Bloch Zener oscillations in ribbon shaped optical lattices

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(2) December 2011 EPL, 96 (2011) 60011 doi: 10.1209/0295-5075/96/60011. www.epljournal.org. Bloch-Zener oscillations in ribbon-shaped optical lattices E. Arévalo1,2 and L. Morales-Molina1,3 1. Max-Planck-Institut für Physik Komplexer Systeme - Nöthnitzer Str. 38, D-01187 Dresden, Germany, EU Friedrich Schiller University, Institute of Condensed Matter Theory and Optics - D-07743 Jena, Germany, EU 3 Departamento de Fı́sica, Facultad de Fı́sica, Pontificia Universidad Católica de Chile Casilla 306, Santiago 22, Chile 2. received 18 August 2011; accepted in final form 8 November 2011 published online 14 December 2011 PACS PACS PACS. 03.75.Lm – Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations 52.35.Mw – Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.) 05.60.Gg – Quantum transport. Abstract – We theoretically study the topological effects of zigzag-edged ribbons on the motion of collective excitations. Pristine geometries resembling those of narrow graphene nanoribbons as well as edge structures with dangling bonds are considered with the help of photonic lattices and/or Bose-Einstein condensates trapped in deep optical lattices. We observe an irrational fractionalization of the excitations when moving under the action of an external field. This effect is discussed in terms of the interband dynamics associated with the systems in consideration. c EPLA, 2011 Copyright . Introduction. – Recent progress in the fabrication of graphene, a truly two-dimensional solid-state material, has enormously stimulated the progress of the research on honeycomb-like systems. These systems show proximity to various carbon-based nanomaterials with exceptional properties owing to its band structure. The hexagonal honeycomb structure occurs in a vast variety of coupled systems and can be artificially made with Bose-Einstein condensates (BECs) trapped in a deep optical lattice [1,2] or with a photonic lattice [3,4]. In fact, these systems constitute a rich experimental playground, where the behavior associated with the band structure of condensedmatter systems can be mimicked with great versatility, often beyond the possibilities of the real materials. So, new interesting scenarios can be envisaged. Our interest here is to study the collective effects arising from coupled systems, such as photonic lattices [2] and/or BECs trapped in deep honeycomb optical lattices [1], mimicking the geometry of graphene nanoribbons. In particular our aim is to consider the geometries of the ultrashort zigzag-edged graphene-like ribbons (ZZEGLRs) depicted in fig. 1(a). These are nonlinear band-gap systems where collective excitations, such as solitary waves (or solitons in short, without implying integrability of the underlying model), can move along the lattice. Notice that the motion of collective excitations implies transfer of either energy or matter between the nearestneighbor sites. This transfer occurs due to either the. (a). (b). Fig. 1: (Color online) (a) Ultranarrow ZZEGLRs with width N = 1 (upper structure) and N = 2 (lower structure); the arrow indicates the direction of the excitation wave number vector k. (b) ω band structures vs. k and the fraction ty /tx for N = 1 (upper panel) and N = 2 (lower panel).. tunnel effect in BEC arrays or evanescent-wave coupling in photonic lattices. Here it is worth noting that the coupled systems we consider allow us to consider only nearest-neighbor interaction. In contrast, graphene systems have been shown to have next-nearest-neighbor. 60011-p1.

