1. Introduction

Flatband systems, which are characterized by the existence of non-dispersive band over the whole Brillouin zone (BZ), have attracted considerable interest in the past decade [1–8]. The associated eigenstates, i.e., compact localized states (CLSs), are perfectly localized to several lattice sites, with vanishing amplitude on all other sites caused by destructive interference [9–11]. Up to now, flatband structures and CLSs have been realized in a variety of artificial physical systems, ranging from cold-atomic optical lattices [12,13] to polariton-exciton condensates [14,15] and photonic waveguide arrays [16–19]. More recently, twisted bilayer graphene that is finely tuned to a magic angle where a flatband emerges has been experimentally realized for studying insulating phases and unconventional superconductivity [20,21]. In addition, photonic moiré lattices and the localization of light based on flatband have also been experimentally realized [22]. Though flatbands have constant eigenvalues, interestingly they can be categorized into two distinct classes, that is, chiral and non-chiral flatbands, according to the symmetry properties in momentum space [23–26]. The chiral flatbands are characterized by the zero eigenenergy and can emerge in bipartite lattices (majority sublattice, which contains more sites per unit cell than the complementary minority sublattice) [4,5]. For instance, one-dimensional (1D) rhombic lattices and two-dimensional (2D) Lieb lattices possess zero-energy flatbands and CLSs supported by one of the two sublattices [16–18,27–30]. Such flatbands are appealing as they provide a unique platform for studying fundamental phenomena, including fractional quantum Hall effects and topological phases [31–34]. In other conditions where the chiral symmetry is absent, a system can also possess a flatband but with nonzero energy. For example, 1D sawtooth lattices and 2D kagome lattices belonging to a family of frustrated tight-binding models support an accidental non-chiral flatband [35–38]. Very recently, the study of the interplay between strain and flatband has brought many new developments. The possibility of achieving partial flatband with strain has been studied in graphene [39]. The high density of states of strain-induced flatband can give rise to interface superconductivity in topological crystalline insulators [40]. It has also shown that tilted Dirac cones may emerge from flatbands when a strain is introduced in photonic orbital graphene or Lieb-kagome lattices [41,42]. Besides, topological band evolution between Lieb and kagome lattices has manifested that the two lattices are interconvertible [43].

On the other hand, Bloch oscillations (BOs), the oscillatory motion of particles in a periodic potential driven by an external force, have intrigued scientists for decades as fundamental phenomena predicted from quantum mechanics [44]. When the external driving field is comparable with the band gap between adjacent Bloch bands, Landau-Zener tunneling (LZT) between Bloch bands, a counteracting effect limiting BOs, appears at the edges of the BZ [45]. Thus far, observations of Landau-Zener Bloch (LZB) oscillations have been achieved in a variety of systems, particularly, in both 1D and 2D photonic lattices formed by coupled waveguide arrays [46–50]. Interestingly, the transport behavior of LZB oscillations is directly governed by the band structure of the system, as the trajectory of the wave-packet during oscillations exactly follows the shape of the band. For example, a broad input beam follows a cosine-like oscillatory path due to the interplay between Bragg refractions and total internal reflections [51]. Investigations of LZB oscillations have also extended to flatband lattices. The combination of the physics of LZB oscillations and flatbands opens promising directions for control of unconventional photonic transport through properly designed lattice structures. Triply LZT between flatband and dispersive bands has been predicted [52]. Moreover, the geometrical phases raised from topological transition can make a complicated interference with the conventional phase of LZB oscillations in topological non-trivial flatband systems [53]. In non-Hermitian flatband settings, LZB oscillations show interesting characteristics including pseudo-Hermitian propagation and secondary emissions at higher-order exceptional points [54,55].

As mentioned above, a strain can break the symmetry and affect the properties of the flatband lattices. Nevertheless, a very intriguing and fundamental question is whether the chiral and non-chiral flatbands behave qualitatively the same when a strain is applied, and how the LZB oscillation dynamics change with the interplay of flatband, lattice strain and the driving field?

