related Localization in Modulated Lattice Array - Wiley Online Library

ORIGINAL PAPER Chaotic Dynamics

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Chaos-related Localization in Modulated Lattice Array Liping Li,* Bo Wang, Xin-You Lü, and Ying Wu* localization-delocalization has been observed in models where only single elements (only driving, only nonlinearity) were considered.[26,27] To our knowledge, it has been clear that the localization and delocalization could coexist in a chaotic system and the transition could be realized by adjusting the driving frequency or other parameters.[28,29] In particular, the chaos-related localization is a significant issue in a driven optical lattice system. Recently, the chaos which may aid the localization in some chaotic parameter regions is reported by investigating quantum transport of a single particle held in a one-dimensional amplitude-modulated and tilted optical lattice.[28] It has been demonstrated experimentally that the degree of localization may closely related to the degree of disorder and Anderson localization occurs at a higher level of disorder in a two-dimensional photonic lattice.[30] Furthermore, as a substitute for disorder, it has been pointed out that highchaoticity could replace higher disorder and assist in Anderson localization.[31] However, here, it was found out that the chaosrelated localization could occur at both higher and lower levels of chaoticity in our modulated lattice array. So, what s the properties and factors associated with such chaos-related localization? In this text, the chaos-related localization is discussed by studying a frequency-modulated lattice array without the bias. Comparing with previous works, physically, such chaos-related localization in our proposal can be divided into two different types: short-term and long-term localization. When the localization occurs in a shorter (longer) evolution time, it is called short-term (longterm) localization here. The short-term chaos-related localization with a shorter lifetime may transform into chaos-assisted tunneling dramatically in the process of evolution, while the longterm chaos-related localization with a longer lifetime has good stability for an enough longer evolution time. Unexpectedly, the chaos-assisted tunneling and chaos-related localization may coexist even under the same system parameters. More interestingly, the short-term chaos-related localization is closely related to a higher degree of chaos and a lower degree of chaos may eventually lead to a long-term localization. So, the chaotic localization could be controlled by the degree of chaos. Furthermore, the chaos-related localization could be manipulated by adjusting the nonlinearity, external driving frequency or the second-order coupling through which the switching between the chaos-related localization and the chaos-assisted tunneling is realized. Besides, due to the sensitivity of chaos to initial conditions, dependence of chaotic dynamics on the phase is discussed. These results provide a new avenues of manipulating the chaotic signal in driven

This paper will discuss the chaos-related localization in a lattice array with an external periodical field acted on a boundary site that allows us to realize the controllable chaotic dynamics with a tunable driving frequency. Two types of chaos-related localization, short-term and long-term localization, which are closely related to the degree of chaos are reported and may provide a way to realize switching from chaos-related localization to chaos-assisted tunneling. Interestingly, with the increase of nonlinearity, driving frequency or even second-order coupling, there always exists a parameter window with sharp edges for long-term localization which facilitates us to find the thresholds to control the system into or out of localization region. In addition, the numerical results further demonstrate that the initial phase of the driving field may greatly influence the degree of the chaos. These results can be extended to finite driven N-site system and may deepen our understanding of chaos-related localization in nonlinear driving system.

1. Introduction Chaotic dynamic has been intensively studied for its importance of quantum effect to deepen the understanding of the complexity of nonlinear dynamics [1–5] and offers an alternative platform for manipulating chaos-based algorithm,[6] communication and quantum transport,[7–9] to name but a few.[10,11] Chaos-assisted tunneling and chaos-related localization, which are defined as the tunneling (localization) in chaotic regions, are two representative problems which have attracted broad interests for their interesting contradiction physics.[12–16] In the previous works, theoretically and experimentally, the nonlinear driven system has exhibited some novel quantum tunneling and localization phenomena including nonlinear Landau-Zener tunneling,[17,18] Anderson localization (AL),[19,20] nonlinear CDT (NCDT)[21] and optical solitons.[22] Most importantly, Anderson localization in disordered systems has been studied intensively combining numerical and analytical arguments [23–25] and the transition of

L. Li, B. Wang School of Information Engineering Huanghe Science and Technology College Zhengzhou 450063, People’s Republic of China E-mail: [email protected] L. Li, X.-Y. L¨ u, Y. Wu School of Physics Huazhong University of Science and Technology Wuhan 430074, People’s Republic of China E-mail: [email protected] The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/andp.201700218

DOI: 10.1002/andp.201700218

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www.ann-phys.org work, when Min(P j ) > 0.5, the atoms are considered to be localized at jth site.

