• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Nonlinear dynamical stability of gap solitons in Bose–Einstein condensate loaded in a deformed honeycomb optical lattice?

    2021-12-22 06:50:06HongjuanMeng蒙紅娟YushanZhou周玉珊XuepingRen任雪平XiaohuanWan萬曉歡JuanZhang張娟JingWang王靜XiaobeiFan樊小貝WenyuanWang王文元andYurenShi石玉仁
    Chinese Physics B 2021年12期
    關(guān)鍵詞:張娟王靜小貝

    Hongjuan Meng(蒙紅娟) Yushan Zhou(周玉珊) Xueping Ren(任雪平)Xiaohuan Wan(萬曉歡) Juan Zhang(張娟) Jing Wang(王靜)Xiaobei Fan(樊小貝) Wenyuan Wang(王文元) and Yuren Shi(石玉仁)

    1College of Physics and Electronic Engineering,Northwest Normal University,Lanzhou 730070,China

    2Key Laboratory of Atomic and Molecular Physics&Functional Materials of Gansu Province,Northwest Normal University,Lanzhou 730070,China

    Keywords: gap soliton,Bose–Einstein condensate,deformed honeycomb optical lattice

    1. Introduction

    Bose–Einstein condensate (BEC) in optical lattices has generated considerable interesting research for studying many quantum phenomena in periodic potentials.[1–4]It arises in many applications ranging such as quantum simulators[5–7]and quantum computation.[8]The cubic lattice is a simple representative example of an optical lattice. Experiments with ultracold atoms have utilized exotic lattices such as triangular,[9]honeycomb,[10]and Kagome lattices.[11]The honeycomb optical lattices can lead to significantly different results due to the underlying symmetries. A special feature of a perfect honeycomb lattice is the two-lowest-band-formed Dirac points.These unique characteristics could provide new opportunities to investigate many interesting physical phenomena in many systems such as monolayer graphene,[12,13]electron gases in semiconductors,[14]coupled photonic lattices,[15,16]and photonic waveguides.[17]

    On the other hand,the interplay between lattice periodicity and atomic interaction brings a large number of interesting nonlinear effects.[18–20]In many cases, periodicity relates to the band structure of the dispersion relation and Bloch waves.Meanwhile,the nonlinearity can induce self-phase modulation and can also couple waves between different modes or Bloch bands. The existence of the nonlinearly localized structures that reside in the linear energy gaps has attracted considerable interest in recent years.[21–23]They are sometimes referred to as gap solitons,which are often considered theoretically as bifurcations from the Bloch-band edges into the band gaps.[24,25]The general properties of gap solitons have been intensively studied in many different nonlinear periodic systems, examples include but are not limited to optical waveguide arrays and photonic lattices,[26,27]semiconductor microcavities,[28]and BEC in optical lattices.[1,23,29,30]

    Gap solitons in honeycomb optical lattice have been studied both theoretically and experimentally.[17]The authors of Ref. [31] demonstrated the gap solitons residing in the gap between the second and the third bands in honeycomb photonic lattices. In Ref. [32], the authors showed solitons and necklaces in the first band gap of a two-dimensional optically induced honeycomb defocusing photonic lattice. In Ref.[33],the authors studied gap solitons in honeycomb dynamical lattices with the cubic nonlinearity. However,there is still great significance to study gap solitons of BEC in honeycomb optical lattice,especially for the interplay between the honeycomb optical lattice and the nonlinearity affects on the existence and dynamical stability of multipole gap solitons.

    Here, we investigate gap solitons of BEC in deformed honeycomb optical lattices with atomic interaction and study their nonlinear dynamical stability. Deformed honeycomb lattices are promising platforms. With the deformation of the honeycomb lattice,the symmetries can be broken,then a gap opens between the first two bands,and the lattice can support a gap soliton.[34,35]The existence of the gap soliton provides a way to detect the symmetry breaking. The deformed honeycomb lattices have been studied extensively,such as nonlinear wave packets,[36]Klein tunneling,[37]fractional topological quantum states,[38]nondegenerate chiral phonons,[39]and topological valley transport.[40]

    In this paper, we show the linear band-gap structure of the deformed honeycomb optical lattices and the existence of the gap solitons. We find that the system admits dipole gap solitons both in the first gap and in the semi-infinite gap, depending on the repulsive or attractive properties of atomic interactions. The dipole gap soliton can have their bright solitary structures be in-phase or out-of-phase. Then,we analysis the nonlinear dynamical stability of these dipole gap solitons by using direct simulations of the Gross–Pitaevskii(GP)equation.Finally,we derive the envelope equations of Bloch waves and analyze these gap solitons.

    The paper is organized as follows: In Section 2,we introduce the model of BEC in deformed honeycomb optical lattices. In Section 3,we show the linear band-gap structure and the existence of the gap solitons. Following this,the analysis of nonlinear dynamical stability of the gap solitons is given in Section 4.Finally,we give summary and conclusion in Section 5.

    2. Model of BEC in a honeycomb optical lattice

    We consider that BEC is loaded in a honeycomb optical lattice. The honeycomb optical lattice can be experimentally generated by superposing three coplanar traveling laser beams.[41,42]The three coplanar traveling laser beams have the same angular frequencyωL=ck0withk0being the wave vector of the radiation. Then, the lattice potential is denoted byV(r)with[35]

    Fig. 1. Contour plot of the potential energy for (a) η =1 (the optical lattice has a perfect standard graphene-like honeycomb),and for(b)η=0.6(the optical lattice becomes a deformed honeycomb).The white dotted line in the figure is a schematic diagram of the corresponding lattice sites.

    3. Gap solitons and theoretical analysis

    3.1. Band-gap structure

    In this subsection, we show the band-gap structure and the existence of the gap solitons. We first investigate the linear band-gap structure of the deformed honeycomb optical lattices. Let us consider quantum mechanical of BEC in such periodic potentials. Gap solitons are stationary solutions to Eq. (2), we can apply Bloch’s theorem, which suggests that the Bloch waves are the nonlinear eigenstates of the following formΨ(r,t)=φk(r)exp(i(k·r ?μt)).[44]Hereμis a real chemical potential, andφkhas the same period as the potential.Then we can obtain the following equation for each Bloch wave stateφk:The set of eigenvaluesμ(k)then forms Bloch bands.

