Dissipative Preparation of Spatial Order in Rydberg-Dressed Bose-Einstein Condensates

We propose a technique for engineering momentum-dependent dissipation in Bose-Einstein condensates with nonlocal interactions. The scheme relies on the use of momentum-dependent dark states in close analogy to velocity-selective coherent population trapping. During the short-time dissipative dynamics, the system is driven into a particular finite-momentum phonon mode, which in real space corresponds to an ordered structure with nonlocal density-density correlations. Dissipation-induced ordering can be observed and studied in present-day experiments using cold atoms with dipole-dipole or off-resonant Rydberg interactions. Because of its dissipative nature, the ordering does not require artificial breaking of translational symmetry by an optical lattice or harmonic trap. This opens up a perspective of direct cooling of quantum gases into strongly interacting phases.

Figure 1

(a) Two metastable components of the ground electronic state, $|g⟩$ and $|s⟩$, are coupled to the electronically excited state, $|e⟩$, decaying at rate $\gamma$. State $|s⟩$ is coupled to a highly excited Rydberg level, $|R⟩$, using a far-detuned field, ${\mathrm{\Delta }}_R\gg {\mathrm{\Omega }}_R$. This results in the soft-core interaction between the atoms in the $|s⟩$ states [see Fig. 2], while states $|g⟩$ and $|e⟩$ are noninteracting. (b) Schematics of the setup: BEC of Rydberg-dressed atoms is confined in a three-dimensional optical dipole trap, fields ${\mathrm{\Omega }}_{g,s}$ are counterpropagating.

Figure 2

(a) Soft-core interaction potential, $V(r)$, between two Rydberg-dressed atoms separated by the distance $r$, and (b) its Fourier transform, $\tilde{V}(k)$. (c) The real part $E(k)$ (blue), and the imaginary part $\mathrm{\Gamma }(k)$ (red), of the dispersion relation $ϵ(k)$, Eq. (6), possessing a roton minimum, obtained using the self-consistent HFB theory. The black dotted line shows $ϵ(k)$ in the absence of dissipation. Parameters correspond to Fig. 3.

Figure 3

Time evolution of the pair correlation function, Eq. (8), from time $t=0$ (blue dotted lines) to $t=t_f$ (red solid lines), obtained within the self-consistent HFB approach. The initial BEC temperatures are (a) $T\approx 0.05T_c$, (b) $T\approx 0.4T_c$. The results correspond to the density $\rho =10^{13}{\mathrm{cm}}^{-3}=10R_c^{-3}$, the interaction constant $C_6=5\times 10^9\mathrm{a}.\mathrm{u}.$, and the Rabi frequency $\mathrm{\Omega }=\gamma /10$, with $\gamma =2\pi \times 5.2\mathrm{MHz}$ (the $6\hspace{0.17em}{}^2{S}_{1/2}\leftrightarrow 6\hspace{0.17em}{}^2{P}_{3/2}$ transition of a Cs atom). The mixing angles are $\theta =0.95$ (a) and $\theta =0.65$ (b).