To address this question, we theoretically investigate band evolution of chiral and non-chiral symmetric flatband photonic rhombic lattices by applying a strain along the diagonal direction and thereby demonstrating Landau-Zener Bloch (LZB) oscillations in the presence of a refractive index gradient. The chiral and non-chiral symmetric flatband lattices are obtained by applying different lattice detuning to uniform lattices. The applied strain introduces a next-nearest-neighbor (NNN) hopping that dramatically changes the band structure. For the chiral symmetric lattices, the strain-induced NNN hopping breaks the chiral symmetry, and therefore destroying the band flatness. While, a nearly flatband may emerge in the bottom of the spectrum. A triply LZT can be observed in such a system and we present the suppression of LZB oscillations due to the band flattening. For the lattices in the absence of chiral symmetry, flatband survives as a result of the retained particle-hole symmetry, and continuously transforms from the middle band to the bottom band with the increase of the strain-induced NNN hopping. It is shown that LZB oscillations only happen between two dispersive bands. More importantly, band gap of the spectrum can be readily tuned, which allows controlling the amplitude of LZT rate, and we observe a period doubling of BOs owing to complete LZT at the BZ edge.

2. Band structure of chiral and non-chiral symmetric rhombic lattices

A photonic rhombic lattice formed by coupled waveguide arrays is shown in Fig. 1(a). Each unit cell consists of one central site B (minority sublattice) and two edge sites A and C (majority sublattice). Such a lattice can be seen as one of the simplest flatband structures and has been well studied to demonstrate phenomena associated with flatband [27–29,56]. The Hamiltonian can be written as:

The chiral symmetry requires that all the energy eigenvalues appear in pairs. Meanwhile, from an eigenstate $\Psi$ with energy $\beta$, one can obtain another state with the opposite energy $-\beta$ by a unitary transformation $C\Psi$. Such a constraint makes the middle band flat with zero energy. The irreducible compact localized state (CLS) of the flatband occupies A, C sites within one unit cell, with equal amplitude and opposite phase, ensuring destructive interference in the neighboring B site. In one-dimensional settings, the CLSs can be classified by the integer number U of unit cells occupied by each state [10]. Thus, the CLS shown in Fig. 1(a) belongs to the U=1 class. Adding a propagation constant detuning can open up a gap while respecting the translational symmetry of the lattices and preserving the existence of the flatband. For simplicity, we set $\Delta _{2}=0$ and depending on the relative values of $\Delta _{1}$ and $\Delta _{3}$, one can get a chiral or non-chiral symmetric flatband lattice. For a detuned lattice $\Delta _{2}=0,\Delta _{1}=-\Delta _{3}=\Delta$, the Hamiltonian still satisfies the chiral symmetry $CH_{k}C^{-1}=-H_{k}$. The dispersion relation is given by: $\beta _\textrm {FB}=0$, $\beta _{\pm }=\pm \sqrt {\Delta ^{2}+8{\rm cos}^{2}\left ( k/2\right )}$. As shown in Fig. 1(c) for $\Delta =-0.45$, the zero-energy flatband is protected by the chiral symmetry and becomes isolated from the dispersive bands, which are gapped symmetrically. The gap width between the flatband and a dispersive band is $\vert \Delta \vert$. In fact, such a lattice can be created by applying a refractive index gradient perpendicular to the waveguides as experimentally demonstrated in our previous work Ref. [56], where we have also shown that the CLS becomes a $U=2$ state occupying two unit cells. When the on-site potential of A site is equal to that of C site (i.e., $\Delta _{2}=0,\Delta _{1}=\Delta _{3}=\Delta$), the chiral symmetry is broken with $CH_{k}C^{-1}\neq -H_{k}$. However, the flatband is preserved due to the particle-hole symmetry and transforms into a non-chiral symmetric band $\beta _\textrm {FB}=\Delta$, and the other two dispersive bands become $\beta _{\pm }=\Delta /2\pm \sqrt {\left ( \Delta /2\right ) ^{2}+8{\rm cos}^{2}\left ( k/2\right )}$. A band gap with gap width equalling $\vert \Delta \vert$ opens, and the flatband only touches one of the two dispersive bands either above or below at $M$ point, depending on the sign of $\Delta$. In our study, we set $\Delta < 0$, and the flatband touches the lowest dispersive band as shown in Fig. 1(d) for $\Delta =-0.7$. Moreover, one can easily find that the CLS in Fig. 1(a) is still the eigenmode of the flatband, i.e., CLS remains invariant in the presence of such a detuning.