    Since gap solitons only exist in the linear energy gaps between the Bloch bands,it is important to first identify the positions of these gaps. To do this,we can neglect the nonlinear terms, i.e.,g=0, and calculate the linear Bloch spectrum by numerical diagonalization of Eq.(3). The band gap spectrum generated by the solution of the linearized version of the equations is shown in Fig. 2. Due to the expected periodicity of the solution to Eq. (3), there are several numerical methods to find the Bloch waveφk.[45–47]We use Fourier collocation method[48]for numerically computing these Bloch bands.

    In Fig.2,we only show the lowest five Bloch bands. One can see that a band gap[49]is separated by the two Bloch bands.The lowest gap below the lowest Bloch band is called the semiinfinite gap and the higher gaps are called the first band gap,the second band gap,and so on.

    Fig.2. (a)The linear Bloch band gap structure of BEC in a deformed honeycomb optical lattice as a function of k(left). (b)Axial side view of the linear Bloch band gap structure as a function of kx,shaded areas correspond to the linear bands.

    The band gap structure of BEC in honeycomb optical lattices depends not only on the depth of the optical lattices but also on the degree of deformed honeycomb. Figure 2 shows how the band gap structure of BEC in honeycomb optical lattices is affected by lattice depthV0and deformation parameterη.

    Figure 3(a) shows the energies of different Bloch bands overlaping for a suffciiently small lattice depthV0,in this case no other band gap appears except the semi-infniite gap. With increasing lattice depthV0, the window of band gap will become more and more obvious.Figure 3(b)shows that the band gap structure depends on the deformation of the honeycomb.Forηchanging in the range<η ≤1,the smaller theη,the more deformed the honeycomb lattice. With increasingη,the frist gap becomes narrower and narrower and eventually disappears. Figure 3(b) shows an interesting phenomenon, that is,the more deformed the honeycomb lattice,the more prominent the frist gap. For a standard honeycomb lattice, the frist gap does not exist.Therefore,in order to obtain gap solitons in the frist gap,one must choose a deformed non-standard honeycomb lattice.In this article,we choose a deformed honeycomb lattice forη=0.6 as an example.

    Fig. 3. The band gap structure of BEC in honeycomb optical lattices with increasing lattice depth(a)and deformation parameter η (b).

    3.2. The gap solitons

    The interplay between spatial periodicity and nonlinear dynamical evolution harbors a large number of interesting physical many-body effects. Gap solitons is the existence of unique soliton solutions that reside in the linear energy gaps between the Bloch bands. Gap solitons spatially localized in gap modes may appear as a result of the balance between the dispersion and the nonlinearity. For BECs in an optical lattice,the hoppings between different lattice sites play the role of dispersion,and the atom-atom interaction brings the nonlinearity.

    In what follows, we consider the nonlinearity brought from the atom-atom interaction and stationary solutions in the linear Bloch bands gaps. We look for gap solitons by numerical methods such as the Newton conjugate-gradient (NCG)method,[50]which is based on iterating certain time evolution equations associated with the linear stability eigenvalue problem and is one of the powerful numerical techniques to find the solitary wave solutions of a nonlinear evolution equation.The methods are applied to compute both the ground states and excited states in a large number of physical systems with and without periodic potentials. We take the Gaussian profile as the initial condition for iterating. One can adjust the amplitude,width and center position of the initial Gaussian profiles to get the wanted gap solitons. For example, in order to obtain unipole gap solitons, the initial Gaussian wave packet is mainly distributed in a certain potential well.In order to obtain the dipole gap solitons,the initial two Gaussian wave packets are mainly distributed in two adjacent potential wells.

    After numerical techniques mentioned above,we find the unipole gap solitons in the semi-infinite gap. Figure 4(a)shows gap solitons in the semi-infinite gap withμ=4.8 andg=?1 as an example. Figure 4(b)shows gap solitons in the semi-infinite gap withμ=5.01 andg=?1 as an example.The gap solitons in the first gap can be obtained withg>0.

    Fig.4.Gap solitons in the semi-infinite gap with attractive atomic interaction g=?1.0:(a)forμ=4.8,(b)forμ=5.01.The white dotted line in the figure is a schematic diagram of the corresponding lattice sites.

    In order to study the properties of gap solitons more clearly,we pay attention to effect of nonlinear eigenenergyμon the amplitude of the gap solitons and the total power of gap solitons. In order to see this, we define the amplitudeAand the total powerPof the gap solitons given as follows:

    In Fig. 5, we show amplitudeAand powerPof the gap solitons for nonlinear eigenenergyμwith respect to on-site interactiongin the range of the band gap. Here, we modulate the nonlinear eigenenergyμfrom semi-infinite band gap to first band gap. Here,we takeg=?1 andg=1 as examples for gap solitons in semi-infinite band gap and first band gap,respectively. When the nonlinear eigenenergyμmoves toward the band,numerical results indicate that the amplitude of gap solitons both in semi-infinite gap and in first gap approach to infinitesimal. However, when the nonlinear eigenenergyμmoves toward the band,the power of gap solitons both in semiinfinite gap and in first gap approach to infinity.

    There is a very interesting phenomenon shown in Fig. 5 for the powerPof the gap solitons versus eigenenergyμin the range of the band gap. When the nonlinear eigenenergyμmoves toward the band, the powerPof gap solitons both in semi-infinite gap and in first gap decreases slowly first, then increases gradually. This means that the relationship between the amplitudeAand powerPof the gap solitons is inconsistent with the eigenenergyμ,they move in step whenμfarther away from the band,and they move out of step whenμcloser to the band. Thus, they are bifurcated from the Bloch band edges.

    Fig.5.The amplitude A and power P of the gap solitons versus eigenenergy μ in the range of the band gap. Shaded areas correspond to the linear bands.