(3) E. Arévalo and L. Morales-Molina (NNN) effects [5,6]. However, for a qualitative comparison here we neglect the NNN effects since the associated NNN hopping energies are in general small [5,6]. We note that the presence of NNN interactions slightly distort the symmetry between the upper and the lower band in the spectral regions far from the band-crossing points in graphene sheets. Similar slight deviations far from the avoided crossings (k = π/2) can be expected in the band structures shown in fig. 1(b) if NNN interactions were also considered here. The collective behavior of the ultrashort-ZZEGLR depicted fig. 1(a) becomes most obvious when looking, in fig. 1(b), at the band structures connecting propagation constants (chemical potentials in BEC arrays), quasimomentums and coupling strengths. The most prominent feature in fig. 1(b) is the existence of two bands mimicking the valence and conduction bands in graphene nanoribbons and separated by the energy gap. So, interband effects can be expected depending on the relative value of the coupling strengths (as shown in fig. 1(b)). Interband Landau-Zener tunneling (LZT) is an important physical mechanism used in some of the graphenebased semiconductor devices currently in development [7,8]. Though some interband tunneling ratios have been estimated [8], the dynamical process at nanometric detail in graphene nanoribbons is still poorly understood. We note that LZT in combination with Bloch oscillations (BO) have been studied in the context of two-dimensional (2D) photonic lattices [9], coupled waveguide arrays [10,11], perturbed Hamiltonian system [12], or 1D BEC arrays [13], among others. Applications such as wave beam splitters or Mach-Zenhder interferometers have been also envisaged [12]. ZZEGLRs can be indexed by its width N , the number of zigzag backbone chains across the ribbon [14]. When N approaches infinity, the ZZEGLR becomes a honeycomb structure similar to a graphene sheet. If N becomes 1 (1-ZZEGLR), as in fig. 1(a) (upper structure), the ZZEGLR takes the form of a polyacetylene [14], where phenomena such as charge fractionalization has been theoretically predicted [15]. Usually, the effect of the carbon dangling bones is neglected when a termination with hydrogen is chosen [16]. However, if we assume that some other molecular termination bearing also a π electron is chosen, the dangling-bond effect can be taken into account, as in fig. 1(a) (upper structure). In this case the band-structure resembles that of the problem of fractional quantum numbers in polyacetylene [15,17] (see fig. 1(b), upper panel). On the other hand, it has been shown that dangling bonds at the edges of zigzag graphene nanoribbons [18] strongly affect the transport properties. However, the full effect on the motion of collective excitations is not well understood. When N becomes 2 (2-ZZEGLR), as in fig. 1(a) (lower structure), the geometry associated with the polyacene molecule chain is obtained. Polyacene is the narrowest version of graphene nanoribbons with a 4-band. structure [19]. However, there are always two of the four expected bands contributing to the transport [19]. Thus only the energy gap between the bands contributing to transport (as shown in fig. 1(b) lower panel) is important in our analysis. In this regard, it was shown recently that upon increasing the number N the band-gap between the band contributing to transport can be reduced and eventually closed [20,21]. In the following we consider collective excitations with a vector wave number k, as shown in fig. 1(a). We also use the concept of the quasimomentum k, defined as the k component along the direction of the longitudinal bonds (bonds joining nearest-neighbor sites along the chain). The coupling strength of the longitudinal and transverse bonds are the positive parameters tx and ty , respectively. Transverse bonds are those whose directions are perpendicular to k in fig. 1(a). In fig. 1(b) the 1- and 2-ZZEGLR band structures of the propagation constant (chemical potential in BEC arrays) ω are shown as functions of k and the fraction ty /tx . By solving the associated linear problem (U = Fl,n = 0) of the nonlinear discrete system (see eqs. (3) and (4)), the analytical expressions of the band structures can be estimated as (1) ω(k)/tx = − cos(k) ± (ty /tx )2 + cos2 (k) for the 1-ZZEGLR (fig. 1(b), upper panel) and 1 ω(k)/tx = ± (ty /tx )2 + 16 cos2 (k), 2. (2). for the 2-ZZEGLR (fig. 1(b), lower panel). In eqs. (1) and (2) the upper (lower) sign corresponds to the upper (lower) band structure in each case (1- and 2-ZZEGLR). From fig. 1(b) we observe that the band-gap reduces as the fraction ty /tx gets smaller. Notice that the control of the coupling strength can be experimentally achieved with great precision in photonic and/or BEC arrays. Besides, the 2-ZZEGLR-band-gap behavior (fig. 1(b), lower panel) with respect to the fraction ty /tx resembles that from graphene nanoribons as a function of the width N [20,22]. The model. – Our starting points are the discrete equations proposed in refs. [1,2] for graphene-like structures with nearest-neighbor interaction, reading as (m+1) al,m+1 i∂t bl,m = −t(m) x (al+1,m + al−1,m ) − ty. +U |bl,m |2 bl,m + Fl,m bl,m ,. (3). i∂t al,m+1 = −t(m+1) (bl+1,m+1 + bl−1,m+1 ) x bl,m + U |al,m+1 |2 al,m+1 + Fl,m+1 al,m+1 , (4) −t(m+1) y where al,n and bl,n are the values of the collectiveexcitation amplitude at the sites {l, n} of the two sublat(n) (n) tice system [1,2]. In eqs. (3), (4) terms tx and ty are couplings and the term U is the nonlinear coefficient associated with the Kerr nonlinearity in photonic lattices, or. 60011-p2.

