Phase-Imprinting of Bose-Einstein Condensates with Rydberg Impurities

We show how the phase profile of Bose-Einstein condensates can be engineered through its interaction with localized Rydberg excitations. The interaction is made controllable and long range by off-resonantly coupling the condensate to another Rydberg state with laser light. Our technique allows the mapping of entanglement generated in systems of few strongly interacting Rydberg atoms onto much larger atom clouds in hybrid setups. As an example we discuss the creation of a spatial mesoscopic superposition state from a bright soliton. Additionally, the phase imprinted onto the condensate using the Rydberg excitations is a diagnostic tool for the latter. For example, a condensate time-of-flight image would permit reconstructing the pattern of an embedded Rydberg crystal.

Figure 1

(a) The setup. Several Rydberg excited impurities (the red balls) are embedded in a Bose-Einstein condensate. The condensed atoms (the green balls) mainly interact with impurities through dressing induced long-range interactions of characteristic range $r_c$ (the red dashed circles). At distances $d$ within the Rydberg orbit, $d<r_0$, collisions with Rydberg electrons are also relevant (the blue dashed circles). After an imprinting period, the condensate phase (blue shades) will be modified only in the vicinity of impurities (dark blue). (b) Level scheme for three representative atoms. Interactions $V_{rr}$ give rise to long-range interactions between dressed condensate atoms, as in Refs. [6, 7]. Interactions $V_{rR}$ and $V_{gR}$ give rise to stronger potentials between dressed condensate atoms and impurities, which are the main focus here. (c) Sketch of these potentials $U_{\mathrm{eff}}^{(2)}$ (red, see text) and $V_{gR}$ (blue) near one of the impurities. $\alpha =\mathrm{\Omega }/(2\mathrm{\Delta })$ quantifies the degree of the Rydberg admixture.

Figure 2

Entanglement transfer from two control Rydberg atoms onto a mesoscopic BEC cloud. The position space density of control atoms is shown as shaded curves: blue for state $|R⟩$ and gray for state $|g⟩$. The condensate density for a soliton is shown as a thick black line. (Blue) Dressing potential $|U_{\mathrm{eff}}^{(2)}|$. (Red) Condensate phase $\phi$ after imprinting. We show all quantities with an arbitrary normalization to fit the same axis. (a),(b) Initial state at $t={\tau }_{\mathrm{imp}}$, (c),(d) final state at $t={\tau }_{\mathrm{imp}}+{\tau }_{\mathrm{mov}}$. (a),(c) show the control atom configuration $|Rg⟩$, (b),(d) show $|gR⟩$. For each configuration, we model condensate evolution separately using Eq. (2) [55]. The dotted line in (c),(d) is for shifted control atom positions $x_i+2{\sigma }_{\mathrm{con}}$.

Figure 3

Determining the spatial arrangement of Rydberg impurities (shown as blue crosses) via phase-imprinting in a 2D ${}^{87}\mathrm{Rb}$ BEC of $N=100$ atoms in a trap with ${\omega }_r=(2\pi )2\mathrm{Hz}$ and ${\omega }_z=(2\pi )100\mathrm{Hz}$. Color bar for (a),(c),(d) [see (d)], where $n_0$ is the respective peak density, color bar for (b),(e) [see (e)]. (a) Initial condensate density $\rho =|\varphi (\mathbf{R}){|}^2$. (b) Condensate phase $\phi$ following phase-imprinting ($t={\tau }_{\mathrm{imp}}$). (c) Condensate density shortly after imprinting ($t={\tau }_{\mathrm{imp}}+12\mathrm{ms}$). (d) Final momentum spectrum $|\tilde{\varphi }(k)|$ ($t=60\mathrm{ms}$). (e) Phase profile reconstructed from (d) as described in the text.