    It is worth emphasizing that the appearance of gap solitons both in the first gap and in the semi-infinite gap depends on the repulsive or attractive properties of atomic interactions.The first gap solitons appear only in repulsive atomic interactions regime (g> 0). On the other hand, semi-infinite gap solitons appear only in attractive atomic interaction regime(g<0). The gap solitons are intrinsically related to the extended Bloch waves,[51–54]which can be understood from this perspective of the composition relation between Bloch waves and gap solitons.[52,55]With this composition relation, many conclusions can be drawn for gap solitons from Bloch waves in nonlinear periodic systems. The reason why the appearance of gap solitons in different gaps depends on the properties of atomic interactions is that the atomic interactions of different properties have different effects on the Bloch waves.[52,55]

    Next,we show dipole gap solitons in the semi-infinite gap withμ= 4.8 andg=?1 in Fig. 6 as an example, (a) for the out-of-phase gap solitons, (b) for the in-phase gap solitons. The dipole gap solitons in the first gap which are similar to solitons of the semi-infinite gap can also be obtained withg>0. In order to study the properties of dipole gap solitons more clearly,we pay attention to effect of nonlinear eigenenergyμon the total power of dipole gap solitons in Fig.7. The amplitudes of the dipole gap solitons in the first gap and in the semi-infinite gap are similar to unipole gap solitons. In Fig.7(a),when the nonlinear eigenenergyμmoves toward the band, the powerPof gap solitons of in-phase in the semiinfinite gap and out-of-phase in the first gap decreases slowly first, then increases gradually. We call gap solitons such that they are bifurcated from the Bloch band edges. In Fig. 7(b),when the nonlinear eigenenergyμmoves toward the band,the powerPof gap solitons of out-of-phase in the semi-infinite gap and in-phase in the first gap decreases slowly. We call gap solitons such that they are not bifurcated from the Bloch band edges.

    Fig. 6. Dipole gap solitons in the semi-infinite gap with attractive atomic interaction g=?1.0 andμ =4.8(a)for out-of-phase gap solitons and(b)for in-phase gap solitons.

    Fig.7. The power P of the dipole gap solitons versus eigenenergyμ in the range of the band gap. Shaded areas correspond to the linear bands.

    3.3. Theoretical analysis of gap solitons

    In this subsection, we derive the envelope equations of small-amplitude Bloch-wave packets from edges of Bloch bands. Considering the stationary solutionsψ(r) of Eq. (2)with the formΨ(r,t)=ψ(r)e?iμt. When the solutionψ(r)is infinitesimal,this solution is a linear superposition of these two Bloch modes. Whenψ(r) is small but not infinitesimal,this solution then becomes a combination of these two Blochwave packets and can be expanded into a multiscale perturbation series:

    Atε3,the equation forψ2is

    The solution to Eq.(7)has the following form:

    Equation (10) is the leading-order two-Bloch-wave packets,whereX=εxandY=εyare slow spatial variables of envelope functionsA1andA2.

    The solution to Eq.(8)is

    Substituting theψ0,ψ1into Eq. (9) and using the Fredholm condition (which requires being orthogonal to the homogeneous solutionsp1(x)p2(y)andp1(y)p2(x)),we can obtain

    Here,the superscript?represents complex conjugation,Assume that the modulation wave is a Gaussian profile one,then it can be expressed as

    whereAandW(W>0) are the amplitude and width of the solitons, respectively. Note that Eq. (16) is not an exact solution to Eq. (15). Substituting Eq. (16) into Eq. (15) yields the residual error. We minimize the magnitude of the residual error and then have the equation. We solve the equation in a semi-numerical manner,which can be performed easily in Mathematica software by the NSolve command. The results read

    It can be seen from Eq.(17)that the amplitudeAof the soliton depends on the parametersμ1,g,andα,whereas the widthWof the soliton depends on the parametersD1andμ1. Equation(17)also gives the condition for the existence of gap solitons,that is,μ1andgmust have the same sign,butμ1andD1have the opposite signs.In the semi-infinite gap,ifD1>0,one will haveμ1<0 andg<0,with the increase of|μ1|,the value ofμdecreases, and the value ofμis farther away from the band. The amplitude of the soliton increases,which is consistent with the previous numerical results shown in Fig.5. For the first gap,ifD1<0,one will haveμ1>0 andg>0. With the increase of|μ1|,the value ofμincreases,and the value ofμis also farther away from the first band. The amplitude of the soliton increases, which is also consistent with the previous numerical results shown in Fig.5.

    4. Linear stabilities and nonlinear dynamical stabilities of gap solitons

    Until now, we have shown the existence of unipole and dipole gap solitons of BEC in a deformed honeycomb optical lattice. One unique feature of BEC in optical lattices is the occurrence of instability. To do this,one can analyze the nonlinear and the linear stability of these gap solitons using direct simulations of the GP Eq.(2)and its linearized equation.

    4.1. Linear stabilities of gap solitons

    The linear stability (so-called Bogolyubov-de Gennes or BdG) analysis is employed in order to consider the fate of small amplitude perturbations and the potential robustness of the solutions. The linear stability of these gap solitons can be modified state and inserted into the time-dependent GP equation. By keeping only the linear term in the perturbation, one can describe the time evolution of the small perturbation,[46,56–58]which yields the linear eigenvalue problem.If the corresponding eigenvalues are real,the perturbation will grow exponentially,which leads to dynamic instability.

    Now we study the linear stability of these gap solitons by numerical methods. We perturb these solitons as

    The above linear eigenvalue problem (19) can be solved numerically by the finite difference method or the Fourier collocation method.[48]To illustrate,we consider unipole solitons in the semi-infinite gap and in the first gap,as shown in Fig.8.Also,we consider dipole gap solitons in the semi-infinite gap as an example,as shown in Fig.9.Stability spectra for unipole solitons are shown in Fig. 8. It is seen that unipole gap solitons far away from the band have stable eigenvalues and thus are linearly stable in Figs. 8(a) and 8(d). However, unipole gap solitons near the band have unstable eigenvalue and thus are linearly unstable in Figs. 8(b) and 8(c). Figure 9 shows dipole gap solitons of stability spectra in the semi-infinite gap.It is seen that in-phase dipole gap solitons are all linearly unstable in Figs.9(a)and 9(b). Figures 9(c)and 9(d)show that out-of-phase dipole gap solitons far away from the band are linearly stable,and near the band are linearly unstable.

    Fig.8. The linear stability analysis of unipole solitons: (a)and(b)for unipole solitons in the semi-infinite gap;(c)and(d)for unipole solitons in the first gap.

    Fig. 9. The linear stability analysis of the dipole gap solitons in the semi-infinite gap:(a)and(b)for in-phase solitons bifurcated from band edges; (c) and (d) for out-of-phase solitons not bifurcated from band edges.

