Superfluid-Quasicrystal in a Bose-Einstein Condensate

A quasicrystal is a class of ordered structures defying conventional classification of solid crystals and may carry classically forbidden (e.g., fivefold) rotational symmetries. In view of long-sought supersolids, a natural question is whether a superfluid can spontaneously form quasicrystalline order that is not possessed by the underlying Hamiltonian, forming “superfluid-quasicrystals.” Here we show that a superfluid-quasicrystal stripe state with the minimal fivefold rotational symmetry can be realized as the ground state of a Bose-Einstein condensate within a practical experimental scheme. There exists a rich phase diagram consisting of various superfluid-quasicrystal, supersolid, and plane-wave phases. Our scheme can be generalized for generating other higher-order (e.g., sevenfold) quasicrystal states, and provides a platform for investigating such new exotic quantum matter.

Figure 1

(a) Proposed experimental scheme for generating superfluid-quasicrystals with fivefold rotational symmetry. The superlattice is generated by two optical lattices with different periods. A potential gradient can be generated using a magnetic field gradient. One Raman beam $R_z$ in the $z$ direction and four others ($R_j$, $j=1$, 2, 3, 4) in the $x\text{−}y$ plane generate the Raman coupling between neighboring wells. (b) SOC in one unit cell. Each Raman process only couples well $j$ to its adjacent neighbor $j+1$. The hopping between neighboring unit cells is irrelevant because of the large bias between $|1⟩$ and $|5⟩$. Only Raman-assisted interwell tunneling in a unit cell is considered.

Figure 2

Single particle lowest band dispersion. (a) $\mathrm{\Omega }=0$. The minima form a regular pentagon structure in the momentum space. (b) $\mathrm{\Omega }=0.4$. The minima are strongly coupled and all spin components are mixed. Wells 1 and 5 are uncoupled and a barrier between them can be clearly observed. The five minima are axisymmetric to ${\widehat{p}}_3$ for any $\mathrm{\Omega }$.

Figure 3

(a) Phase diagram from the variational ansatz analysis when all the detunings ${\delta }_j$ are set to be zero. The color represents the value of $|C_3{|}^2$. The white region is the plane-wave phase. The three symbols (cross, circle, and square stand for $\mathrm{\Omega }=0.06$, 0.77, and 0.95) along $\overline{n}{c}_0=0.1$ are examples for superfluid-quasicrystal, supersolid, and plane-wave phases. (b) Phase transitions between different phases. The blue circles (green rhombus) and orange squares (red triangles) show how $|C_1{|}^2+|C_5{|}^2$ ($|C_2{|}^2+|C_4{|}^2$) varies with Raman coupling for $\overline{n}{c}_0=0.01$ and 0.2, respectively.

Figure 4

(a)[(b)] Real-space (momentum-space) distribution for the superfluid-quasicrystal phase from GPE simulation. The parameters are the same as the black cross marked in Fig. 3. The five minima in momentum space are evenly populated. (c) Spatial distribution for a supersolid phase [parameters chosen as black circle in Fig. 3] with three minima populated. The top and bottom insets show the real-space density of state $|3⟩$ and the momentum-space distribution, respectively. (d) Phase distribution for state $|3⟩$ at a nonzero momentum plane-wave phase, and the insets give real (top) and momentum (bottom) space distributions. The parameters correspond to the black square in Fig. 3. All real-space density distributions (including following panels) are obtained through subtracting the real-space density at zero Raman coupling from those of finite $\mathrm{\Omega }$ to rule out the large density variation across the harmonic trap. The plots are rescaled over the average density in the trap. The total momentum-space distribution is the direct summation of that for each pseudospin component.

Figure 5

(a)[(b)] Real-space (momentum-space) distribution for a supersolid phase when one minimum is knocked off with a small detuning ${\delta }_3\sim 0.01E_R$ in the superfluid-quasicrystal phase in Figs. 4 and 4.