2.1. Nonlinear Dynamics in Modulated Three-Site System

Figure 1. Schematic diagram of the periodically modulated zigzag system. Here the left boundary site ( site 1 ) is driven with amplitude A, frequency ω and phase φ, and all other sites are undriven. and ν is the coupling intensity between nearest-neighboring and next-nearest-neighboring sites. Due to the zigzag structure, ν is adjustable and could be much larger than in a linear array structure.

The three-state-driven system, despite its simplicity, is a realistic model to describe a variety of fascinating quantum effects with second-order coupling. For facilitating discussion about the chaotic properties of atomic localization and calculating the Lyapunov exponent of the dynamical evolution, the atomic probability amplitude is defined as a j = ar j + iai j (ar j and ai j are real numbers). Therefore, the atomic evolution equations Eq. (1) for a modulated three-site system could be rewritten, dar 1 = A cos(ωt + φ)ai1 + U|ar 1 |2 ai1 + U|ai1 |2 ai1 dt

optical devices and could effectively clear the crosstalk between chaos-assisted tunneling and chaos-related localization.

+ ai2 + νai3 dai1 = −A cos(ωt + φ)ar 1 − U|ar 1 |2 ar 1 − U|ai1 |2 ar 1 dt

2. Model and Dynamic

− ar 2 − νar 3

A model of M identical bosons in a N sites zigzag lattice array with site 1 modulated periodically is discussed, as shown in Figure 1. In such a structure, the second-order coupling (ν) between next-nearest-neighboring sites, which is determined by the separation between site j − 1 and j + 1, could be much less than or comparable with the coupling intensity () of nearest-neighborsite. In the large-particle-number limit and taking into account the interaction between bosons, the dynamic can be described by the discrete nonlinear Schr¨ oding equation (DNLSE), i

da1 = γ (t)a1 + U|a1 |2 a1 + a2 + νa3 , dt

i

da j = a j −1 + a j +1 + U|a j |2 a j + ν a j −2 + a j +2 , dt

(1)

where γ (t) denotes an external energy bias and a harmonicoscillating field is assumed to apply in the form of γ (t) = A cos(ωt + φ) on site 1 while other sites are undriven. A, ω and φ are the driving amplitude, frequency and the initial phase of the external driving field and rich system properties emerge if the driving field is turned on. U characterizes the nonlinear atomatom interaction and the strength is controllable via Feshbach resonance.[32,33] The system has now been realized experimentally in numerous ways. For example, two weakly linked BoseEinstein condensates (BEC) including about 1150 atoms in a double-well potential is researched by DNLSE and a simple estimate gives ≈ π · 10.4s −1 and U/ ≈ 30.[34] Tunnelling times corresponding to such a value of the nearest-neighbor-site coupling intensity are on the order of several hundred milliseconds, which are well within the reach of the related experimental setup of a BEC with an optical lattice. It might be more feasible in carrying out such system implemented by the light in coupled 2 nonlinear optical waveguides.[21] P j = a j (t) represents the occupation probabilities at jth site and the minimum value of P j is used to estimate the inhibition of tunneling oscillation. In our

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dar 2 = U|ar 2 |2 ai2 + U|ai2 |2 ai2 + ai1 + ai3 dt dai2 = −U|ar 2 |2 ar 2 − U|ai2 |2 ar 2 − ar 1 − ar 3 dt dar 3 = U|ar 3 |2 ai3 + U|ai3 |2 ai3 + ai2 + νai1 dt dai3 = −U|ar 3 |2 ar 3 − U|ai3 |2 ar 3 − ar 2 − νar 1 dt

(2)