    4.2. Nonlinear dynamical stabilities of unipole gap solitons

    In the above section, we have analyzed the linear stability problem for unipole and dipole gap solitons. However,the important issue is the nonlinear dynamical stability of these unipole gap solitons and dipole gap solitons,which can be induced by the presence of nonlinear interactions and periodic potential. Dynamical instability implies that small deviations from the stationary solution grow exponentially in time. In order to examine the nonlinear dynamical stability of these unipole gap solitons and dipole gap solitons found above,we directly compute Eq. (2) over a long time. The initial conditions are taken to be the gap soliton with 1%perturbation in the amplitude and phase. We consider a stable gap soliton satisfying the two conditions as follows: it should preserve its peak amplitude,and it should preserve its position on the lattices.

    Fig.10. Nonlinear evolution of unipole gap solitons in the semi-infinite gap with μ =4.8 (corresponding to the point a in Fig. 5). The initial conditions were taken with 1%perturbation in the amplitude and phase to trigger potential instabilities: (a)t=0,(b)t=100.

    We first consider the nonlinear dynamical stabilities of the unipole gap solitons, and takeμ=4.8 in semi-finite gap as an example (corresponding to the pointain Fig. 5). In Fig.10,we show time evolution of unipole gap solitons in the semi-infinite gap fort=0 and 100. It can be seen from this figure that the amplitude and position on the lattices of the unipole gap solitons are preserved. The gap soliton is nonlinear dynamical stable. This is confirmed by the linear stability analysis in Fig.8(a).

    Then, the nonlinear dynamical stability properties of the unipole gap solitons withμ=5.01 is examined, as shown in Figs.11(a)and 11(b)fort=0 and 100,respectively. We cam see the maximum amplitude and the location of the center of mass versus the evolution timet. During the nonlinear dynamical evolution, the peak amplitude undergoes finite-time collapse, its position on the lattice drift-unstable solitons is typically characterized by“humps”that drift from lattice maxima toward nearby minima. Obviously, the unipole gap soliton shows nonlinearly dynamical unstable,which is also confirmed by the linear stability analysis in Fig.8(b).

    Fig.11. Nonlinear evolution of unipole gap solitons in the semi-infinite gap with μ =5.01(corresponding to the point b in Fig.5). The initial conditions are taken with 1% perturbation in the amplitude and phase to trigger potential instabilities: (a)t=0 and(b)t=100.

    In order to examine the nonlinear dynamical stability of unipole gap solitons in the first gap,we also directly compute Eq. (2) over a long time. The initial conditions were taken to be a unipole gap soliton with 1%perturbation in amplitude and phase in Fig.12. It can be seen that the unipole gap soliton shows nonlinearly dynamical unstable and (b) nonlinear dynamical stable. These are also confirmed by the linear stability analysis in Figs.8(c)and 8(d),respectively.

    Fig. 12. Nonlinear evolution of the unipole gap solitons in the first gap with repulsive atomic interactions g=1. The initial conditions are taken with 1%perturbation in the amplitude and phase to trigger potential instabilities:(a)μ=5.36 and(b)μ=5.56.The unipole gap soliton corresponds to the points c and d in Fig.5,respectively.

    It is well known that the power versusμ, i.e., dP/dμ,plays an important role in determining the stability properties of the solitons. This connection between the slope of the power curve and the linear stability is the so-called Vakhitov–Kolokolov stability criterion.[59]This criterion states that with the continuous change ofμ, a pair of real eigenvalues move towards the origin, collide there,and then bifurcate along the imaginary axis. The power minimum point turns out to be precisely the bifurcation point,which indeed suggests that linear stability of these solitary waves is intimately related to the sign of the slope of the power curve.[48]In a large number of simulations,we have ever plotted the total power versusμfor unipole gap solitons corresponding to the semi-infinite gap and the first gap,and have observed two recurring features:

    (a)In the semi-infinite gap,the unipole gap soliton is nonlinear dynamical stable only if its power decreases with increasingμ, i.e., dP/dμ<0(as the nonlinear dynamical evolution of the pointain Fig.5).On the contrary,the unipole gap soliton will be nonlinear dynamical unstable only if its power increases with increasingμ,i.e.,dP/dμ>0(as the nonlinear dynamical evolution of the pointbin Fig.5).

    (b) In the first gap, the unipole gap soliton will be nonlinearly dynamically stable only if its power increases with increasingμ,i.e.,dP/dμ>0(as the nonlinear dynamical evolution of the pointdin Fig.5). Meanwhile,the unipole gap soliton will be unstable only if its power decreases with increasingμ,i.e.,dP/dμ>0(as the nonlinear dynamical evolution of the pointcin Fig.5).

    The nonlinear dynamical stabilities of unipole gap solitons depend on dP/dμ,which are opposite in semi-infinite gap and first gap. The reason for this phenomenon is that there are different requirements for the properties of atomic interaction when solitons occur in two gaps.

    4.3. Nonlinear dynamical stabilities of dipole gap solitons

    In the semi-infinite gap, the out-of-phase and in-phase dipole gap solitons exist (shown in Fig. 6) for the attractive atomic interactions regime. Now,we investigate the nonlinear stability properties of the lattice soliton. The in-phase dipole gap soliton corresponds to Fig.6(b). The dynamics of nonlinear evolution of the in-phase dipole gap soliton withμ=4.8 is shown in Fig. 13. Figures 13(a) and 13(b) depict the density profiles of the in-phase dipole gap soliton fort=0 and 100, respectively. In fact, in the semi-infinite gap, for all inphase dipole gap solitons their nonlinear dynamic evolution will show a similar phenomenon nonlinearly dynamical unstable,as seen in Fig.7(a).In the first gap,the in-phase dipole gap solitons satisfy the condition dP/dμ>0. Solutions near the band have unstable eigenvalues and thus are linearly unstable.However, solitons far away from the band have no unstable eigenvalues and thus are linearly stable.

    Fig. 13. Nonlinear evolution of the in-phase dipole gap soliton in the semi-infinite gap with repulsive atomic interactions g=?1. The inphase dipole gap soliton corresponds to Fig.6(b). The initial conditions are taken with 1% perturbation in the amplitude and phase to trigger potential instabilities.