3. Band evolution and LZB oscillations in strained chiral symmetric rhombic lattices

Adding a strain along the diagonal direction introduces a NNN hopping between A and C sites, and therefore moves the flatband in relation to the dispersive bands. We first investigate the band evolution of the chiral symmetric detuned lattices in Fig. 1(c) by tuning the strain-induced NNN hopping $t_{1}$. As depicted in Fig. 2(a), flatband is distorted with the central part bulged a bit and therefore becomes slightly dispersive once the NNN hopping is introduced. In this case, the Hamiltonian in the momentum space no longer satisfies the chiral symmetry, i.e., chiral symmetry required for the flatband is broken. Further increase the magnitude of $t_{1}$, the distortion of the central band gets more pronounced as depicted in Figs. 2(b) and 2(c). On the contrary, the corresponding part of the lowest band becomes partially flat. It also should be pointed out that the band gap width (0.49, 1 and 1.76 in Figs. 2(a), 2(b) and 2(c), respectively) between the upper two bands is increased during the evolution. Meanwhile, our numerical results show that the widths of the two gaps have the same values at $M$ point. For $t_{1}=2.5$, the central flatband is completely destroyed and evolves into a dispersive band as shown in Fig. 2(d). The gap width between the two dispersive bands is 2.54. Remarkably, we find that the lowest band is very flat compared with the upper two bands. We denote $W$ as the maximum bandwidth of the lowest band, $R$ as the maximum bandwidth of the upper two bands. We obtain a nearly flatband with a large flatness ratio (the ratio of $R$ and $W$, $R/W$) about 200. For $t_{1}>2.5$, the band gets narrower with the increase of $t_{1}$ and eventually it is approximately equal to $t_{1}+0.1$.

 

Fig. 1. (a) Schematic diagram of a photonic rhombic lattice formed by coupled waveguide arrays with each unit cell consisting of three sites (A, B, and C). $t$ is the NN hopping and $t_{1}$ is the strain-induced NNN hopping between A and C sites. The ratio of $t_{1}$ and $t$ (i.e., $t_{1}/t$) can be tuned by applying a strain along the diagonal direction as represented by the two solid arrows. $\Delta \beta$ represents a linear gradient of the refractive index (wave-number difference between adjacent waveguides) applied to the lattice. Colored waveguides show a CLS with equal amplitude but opposite phase. (b-d) Corresponding band structure of (b) a uniform lattice ($\Delta _{1}=\Delta _{2}=\Delta _{3}=0$), (c) a chiral symmetric detuned lattice ($\Delta _{2}=0,\Delta _{1}=-\Delta _{3}=-0.45$), and (d) a non-chiral symmetric detuned lattice ($\Delta _{2}=0,\Delta _{1}=\Delta _{3}=-0.7$) in the tight-binding approximation where only NN hopping is considered. For simplicity, we set $t=1$ and the lattice period $d=1$.

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Fig. 2. Band evolution of the strained chiral symmetric rhombic lattices. (a) $t_{1}=0.2$, flatband is bulged a bit at the BZ edge ($M$ point, $k=\pi$). (b) $t_{1}=0.9$, flatband is destroyed with the central part becoming slightly dispersive, and on the contrary, the lowest band becomes partially flat around the $M$ point. (c) $t_{1}=1.7$, the middle band keeps flat only around $\Gamma$ point ($k=0$), whereas most parts of the lowest band become flat. (d) $t_{1}=2.5$, the middle band transforms into a dispersive band and a nearly flatband is generated as the bottom band. Inset shows the zoom-in picture of the nearly flatband with band width less than 0.1. The flatness ratio can reach a high value of about 200.

