Emergent flat bands and nonlinear phenomena of Bose−Einstein condensates in two-dimensional trilayer moiré optical lattices

Abstract

Moiré superlattices, classified as a supercell periodic structure configured from a periodic lattice overlaps with its twisted counterpart, have been demonstrated to exhibit many emergent linear and nonlinear phenomena like moiré induced flat bands, unconventional superconductivity, unique linear classical wave localization and nonlinear localized modes. Recent studies are, however, mainly focused on twisted bilayer structures, the static and nonequilibrium physics of trilayer moiré superlattices have remained largely unknown. We here consider trilayer moiré optical lattices by which Bose−Einstein condensates are trapped, and demonstrate theoretically the emergent flat bands and nonlinear phenomena — localized gap modes in form of gap solitons and vortices with a topological charge s=1. We give a unified picture for constructing optimal (largest) photonic forbidden gaps in such trilayer moiré superlattices and reveal nonlinear localization mechanism therein. Computational studies demonstrated the robustness of these localized modes, enabling insightful inspections of moiré physics and exploring ongoing moiré photonics applications in twisted trilayer superlattices in optics and ultracold atoms.

Graphical abstract

Keywords

moiréoptical lattices / emergent flat bands and nonlinear phenomena / gap solitons and vortices / twisted trilayer superlattices

1 Introduction

Emergent nonlinear phenomena are usually referred to the spontaneous formation (self-organization) of diverse spatial, temporal, or spatiotemporal nonlinear-wave patterns in complex nonlinear dynamical systems [1]. They are omnipresent in nature [1-4] by appearing, for example, as spatial, temporal, and spatiotemporal optical solitons in nonlinear optics, and as dark and bright matter-wave solitons, gap solitons and vortices in ultracold atomic media context [5-13]. Solitons, by definition, are solitary waves occurred in nonlinear systems if an elegant balance is reached between spatial diffraction/temporal dispersion and nonlinearity. Particularly, Bose−Einstein condensates (BECs) are clear, controllable, easy-to-realize nonlinear physical medium with intrinsic self-defocusing or focusing nonlinearity induced by (repulsive or attractive) atom−atom interactions, providing a practical, effective and powerful platform to simulate and study diverse nonlinear phenomena and quantum matter-wave dynamics with or without linear potentials including optical lattices [1, 14, 15].

During the past few years, moiré superlattices, classified as a supercell periodic structure constructed from two overlayed periodic lattices with a relatively twist, have garnered a keen research interest of scientists from different branches of sciences in the progress of a plethora of emergent linear, nonlinear, and quantum phenomena [16-24], including moiré induced flat bands, unconventional superconductivity and correlated insulators under a magic angle, unique linear classical wave localization [27] and nonlinear localized modes [28]. The twisted bilayer photonics moiré or optical lattices in photonics and BECs have been fabricated successfully by femtosecond laser writing (optical induction) method [27-29] and coherent optical interferences [30-32] respectively; and notably, enormous emergent nonlinear phenomena in both physical contexts are being discovered, such as low-threshold nonlinear optical soliton formation [28], multifrequency solitons [33], light bullets [34], vortex solitons [35], nonlinear localizations of light at the edges and in the corners (of truncated moiré waveguides) [36], and magic-angle lasers [37] and reconfigurable moiré nanolaser array [38], localized gap modes of various types [39-45].

Compared to the twisted bilayer superlattices, the trilayer moiré lattices offer another degree of twisting diversity to reveal the underlying linear and nonlinear properties of classical waves (optical, acoustics, mechanical waves, etc.) and atomic matter waves in ultracold degenerate quantum gases. The structural complexity (e.g., lattice mismatch, periodicity and the relative potential depths between the sublattices) of the twisted trilayer periodic potentials can be precisely tuned and arbitrarily configured to some extent, therefore, such higher-order moiré systems are also called as multi-moiré heterostructures or “moiré of moiré” superlattices in literature [46-53]. However, to our knowledge, linear band properties and moiré physics for such twisted superlattices but with trilayer stacking (configuration) have been poorly understood, and the underlying nonlinear localization mechanism of optical and matter waves is hanging.