    Fig. 14. Nonlinear evolution of the out-of-phase dipole gap soliton in the semi-infinite gap with repulsive atomic interactions g=?1 andμ =4.8. The out-of-phase dipole gap soliton corresponds to Fig.6(a).The initial conditions are taken with 1% perturbation in the amplitude and phase to trigger potential instabilities: (a)t=0 and(b)t=100.

    Figure 14 shows nonlinear evolution of the out-of-phase dipole gap soliton in the semi-infinite gap atμ=4.8, corresponds to the soliton in Fig.6(a). Figures 14(a)and 14(b)fort=0 and 100 respectively. It can be seen from this figure that the amplitude and position on the lattices of the out-of-phase dipole gap solitons are preserved.The out-of-phase dipole gap soliton is nonlinear dynamical stable. Figure 15 shows nonlinear evolution of the out-of-phase dipole gap soliton corresponding to the semi-infinite gap atμ=5.0.The dipole gap soliton also corresponds to the Fig. 6(a). Figures 15(a) and 15(b)fort=0,100 respectively. It can be seen from this figure that the maximum amplitude and the location of the center of mass versus the evolution timet. During the nonlinear dynamical evolution, the peak amplitude undergoing finite-time collapse, its position on the lattice drift-unstable solitons are typically characterized by“humps”that drift from lattice maxima toward nearby minima. Obviously the dipole gap soliton shows nonlinearly dynamical unstable. That is, in the semiinfinite gap,when solutions near the band have unstable eigenvalue and thus are linearly unstable.Solitons far away from the band have no unstable eigenvalues and thus are linearly stable.However,for all out-of-phase dipole gap solitons show nonlinearly dynamical unstable in the first gap. It can be seen from Fig.7.

    Fig. 15. Time evolution of the out-of-phase dipole gap solitons in the semi-infinite gap withμ=5.0,g=?1. The initial conditions are taken with 1% perturbation in the amplitude and phase to trigger potential instabilities: (a)t=0 and(b)t=100.

    5. Summary

    We have investigated the existence and dynamical stability of multipole gap solitons in BEC loaded in a deformed honeycomb optical lattice. Honeycomb lattices possess a unique band structure, the first and second bands intersect at a set of so-called Dirac points. We find that,when the lattice is gradually deformed, at some strong enough deformation, a gap forms between the first and second bands. We show that the gap in such a deformed lattice can form gap soliton. Both the unipole gap soliton and dipole gap soliton are found. The dipoles can have their bright solitary structures being in-phase or out-of-phase. The linear stabilities and nonlinear stabilities of these gap solitons are investigated.

    The stabilities of unipole gap solitons depend on dP/dμ.In the semi-infinite gap,the unipole gap soliton will be stable if its power and eigenenergy satisfy dP/dμ<0, whereas the unipole gap soliton will be unstable if its power and eigenenergy satisfy dP/dμ>0.In the first gap,the unipole gap soliton will be stable if its power and eigenenergy satisfy dP/dμ>0,

    whereas the unipole gap soliton will be unstable if its power and eigenenergy satisfy dP/dμ>0. The stabilities of dipole gap solitons are independent of the dP/dμ. For the in-phase dipole gap soliton in semi-infinite gap and the out-of-phase dipole gap solitons in first gap,they are always unstable. For the out-of-phase dipole gap soliton in semi-infinite gap and the in-phase dipole gap solitons in first gap,they are stable far away from the band,while unstable near the band.

    Our work provides a qualitative and quantitative guideline towards the more detailed understanding of the possible dynamical features and of the interactions between the coherent structures of strongly correlated states of BEC in 2D optical lattices.

