Nonlinear localization of ultracold atomic Fermi gas in moiré optical lattices

. Moiré superlattices, a twisted functional structure crossing the periodic and nonperiodic potentials, have recently attracted great interest in multidisciplinary fields, including optics and ultracold atoms, because of their unique band structures, physical properties, and potential implications. Driven by recent experiments on quantum phenomena of bosonic gases, the atomic Bose – Einstein condensates in moiré optical lattices, by which other quantum gases such as ultracold fermionic atoms are trapped, could be readily achieved in ultracold atom laboratories, whereas the associated nonlinear localization mechanism remains unexploited. Here, we report the nonlinear localization theory of ultracold atomic Fermi gases in two-dimensional moiré optical lattices. The linear Bloch-wave spectrum of such a twisted structure exhibits rich nontrivial flat bands, which are separated by different finite bandgaps wherein the existence, properties, and dynamics of localized superfluid Fermi gas structures of two types, gap solitons and gap vortices (topological modes) with vortex charge S ¼ 1, are studied numerically. Our results demonstrate the wide stability regions and robustness of these localized structures, opening up a new avenue for studying soliton physics and moiré physics in ultracold atoms beyond bosonic gases.


Introduction
][32][33][34][35][36][37][38][39][40][41][42][43][44] Particularly, the linear localization of light and transition between localization and delocalization due to the unique flat-band property has been first confirmed in photonic moiré lattices; 31 in such a setting but adding nonlinearity, an extremely low threshold of optical soliton formation via twisted angle was also observed experimentally by the same group. 326][47][48][49][50] These include, but are not limited to, the generation of multifrequency solitons in quadratic nonlinear media, 45 light bullets, 46 and vortex solitons 47 as well as 2D Thouless pumping 33 of light in photonic moiré lattices; linear and nonlinear light localizations at the edges of moiré arrays; 48 2D gap solitons in parity-time symmetric and regular moiré optical lattices; 49 matter-wave gap solitons; and vortices in dense Bose-Einstein condensates with moiré optical lattices. 50However, one can find that all these soliton studies are in the Boson settings; 33,[45][46][47][48][49][50] it is instructive to study soliton formations and disclose the nonlinear localization theory in other physical settings such as degenerate Fermi gases.
This work is intended to explore how the nonlinear localization of superfluid Fermi gases trapped by moiré optical lattices works and what kinds of robust localized gap modes can be created and how to launch them.By means of linear Bloch theory, we find that such 2D twisted bilayer optical lattices have abundant extremely flat bands and that the widths of numbered finite gaps vary differently with the change of twisted angle and strength contrast of the two sublattices.Our systematic numerical simulations and theoretical analysis demonstrate that the band flatness could enable us to construct robust 2D localized gap modes as fundamental gap solitons and vortices (with imprinted vorticity S ¼ 1) inside the several finite gaps (whose first, second, third, fourth, and fifth finite gaps are examined).The robustness of both localized gap modes implies that the moiré optical lattices can be set as an ideal platform for predicting and observing emergent nonlinear wave structures in the settings of ultracold atomic gases with shallow (small strength) lattices, providing a new degree of freedom in exploring and tuning the nonlinear physics of ultracold atom settings with diverse gases.We would like to emphasize that while flat bands appear also in conventional optical lattices; the condition is that the optical lattices should be fabricated deep enough (very large lattice strength, hard to realize) or in special spatial configurations.The nonlinear localization physics revealed here is positive, timely, and meaningful, given the fact that the atomic Bose-Einstein condensate has recently been prepared in twisted bilayer optical lattices, 51 and its Fermi counterpart (superfluid fermion gas in moiré optical lattices) can be readily implemented in the current state-of-the-art ultracold atomic laboratories.
2 Model and Its Linear Bloch Spectrum

Theoretical Model
Nonlinear evolutional dynamics of a 2D ultracold Fermi gas (Bardeen Cooper Schrieffer superfluid of spin-1/2 Fermionic atoms) loaded onto the moiré optical lattices in mean-field theory is governed by the density-functional equation (theory), whose scaled wave function (complex order parameter), ψ, yields [16][17][18] where R ¼ ðx; yÞ and Laplacian , the nonlinear coefficient, g, represent repulsive (self-defocusing) nonlinearity induced by atom-atom collisions, set as g ¼ 1 (for discussion), which, in reality, can be tuned using the Feshbach resonance technology. 52It is relevant to remark that the nonlinearity represents a repulsive nonlinear term of power 7/3, which is unique for degenerate Fermi gases.The 2D moiré optical lattices, consisting of two uniform square lattices with a twisted angle θ to each other, follow where V 1,2 > 0 represents the modulation depth (amplitude) of the two optical lattices with periodicity π.For convenient discussion, we take the strength difference as The transformation between the ðx; yÞ and ðx 0 ; y 0 Þ planes is, in mathematics and real spaces, defined by a rotating angle θ: One can know from Eq. ( 2) that the moiré optical lattice returns to the normal square lattice if we set V 2 ¼ 0 or θ ¼ 0. The extraordinary feature of such a moiré pattern is that, when gradually changing the rotating angle θ (between the two sublattices), the moiré optical lattice will be continuously transformed into the periodic and aperiodic structures, thereby filling a gap crossing of perfect periodic potentials and aperiodic structures and opening a new strategy for light-field manipulation (bandgap engineering) and its nonlinear dynamic control dependent merely on a new rotating degree of freedom of the sublattices even though under shallow lattice modulation.As pointed out in previous publications, [30][31][32][33]44,49,50 such a moiré optical lattice allows the existence of the widest finite gaps separated by neighboring extremely flat Bloch bands provided that it is the perfect periodic pattern which, in fact, is achieved only if we consider the special case of Pythagorean angles θ ¼ arctanða∕bÞ, with cos θ ¼ a∕c, sin θ ¼ b∕c, and natural numbers ða; b; cÞ constitute Pythagorean triples, The stationary solitary matter-wave solution ϕ under chemical potential μ can be constructed as ψ ¼ ϕe −iμt , and after substituting it into the dynamical fundamental equation [Eq.( 1)], one can get the stationary equation, Equation ( 4) holds the norm (number of atoms) N ¼ RR jϕj 2 dx dy.