Herein, we try to capture a unified picture of what the associated linear band structures look like and how to produce periodic patterns (commensurate structures) with optimal (largest) photonic forbidden gaps in two-dimensional trilayer moiré optical lattices with two twisted angles, and demonstrate that the appearance of unique flat Bloch bands bridged by finite gaps within where emergent nonlinear phenomena of localized gap modes in terms of gap solitons and vortices are resided, theoretically and numerically. The existence, formation, and robust dynamics of these localized gap modes are verified by perturbed evolutions, revealing the nonlinear optical and matter-waves localization mechanism of trilayer moiré systems with both square and hexagonal optical lattices, providing new insights into moiré physics and potential moiré photonics applications in higher-order moiré systems in the contexts of optics and ultracold atoms.

2 Results

2.1 Theoretical model of 2D trilayer moiré superlattices

Our mean-field approximated model, described evolutional dynamics of BECs loaded onto two-dimensional trilayer moiré optical lattices, is in the framework of generalized Gross−Pitaevskii equations for macroscopic atomic-wave functions \mathrm{\Psi }:

(1)\mathrm{i}{\mathrm{\partial }}_t\mathrm{\Psi }=-{\displaystyle \frac{1}{2}}({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)\mathrm{\Psi }+V_{\mathrm{O}\mathrm{L}}(x,y)\mathrm{\Psi }+\gamma |\mathrm{\Psi }{|}^2\mathrm{\Psi },

where t is dimensionless time, x and y are cartesian coordinates, coefficient \gamma =1 denotes nonlinear strength of repulsive atom-atom interactions, which can be experimentally tuned by Feshbach-resonance techniques in various ultracold atoms like {}^{87}Rb atoms [54]. The trilayer moiré optical lattices are made using the same way with that of bilayer ones in BECs in recent experiments, and use it twice, so that the resulting trilayer composite structure yields

(2)\begin{array}{rl}{V}_{\mathrm{S}}=& V_3[\sin {}^2(T_2{x}^{″})+\sin {}^2(T_2{y}^{″})]\\ & +V_1[\sin {}^2(Tx)+\sin {}^2(Ty)]\\ & +V_2[\sin {}^2(T{x}^{\prime })+\sin {}^2(T{y}^{\prime })],\end{array}

here T=\pi /{\omega }_0=\pi /4 with {\omega }_0=4, V_{1,2,3} being the depth (strength) of the three square optical lattices with periods \pi /T and \pi /T_2, we set V_1=V_2=V for simplicity otherwise stated, and define the relative strength P=V_3/V_1. As for the bilayer moiré lattice, the twisted angle {\theta }_1 relates two-dimensional mutual rotation between (x,y) plane and the (x^{\prime },y^{\prime }) plane:

(3)\left(\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos ({\theta }_1)& -\sin ({\theta }_1)\\ \sin ({\theta }_1)& \cos ({\theta }_1)\end{array}\right)\left(\begin{array}{l}x\\ y\end{array}\right).

It is the principle to abide by to write down the one for trilayer moiré superlattices, since it applies to the second twisted angle {\theta }_2. As highlighted in literature, the bilayer moiré square lattice conforms with an elementary Bravais square lattice provided that a Pythagorean angle is achieved by setting \theta = arctan[(m^2-n^2)/(2mn)], where the Pythagorean triples (m^2-n^2,2mn,m^2+n^2) at natural numbers (m,n); it is called Pythagorean (commensurate) bilayer moiré square lattice accordingly. In Fig. 1(a), we show such bilayer moiré lattice composed of two square sublattices with a rotation angle {\theta }_1; then the bilayer twisted lattice should be considered as a new sublattice for constructing trilayer moiré lattice, the configuration of which is formed by adding another (the third) sublattice with a second rotation angle {\theta }_2. In order to produce commensurate patterns with optimal (largest) forbidden band gaps, we must set two Pythagorean angles {\theta }_1=\alpha and {\theta }_2=\beta, the third sublattice then has a rotation angle \alpha /2+\beta as depicted in Fig. 1(b), the commensurate trilayer moiré square lattice exhibits a larger supercell and a total equivalent rotation angle \theta =(\alpha +\beta )/2 [see Fig. 1(c)]. Deserved to be noted that the two smallest Pythagorean angles are \theta = arctan(3/4) and \theta = arctan(5/12) with natural numbers (m,n) being (2, 1) and (3, 2), respectively.