With the absence of imaginary number, Eq. (2) could also give the evolution properties and the atomic occupation probability at site 1 could be rewritten as P1 = |ar 1 |2 + |ai1 |2 . In addition, defining the evolution of a perturbation as ε = (εr 1 , εi1 , εr 2 , εi2 , εr 3 , εi3 ) and by linearizing Eq. (2), the adjacent trajectories of two nearby points moving forward in phase space yield, dεr 1 = A cos(ωt + φ)εi1 + U|ar 1 |2 εi1 + 3U|ai1 |2 εi1 dt + 2Uar 1 ai1 εr 1 + εi2 + νεi3 dεi1 = − A cos(ωt + φ)εr 1 − U|ai1 |2 εr 1 − 3U|ar 1 |2 εr 1 dt − 2Uar 1 ai1 εi1 − εr 2 − νεr 3 dεr 2 = U|ar 2 |2 εi2 + 3U|ai2 |2 εi2 + 2Uar 2 ai2 εr 2 dt + εi1 + εi3 dεi2 = − U|ai2 |2 εr 2 − 3U|ar 2 |2 εr 2 − 2Uar 2 ai2 εi2 dt

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ε

ε


Figure 2. The short-term and long-term localization. The population distribution P1 versus the evolution time t with different nonlinearity. (a) displays the short-term localization with a lifetime tc under the conditions of U = 8.9, while (b) shows the long-term localization and the particle always be localized at 1-site over all the observation time when U = 15. Other parameters are chosen as A = 10, ω = 10, = 1, ν = 0 and φ = 0.

dεr 3 = U|ar 3 |2 εi3 + 3U|ai3 |2 εi3 + 2Uar 3 ai3 εr 3 dt + εi2 + νεi1 dεi3 = − U|ai3 |2 εr 3 − 3U|ar 3 |2 εr 3 − 2Uar 3 ai3 εi3 dt − εr 2 − νεr 1

(3)

Here ε P1 = (ar 1 + εr 1 )2 + (ai1 + εi1 )2 − P1 is used to characterize the divergence of nearby trajectories in phase space and its logarithm ln(εP1 ) shows the tendency of the perturbation. Then the logarithmic slope of the perturbation ε P1 versus time t is defined as Lyapunov exponent (L E ). The negative and positive values of the Lyapunov exponent, respectively denote states of the dynamical system, being out and into the chaotic motions. The nonlinear differential equations (1–3) are numerically solved by Runge-Kutta method in the following sections with the time steps, relative and absolute errors being 10−2 , 10−3 and 10−6 . Besides, it should be pointed out that symplectic integration methods [35,36] can also work well for such problems and present good stability in particular for long integration times and large system sizes.

2.2. Short-Term and Long-Term Localization in Modulated Three-Site System. To investigate the dynamics, the time-dependent nonlinear Schr¨ odinger equation (1) is solved numerically under the initial condition (a1 (0), a2 (0), a3 (0)) = (1, 0, 0). The evolution of the 2 probability distribution P1 = a1 (t) is presented in Figure 2 for A = 10, ω = 10, = 1, ν = 0 and φ = 0. In the case of U = 8.9, as shown in Figure 2(a), the atomic localization is observed in the initial period of evolution before time tc and then tunneling property dominates, demonstrating no suppression of tunneling. When U = 15, Figure 2(b) indicates that the oscillations of P1 are limited between 0.96 and 1 at site 1 during all our observation time, signaling a complete suppression of tunneling

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Figure 3. Chaotic property of short-term localization. The left column shows chaos-related localization with 30 time-units before tc and the right column shows chaos-assisted tunneling with 500 time-units after tc . (a) and (b): The deviation of nearby points l n(ε P1 ) versus the time t. (c) and (d): The atomic trajectory (the first derivation of P1 versus P1 ) in the phase space. The effect of increasing trend of the deviation and the disorderly phase diagrams may be explained as the existence of chaos. The parameters are chosen as U = 8.9, A = 10, ω = 10, = 1, ν = 0 and φ = 0.

between the energy states. P1 remains near unity even when the evolution time is extended to two orders of magnitude (see the inset in Figure 2(b)). Here, in terms of the temporal duration, the localization is divided into two types of short-term and longterm localization. For short-term localization, there would be a sharp switch from localization to tunneling at the critical point tc which is used here to determine the lifetime of localization approximately, while for long-term localization, all particles would be localized at 1th site over all the evolution time. Further study shows that the lifetime of atomic localization tc is just a reference value which is determined by the system parameters and calculation accuracy. Then, in order to understand the connection between shortterm localization and the chaos, a perturbation ε is introduced to describe the adjacent trajectory of two nearby points. Fixing U = 8.9 (see Figure 2(a)), Figure 3 displays our numerical results of the atomic chaotic properties when ε = (10−10 , 10−10 , 10−10 , 10−10 , 10−10 , 10−10 ). The left column corresponds to the atomic localization with 30 time-units before tc . Figure 3(a) shows the dynamic of nearby points (P1 + ε P1 ) of P1 versus time t characterizing the deviation of the population distribution and the increasing trend of the curve implies that the atomic trajectory may be chaotic. The phase diagram which is defined as the first derivation of P1 versus P1 shown in Figure 3(c), reveals the chaotic trajectory in phase space. So, the effect of