Linear Bandgap Properties
Although the fact that the moiré patterns could be twisted with arbitrary angles θ, allowing ample band diagrams to emerge accordingly, the Pythagorean lattices (the moiré optical lattice under the unique case of Pythagorean angles θ, as stressed above) support the optimal bandgap tuning, which is the spatial tailoring of the largest width of the finite gaps, within which the localized gap modes of diverse types to be excited are of our fundamental interest.Our attention here is only paid to the moiré optical lattices at different Pythagorean twisted angles θ, which, in fact, are equipped with periodic translational symmetry, and therefore, the general linear Bloch bandgap theory widely used in solid-state physics and other periodic potentials, including photonic crystals and optical lattices, can be readily applied here.Bearing this in mind, one can produce the underlying linear Bloch spectrum after a routine operation.Contour profiles of Pythagorean moiré optical lattices under three different Pythagorean twisted angles θ ¼ arctanð3∕4Þ, θ ¼ arctanð5∕12Þ, and θ ¼ arctanð8∕15Þ are depicted in Figs.1(a)-1(c), respectively.The three twisted angles enlarge the lattice period of moiré optical lattices as compared with the original square lattices (ε 2 ¼ 0 or θ ¼ 0).Ignoring the last nonlinear term of Eq. ( 4), calculating the corresponding eigenvalues within the first Brillouin zone in the reciprocal space [which is depicted in the inset of Fig. 1(d)], one can get the linear Bloch spectrum under the same strength difference P ¼ 0.5, as displayed separately in Figs.1(d)-1(f) and in the bottom line of the figure with red dashed vertical lines.Once again, we can see that, as those discovered in other Pythagorean moiré lattices, despite being shallow, there are abundant flat Bloch bands for the Pythagorean moiré optical lattices; strikingly, the former features a very narrow gap (which we name minigap "Q") between the first and second gaps; the latter two cases exhibit more flat bands, and the higher finite gaps can be wider than their first one.Further research revealed that, for the former case, the widths of both the first gap and minigap decrease when increasing P, and the minigap (Q) disappears at P > 0.53, while the second gap's width grows gradually [Fig.1(g)]; by comparison, the first gap's width increases, whereas the decrease is only for the fourth and fifth gaps for the latter two cases, according to Figs. 1(h) and 1(i).These rich linear bandgap properties of moiré optical lattices indicate that very flexible and tunable bandgap engineering can be achieved by simply changing the rotating angle θ and strength difference P, providing a new linear (flat-band) localization mechanism (different from that of Anderson localization) of optical and matter waves, 31    nonlinear localization of various localized gap modes within those numbered finite gaps of the associated linear Bloch spectra, as will be presented below.Before going further, let us introduce the numerical calculation recipes for launching the localized gap modes and testing their stability, which yield the following: the stationary solutions of the localized gap modes under consideration are first constructed through Eq. ( 4) using the widely used methodmodified squared-operator iteration method; 53 then, the stability and instability properties of the gap modes thus found will be scrutinized in direct perturbed numerical computation of the dynamical equation [Eq.( 1)] by taking a converged and highly accurate computational approach called the fourth-order Runge-Kutta method.