On another side, the general hexagonal lattice (also called triangular lattice) yields

(4)V_{\mathrm{H}}=V_0|{\displaystyle \sum _{j=1}^3{\eta }_j{\mathrm{e}}^{\mathrm{i}[\sin ({\phi }_j)x+\cos ({\phi }_j)y]}{|}^2,}

where V_0=4, {\eta }_1={\eta }_2={\eta }_3=3 and {\phi }_1=0, {\phi }_2=2\pi /3, {\phi }_3=4\pi /3. Then the bilayer moiré hexagonal lattice can be produced by placing two hexagonal sublattices with a rotation angle \theta through the transformation Eq. (3) that used for the square one. In order to get a periodic (commensurate) bilayer moiré hexagonal lattice, a similar 120 degree triple (m^2, n^2, m^2+n^2-2mn\cos ({120}^{\circ })=m^2+n^2+mn) or (m, n, \sqrt{m^2+n^2+mn}) should be adopted, which corresponds to a specific rotation angle \theta = arccos[(2m^2+mn)/(2m\sqrt{m^2+n^2+mn})] (or \theta = arctan[\sqrt{3}n/(2m+n)]) at natural numbers (m,n). Likewise, to get the optimal forbidden band gaps for commensurate trilayer moiré hexagonal lattice, once again, we can use the similar strategy as that for constructing trilayer moiré square lattice in Figs. 1(a)−(c), such strategy is formulated in the bottom of Fig. 1 [see Figs. 1(d)−(f)]. The two smallest rotation angles are then yield \theta = arccos(11/14) and \theta = arccos(13/19) with respective natural numbers (m,n) as (3, 5) and (5, 16).

It should be pointed out that, the tunable trilayer moiré superlattices with diverse configurable structures (including the square and hexagonal optical lattices) can be readily created in ultracold and hot atoms by interfering multiple counterpropagating laser beams that form standing waves [30, 31]; besides, they can also be imprinted in photorefractive crystals via optical induction [27-29]. It is also deserved to be noted that the moiré period (lattice constant) of a periodic twisted trilayer optical lattice can be calculated in the same way with that of twisted bilayer ones [39-44], but just doing twice.

2.2 Linear Bloch-wave spectrum

Our first priority is to ascertain how linear Bloch band structures of the trilayer moiré optical lattices are distributed. For this, we consider the linear model Eq. (1) through abandoning its last nonlinear term, and adopt the conventional linear Bloch theory for the case of commensurate (Pythagorean) trilayer moiré square optical lattices with two rotation Pythagorean angles {\theta }_1={\theta }_2= arctan(3/4), whose top view is displayed in Fig. 2(a). It exhibits a much larger supercell [as emphasized in Fig. 1(c)] and the associated Bloch-wave structure is depicted in Fig. 2(b), showing an emergence of many flat bands and unconventional topological forbidden gaps — the width of the second finite gap is wider than that of the first gap. We emphasize that the energy (chemical potential or frequency in optics) takes constant reflecting an emergence of flat band which is induced by a larger supercell in moiré superlattices. For the incommensurate trilayer moiré square lattice, we can use the approximate Pythagorean (incommensurate) lattice to obtain its Bloch-wave spectrum according to the previous literature [27]. Such case for {\theta }_1= arctan(3/4) and {\theta }_2= arctan(3/4)+5\pi /207 is shown in Fig. 2(c), where the first finite gap narrows quickly after a small twist around Pythagorean angle. Similarly, for the commensurate trilayer moiré hexagonal optical lattice with two rotation angles {\theta }_1={\theta }_2= arccos(11/14) in Fig. 2(d), it has a wide first forbidden gap [see Fig. 2(e)]; its Bloch-wave spectrum changes drastically with the change of two rotation angles {\theta }_1= arccos(11/14) and {\theta }_2= arccos(13/19), according to Fig. 2(f), although the extremely flat Bloch bands are still there. As we can see from Fig. 2(g), increasing the strength difference p within the range (0,1] indicates the appearance, expansion, shrinkage and disappearance of gap I, whereas the gap II narrows firstly and then widens; for another pairs of different rotation angles in Fig. 2(h), the width of gap I develops in different way, demonstrating the highly tunability and flexibility of the twisting trilayer structures in band gap engineering. Further, when we increase the lattice depth, widths of both the gap I and gap II broaden gradually, as indicated in Fig. 2(i).

