.

Figure 6

(a) Energy levels of the initial $37P+37P$ and final $37S+38S$ collective states of two interacting Rb Rydberg atoms in an electric field. The arrows indicate rf-induced Förster resonances $\mathrm{Rb}\left(37P_{3/2}\right)+\mathrm{Rb}\left(37P_{3/2}\right)\to \mathrm{Rb}\left(37S\right)+\mathrm{Rb}\left(38S\right)$ of various orders at rf frequency 15 MHz. (b) The same for the rf-induced Förster resonances $\mathrm{Rb}\left(39P_{3/2}\right)+\mathrm{Rb}\left(39P_{3/2}\right)\to \mathrm{Rb}\left(39S\right)+\mathrm{Rb}\left(40S\right)$ at rf frequency 90 MHz. (c) Energy levels of the $37P+37P$ and $37S+38S$ collective states of two Rydberg atoms in an electric field in the presence of the first Floquet sidebands at $\pm 15$ MHz. The red (gray) circles indicate the intersections of the Floquet sidebands corresponding to rf-assisted Förster resonances. (d) The same for the $39P+39P$ and $39S+40S$ collective states at rf frequency 90 MHz. (e) Experimental record of the rf-assisted Förster resonances on the $37P$ state for $N=2–5$ detected Rydberg atoms at a 15 MHz rf frequency and 200 mV amplitude. (f) The same for the “inaccessible” Förster resonances on the $39P$ state at 90 MHz and 100 mV.

Figure 7

Comparison between experimental (left-hand column) and theoretical (right-hand column) line shapes of the rf-induced Förster resonances $\mathrm{Rb}\left(37P\right)+\mathrm{Rb}\left(37P\right)\to \mathrm{Rb}\left(37S\right)+\mathrm{Rb}\left(38S\right)$ for two Rydberg atoms at various amplitudes of the 15 MHz rf field and interaction time $t=2.975\mu \mathrm{s}$.

Figure 8

(a)–(d) Comparison between experimental and theoretical line shapes of the inaccessible Förster resonance $\mathrm{Rb}\left(39P\right)+\mathrm{Rb}\left(39P\right)\to \mathrm{Rb}\left(39S\right)+\mathrm{Rb}\left(40S\right)$ induced by the 95 MHz and 100 mV rf field for two Rydberg atoms at various interaction times. (e) Comparison between experiment and theory for the resonance amplitude at various interaction times. (f) Comparison between experiment and theory for the resonance width at various interaction times.