Creating a supersolid in one-dimensional Bose mixtures

We identify a one-dimensional supersolid phase in a binary mixture of near-hard-core bosons with weak, local interspecies repulsion. We find realistic conditions under which such a phase, defined here as the coexistence of quasisuperfluidity and quasi-charge-density-wave order, can be produced and observed in finite ultracold atom systems in a harmonic trap. Our analysis is based on Luttinger liquid theory supported by numerical calculations using the time-evolving block decimation method. Clear experimental signatures of these two orders can be found, respectively, in time-of-flight interference patterns and the structure factor $S\left(k\right)$ derived from density correlations.

Figure 1

Phase diagram of a Bose mixture, from LL theory, as a function of the interspecies interaction $U_{12}$ (in units of $v$, see text) and the LL parameter $K$ of the uncoupled system [Eq. (4)]; on the left the full diagram, on the right (corresponding to the shaded area) the near-hard-core, repulsive regime with the supersolid (SS) phase. For attractive interactions, in the paired regime, paired SF (PSF) is dominant, with CDW QLRO being subdominant in parts of it. Outside of that regime, SF is the dominant quasiorder, but for the repulsive, near-hard-core regime we have CDW QLRO as well, which constitutes a supersolid phase. For large repulsive interactions, the system phase separates (PS); for large attractive ones, it collapses (CL).

Figure 2

(Color online) The single-atom density of species 1 (red) and 2 (blue) of a system of 90 sites, with 17 atoms of each type, with $t∕U=0.005$, $U_{12}∕U=0.04$, and a trap parameter $\Omega ={10}^{-5}U$.

Figure 3

LL parameters $K_s$ and $K_a$, as a function of $U_{12}$, for the same parameters as in Fig. 2, from numerical fitting.

Figure 4

Interference pattern of the atomic mixture, with the parameters of Fig. 2, when interpreted as ${}^{87}\mathrm{Rb}$, released from an optical lattice with lattice constant $a=400\mathrm{nm}$, localized states with a length scale $d=60\mathrm{nm}$, after an expansion time $t=30\mathrm{ms}$. This could be realized with an optical lattice potential of $18E_R$ in the longitudinal and $30E_R$ in the transversal directions, $E_R$ being the recoil energy. In (a) we see the 1D density $n$ of the full pattern (in units ${\mathrm{mm}}^{-1}$), as a function of the spatial coordinate $x$ (in mm). In (b) we fit the central interference peak with a power law. The good numerical fit indicates the presence of single-particle quasisuperfluidity.

Figure 5

Structure factor of the atomic mixture, with the same parameters as in Fig. 4, in $1∕\mu \mathrm{m}$, as a function of $k$ in units of $2\pi ∕a$. (a) corresponds to very weak interactions $U_{12}∕U=0.0025$ and shows no particular structure beyond a SF signature; (b) corresponds to $U_{12}∕U=0.04$. We see additional peaks, which indicate CDW QLRO at a momentum that is consistent with the density at the center of the trap.