Emergent Quasicrystalline Symmetry in Light-Induced Quantum Phase Transitions

The discovery of quasicrystals with crystallographically forbidden rotational symmetries has changed the notion of the ordering in materials, yet little is known about the dynamical emergence of such exotic forms of order. Here we theoretically study a nonequilibrium cavity-QED setup realizing a zero-temperature quantum phase transition from a homogeneous Bose-Einstein condensate to a quasicrystalline phase via collective superradiant light scattering. Across the superradiant phase transition, collective light scattering creates a dynamical, quasicrystalline optical potential for the atoms. Remarkably, the quasicrystalline potential is “emergent” as its eightfold rotational symmetry is not present in the Hamiltonian of the system, rather appears solely in the low-energy states. For sufficiently strong two-body contact interactions between atoms, a quasicrystalline order is stabilized in the system, while for weakly interacting atoms the condensate is localized in one or few of the deepest minima of the quasicrystalline potential.

Figure 1

Schematic view of a quasi-two-dimensional driven BEC inside four identical crossed linear cavities, which make a 45° angle with one another. The driving laser with the right circular polarization ${\widehat{\sigma }}_{+}$ propagates along the $z$ direction and it is not shown explicitly in the figure. The inset depicts the internal atom-photon couplings.

Figure 2

The phase diagram of the system in the $\{N{g}_0/\hslash {\omega }_r{\lambda }_0^2,\sqrt{N}{\eta }_0/{\omega }_r\}$ parameter plane. The system exhibits three distinct phases: normal homogenous, superradiant localized, and superradiant quasicrystal states. The inverse participation ratios $\{I_{\mathbf{r}},I_{\mathbf{p}}\}$ as well as the field amplitudes $|{\alpha }_j|$ display nonanalytical behavior on the onset of the superradiant phase transition as shown in (a)–(c). The order and the nature of the transitions are indicated in (c). A typical emergent superradiant quasicrystalline optical potential with the eightfold rotational symmetry $C_8$ located at the origin is presented in (d). The parameters are set to $({\mathrm{\Delta }}_c,\kappa ,N{\mathcal{G}}_0^2/{\mathrm{\Delta }}_a)=(-10,10,-1){\omega }_r$.

Figure 3

Typical atomic momentum distribution in the SRL (a) and SRQC (b) phases. The momentum distribution in the SRQC state exhibits a clear eightfold rotational symmetry, with occupied fractional momentum states. In the SRL phase, it is, however, a Gaussian with remnants of the eightfold rotational symmetry on top of it. The parameters are set to $\sqrt{N}{\eta }_0/{\omega }_r=4$ and $N{g}_0/\hslash {\omega }_r{\lambda }_0^2=0$ (a) and 5 (b), with the rest being the same as Fig. 2.