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Figure 4. Chaotic property of long-term localization. (a)–(c)The population distribution P1 versus t under three different initial √ conditions. (a1 (0), a2 (0), √ a3 (0)) is set to be (a)(1, 0, 0), (b)(0.997, 1 − 0.9972 , 0) and (c)(0.99, 1 − 0.992 , 0) separately. The long-term localization is sensitive to the initial condition such that the long-term localization is chaotic. (d)–(f): the first derivation of P1 versus P1 in phase space according to three different initial conditions. The disorderly phase diagrams may further prove the existence of chaos. The parameters are chosen as U = 15, A = 10, ω = 10, = 1, ν = 0 and φ = 0.

increasing trend of the deviation and the disorderly phase diagram may be interpreted as the existence of chaos and such short-term localization in our proposal is the chaos-related localization. Furthermore, the conditions with 500 time-units are plotted in the right column. Figure 3(b) presents the growing trend of the deviation with the increase of evolution time. The parameter region around tc is amplified shown in the inset of Figure 3(b) which displays that the deviation begins to rise slowly and then rapidly about after tc . Comparing with Figure 2(a), the slow rise corresponds to the short-term localization and the rapid rise is related to the atomic tunneling. Figure 3(d) reveals that trajectory of the atomic tunneling dynamic in phase space after time tc at site 1 is chaotic. Naturally, the tunneling is chaos-assisted tunneling. Briefly, Figure 3 has revealed a fact that the chaos-related localization and chaos-assisted tunneling could coexist even under the same parameters condition and the transition occurs on the instant at the critical time point tc . To clear the relation between long-term localization and chaos, the sensitivity of chaos to initial conditions is discussed in Figure 4. For U = 15 (see Figure 2(b)), Figure 4(a)–(c) present the population distribution P1 versus time t with three a2 (0), a3 (0)) being (1, 0, 0), different √ initial conditions (a1 (0), √ (0.997, 1 − 0.9972 , 0) and (0.99, 1 − 0.992 , 0) respectively. When a1 (0) = 1 (Figure 4(a) and Figure 2(b)), P1 is approximately equal to 1, signaling the existence of long-term localization. In the case of a1 (0) = 0.997 (Figure 4(b)), the short-term localization is

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Figure 5. System dynamics with different evolution time controlled by the nonlinearity. The left column shows the system dynamics with 30 timeunits in contrast to the right column with 100 time-units. (a) and (b): The minimum value of population distribution at site-1 Mi n(P1 ) versus nonlinearity U . (c) and (d): The Lyapunov exponent versus nonlinearity corresponding to (a) and (b). The short-term localization may possess a higher degree of chaos than the long-term localization. Other parameters are chosen as A = 10, ω = 10, = 1, ν = 0 and φ = 0.

observed and when t increases beyond the critical point tc , the complete tunneling emerges. When a1 (0) = 0.99 (Figure 4(c)), there is no inhibition of large amplitude tunneling. Clearly, as a1 (0) decreases gradually, the system’s state is found to follow the change of long-term localization, short-term localization and completely tunneling. Accordingly, the associated phase diagram is plotted in Figure 4(d)–(f) and a relatively small variation of initial probability can induce, in effect, great influence on atomic trajectory in phase space, in which the irregular trajectory indicates that both the localization and tunneling in Figure 4(a)–(c) are chaotic. Therein the long-term localization in our proposal is chaos-related localization actually. We stress that chaos-related long-term localization is not so sensitive to P1 (0) as the shortterm localization and generally the sensitivity will decrease with the increase of nonlinearity. Further numerical simulation (not shown here) demonstrates that when U = 30, the long-term localization will keep stable if the initial P1 (0) changes within a few percent. This implies that it is more realistic to observe experimentally the chaos-related long-term localization in the parameter window with stronger nonlinearity. To explore the chaos-related localization on a deeper level, the minimum value of P1 versus nonlinearity U with different length of evolution time t has been plotted in Figure 5(a)–(b). For the experiment in Ref. [34] , fixing U/ = 30, the left column represents