To elucidate the nonlinear localization mechanism of matter waves controlled by trilayer moiré optical lattices, we seek stationary mode \varphi as \mathrm{\Psi }=\varphi {\mathrm{e}}^{-\mathrm{i}\mu t+\mathrm{i}s\vartheta }, here \mu being chemical potential, and s the topological charge of vortex condensate (s=0 for non-vortex mode), substituting such solution into Eq. (1) and after simplification leads to the stationary wave-function \varphi:

(5)\mu \varphi =-{\displaystyle \frac{1}{2}}({\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2)\varphi +V_{\mathrm{O}\mathrm{L}}(x,y)\varphi +{\left|\varphi \right|}^2\varphi .

The number of ultracold atoms N=\int \int \mid \varphi {\mid }^2\mathrm{d}x\mathrm{d}y is conserved in Eq. (5). To excite localized gap modes based on Eq. (5), we take a powerful and quick-convergent numerical way— modified squared-operator iteration method [56], and then find their stability regions via direct perturbed simulations of dynamical model Eq. (1) by means of fourth-order Runge−Kutta method.

2.3 Emergent fundamental gap solitons

The typical type of isotropic localized gap mode is fundamental gap solitons excited within the forbidden gap of the underlying Bloch-wave spectrum. Figures 3(a) and (b) show two characteristic profiles of gap solitons, which are prepared in the first and second gap respectively, supported by the trilayer moiré square optical lattices with two equal rotation Pythagorean angles {\theta }_1={\theta }_2= arctan(3/4); almost no side peaks are exhibited in both solitons, showing a highly nonlinear localization capability of the trilayer superlattices. The dependence between number of atoms (N) and chemical potential (\mu) of such case is summed in Fig. 3(d), which shows the abidance by “anti-Vakhitov−Kolokolov” (anti-VK) criterion (\mathrm{\partial }N/\mathrm{\partial }\mu >0)— an essential but not sufficient condition for the formation of gap solitons in repulsive nonlinear media [40-43, 55, 57]. It is obvious from the Fig. 3(d) that the stability regions are wide and unstable solitons exist only close to the edges; the situation remains for the case with different rotation Pythagorean angles {\theta }_1= arctan(3/4) and {\theta }_2= arctan(5/12) in Fig. 3(e), its typical soliton profile with single peak is depicted in Fig. 3(c). For the latter case, the gap solitons are exceptional stable in the first gap, and are partially unstable around the both edges of the second gap, indicating that a weak regulation of the second rotation angle {\theta }_2 of the trilayer moiré lattices not only shapes its linear Bloch-wave structure (see Fig. 2), but also modulates the stability conditions of the excited nonlinear localized modes. We have also disclosed variation of the third lattice depth V_3=V_0 by setting V_1+V_2+V_3=4 (recall that V_1=V_2) and under a defined chemical potential (\mu =3) on the matter-wave gap soliton’s number N in Fig. 3(f), which yields a growth tendency.

2.4 Emergent vortex gap solitons

In addition to the topological-free nonlinear localized gap mode — fundamental gap solitons reported above, it is a more interesting issue to check the formation and stability of another type of localized gap mode that takes the topological charge s (non-zero natural number). The latter mode is the gap vortex (or vortex gap soliton) with a central flow (or phase) singularity, which is usually constructed as an arrangement of four fundamental gap solitons surrounded by the entire 2\pi \times s phase circulation. For the most common pattern under vorticity s=1, three typical examples of them and their relevant phase distributed structures are respectively displayed in the top and second lines of Fig. 4 [see Figs. 4(a)−(c)], wherein shows the twisted composite localized modes and complex phase structures induced by the twisting trilayer moiré optical lattices. Their dependencies N(\mu ) under the scenarios of two equal rotation Pythagorean angles {\theta }_1={\theta }_2= arctan(3/4) and the unequal one with {\theta }_1= arctan(3/4), {\theta }_2= arctan(5/12) are severally summarized in Figs. 4(d) and (e), showing a contracting trend of the stability regions in both the first forbidden gap and the second one [compared to that for fundamental gap solitons in Figs. 3(d) and (e)]. Again, for the gap vortices supported by trilayer moiré optical lattices by increasing the third lattice depth, the necessary number of atoms N grows steadily, according to Fig. 4(f).