and robust
3 Numerical Results and Discussions

Fundamental Gap Solitons
We first survey the excitation of the simplest matter-wave localized gap modes with a single peak and isotropic shapes called fundamental gap solitons, which can be upheld by the Pythagorean moiré optical lattices [Eq.( 2)], as done in conventional periodic potentials.As far as the aforementioned three Pythagorean twisted angles [θ ¼ arctanð3∕4Þ, θ ¼ arctanð5∕12Þ, and θ ¼ arctanð8∕15Þ] are concerned, the corresponding relationships among the number of atoms, N, and chemical potential μ for the fundamental gap solitons are, respectively, depicted in Figs.0][11][12][13][14][15]54 Detailed numerical perturbed calculations in the dynamics equation [Eq.( 1)] reveal that these fundamental gap solitons are robustly stable localized modes existing in the middle of the finite gaps; they tend to be unstable near the edges of Bloch bands, as underlined in the figures.Moreover, we have examined the dependence between N and the same sublattice depth which is summed up in Fig. 2(d), demonstrating a decreasing trend with the increase of V 0 ; it coincides with the significant experimental finding of low threshold (of the laser power in nonlinear optics) for the formation of optical solitons in photonic moiré lattices.It is necessary to note that the typical profiles of the so-found fundamental gap solitons are shown in Figs.2(e)-2(g), where the former is a stable gap soliton owning negligible side peaks, while the latter two are unstable ones accompanied by multiple side peaks that are induced by strong Bragg scattering.Such side peaks of the latter two solitons may cause instability in the course of time evolution.
In the perturbed time evolution, as shown in Fig. 3, the stable fundamental gap soliton sustains its shape [Fig.3  A more interesting thing is that whether such Pythagorean moiré optical lattices can give rise to robust vortex gap solitons and how to stabilize them at will.It is thus the target in this section to reveal these issues for gap vortices, the vortex gap solitons with topological charge S; for the sake of discussion, we only consider the case of S ¼ 1.As reported elsewhere, such gap vortices are usually featured by a null value in the middle and consist of four bright (fundamental gap solitons) modes/peaks that are entangled with π∕2 phase shifts between them, enabling the whole 2π phase circulation (S ¼ 1).The stability regions of the gap vortices reduce a little bit compared with their fundamental gap solitons counterpart because of the existence of destructive interaction among the four solitons, as can be seen from the curve NðμÞ and the associated stability regions portrayed in Figs.4(a   In sum, we have presented numerical investigations and theoretical analysis of nonlinear localizations of 2D ultracold atomic Fermi gas in moiré optical lattices with repulsive (self-defocusing) nonlinearity.A noteworthy finding is that, although being formed in the systems with shallow optical lattices, the tightly localized gap modes, including fundamental gap solitons and vortices with topological charge S ¼ 1, laid inside the finite gaps spaced apart by extremely flat Bloch bands of the underlying linear Bloch spectrum, were shown to have strong localization and robust stability properties.Our nonlinear localization mechanism of ultracold atomic Fermi gas may be readily realized in ultracold atomic laboratories worldwide by means of a similar technique used in recently observed atomic Bose-Einstein condensate and quantum phase transition in 2D twisted bilayer optical lattices.We therefore expect the results predicted here to be of great interest in nonlinear sciences, since we suggest a new way for the creation and control of strongly localized matter-waves beyond atomic Bose-Einstein condensate in moiré optical lattices with shallow depth but exhibiting tunable flat bands and numbered finite gaps.Given the fact that the atomic Bose-Einstein condensate has recently been prepared in twisted bilayer optical lattices, 51 its Fermi counterpart (superfluid fermionic gases in moiré optical lattices) can be readily implemented in the current state-of-the-art ultracold atomic laboratories.

Disclosures
The authors declare no conflicts of interest.

Fig. 3
Fig. 3 Perturbed evolutions of fundamental gap solitons.Top-view profiles (a)-(c) and the perturbed evolution in real time (d)-(f) of typical fundamental gap solitons for the marked points (A, B, and C) in Fig. 2(a): stable (a), weakly unstable (b), and strongly unstable (c) gap solitons.
(a)], while there are oscillating or growing side peaks (lobes) for the unstable fundamental modes [Figs.3(b) and 3(c)].Further insights into their evolutions confirm that the unstable fundamental gap solitons are only in the weakly unstable mode; this once again verifies the excellent platform for soliton formation and control in physical settings with moiré optical lattices.

3. 2
Vortex Gap Solitons with Topological Charge S ¼ 1 )-4(c) for the three Pythagorean angles.Also, a decreasing relation NðV 0 Þ proceeds for the vortex gap solitons, according to Fig. 4(d).The schematic shapes of the vortex gap solitons are displayed in Figs.4(h)-4(j), the cross-sectional cut at y ¼ 1.5 is displayed in Figs.4(e)-4(g), and the associated spatial phase structures are depicted in the bottom line of Fig. 4 [see Figs.4(k)-4(m)], showcasing the twisted modes unique for moiré lattices.In direct perturbed simulations by solving the dynamical equation [Eq.(1)], we can see that the stable vortex gap soliton remains unchanged in long time evolution [Fig.5(a)]; and for the unstable ones, as those displayed in Figs.5(b) and 5(c), only their side peaks undergo regular oscillations and small strength growth, demonstrating weak instability.

Fig. 5
Fig. 5 Perturbed evolutions of gap vortices.Top-view profiles (a)-(c) and the perturbed evolution in real time (d)-(f) of typical vortex gap solitons for the marked points (D, E, and F ) in Fig. 4(a): stable (a), weakly unstable (b), and strongly unstable (c) gap vortices.