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the cases with shorter evolution time (30 time-units) and the right column with a longer evolution time (100 time-units). Comparing with Figure 5(a) and Figure 5(b), there are more peaks in Figure 5(a) especially in the parameter range 4 < U < 13, denoting that a number of localization is short-term and will disappear when the evolution time is extended longer. Particularly, there exists a parameter region about 14 < U < 20 with sharp edges where the strong localization occurs both in Figure 5(a) and Figure 5(b) corresponding to the long-term localization. The commonality is that when the nonlinearity is small, the system transforms between localization and delocalization repeatedly. When the nonlinearity increases to near U = 15, however, the longterm localization is observed and the particle will be always localized at site 1 even if the observation time is longer enough. If the nonlinearity continues to increase, the particle may transform between localization and delocalization again in the parameter region around U = 21 and ultimately the long-term localization dominates when U > 23. Therefore, the lifetime of the localization at site 1 is greatly different with the change of nonlinear coefficient and there exists a parameter window for longterm localization. To get further evidence of chaotic dynamic, the logarithmic slope of l n(ε P1 ) versus time t, known as Lyapunov exponent (L E ), is used here to describe the chaotic properties. Lyapunov exponent versus nonlinear coefficient U is plotted in Figure 5(c)–(d). When L E > 0, the system can be considered to be chaotic and a higher value of L E in our model usually means a higher degree of chaos. The simulation results of L E is in good absolute agreement with the evolution properties of Min(P1 ) in which the L E value of chaos-assisted tunneling is always much larger than zero. Due to chaos being hindered by strong localization, chaos-related localization have, therefore, a weaker degree of chaos. Relatively, the short-term localization, as shown in Figure 2(a), can possess a larger value of L E (a larger degree of chaos ) than the long-term localization, as shown in Figure 2(b). It should be clarified that the lower (higher) degree of chaos is different from the long-term (short-term)localization and the relation is correct just in our proposal. In a nutshell, there are two types of chaos-related localization in our model and the lifetime of localization may greatly determined by the degree of chaos. It should be noticed that (i) the long-term localization features a very low L E because of suppression of the chaotic dynamics by high degree of localization, and (ii) the relative stable parameter window with sharp edges for chaos-related long-term localization around 14 < U < 20 provides a reference value to achieve the transition between chaosassisted tunneling and chaos-related localization.

3. Controllable Chaos-related Localization in Modulated Three-site System. This section will focus on the control of chaotic dynamic in modulated three-site system and try to realize the controllable transition between chaos-related localization and chaos-assisted tunneling. Firstly, the driving frequency is treated as a variable. Fixing parameters U = 15, A = 10, = 1, ν = 0 and φ = 0, the minimum value of population distribution Min(P1 ) versus ω with 100 time-units is plotted in Figure 6(a). The effective inter-site tun-

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ω

ν

Figure 6. Controllable chaotic dynamics in modulated three-site system. (a)–(b)Mi n(P1 ) and the Lyapunov exponent L E versus ω with 100 timeunits when ν = 0. The insert above in Figure 6(b) shows that L E > 0 within the parameters range 9 < ω < 11 . When ω > 15, L E > 0 at most parameters points (see the lower insert in Figure 6(b)). (c)–(d)Mi n(P1 ) and the Lyapunov exponent L E versus ν with 100 time-units when ω = 15 and L E > 0 within the parameter range 0.4 < ν < 0.8 (see the insert in Figure 6(d)). Other parameters are chosen as U = 15, A = 10, = 1 and φ = 0.

