Dynamical Creation of a Supersolid in Asymmetric Mixtures of Bosons

We propose a scheme to dynamically create a supersolid state in an optical lattice, using an attractive mixture of mass-imbalanced bosons. Starting from a “molecular” quantum crystal, supersolidity is induced dynamically as an out-of-equilibrium state. When neighboring molecular wave functions overlap, both bosonic species simultaneously exhibit quasicondensation and long-range solid order, which is stabilized by their mass imbalance. Supersolidity appears in a perfect one-dimensional crystal, without the requirement of doping. Our model can be realized in present experiments with bosonic mixtures that feature simple on-site interactions, clearing the path to the observation of supersolidity.

Figure 1

Dynamical onset of supersolidity by quantum quenching a mixture of light and heavy bosons. (a) A product state of bosonic trimers is the initial state of the evolution (larger symbols represent the $\downarrow$ bosons); switching off one of the superlattice components leads to a supersolid state in which the particles delocalize into a (quasi)condensate while maintaining the original solid pattern without imperfections. (b) Momentum profile of the $\downarrow$ bosons, $⟨n_k^{\downarrow }⟩$ versus time in units of hopping events $\hslash /J_{\downarrow }$. A quasicondensate peak develops rapidly. Inset: Density distribution $⟨n_i^{\downarrow }⟩$ averaged over the last third of the evolution time, showing that crystalline order is conserved in the system. (The simulation parameters are $L=28$, $N_{\downarrow }=18$, $N_{\uparrow }=9$, $J_{\downarrow }/J_{\uparrow }=0.1$, $U/J_{\uparrow }=3.0$.)

Figure 2

Phase diagrams in and out of equilibrium. (a) Equilibrium phase diagram (empty circles). The dash-dotted line represents the points where the hopping of the $\downarrow$ bosons, $J_{\downarrow }$, overcomes the energy gap to crystal dislocations, giving rise to the solid–super-Tonks ($s$-Tonks) transition. The dashed line marks the points where a single-trimer wave function spreads over 2.8 sites. (b) Out-of-equilibrium phase diagram. An extended supersolid phase exists in the transient state attained after the quantum quench. Blue symbols (solid lines) delimit the boundaries of the solid phase, red symbols (dashed lines) mark the lower boundary for the quasicondensed ($q\mathrm{\text{−}}c$) phase. The overlap of both phases (blue shaded region) is identified as the supersolid phase. The yellow (light) filled symbols correspond to equilibrium data points. The lower boundary of the superfluid–super-Tonks region of the equilibrium phase diagram is seen to coincide with the lower boundary of the supersolid region out of equilibrium.

Figure 3

Coexistence of solid order and quasicondensation in the supersolid phase. (a) The structure factor peak $S(q_{\mathrm{tr}}=2\pi /3)$ scales linearly with system size $L$, demonstrating solid order for both bosonic species. (b) The density peak in momentum space $⟨n_{k=0}^{\downarrow }⟩$ is plotted versus $L$ on a log-log scale, showing algebraic scaling and thus quasicondensation. Boxes (diamonds) stand for particle species $\downarrow$ ($\uparrow$), respectively. The data represented by dark (blue) boxes in (a) are offset by $-0.2$ for better visibility. Parameters: $J_{\downarrow }/J_{\uparrow }=0.1$, $U/J_{\uparrow }=3.0$ [dark (blue) symbols] and $J_{\downarrow }/J_{\uparrow }=0.15$, $U/J_{\uparrow }=2.5$ [light (red) symbols]. (c) Square modulus of the natural orbital ${\chi }_i^{(0)}$ corresponding to the largest eigenvalue of the OBDM, calculated at final time $\tau$. In the supersolid regime [dark (blue) symbols and light (red) symbols for $\downarrow$ and $\uparrow$ bosons], the natural orbital shows the characteristic crystalline order. This pattern is washed out in the purely quasicondensed regime (dashed curves and solid curves for $\downarrow$ and $\uparrow$). The supersolid data are offset by $+0.02$ for the sake of visibility. Parameters: $J_{\downarrow }/J_{\uparrow }=0.1$ (supersolid), $J_{\downarrow }/J_{\uparrow }=0.8$ (quasicondensed), $U/J_{\uparrow }=3.0$, $N_{\downarrow }=18$, $N_{\uparrow }=9$, $L=28$.

Figure 4

Overlap of the equilibrium ground state with the initial trimer-crystal state. The overlap $|c_0{|}^2$ (contour plot) agrees well with the boundaries of the nonequilibrium supersolid phase [black symbols, cf. Fig. 2b]. This suggests a superfluid ground state as a necessary condition for supersolidity to dynamically set in. The overlap $|c_0{|}^2$ has been calculated via exact diagonalization on a $L=10$ chain containing three trimers.