(4) Bloch-Zener oscillations in ribbon-shaped optical lattices the self-interaction effect in BEC arrays. In the latter system the interaction strength can be tuned by using Feschbach resonances [23]. In what follows U is set to be the unity for a self-focusing regime with a bright soliton solution. We note that the following analysis can be also done for a repulsive interaction and dark soliton solutions. Here it is worth mentioning that the role of the nonlinear coefficient U in eqs. (3), (4) should not be confused with the Hubbard correction present in the fermionic tightbinding Hamiltonian describing carbon nanoribbons [22]. Notice that the Hubbard term in bosonic systems (e.g. BEC arrays) leads to an on-site self-interaction effect. In contrast, in fermionic systems it is associated with the on-site Coulomb repulsion [22], and its main effect is to increase the gap between the bands near to the avoiding crossing. This latter effect is not captured by the nonlinearity in eqs. (3), (4). In this regard, it is also important to mention that nonlinearity, as shown in eqs. (3), (4), can emerge in graphene systems when the dynamical behavior of one-electron wave packets takes into account the electron-phonon interaction in the limit of an adiabatic coupling [24]. Besides, the systems we consider here (photonic lattices and/or BEC arrays) contain a similar nonlinearity. On the other hand, the nonlinearity allows one to consider the presence of moving localized excitations, i.e. low-amplitude excitations whose motion is governed by the linear band structure of the system. These excitations are very suitable for studying linear band effects arising in diverse lattice geometries. Moreover, they are stable against either collisions or the presence of small perturbations in the lattice. We note also that these excitations are prone to diffraction, however, for the range of time considered here, this effect does not play any important role in our analysis. In fact, several BO have been experimentally observed in diffractive waveguide arrays with inter-guide separation distance of few tens of micrometers and BO periods of tents of millimeters [10]. In order to study the effect of the band structures, we consider an external constant force α. This force is included in the function Fl,n = α l in eqs. (3), (4), where l is the discrete position coordinate along the chain. α mimics the effect of an external electric field in carbonnanostructures [8], or the effect of an external tilted trap in BEC arrays [25], or also the effect of a refractive-index gradient in photonic lattices [10]. In eqs. (3), (4) the phases associated with the projection of the excitation wave number k on the position vector of the lattice sites [1,2] are not explicitly written, since they are only required for the derivation of the nonlinear Dirac equation (NDE) [1,2]. The NDE is only valid in the vicinity of the band-crossing points (also called diabolical or Dirac points) and not for ultranarrow ZZEGLRs [26]. Moreover, since our purpose is to consider the whole first Brillouin zone, we remain with the general description given by eqs. (3), (4).. In order to proceed, we first determine soliton solutions associated with the zigzag structures in fig. 1(a). These structures can be described by the general eqs. (3), (4) when the couplings are defined for 1-ZZEGLR as t(n) x = tx δm,n ,. t(n) y = ty (δn,m+1 + δn,m ),. (5). and for 2-ZZEGLR as t(n) x = tx (δm,n + δm+1,n ),. t(n) y = ty δn,m+1 .. (6). Here δm,n is the Kronecker delta. By substituting eqs. (5) or (6) in eqs. (3), (4), assuming that α = 0, and using the method proposed in refs. [27,28], we can obtain analytical bright (sech-type) and dark (tanh-type) soliton solutions for the two band structures. The amplitude and width of these solutions depend on the form of the band structure and present nontrivial brightand dark- soliton regions. However, since our interest is the interband effect on collective excitations, we simplify our discussion without losing generality by fixing our initial ansatz to a sech-like excitation with some suitable amplitude and width values. Here it is worth mentioning that due to the presence of cubic nonlinearity, the modulation stability presents an amplitude threshold above which the solitary excitations are pinned to the lattice or even destroyed when LZT occurs. The destruction of these excitations occurs more in the fashion of solitons in ref. [13]. Moreover, it is known that Bloch oscillations become unstable for relatively large amplitude excitations; thus further external control is needed to stabilize the wave packet (see, e.g. [29]). Here we choose amplitudes below this threshold (see values at the captions of figs. 2), so only motion effects associated with band structure are expected and Bloch oscillations are barely affected by the nonlinear perturbation. Results. – In the following by using a Heun’s method we solve numerically eqs. (3), (6) with open boundary conditions at the lattice extremes. In principle, we can start our solitary excitations in any point of the band structures, however, for the sake of simplicity we choose k = 0 at the lower band as shown in figs. 2(a) and 3(a). By setting α to some small finite value the quasimomentum k changes some amount αt, i.e. k → k + αt, where t is time. In doing so, a LZT may happen, which results from the trade-off between α and the gap width ∆ at k = π/2. This gap in turn depends on the fraction ty /tx , as shown in fig. 1(a). Since LZT is more likely for small ∆, we have considered in all cases an small fraction value ty /tx = 0.1. In particular figs. 2(a) and 3(a) schematically show how an initial excitation (at k = 0) is driven along the lower band, then undergoes LZT (around k ∼ π/2) and, finally, how a second excitation in the upper band appears. We remark that this second excitation is a fraction of the initial one. So, the LZT can be seen as a fractionalization process of the excitation amplitude distribution, where the two final fractions move apart from each other. Here,. 60011-p3.