neling is observed when the driving frequency is small, and then the tunneling is suppressed and the particle is frozen at site 1 within the parameter range 9 < ω < 11. When ω increases more than 12, the fast conversion between localization and delocalization occurs and the final state of the system tends to long-term localization if ω becomes larger than 18. The driving frequency is found to have great and complex influence on the system dynamics. The Lyapunov exponent versus driving frequency with 100 time-units is illustrated in Figure 6(b) and obviously the longterm localization possesses a lower value of L E within the parameters range 9 < ω < 11, in which L E is more than zero actually as shown in the upper insert in Figure 6(b) by amplifying L E from 0 to 0.05. So, there exists a parameter window for the chaosrelated localization around ω = 10 with sharp edges, thereby providing a controllable window into or out of chaos-related localization regime. Moreover, the chaos-related localization could also be observed when about ω > 18, as shown in the following insert in Figure 6(b). In the diagram, Lyapunov exponent may be greater than zero at most parameter points, however, at some special points, L E ≤ 0 represents a fact that localization in our proposal is not always chaotic. Subsequently, the chaos-related localization is controlled via second-order coupling, which could also be treated as a variable due to the special zigzag structure. It is worth noting that ν = 1 represents a equilateral triangle structure. Initially, fixing

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the case of φ = 0. Although the initial driving phase can influence the degree of chaos dramatically, it should be stressed that the increase of the initial driving phase may not always lead to the increase of the slope. Such results illustrate that the atomic chaotic dynamics in our driving system are intensively dependent on the initial driving phase. We come to a conclusion that, in the modulated three-site system, the driving frequency and second-order coupling may influence the chaos-related localization drastically. The parameter windows of chaos-related localization in Figure 6(a)–(b) and Figure 6(c)-(d) suggest a critical value for the transformation from localization to delocalization or the contrary process. The controllable chaos-related localization is achieved just by adjusting the driving frequency or second-order coupling. Moreover, the initial phase of the external field may also influence the degree of the chaos.

π π

ε

φ φ φ

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4. Chaos-related Localization in Modulated N-site System.

Figure 7. The dependence of the perturbation evolution on the initial phase of driving field. The evolution of ln(ε P1 ) for different values of φ (from bottom to top): φ = 0 (black dotted line), φ = 3π/8 (blue dotted line) and φ = 5π/8 (red line). The insert illustrates the evolution of P1 versus t when φ = 5π/8 and the short-term localization is observed. Other parameters are chosen as U = 15, A = 10, ω = 15, = 1 and ν = 0.

the nonlinear coefficient U = 15 and other parameters A = 10, = 1, ω = 15 and φ = 0, the minimum value of P1 versus ν with 100 time-units is plotted in Figure 6(c). With the increase of second-order coupling, the dynamic displays multiple transformations between localization and delocalization. Since all the value of Lyapunov exponent L E > 0, shown in Figure 6(d), three main features are outlined below: (i)there indeed exists chaosrelated localization when ν varies from 0 to 1; (ii)within the parameter range 0.4 < ν < 0.8 (see the insert in Figure 6(d)), atomic stable localization is observed with a lower degree of chaos; (iii)the transition between chaos-assisted tunneling and chaos-related localization could be realized through controlling the parameter ν. Finally, the influence of the phase on chaos-related localization is discussed in this modulated three-site system. Under the condition of U = 15, A = 10, ω = 10, = 1and ν = 0, corresponding to the long-term localization discussed in Figure 2(b) and Figure 6(a-b) with ω = 10, φ = 0, the dependence of the l n(ε P1 ) on the initial driving phase φ is presented in Figure 7. The slope of ln(ε P1 ) versus t curve is greatly different as the initial phase changes and the degree of chaos in order from the large to the small are: φ = 5π/8 , φ = 3π/8 and φ = 0. When φ = 5π/8, the fast rising curve means a transition from localization to delocalization and the insert in Figure 7 further proves that the longterm localization when φ = 0 has been transformed into shortterm localization due to the change of initial phase. When φ = 3π/8, the very slow growth of the curve means a chaos-related long-term localization with a slightly higher degree of chaos than