    猜你喜歡
    張娟王靜小貝
    貝小貝哭了
    Clinical study of warm needling moxibustion combined with entecavir in the treatment of compensated cirrhosis due to chronic hepatitis B
    夢寐以求的生日禮物
    The Management Methods And Thinking Of Personnel Files
    客聯(lián)(2021年9期)2021-11-07 19:21:33
    The Development of Contemporary Oil Painting Art
    青年生活(2019年16期)2019-10-21 01:46:49
    小貝趕集記
    兒童時代(2017年5期)2017-07-18 11:55:35
    Income Inequality in Developing Countries
    商情(2017年17期)2017-06-10 12:27:58
    Let it Go隨它吧
    小貝螺的大世界
    小貝當(dāng)刮臉
    精品午夜福利在线看| 国产成人精品婷婷| 少妇高潮的动态图| 国产乱人偷精品视频| 久久午夜福利片| 成人漫画全彩无遮挡| 18禁裸乳无遮挡动漫免费视频| 国产爱豆传媒在线观看| 国产黄片美女视频| 日韩欧美一区视频在线观看 | 男女国产视频网站| 日韩中文字幕视频在线看片 | 亚洲四区av| 国国产精品蜜臀av免费| 亚洲国产欧美在线一区| 亚洲精品视频女| 免费人成在线观看视频色| 午夜福利视频精品| 欧美性感艳星| 2018国产大陆天天弄谢| 欧美日韩精品成人综合77777| 两个人的视频大全免费| 亚洲久久久国产精品| 免费黄色在线免费观看| 亚洲自偷自拍三级| 夫妻性生交免费视频一级片| 国产精品久久久久久精品古装| 免费观看性生交大片5| 久久99热这里只有精品18| 久久99蜜桃精品久久| 人体艺术视频欧美日本| 99re6热这里在线精品视频| 成人综合一区亚洲| 老女人水多毛片| 观看免费一级毛片| 亚洲va在线va天堂va国产| av又黄又爽大尺度在线免费看| 亚洲美女黄色视频免费看| 久久久久久久久久久免费av| 免费黄频网站在线观看国产| 免费观看的影片在线观看| 18禁在线播放成人免费| 我的女老师完整版在线观看| 午夜日本视频在线| 一个人看的www免费观看视频| 久久久精品免费免费高清| 亚洲人成网站在线播| videossex国产| 狂野欧美激情性xxxx在线观看| 国产真实伦视频高清在线观看| 久久久久国产网址| 老女人水多毛片| 极品教师在线视频| 国产人妻一区二区三区在| 国产成人免费无遮挡视频| 亚洲综合色惰| 欧美三级亚洲精品| 久久鲁丝午夜福利片| 中文字幕久久专区| 亚洲精品一区蜜桃| 中文字幕亚洲精品专区| 国产中年淑女户外野战色| 亚洲精品国产成人久久av| 亚洲怡红院男人天堂| 永久网站在线| 少妇高潮的动态图| 伦精品一区二区三区| 国产精品国产av在线观看| av黄色大香蕉| 欧美极品一区二区三区四区| 麻豆成人午夜福利视频| 高清不卡的av网站| 嫩草影院新地址| 亚洲欧洲日产国产| 自拍欧美九色日韩亚洲蝌蚪91 | 日韩在线高清观看一区二区三区| 久久久久久久亚洲中文字幕| 简卡轻食公司| 免费观看性生交大片5| 国产精品三级大全| 亚洲精品日韩在线中文字幕| 久久久久久久久久久免费av| 一个人看视频在线观看www免费| 国产欧美日韩精品一区二区| 成年人午夜在线观看视频| 一级a做视频免费观看| 亚洲av中文av极速乱| 亚洲最大成人中文| 又爽又黄a免费视频| 在线观看免费高清a一片| 只有这里有精品99| 在线看a的网站| 亚洲国产高清在线一区二区三| 夫妻性生交免费视频一级片| 亚洲成色77777| 波野结衣二区三区在线| 精品熟女少妇av免费看| 亚洲欧美一区二区三区黑人 | 亚洲最大成人中文| 91精品一卡2卡3卡4卡| 日日摸夜夜添夜夜添av毛片| 在线亚洲精品国产二区图片欧美 | 黄色一级大片看看| 午夜视频国产福利| 亚洲av成人精品一区久久| 女的被弄到高潮叫床怎么办| 国产亚洲一区二区精品| 男人和女人高潮做爰伦理| 亚洲综合精品二区| 一级二级三级毛片免费看| 高清av免费在线| 日韩中文字幕视频在线看片 | 一级av片app| 丝袜脚勾引网站| 国产成人免费无遮挡视频| 天堂8中文在线网| 一级a做视频免费观看| 免费不卡的大黄色大毛片视频在线观看| 久久精品久久久久久噜噜老黄| 日日啪夜夜撸| 亚洲激情五月婷婷啪啪| 中文天堂在线官网| 久久人妻熟女aⅴ| 亚洲中文av在线| 五月开心婷婷网| 女人久久www免费人成看片| 亚洲国产日韩一区二区| 大香蕉久久网| 亚洲欧美日韩东京热| 麻豆成人av视频| 久久亚洲国产成人精品v| 亚洲欧美成人精品一区二区| 国产毛片在线视频| 中国美白少妇内射xxxbb| 精品酒店卫生间| 看免费成人av毛片| 精品少妇黑人巨大在线播放| 久久鲁丝午夜福利片| 99精国产麻豆久久婷婷| 国产精品久久久久久久电影| 夜夜看夜夜爽夜夜摸| 波野结衣二区三区在线| 中文字幕人妻熟人妻熟丝袜美| 九九爱精品视频在线观看| 国产精品伦人一区二区| 中文字幕精品免费在线观看视频 | 亚洲精品自拍成人| 欧美bdsm另类| 久久精品久久久久久噜噜老黄| 99久久中文字幕三级久久日本| 国产欧美另类精品又又久久亚洲欧美| 亚洲内射少妇av| 国产精品爽爽va在线观看网站| 成人影院久久| 亚洲欧美日韩卡通动漫| 国产成人a∨麻豆精品| 亚洲高清免费不卡视频| 久久 成人 亚洲| 亚洲精品成人av观看孕妇| 麻豆乱淫一区二区| 建设人人有责人人尽责人人享有的 | 九九在线视频观看精品| 男女边摸边吃奶| av在线蜜桃| 欧美xxⅹ黑人| av国产免费在线观看| 最近最新中文字幕免费大全7| 有码 亚洲区| 舔av片在线| 国产视频首页在线观看| 在线观看三级黄色| 国产免费一区二区三区四区乱码| 亚洲不卡免费看| 国产精品.