Tracking Rydberg atoms with Bose-Einstein condensates

We propose to track the position and velocity of mobile Rydberg excited impurity atoms through the elastic interactions of the Rydberg electron with a host condensate. Tracks first occur in the condensate phase, but are then naturally converted to features in the condensate density or momentum distribution. The condensate thus acts analogously to the cloud or bubble chambers in the early days of elementary particle physics. The technique will be useful for exploring Rydberg-Rydberg scattering, rare inelastic processes involving the Rydberg impurities, coherence in Rydberg motion, and forces exerted by the condensate on the impurities. Our simulations show that resolvable tracks can be generated within the immersed Rydberg lifetime and condensate heating is under control. Finally, we demonstrate the utility of this Rydberg tracking technique to study ionizing Rydberg collisions or angular momentum changing interactions with the condensate.

Figure 1

Sketch showing a one-dimensional cut through a Bose-Einstein condensate that is tracking two repelling Rydberg impurities (black $•$). (a) Condensate density $\rho$ before (dashed black line) and after (solid green line) Rydberg motion, with corresponding phase signal (oscillating red line) and sketched Rydberg electron wave functions (blue bottom near locations $x_1$ and $x_2$). (b) Interaction potential $V_{gR}\left(\mathbf{R}\right)$ (thin black line) between condensate atoms and a single Rydberg impurity at $x=0$ for principal quantum number $\nu =20$. The classical approximation $V_c$ (thick red line) averages over fast oscillations. We also employ the indicated cutoff $V_{max}$ (dotted blue line).

Figure 2

Sketch of the Rydberg orbital with a larger $z$ extension than the host BEC, justifying the replacement of the BEC-Rydberg interaction by a 2D cut at $z=0$ through the complete potential.

Figure 3

Tracking Rydberg atoms in a BEC through density or phase information. Five $\nu =80$ Rydberg excitations are in a ${}^{87}\mathrm{Rb}$ BEC of $N=1500$ atoms in a ${\omega }_r/2\pi =4$ Hz pancake trap, with transverse strength ${\omega }_z/2\pi =1$ kHz. (a) Phase imprinted by Rydberg atoms after $t={\tau }_{\text{imp}}=7\mu \mathrm{s}$. (b) The density signal at this time is weak. (c) and (d) Assuming impurities are removed at ${\tau }_{\text{imp}}$, the BEC evolution (1) converts the phase signal into clear density signals on the longer timescale ${\tau }_{\text{mov}}=0.54$ ms. See also the Supplemental Material [29].

Figure 4

(a) Condensate phase after motion of a single mobile Rydberg impurity in a homogeneous background using the real potential $V_{gR}\left(\mathbf{R}\right)$ [thin black line in Fig. 1]. (b) The difference compared to the phases caused by the classical approximation $V_c\left(\mathbf{R}\right)$ [red line in Fig. 1] is small (see also movie in the Supplemental Material [29]). (c) Modifying the potential cutoff only affects the less relevant central part of the track. We show the difference between phases resulting from $V_{max}=2$ and 6 MHz. (d) Number of uncondensed atoms $N_{\text{unc}}$, due to Rydberg impurity determined from the TWA for $V_{max}=2$, 4, and 6 MHz (from the bottom line to the top line). Dashed lines match the linear rate equation $d{N}_{\text{unc}}/dt=\mathrm{\Gamma }$, with $\mathrm{\Gamma }\approx 280300$ atoms $/\mu \mathrm{s}$. In contrast, for Fig. 3, heating remains negligible.

Figure 5

Incoherent response of host Bose gas to mobile Rydberg impurity. (a) Uncondensed atom number $N_{\text{unc}}\left(t\right)$, starting from a vacuum state for different $V_{max}$. Here $N_{\text{unc}}$ is increasing for larger $V_{max}$ (from the bottom line to the top line). The initial total number in the simulation box was $N=70000$ atoms. (b) $N_{\text{unc}}\left(t_0\right)$ at a fixed time $t_0=35$ $\mu \mathrm{s}$ as a function of $V_{max}$, showing saturation. (c) The heating actually decreased for increasing principal quantum number $\nu$ (from the top line to the bottom line). (d) Scaling of final heating rate $\mathrm{\Gamma }$ with $\nu$, deduced from linear fits as shown by dashed lines in (c).

Figure 6

Backaction of condensate background on dynamic Rydberg atoms. (a) Rydberg velocity as a function of time, with (dashed line) and without (solid line) condensate backaction. (b) Energies with backaction, showing conservation of the total energy $E_{\text{tot}}$ (solid green line). Its components are BEC energy (dotted red line), impurity energy (dashed magenta line), and BEC-Rydberg interaction energy $E_{\text{int}}$ (black $\diamond$'s). We also show $E_{\text{tot}}-E_{\text{int}}$ (blue $•$'s).

Figure 7

Condensate tracking of ultracold Rydberg processes (see also [29]). (a) Shown on top is the ionizing attractive collision of Rydberg atoms in states $|\nu =80,l=2,m=2\rangle$ for quantization along the $y$ axis. We assume instant ionization at $d_{\text{ion}}=5$ $\mu \mathrm{m}$ and included ${\sigma }_{\text{opt}}=0.5$ $\mu \mathrm{m}$ optical resolution (see Appendix pp4). Shown on the bottom are atoms in the same states but moving repulsively. (b) Angular momentum $l$-changing collision with a background atom from $l=0$ to $l^{\prime }=1$ at the white dashed line, clearly changing the character of the track.

Figure 8

Same as in Fig. 3 but using the cutoff $V_{max}=20$ MHz instead of 2 MHz.

Figure 9

Same as Fig. 3 but assuming an optical resolution ${\sigma }_{\text{opt}}=0.5\mu \text{m}$, as used for Fig. 7.

Figure 10

Same as Fig. 7 but without finite resolution effects.