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By introducing a gradient of the refractive index $\Delta \beta$ to the waveguide arrays as shown in Fig. 1(a), one can mimic the linear potential associated with a static electric field applied to a crystal and thus resulting in optical BOs. As mentioned above, an intriguing feature of BOs is that the trajectory of the wave-packet during propagation reveals the profile of the band. Thus, it is interesting to demonstrate the significant impacts of strain-induced NNN interaction on band structure and wave propagation dynamics by investigating the BOs of the system. In the following, we study BOs of both the flatband and the lowest dispersive band with a transverse index gradient $\Delta \beta = 0.15$ applied. The Schrödinger equation describing the dynamics of optical wave-packet takes the form $i\left ( d/dz\right )\Psi _{n}=\left ( H+D\right )\Psi _{n}$, where $z$ denotes the propagation coordinate, $\Psi _{n}=\left ( a_{n},b_{n},c_{n}\right )^{T}$ is the three-component wave function describing the field amplitude in unit cell n and

 

Fig. 3. LZB oscillations of the strained chiral symmetric rhombic lattices for the excitation of the flatband. (a) An initial excitation beam $a_{n}=-c_{n}=e^{-n^{2}/(2\sigma ^{2})}$ ($\sigma =5$, the index $n$ represents the cell number and ranges from -50 to 50) formed by combinations of CLSs with a broad Gaussian distribution (red line) at normal incidence. Colored sites represent nonzero amplitudes with blue and green sites having $\pi$ phase difference. A small refractive index gradient $\Delta \beta = 0.15$ is applied. (b) $t_{1}=0$, triply LZB oscillations between the flatband and two dispersive bands. The energy distributes symmetrically during the oscillations, reflecting the symmetric band structure. (c) $t_{1}=0.2$, asymmetric LZB oscillations with LZT mainly happens between the flatband and the highest dispersive band due to the asymmetry of the band structure. (d) $t_{1}=0.9$, the input stays almost unaltered with a sharp mini-curve accompanied by slight LZT emerging at the BZ edge. (e) $t_{1}=1.7$, cosine-like BOs without LZT, indicating that the middle flatband evolves into a nearly dispersive band and a large band gap appears. The energy plotted in all figures is $Q_{n}=a_{n}^{2}+b_{n}^{2}+c_{n}^{2}$. The oscillation period is $l=2\pi /\Delta \beta$ and the propagation length $L=4l$ is divided into 160 steps.

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In order to reveal the changes of the dispersive bands, we investigate the LZB oscillations for the excitation of the lowest dispersive band. Different form that in Fig. 3, all the sublattices including B are excited with $a_{n}=-0.75b_{n}=c_{n}=e^{-n^{2}/(2\sigma ^{2})}$ ($\sigma =5$) [Fig. 4 (a)]. Our results show that the undergoing BOs also clearly verify the band evolution process. As can be seen in Fig. 4(b), a triply LZB oscillation pattern is obtained. Note that most of the energy locates in the lower two bands with only weak energy distributed in the highest dispersive band. Such a feature becomes more distinct for $t_{1}=0.2$ because of the larger gap width between the two dispersive bands [Fig. 4(c)]. For $t_{1}=0.9$, LZT disappears and the input beam exhibits periodically switched accelerating direction and recurrences of beam profiles [Fig. 4 (d)]. Meanwhile, the trajectory of the wave-packet is partially flat around the reflection area, which indicating that the lowest dispersive band starts to become flat at the BZ edge. Therefore, the maximal expansion of the wave-packet becomes smaller. The amplitude of oscillations decreases with further increase of $t_{1}$, and an almost complete suppression of the BOs is illustrated in Fig. 4(e) for $t_{1}=1.7$. Such a pattern exactly confirms the band flatting feature of the lowest dispersive band shown in Fig. 2(c).

 

Fig. 4. Same as in Fig. 3 but for the excitation of the lowest dispersive band. Colored sites in (a) represent nonzero amplitudes with blue and green sites having $\pi$ phase difference. All the waveguides are excited with $a_{n}=-0.75b_{n}=c_{n}=e^{-n^{2}/(2\sigma ^{2})}$ ($\sigma =5$) at normal incidence to excite $\Gamma$ point ($k=0$) and other parameters are the same as in Fig. 3.