2.5 Perturbed evolution of emergent nonlinear gap modes

The perturbed dynamical evolution is a key indicator to judge the robustness and fragility (noise immunity) of the emergent nonlinear gap modes. To this, we check the perturbed dynamics of the so-found gap solitons and vortical ones, and obtain the corresponding stability and instability regions via a great number of direct perturbed simulations in the framework of the governing dynamical model Eq. (1). Displayed in the first line of Fig. 5(panels a−f) are the marked points mentioned above for both stable and unstable fundamental and vortex gap solitons, whose respective perturbed evolutional processes over time are shown in the bottom line of Fig. 5. It can be seen that the stable localized gap modes in Figs. 5(a, d, f) exhibit robustly coherence, whereas the unstable ones in Figs. 5(b, c, e) evolve either into weakly oscillating localized mode [Figs. 5(b, e)] or a breathing mode [Fig. 5(c)], demonstrating once again the strongly localized power of the trilayer moiré optical lattices and rich and interesting dynamical properties of the emergent nonlinear gap modes supported by such two-dimensional reconfigurable twisted superlattices.

As stated above, the emergence of flat bands owning to the formation of a much larger supercell (which exhibits a stronger Bragg scattering) in twisted configuration, the flat bands can provide a strong localization and thus the emergent nonlinear gap modes can be robustly enough.

3 Conclusion

We have revealed theoretically the existence of flat Bloch bands for Bose−Einstein condensates trapped in trilayer moiré superlattices with two rotation angles {\theta }_{1,2} (by contrast, there is only one twisted degree of freedom for bilayer moiré superlattices), and formulated a unified picture for searching optimal atomic (or photonic) forbidden gaps in such trilayer moiré square and hexagonal optical lattices. Notably, an emergence of flat bands in relatively shallow trilayer moiré lattices owning to a larger supercell is opened compared to the constitute sublattice and to that of their bilayer counterparts, suggesting an effective way for matter-wave (or light-field) manipulation in twisted photonics with small lattice depth. Further, using theoretical analysis and numerical simulations, we have also unveiled the nonlinear localization mechanism of ultracold atoms loaded onto such trilayer moiré superlattices by presenting the emergent nonlinear phenomena as localized gap modes — fundamental gap solitons and gap vortices with a topological charge. Robustness of both localized gap modes and their stability regions were identified through our numerous direct perturbed computations. The unique flat band properties and soliton physics predicted in trilayer moiré square and hexagonal optical lattices lay a foundation for exploring in-depth understanding of the linear, nonlinear and quantum “moiré of moiré” physics of twisted trilayer superlattices in diverse fields not limited to ultracold atoms and photonics.

Before closing, we state the fact that although most physical properties (such as the flat bands and the excitation of localized modes) already found in bilayer moiré superlattices, adding the third layer is still necessary from at least the following three aspects to think about: (i) compared to its bilayer counterpart, the trilayer moiré superlattice offers a more flexible condition for constructing the associated periodic (commesurate) structure by tuning two twisted angles; (ii) the trilayer moiré superlattice under incommensurate case can be considered as a novel type of quasicrystals-like structure ({\theta }_1={\theta }_2={60}^{\circ }) which is different from that of the bilayer one; (iii) it is interesting to see how the few-body physics and many-body effects work by placing atoms in three layers, the situation can not find its analog in bilayer moiré superlattices. In the context of optics, the creation of novel structured light carried orbital angular momentum [58-60] in moiré superlattices is a new subject deserved to be researched in the future. Accordingly, rich linear, nonlinear, and quantum physics and effects in twisted trilayer superlattices remain to be further dug into.

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