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Now, the chaos-related localization in other N-site system where site 1 is shifted periodically against all of the other sites will be studied. In what follows,the results for four-site system are presented in Figure 8, which are explored by solving the equation (1) numerically with the initial condition (a1 (0), a2 (0), a3 (0), a4 (0)) = (1, 0, 0, 0) and a total integration time of 100 time-units. Under the conditions of A = 10, ω = 10, = 1, ν = 0 and φ = 0 as in Figure 2, the minimum value of P1 and Lyapunov exponent (L E ) as a function of nonlinearity U are plotted in Figure 8(a) and Figure 8(b). In Figure 8(a), multiple conversions between localization and delocalization appear in the parameter regions U < 14 and 20 < U < 23, in which the L E is much larger than zero. Differently, in the cases of 14 < U < 20 and U > 23, Min(P1 ) remains nearly 1 and the L E takes extremely low value about zero. Despite its relatively small value, detailed examination of the L E reveals that L E > 0 and the dynamic is chaotic. The impact of ω on the chaotic dynamics in the four-site system has also been investigated when U = 15, A = 10, = 1, ν = 0 and φ = 0, as discussed in Figure 6. Min(P1 ) and L E versus the driving frequency ω are depicted in Figure 8(c) and Figure 8(d). As ω is increased from zero, Min(P1 ) decreases from value of 1 to zero fleetly before it peaks near ω = 8. The long-term localization occurs within 8 < ω < 11 in which the L E is more than zero at most parameter points. Such parameter region with sharp edges may facilitate us to achieve the transition between chaos-related localization and chaos-assisted tunneling. The influence of second-order coupling on a four-site system is plotted in Figure 8(e) and Figure 8(f) when U = 15, ω = 10, = 1, ν = 0 and φ = 0. Figure 8 (e) shows the minimum value of P1 as a function of ν and a similar localization-delocalization transition is observed. Like the case of the three-site system, this four-site system possesses a parameter region 0.2 < ν < 0.5, in which the particle is localized at site 1. From Figure 8(f), it can been seen that L E is always more than zero and L E is relatively lower when ν < 0.5, where long-term localization can be controlled by the second-order coupling. The modulated lattice array with larger system sizes such as N = 30 is further studied and the numerical results of the

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5. Conclusion

ω

A practical model have been proposed to achieve the controllable chaos-related localization in the driven three site system and the results could also be generalized to N site lattice array. The shortterm localization provides us a way to realize the transition between localization and delocalization just through controlling the evolution time. Besides the degree of chaos-related localization is enhanced dramatically in our model due to the reduced degree of chaos. The perfect localization in nonlinear system could be realized if the degree of chaos can be controlled. Our results reveal that chaos-related localization could be switched on and off by driving the site 1 with a tunable driving frequency or by adjusting the intensity of second-order coupling. In addition, the initial driving phase dependence of chaos-related localization may open up a new way to manipulate atomic dynamic under a phasemodulated regime. These results provide a promising route to control the chaos-related localization, especially to promote the understanding of fundamental physics about chaos-related localization and chaos-assisted tunneling, and may support the potential applications in chaos-based technologies.

ω

Conflict of Interest The authors declare no conflict of interest

Keywords

ν

Chaotic dynamic, Chaos-assisted tunneling, Chaos-related localization, short-term localization, long-term localization

ν

Figure 8. Chaos-related localization in modulated four-site system. (a)– (b)Mi n(P1 ) and the Lyapunov exponent versus nonlinearity U when ω = 10, ν = 0 . (c)–(d)Mi n(P1 ) and the Lyapunov exponent versus the driving frequency ω when U = 15, ν = 0. (e)–(f)Mi n(P1 ) and the Lyapunov exponent versus the second-order coupling ν when U = 15, ω = 10. Similar chaotic dynamics are observed as in modulated three-site system ( see Figure 5 and Figure 6). Other parameters are A = 10, = 1 and φ = 0.

evolution of P1 (not listed here) show a similar dynamic as in three- and four-sites system. The dynamics in 30-site system are as follows: (i)the chaos-related short-term and long-term localization are observed under the same conditions as in Figure 2; (ii)the transition between chaos-related localization and chaosassisted tunneling can be achieved through adjusting the parameters; (iii)more interesting results has been observed when to check the dynamics of other sites Pi (t) as i ≥ 2 in the case of chaos-assisted tunneling measured for site P1 (t). In a shorter initial evolution time, a larger value of Pi (Pi >0.5) appears just in a very few number of previous sites (i=1,2,3 mostly). With the increase of site number and evolution time, the equidistribution among all the sites Pi is observed with an average population distribution Pi < 0.2. Briefly, in the periodically driving N-site system, the numerical results are similar as in the modulated three-site system and the chaos-related localization can be controlled by nonlinearity, the driving frequency and the second-order coupling.

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Received: June 8, 2017 Revised: August 24, 2017 Published online: October 5, 2017

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