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4. Band evolution and LZB oscillations in strained non-chiral symmetric rhombic lattices

Next, we present the band evolution and LZB oscillations of the non-chiral symmetric lattices shown in Fig. 1(d) when the strain is applied. In this case, the dispersion relations become: $\beta _\textrm {FB}=\Delta -t_{1}$, $\beta _{\pm }=\left ( \Delta + t_{1}\right )/2 \pm \sqrt {\left ( \left ( \Delta + t_{1}\right )/2\right )^{2}+8{\rm cos}^{2}\left ( k/2\right )}$. In contrast to the chiral symmetric lattices, the band flatness is retained in the presence of strain-induced NNN hopping. Tuning the NNN hopping will shift the flatband energy with respect to the dispersive bands. As depicted in Fig. 5, the non-chiral flatband is shifted downward and evolves continuously from the middle band to the bottom band with the increase of $t_{1}$. Meanwhile, the flatband decouples from the lowest dispersive band with the band crossing of them appears. More importantly, the band gap width between the two dispersive bands equals $\left | \Delta +t_{1}\right |$. It decreases and then increases during the evolution. Specifically, as $t_{1}$ increases, the band gap becomes narrow for $t_{1}<-\Delta$. The gap value is 0.2 for $t_{1}=0.5$ in Fig. 5(a). The most striking impact of the coexistence of lattice detuning and strain is that the band structure turns gapless with the two dispersive bands touching each other for $t_{1}=-\Delta$ [Fig. 5(b)]. For $t_{1}>-\Delta$, the band gap opens again. In Fig. 5(c), a spectrum similar to Fig. 5(a) is achieved for $t_{1}=0.9$. The gap value of the two cases are also exactly the same. Subsequently, increasing $t_{1}$ results in the separable band structure with the flatband evolving into the bottom band as shown in Fig. 5(d). Note that, flatband CLSs remain unchanged compared with the uniform lattice in Fig. 1(a), regardless of the different parameters of $\Delta$ and $t_{1}$. The particle-hole symmetry of the system is protected during the band evolution, and thus destructive interference in the neighboring B site will always be satisfied. Such a feature will significantly change the LZB oscillation dynamics of the system.

As stated above, flatband CLSs keep unchanged even in the presence of strain-induced NNN hopping for the non-chiral symmetric rhombic lattices. Consequently, the input shown in Fig. 3(a) still can be used to excite flatband modes. However, one can easily find that such a localized initial state will not experience any displacement even in the presence of different strain-induced NNN hopping interactions, which is similar to the cases without NNN hopping demonstrated in Refs. [27,52,56]. This is because, despite the presence of gradient and NNN hoppings, the non-chiral symmetric rhombic lattices always satisfy the particle-hole symmetry $a(c)\rightarrow c(a)$. The excitation with all momenta acquires the same phase shifts upon propagation and therefore does not diffract.

Here, we focus on studying LZB oscillations for the excitation of the lowest dispersive band. A broad beam $a_{n}=-\sqrt {2}b_{n}/2=c_{n}=e^{-n^{2}/(2\sigma ^{2})}$ ($\sigma =5$), which is similar to that shown in Fig. 4(a) and matches the Bloch modes of $\Gamma$ point, is constructed to excite the lattices at normal incidence [Fig. 6(a)]. Figure 6(b) depicts the typical oscillation pattern without NNN hopping. The input experiences perfect BOs and the LZT is almost invisible. In this case, the initial incident beam has a transverse vector $k=0$ located at the center of the BZ and travels towards the BZ edge under the action of the linear index gradient. When the beam arrives at the BZ edge, Bragg reflections without obvious LZT take place, making the transverse $k$ vector reverse direction and deflect to the center of the BZ. Apparent LZT can be seen in the presence of strain-induced NNN hopping for $t_{1}=0.5$ [Fig. 6(c)]. Nevertheless, quite different from the triply LZB oscillations in Fig. 4, the energy only distributes in the two dispersive bands without any energy generated in the flatband. This is because flatband and dispersive bands are completely decoupled and the wave-packet cannot match the flatband CLSs during the propagation. The tunneling probability can be readily obtained using the Landau-Zener formula $P_{LZ}=e^{-\pi \omega ^{2}/2\alpha }$, which equals 0.16 in Fig. 6(b) and 0.86 in Fig. 6(c). Interestingly, by fine tuning $t_{1}$ to approach the critical value $t_{1}\rightarrow -\Delta$ , the wave-packet dynamics changes drastically into an oscillation pattern with a large amplitude and the period is doubled compared with other cases [Fig. 6(d)]. This can be considered as a complete LZT with light escaping totally from the lowest band to the highest dispersive band at the BZ edge. When the wave-packet changes the direction, complete LZT happens again and the processes then repeat periodically. Such a pattern exactly corresponds to the scanning of the gapless dispersive bands shown in Fig. 5(b). For $t_{1}=0.9$, reappearance of LZB oscillations is obtained [Fig. 6(e)]. We observe a same pattern compared with Fig. 6(c) because of the similar band structure and same gap width of the two cases. Likewise, further increase of $t_{1}$ will result in restoration of BOs in Fig. 6(b), indicating that a large band gap width appears again.