久久久| 中文天堂在线官网| 日日撸夜夜添| av国产久精品久网站免费入址| 特大巨黑吊av在线直播| 国产成人精品一,二区| 成人免费观看视频高清| 寂寞人妻少妇视频99o| 91久久精品电影网| 亚洲av日韩在线播放| 亚洲欧美日韩东京热| h日本视频在线播放| 在线天堂最新版资源| 高清黄色对白视频在线免费看 | 老女人水多毛片| 中文字幕免费在线视频6| 在线观看一区二区三区激情| 网址你懂的国产日韩在线| 国产女主播在线喷水免费视频网站| 最黄视频免费看| 精品久久久久久久久av| 久久久久国产精品人妻一区二区| 国产成人aa在线观看| 久久精品久久久久久久性| 中文字幕亚洲精品专区| av线在线观看网站| 亚洲人成网站高清观看| 欧美一级a爱片免费观看看| 国产高清三级在线| 成人午夜精彩视频在线观看| 久久av网站| 国产精品一区二区在线观看99| 国产av精品麻豆| 久久人妻熟女aⅴ| 精品人妻偷拍中文字幕| 成人毛片a级毛片在线播放| 久久精品人妻少妇| 深夜a级毛片| 国产精品秋霞免费鲁丝片| 亚洲精品色激情综合| 成人国产av品久久久| 在线观看国产h片| 一级片'在线观看视频| 国国产精品蜜臀av免费| 日本av手机在线免费观看| 亚洲欧美清纯卡通| 国内精品宾馆在线| 2021少妇久久久久久久久久久| 综合色丁香网| 免费av中文字幕在线| 嫩草影院新地址| 夜夜骑夜夜射夜夜干| 午夜激情福利司机影院| 香蕉精品网在线| 国产欧美日韩一区二区三区在线 | 少妇被粗大猛烈的视频| 只有这里有精品99| 99久久精品热视频| 亚洲自偷自拍三级| 青青草视频在线视频观看| 精品久久久久久久末码| 久久久久网色| 欧美3d第一页| 一个人免费看片子| 日韩成人伦理影院| 黄色配什么色好看| 在线看a的网站| 亚洲熟女精品中文字幕| 亚洲一级一片aⅴ在线观看| 久久久久人妻精品一区果冻| 国产精品久久久久久av不卡| 欧美精品亚洲一区二区| 亚洲av在线观看美女高潮| 国产成人午夜福利电影在线观看| 在线天堂最新版资源| 亚洲精品视频女| 国产伦精品一区二区三区视频9| 国产精品爽爽va在线观看网站| 免费观看的影片在线观看| 丝瓜视频免费看黄片| 18+在线观看网站| 亚洲欧美一区二区三区黑人 | a级毛色黄片| 少妇精品久久久久久久| h日本视频在线播放| 亚洲精品,欧美精品| 国产一区亚洲一区在线观看| 久久6这里有精品| 人妻少妇偷人精品九色| 伊人久久国产一区二区| 亚洲国产日韩一区二区| freevideosex欧美| 一级a做视频免费观看| 欧美日韩国产mv在线观看视频 | 天天躁夜夜躁狠狠久久av| 亚洲av男天堂| 亚洲成人手机| 亚洲精品456在线播放app| 蜜桃在线观看..| 交换朋友夫妻互换小说| 99久久精品热视频| 天美传媒精品一区二区| 韩国av在线不卡| 99热这里只有是精品50| 国产探花极品一区二区| 亚洲精品久久久久久婷婷小说| 又粗又硬又长又爽又黄的视频| videossex国产| 午夜精品国产一区二区电影| 99re6热这里在线精品视频| 亚洲真实伦在线观看| 2018国产大陆天天弄谢| 免费观看的影片在线观看| 晚上一个人看的免费电影| 另类亚洲欧美激情| 三级国产精品欧美在线观看| 日韩中字成人| av免费观看日本| 大片电影免费在线观看免费| 好男人视频免费观看在线| 嘟嘟电影网在线观看| 欧美日本视频| 91午夜精品亚洲一区二区三区| 99re6热这里在线精品视频| 我的老师免费观看完整版| 搡女人真爽免费视频火全软件| 一级二级三级毛片免费看| 乱系列少妇在线播放| 美女中出高潮动态图| 亚洲电影在线观看av| 久久精品久久久久久噜噜老黄| 天堂8中文在线网| 简卡轻食公司| 九九爱精品视频在线观看| 99国产精品免费福利视频| 1000部很黄的大片| 我的女老师完整版在线观看| 久久青草综合色| 最近的中文字幕免费完整| 成年女人在线观看亚洲视频| 国产高清不卡午夜福利| av免费观看日本| 亚洲av电影在线观看一区二区三区| 久久久色成人| 国产高清三级在线| 看十八女毛片水多多多| av国产免费在线观看| 成人漫画全彩无遮挡| 熟女人妻精品中文字幕| tube8黄色片| 久久女婷五月综合色啪小说| av免费观看日本| 美女xxoo啪啪120秒动态图| 久久人人爽人人片av| 国产av一区二区精品久久 | 少妇被粗大猛烈的视频| 深爱激情五月婷婷| 欧美xxxx性猛交bbbb| 久久久久久久亚洲中文字幕| 熟女电影av网| 一区二区av电影网| 91在线精品国自产拍蜜月| 国产精品一二三区在线看| 熟女av电影| 日韩av在线免费看完整版不卡| 中文精品一卡2卡3卡4更新| 嫩草影院入口| 亚洲精品乱码久久久v下载方式| 伊人久久精品亚洲午夜| 一区二区三区精品91| av一本久久久久| 国产精品久久久久久精品电影小说 | 亚洲国产精品国产精品| 亚洲国产日韩一区二区| 一区在线观看完整版| 黄色怎么调成土黄色| 99热这里只有是精品在线观看| 国产成人午夜福利电影在线观看| 中文天堂在线官网| 亚洲内射少妇av| 久久久久网色| 日韩电影二区| 欧美三级亚洲精品| 国产男人的电影天堂91| 一级毛片我不卡| 新久久久久国产一级毛片| 大陆偷拍与自拍| 97超视频在线观看视频| 一级毛片我不卡| 国产大屁股一区二区在线视频| 女人久久www免费人成看片| 最黄视频免费看| 人人妻人人添人人爽欧美一区卜 | 国产日韩欧美亚洲二区| 亚洲经典国产精华液单| 国产精品秋霞免费鲁丝片| 王馨瑶露胸无遮挡在线观看| 亚洲国产成人一精品久久久| 看十八女毛片水多多多| 少妇人妻久久综合中文| 亚洲婷婷狠狠爱综合网| 久久精品国产鲁丝片午夜精品| 欧美区成人在线视频| 各种免费的搞黄视频| 99久久精品国产国产毛片| 日韩电影二区| 国产爱豆传媒在线观看| 97在线视频观看| 三级国产精品片| 久久久久久久久久成人| 最新中文字幕久久久久| 久久国产亚洲av麻豆专区| 日日啪夜夜爽| 成人亚洲欧美一区二区av| 国产精品一及| a 毛片基地| 免费看不卡的av| 汤姆久久久久久久影院中文字幕| 日韩强制内射视频| 国产精品国产三级国产专区5o| 青春草视频在线免费观看| 国产黄色免费在线视频| 久久影院123| 干丝袜人妻中文字幕| 国产一区二区三区综合在线观看 | 成人毛片60女人毛片免费| 日韩av免费高清视频| 国产人妻一区二区三区在| 