Entangling distant atom clouds through Rydberg dressing

In Rydberg dressed ultracold gases, ground-state atoms inherit properties of a weakly admixed Rydberg state, such as sensitivity to long-range interactions. We show that through hyperfine-state-dependent interactions, a pair of atom clouds can evolve into a spin and subsequently into a spatial mesoscopic superposition state: The pair is in a coherent superposition of two configurations, with cloud locations separated by micrometers. The mesoscopic nature of the state can be proven with absorption imaging, while the coherence can be revealed though recombination and interference of the split wave packets.

Figure 1

Schematic of Rydberg dressed atom clouds. (a) All atoms are in either of two hyperfine ground states $|g\rangle$,$|h\rangle$. Dressing lasers can couple one atom per cloud to either of the $|s\rangle$,$|p\rangle$ Rydberg states. The Rydberg states participate in state changing dipole-dipole interactions $|sp\rangle \leftrightarrow |ps\rangle$. (b) Two atom clouds with width $\sigma$, separated by a distance $d$; $r_{\text{bl}}$ indicates the blockade radius. Due to hyperfine-state-dependent intercloud forces, a suitable initial state evolves dynamically into a nonclassical position-space superposition state, with the pair of clouds in either the full shaded (red) or striped (green) configuration.

Figure 2

(a) Born-Oppenheimer surfaces $U_k\left(\mathbf{R}\right)$ as a function of $d$ for the case $2N=8$, $\sigma =1\mu$m, $\alpha =0.15$, and $\mu =5760$ atomic units. Positions $\mathbf{R}$ have been chosen around ${\mathbf{R}}_{\mathbf{0}}$ in accordance with a single realization of a Gaussian distribution (width $\sigma$). The insets show the eigenstates belonging to the highest and lowest (colored) energies. Red, top inset: Coefficients $c_{\mathbf{k}\text{,rep}}=\langle \mathbf{k}|{\Psi }_{\text{rep}}\rangle$ of the most repulsive (highest-energy) state. Blue, bottom inset: $c_{\mathbf{k}\text{,att}}=\langle \mathbf{k}|{\Psi }_{\text{att}}\rangle$, most attractive (lowest energy). A plot of the modulus $|c_{\mathbf{k}\text{,att}}|$ coincides with $c_{\mathbf{k}\text{,rep}}$, since the states differ only by signs of coefficients (blue dashed, top inset). The transparent spheres visualize the $Q$ functions of $|{\Psi }_{\text{rep/att}}\rangle$ in a pseudospin picture, in which the states resemble coherent spin states [37]. (b)–(d) Creation of $|{\Psi }_{\text{rep}}\rangle$ for $d=11\mu$m, using a chirped microwave pulse as described in the text. (b) Time dependence of microwave Rabi frequency ${\Omega }_{\text{rf}}$ (black) and detuning ${\Delta }_{\text{rf}}/10$ (red). (c) Resulting energy spectrum of ${\widehat{H}}_{\text{ini}}$, Eq. (2), the gray (red) line, is the state to be adiabatically followed. (d) Energy gap between the two highest states of (c) (black), compared with analytical prediction Eq. (3) (red dashed).

Figure 3

Evolution of the initial hyperfine state $|{\Psi }_{\text{cat}}\rangle$ into a spatial superposition of cloud locations for $2N=4$ and trap frequency $\omega =\left(2\pi \right)\times 400$ Hz. We compare the total atomic density $n(x,t)$ from quantum-classical simulations (gray shading, black three-dimensional lines), to full quantum solutions (orange dashed). The lines overlayed on the gray shading are exemplary repulsive, marked $O$ (green), and attractive, marked $I$ (red), quantum-classical trajectories. Transparent spheres show the conversion of a spin cat state into a spatial cat state that would underlie this process for $N=20$, spin coordinate axes for spheres as in Fig. 2. Nonadiabatic population loss $P_{na}$ from $|{\Psi }_{\text{cat}}\rangle$ during the initial acceleration phase is shown magnified in the left inset (magenta). The back panel shows the interference signal found in the relative distance distribution $\rho \left(z\right)$ at $t={\tau }_{\text{trap}}/2$ (blue); note the different abscissa used.