 

Fig. 5. Band evolution of the strained non-chiral symmetric rhombic lattices. (a) $t_{1}=0.5$, flatband is shifted downward and crosses the lowest dispersive band. Simultaneously, the gap width decreases. (b) $t_{1}=0.7$, two dispersive bands touch each other and a vanishing band gap appears. (c) $t_{1}=0.9$, the band gap is reopened. (d) $t_{1}=2.0$, the middle flatband transforms into the bottom band. The gap widths between the two dispersive bands in (a), (b), (c), (d) are 0.2, 0, 0.2, 1.3, respectively.

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Fig. 6. LZB oscillations of the strained non-chiral symmetric rhombic lattices for the excitation of the lowest dispersive band. (a) $t_{1}=0$, periodic oscillations with almost invisible LZT. (b) $t_{1}=0.5$, BOs together with LZT between the two dispersive bands. (c) $t_{1}=0.7$, BOs and complete LZT due to the vanishing band gap and as a result, the period of BOs is doubled. (d) $t_{1}=0.9$, a same LZB oscillation pattern as in (b). All the waveguides are excited with $a_{n}=-\sqrt {2}b_{n}/2=c_{n}=e^{-n^{2}/(2\sigma ^{2})}$ ($\sigma =5$) at normal incidence and other parameters are the same as in Fig. 4. Note that the LZB oscillations only happen between two dispersive bands with no energy distributed in the flatband. The probabilities of the LZT are 0.16, 0.86, 1 and 0.86, respectively.

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Before closing, we would like to give some parameters that might be used in experiment so that one can apply the results to a realistic photonic lattice. Silica glass with a refractive index $n_{0} = 1.45$ is used as the supporting media. The waveguide arrays have the refractive index profile $\Delta n=2*10^{-4}$. The effective index gradient can be obtained by designing curved waveguide arrays [27] and is set to be $\delta n=3.2*10^{-6}$. The diameter of every waveguide is $3 \mu m$ and the distance between two adjacent waveguides is $10 \mu m$. These parameters can be implemented experimentally via femtosecond laser-writing technique as shown in the previous studies [16,17,27,47]. Plugging these parameters into paraxial wave equation [16,17] and by exciting a single waveguide, one can estimate the coupling coefficient and here t is about $0.28 mm^{-1}$. The wavelength of the input beam is $\lambda =488 nm$ and the index gradient field $\Delta \beta =2\pi \delta n/ \lambda =0.042 mm^{-1}$. Then, we can find that the maximal oscillation amplitude corresponding to our results in Fig. 6 is about $0.5 mm$, the oscillation period is $l=2\pi /\Delta \beta =150 mm$, and the propagation length $L=4l=600 mm$.

5. Conclusion

We have demonstrated distinctly different band evolutions of the chiral and non-chiral symmetric rhombic lattices by tuning the strain-induced NNN hopping, and presented LZB oscillations in the presence of refractive index gradient. In the chiral symmetric lattices, the middle flatband is destroyed while a nearly flatband is generated as the bottom band. It allows controllable oscillation amplitude and may enable complete suppression of BOs. On the other hand, the non-chiral symmetric flatband lattices provide a flexible route to control the LZT rate and may result in complete LZT. Our findings show insights into the interplay of lattice symmetry and strain in the application of achieving tunable flatband. This may also open the avenue for further interesting studies in 2D flatband lattices hunting for fractional charge transport and topological phases, among others.