(5) E. Arévalo and L. Morales-Molina. Fig. 2: (Color online) (a) The 1-ZZEGLR ω band structure; the arrows indicate the excitation motion along the lower band (organge) as well as in the upper band (green), if LZT occurs. The vertical line at k = 1.1π/2 indicates where (if necessary) α is turned off. (b) Density plot of a soliton Bloch oscillation with negligible LZT for α = −π × 10−3 (red color indicates the confinement of the excitation due to the presence of flat bands and blue color when the excitation moves along the backbone). (c) Density plot of a soliton Bloch oscillation with LZT at t = 10π for α = −5 × 10−2 (α is set to zero at t = 11π (k = 1.1π/2)). The initial soliton is a sech function with amplitude 0.05 and width of 20 lattice sites.. the fractionalization process is irrational and can be characterized by the interband tunneling ratio as shown below. Notice that the excitation fraction undergoing the LZT describes a full BO, whereas the other fraction performs only half an oscillation with a half of the period [7,10,12]. The oscillations resulting from the combination of BO and LZT are usually referred to Bloch-Zener oscillations [30]. Examples for the 1- and 2-ZZEGLR are given in figs. 2(b), (c) and 3(b), (c), respectively. In particular, fig. 2(b) corresponds to a case where no LZT occurs in the 1-ZZEGLR. We observe for initial times (t < 500) that the excitation mostly populates the sites of the central backbone, as schematically shown in the insets of the figure (see also fig. 4(a)). When the excitation is slowly driven along the lower band, an oscillation much larger than the soliton width occurs. This initial oscillation ends at t = 500 when k π/2. Around this time the excitation population rapidly moves from the central backbone to the dangling bonds (see insets in fig. 2(b) and fig. 4(a)). For 500 < t < 1000 the flat nature of the lower band is reflected as a confinement of the. Fig. 3: (Color online) (a) The same as fig. 2 but for 2-ZZEGLR. (b) α = −1.15 × 10−3 . (c) α = −7 × 10−3 .. excitation population to the dangling bonds, so that the excitation gets localized. Conversely, for times t > 1000 (k > π) the excitation is driven back to its initial position in the band structure (fig. 2(a)) while propagating in the position space. That is, the excitation population moves back from the dangling bonds to the central backbone and further oscillating motion in the position space can be observed for small values of α. This navigation process through states of a single band is actually a consequence of the adiabaticity theorem. In particular, moving through avoided crossings at very low speed allow for a significant change of the mobility properties of the system [31]. Here, the existence of flat bands are crucial for achieving a nice control of the mobility of excitations [32,33]. In fig. 2(c) the magnitude of α has been increased, so that LZT is more likely to occur. The initial oscillation, as described for fig. 2(b), occurs faster (t 10π (k π/2)) and with shorter amplitude in comparison with the soliton width. Since in some cases the Bloch-Zener oscillation amplitudes can be smaller than the soliton width, a clear observation of the fractional excitations is impossible without turning off the external force (α = 0) shortly after a LZT. This is shown in fig. 2(c) at t = 11π, when k = 1.1π/2 and signaled with a vertical line in 2(a). So, for later times (t > 11π) we observe that the fractional excitations separate from each other with their own group velocities. The excitation velocities are different, since they are located at different points of the band structure. In fact, only a fraction of the excitation population remains confined at the dangling bonds, while the other fraction propagates with constant speed along the central backbone, as shown in the figure. 60011-p4.

