Engineering interaction potentials for stabilizing quantum quasicrystal phases

TL;DR

This work identifies how engineered multi-length-scale pair interactions stabilize quantum quasicrystals in two-dimensional bosonic systems, revealing that octagonal, decagonal, and dodecagonal patterns require 4, 3, and 2 characteristic momentum scales, respectively. By combining a mean-field spectral variational approach with nonlocal Gross-Pitaevskii simulations, the authors map ground-state phase diagrams for 8-, 10-, and 12-fold QC orders, and show that dodecagonal patterns benefit most from hexagonal-symmetric energy contractions, allowing robust QC formation even at moderate quantum fluctuations. They demonstrate that these quasicrystal phases can coexist with finite superfluid fractions, forming superquasicrystal states, and show that adding extra minima in the Fourier-space potential can dramatically extend QC stability ranges, with strong implications for experimental realization using laser-painted interactions in cavity QED. The results provide design principles for on-demand quantum quasicrystals and highlight cavity-mediated interactions as a versatile platform for exploring modulated quantum phases and supersolidity in strongly controllable settings.

Abstract

We investigate the necessary features of the pair interaction for the stabilization of self-assembled quantum quasicrystals in two-dimensional bosonic systems. Unlike the classical scenario, our results show that two-dimensional octagonal, decagonal, and dodecagonal aperiodic phases require a distinct number of properly tuned characteristic length scales for their stabilization. By using a mean field spectral variational approach and Gross-Pitaevskii numerical calculations, we determine that the dodecagonal quasicrystal structure requires at least two characteristic length scales for its stabilization, while the decagonal and octagonal patterns need at least three and four length scales, respectively. The family of pair interaction potentials considered, albeit simple, is well justified in terms of a novel experimental platform based on laser-painted interactions in a cavity QED setup. Finally, we perform a structural characterization of the quasicrystal patterns obtained and show that these phases coexist with a finite superfluid fraction, forming what can be called a super quasicrystal phase.

Figures (8)

• Figure 1: (a)-(c) Pair interaction potential minima structure in Fourier space for the corresponding phase diagrams in (d)-(f), where the first length scale is positioned at $k=\tilde{k}_{1}=1$ and the secondary characteristic scales are presented in Tab. \ref{['table:secondary_wave_vectors']}. Phase diagram for the minima structure associated with the dodecagonal quasicrystal where (d) $\bar{k}=\tilde{k}_{2}$ (e) $\bar{k}=\tilde{k}_{3}$ and (f) $\bar{k}=\tilde{k}_{4}$ indicates the position of the minimum in reciprocal space for which we modify the depth as a parameter to determine the ground state of the system. The acronym HSF stands for homogeneous superfluid, whereas HS and $12$-QC stands for hexagonal solid and dodecagonal quasicrystal phases, respectively. The hexagonal states HS$_{i}$ refers to a hexagonal phase with characteristic wave vectors dominated by the minimum of $\hat{v}(k)$ at $\tilde{k}_{i}$.
• Figure 2: (a) Real space mapping for the dodecagonal structure in the phase diagram of Fig. \ref{['fig:dodecagonal_grid']}(d) for parameter values $\hat{v}(\tilde{k}_{2})=-2.30$ and $\gamma=0.50$ and its corresponding (b) diffraction pattern. The red circles indicates the two minima and the size of the Fourier modes in blue disks is scaled by a linear function, where the zero harmonic mode has been omitted. (c) Most relevant energy contributions to the total energy of the quasicrystalline structures as a contraction in the $\tilde{k}_{1}=1$ layer for the dodecagonal vectors. (d) Superfluid fraction for a fixed intensity of the pair interaction, $\gamma=0.50$, as a function of $\hat{v}(\tilde{k}_2)$ for the ground state phase diagram in Fig. \ref{['fig:dodecagonal_grid']}(d). The blue points represents the superfluid fraction obtained via mean field spectral variational method while the orange diamonds are determined via the solution of the GPE.
• Figure 3: (a)-(c) Pair interaction potential minima structure in Fourier space for the corresponding phase diagrams in (d)-(f), in which the first length scale is positioned at $k=\tilde{k}_{1}=1$ and the secondary characteristic wave vectors are presented in Tab. \ref{['table:secondary_wave_vectors']}. Phase diagram for the minima structure associated with the decagonal quasicrystal where (d) $\bar{k}=\tilde{k}_{2}$ (e) $\bar{k}=\tilde{k}_{3}$ and (f) $\bar{k}=\tilde{k}_{4}$ indicates the position of the minimum in reciprocal space for which we modify the depth as a parameter to determine the ground state of the system. The notation employed to label the phases is analogous to the one established in Fig.\ref{['fig:dodecagonal_grid']}.
• Figure 4: (a) Real space mapping for the decagonal structure in the phase diagram of Fig. \ref{['fig:decagonal_grid']}(f) for parameter values $\hat{v}(\tilde{k}_{4})=-1.50$ and $\gamma=0.50$ and its corresponding (b) diffraction pattern. The red circles indicates the two minima and the size of the Fourier modes in blue disks is scaled by a linear function, where the zero harmonic mode has been omitted. (c) Most relevant energy contributions to the total energy of the quasicrystalline structures as a contraction in the $\tilde{k}_{1}=1$ layer for the decagonal vectors. (d) Superfluid fraction for a fixed intensity of the pair interaction, $\gamma=0.50$, for the the ground state phase diagram depicted in Figure \ref{['fig:decagonal_grid']}(f). The blue points represents the superfluid fraction obtained via mean field spectral variational method while the orange diamonds are determined via GPE simulations.
• Figure 5: (a)-(c) Pair interaction potential minima structure in Fourier space for the corresponding phase diagrams in (d)-(f), where the first length scale is positioned at $k=\tilde{k}_{1}=1$ and the secondary characteristic scales are presented in Tab. \ref{['table:secondary_wave_vectors']}. Phase diagram for the minima structure associated with the octagonal quasicrystal where (d) $\bar{k}=\tilde{k}_{2}$ (e) $\bar{k}=\tilde{k}_{3}$ and (f) $\bar{k}=\tilde{k}_{4}$ indicates the position of the minimum in reciprocal space for which we modify the depth as a parameter to determine the ground state of the system. The notation employed to label the phases is analogous to the one established in Fig.\ref{['fig:dodecagonal_grid']}.