国产av一区二区精品久久 | 日韩大片免费观看网站| 大片免费播放器 马上看| av线在线观看网站| 在线天堂最新版资源| 免费久久久久久久精品成人欧美视频 | 在线播放无遮挡| 一区在线观看完整版| videossex国产| 亚洲图色成人| 国产熟女欧美一区二区| 日本vs欧美在线观看视频 | 日韩成人av中文字幕在线观看| 亚洲精品中文字幕在线视频 | 最近中文字幕2019免费版| 最近最新中文字幕免费大全7| 亚洲av免费高清在线观看| 久热这里只有精品99| 国产午夜精品久久久久久一区二区三区| 卡戴珊不雅视频在线播放| 伊人久久国产一区二区| 日本-黄色视频高清免费观看| 亚洲欧洲日产国产| 色视频www国产| 黄色日韩在线| 爱豆传媒免费全集在线观看| 久久久久国产精品人妻一区二区| 欧美激情国产日韩精品一区| 国产精品秋霞免费鲁丝片| 亚洲av免费高清在线观看| 午夜免费男女啪啪视频观看| 婷婷色综合大香蕉| 国语对白做爰xxxⅹ性视频网站| 美女视频免费永久观看网站| av福利片在线观看| 国产成人精品婷婷| 久久久久性生活片| 九草在线视频观看| 国产一区亚洲一区在线观看| 国产精品99久久99久久久不卡 | 欧美成人午夜免费资源| 成年美女黄网站色视频大全免费 | 精品视频人人做人人爽| 国产精品一区二区性色av| 精品少妇久久久久久888优播| 国产久久久一区二区三区| 肉色欧美久久久久久久蜜桃| 亚洲内射少妇av| 亚洲av综合色区一区| 在线天堂最新版资源| 男女国产视频网站| 熟妇人妻不卡中文字幕| 男女免费视频国产| 在线观看美女被高潮喷水网站| 国产亚洲av片在线观看秒播厂| 男人舔奶头视频| 亚洲国产最新在线播放| 免费看不卡的av| 亚洲av成人精品一二三区| 热99国产精品久久久久久7| 在线播放无遮挡| 日韩制服骚丝袜av| 麻豆成人午夜福利视频| 久久精品国产亚洲av涩爱| 国产成人91sexporn| 欧美精品亚洲一区二区| 亚洲国产日韩一区二区| 日本vs欧美在线观看视频 | 七月丁香在线播放| 下体分泌物呈黄色| 中国美白少妇内射xxxbb| 精品熟女少妇av免费看| 伦理电影免费视频| 久久久久国产精品人妻一区二区| av在线播放精品| 一本一本综合久久| 亚洲无线观看免费| 七月丁香在线播放| 麻豆精品久久久久久蜜桃| 最近中文字幕2019免费版| 日韩强制内射视频| 久久精品国产亚洲av天美| 国内少妇人妻偷人精品xxx网站| 欧美zozozo另类| 大又大粗又爽又黄少妇毛片口| 国产精品无大码| 99久久综合免费| 97超碰精品成人国产| 最新中文字幕久久久久| 日韩 亚洲 欧美在线| 国产成人精品福利久久| 一级片'在线观看视频| 亚洲不卡免费看| 亚洲精品视频女| 欧美成人a在线观看| 亚洲丝袜综合中文字幕| 日本wwww免费看| 天堂俺去俺来也www色官网| 插逼视频在线观看| 少妇裸体淫交视频免费看高清| 97在线人人人人妻| 亚洲精品自拍成人| 能在线免费看毛片的网站| 亚洲无线观看免费| 五月玫瑰六月丁香| 欧美 日韩 精品 国产| 交换朋友夫妻互换小说| 国产成人a∨麻豆精品| 国产又色又爽无遮挡免| 久久久久视频综合| 特大巨黑吊av在线直播| 亚洲一区二区三区欧美精品| 能在线免费看毛片的网站| 99re6热这里在线精品视频| 我要看日韩黄色一级片| 久久精品熟女亚洲av麻豆精品| 精品午夜福利在线看| 中文在线观看免费www的网站| 久久精品国产自在天天线| 国产精品国产三级国产专区5o| 99视频精品全部免费 在线| 国精品久久久久久国模美| 高清在线视频一区二区三区| 91aial.com中文字幕在线观看| 亚洲欧美一区二区三区国产| 熟女电影av网| 久久国产亚洲av麻豆专区| 国产成人一区二区在线| 亚洲国产欧美在线一区| 一个人免费看片子| 免费观看a级毛片全部| 麻豆精品久久久久久蜜桃| 亚洲精品视频女| 美女福利国产在线 | 亚洲第一av免费看| 免费人妻精品一区二区三区视频| 亚洲av免费高清在线观看| 小蜜桃在线观看免费完整版高清| 欧美精品国产亚洲| 亚洲人成网站高清观看| 国产精品福利在线免费观看| 91精品伊人久久大香线蕉| 亚洲精华国产精华液的使用体验| 免费av中文字幕在线| 久久久a久久爽久久v久久| 国产精品av视频在线免费观看| 嘟嘟电影网在线观看| 日韩一本色道免费dvd| 久久久久网色| 亚洲国产高清在线一区二区三| 日本一二三区视频观看| 黄色配什么色好看| 最近中文字幕高清免费大全6| 亚洲av中文字字幕乱码综合| 精品久久国产蜜桃| 精品视频人人做人人爽| 国产成人精品婷婷| 99热这里只有是精品50| 欧美日韩精品成人综合77777| 高清欧美精品videossex| 亚洲经典国产精华液单| 男男h啪啪无遮挡| www.av在线官网国产| 视频中文字幕在线观看| 中国美白少妇内射xxxbb| 成人漫画全彩无遮挡| 99re6热这里在线精品视频| 人人妻人人添人人爽欧美一区卜 | 亚洲av二区三区四区| 六月丁香七月| 国产黄片美女视频| 国产精品久久久久久av不卡| 波野结衣二区三区在线| 在线观看免费高清a一片| 黄色一级大片看看| 视频中文字幕在线观看| 欧美日本视频| 99九九线精品视频在线观看视频| 亚洲精华国产精华液的使用体验| 久久久精品94久久精品| 久久国产精品男人的天堂亚洲 | 亚洲欧美日韩无卡精品| 婷婷色av中文字幕| 少妇人妻一区二区三区视频| 啦啦啦啦在线视频资源| 日韩av在线免费看完整版不卡| 久久久久精品性色| 午夜福利高清视频| 最近2019中文字幕mv第一页| 人妻夜夜爽99麻豆av| 亚洲四区av| 色网站视频免费| 一区二区三区乱码不卡18| 99视频精品全部免费 在线| 啦啦啦中文免费视频观看日本| 中文字幕久久专区| 18禁裸乳无遮挡动漫免费视频| 丝袜喷水一区| 2018国产大陆天天弄谢| 国产成人a区在线观看| 国产亚洲午夜精品一区二区久久| av专区在线播放| 亚洲av电影在线观看一区二区三区| 久久鲁丝午夜福利片| 美女内射精品一级片tv| 亚洲欧美一区二区三区黑人 | 亚洲精品,欧美精品| 青春草国产在线视频| 亚洲美女黄色视频免费看| 亚洲av综合色区一区| 久久99热这里只有精品18| 国产精品麻豆人妻色哟哟久久| 久久影院123| 大香蕉久久网|