(6) Bloch-Zener oscillations in ribbon-shaped optical lattices. Fig. 5: Numerical estimation of the tunneling ratio PLZT for 1-ZZEGLR (solid circles) and 2-ZZEGLR (solid rhombuses); Analytical LZT estimation given in eq. (8) (solid lines).. Fig. 4: 1-ZZEGLR normalized excitation population vs. time: (a) and (b) correspond to figs. 2(b) and (c), respectively; backbone population (solid line) and dangling-bond population (dashed line). 2-ZZEGLR normalized excitation population vs. time: (b) and (c) correspond to figs. 3(b) and (c), respectively; 2-bond-site population (solid line) and 3-bond-site population (dashed line).. insets. In figs. 4(a) and (b) the excitation-population dynamics for the corresponding figs. 2(b) and (c) are shown, respectively. Interestingly, the fractionalization process observed here is different to that proposed for electronic excitations in polyacetylene [15,17]. In this case both fractional charges move away from each other with half of the charge. In figs. 3(a)–(c) a similar analysis is performed for the 2-ZZEGLR, which is characterized by two backbones, each one with alternating sites of two and three bonds. Besides, the band-gap structure approaches to a conicallike topology for small fraction values ty /tx , as shown in fig. 1(b). Interestingly, though the overall Bloch-Zener oscillation effect can be well observed, the excitationpopulation dynamics is different from the previous case. In fact, since the 2-ZZEGLR does not have dangling bonds, the flat bands are absent. So, the excitations can easily move. In fig. 3(a) it is shown schematically how excitations are driven from k = 0 along the lower band up to k = π/2, where the fractionalization process can take place. For k > π/2 the fractional excitations can also be further driven. In figs. 3(b) and (c) BO are shown with and without LZT, respectively. Unlike the 1-ZZEGLR, in the present case the excitation populations of the 3-bond and 2-bond sites appear similarly distributed with slightly higher excitation population in the 3-bond sites (see fig. 4(c)). For the BO shown in fig. 3(b), the excitation population difference between the 2- and 3-bond sites is small fairly. before and after k = π/2. Only around k = π/2 at t = 1500 almost 80% of the excitation population moves to the 3-bond sites (see fig. 4(c)). In contrast, when a LZT occurs (fig. 3(c)), an strong fluctuation in the excitation population between 2-bond and 3-bond sites is observed (see fig. 4(d)). Shortly before occurs the interband transition (t 225 in fig. 3(c)) the 3-bond sites gain in excitation population. However, around t 225, a large fraction of the excitation population rapidly moves to the 2-bond sites (see also fig. 4(d)). Beyond the LZT (225 < t 450) and after some small fluctuations the excitation populations of both site types become nearly the same. Interestingly, close to the avoiding crossing at k π/2 (t 225), where the group velocity tends to vanish, most of the excitation population accumulates in one type of lattice site. This staggered edge localization of the excitation population somewhat resembles the dynamics predicted for spin-polarized states in real carbon nanoribbons [22,34,35] despite the fact that no increase of the gap between the bands at the avoiding crossing is predicted in our system. The numerical and analytical estimations of the interband tunneling ratio PLZT as a function of α are shown in fig. 5. These tunneling ratios are estimated by measuring the fractional excitation population in the upper band after LZT with respect to the initial excitation population in the lower band. Notice that the Landau-Zener theory for a two-level avoided crossing can be described by the Hamiltonian + ∆ 2 , (7) HT = ∆ 2 − where ∆ is the coupling constant and + and − are the energies for the upper and lower bands in the limit ∆ = 0, respectively. Here, we assume that the energy distance is a linear function of the parameter k, i.e., + − − = βk, where β is a constant. Thus, when the quantity k is changed some amount k = αt, the energy distance changes as + − − = βαt. Hence, it can be shown that the tunneling ratio between the lower and the upper bands is. 60011-p5.

(7) E. Arévalo and L. Morales-Molina given by [36]. . PLZT = exp −. π 2. . ∆2 π ∆2 = exp − . d 2 βα dt (+ − − ). (8). In both scenarios, 1-ZZEGLR and 2-ZZEGR the gap ∆ and β can be estimated from their corresponding spectrum of energy. In a first scenario 1-ZZEGLR the gap ∆ = 0.2 and β ≈ 4/π (for solid circles in fig. 5), whereas in the second scenario 2-ZZEGLR, the gap ∆ = 0.1 and β ≈ 8/π (for solid rhombuses in fig. 5). We observe in the case 1-ZZEGLR that the numerical results for the tunneling ratios (solid circles in fig. 5) are very well fitted by eq. (8) (solid line in fig. 5). In the scenario for 2-ZZEGLR the eq. (8) (solid line in fig. 5) does not show an exact agreement with the simulations (solid rhombuses in fig. 5), nonetheless it reproduces very well the behavior. Small discrepancies are attributed to the fact that numerical simulations were carried out for a wave packet, whereas the analytical expresions were derived for a single plane wave with a well defined Bloch momentum. Here the small nonlinear interaction is negligible. In the strong nonlinear regime, however, the nonlinear contribution have to be summed up to these discrepancies, leading in some scenarios to a complete collapse of the linear theory [37]. Finally, from results in fig. 5, we observe that the LZT is less likely to occur in the 1-ZZEGLR. It means that, in the presence of an external field, excitations trapped in dangling bonds are less prone to get loss in other bands than those trapped in the edges (without dangling bonds, as in 2-ZZEGLR). Conclusions. – In conclusion we have shown that the motion of localized excitations in systems with dangling bonds and characterized by flat bands can be manipulated so that excitations are trapped or set in motion. 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