Does chaos assist localization or delocalization? Jintao Tan, Gengbiao Lu, Yunrong Luo, and Wenhua Hai Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 043114 (2014); doi: 10.1063/1.4898332 View online: http://dx.doi.org/10.1063/1.4898332 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions J. Math. Phys. 54, 123505 (2013); 10.1063/1.4842375 Chaos and Symmetry AIP Conf. Proc. 726, 43 (2004); 10.1063/1.1805913 Quantum And Classical Dynamics Of Atoms In A Magnetooptical Lattice AIP Conf. Proc. 676, 283 (2003); 10.1063/1.1612224 Anderson localization of ballooning modes, quantum chaos and the stability of compact quasiaxially symmetric stellarators Phys. Plasmas 9, 1990 (2002); 10.1063/1.1448344 Microscopic theory of transport phenomenon in finite system AIP Conf. Proc. 597, 375 (2001); 10.1063/1.1427486
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CHAOS 24, 043114 (2014)
Does chaos assist localization or delocalization? Jintao Tan,1 Gengbiao Lu,2 Yunrong Luo,1 and Wenhua Hai1,a) 1
Department of Physics and Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081, China 2 Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004, China
(Received 12 August 2014; accepted 6 October 2014; published online 20 October 2014) We aim at a long-standing contradiction between chaos-assisted tunneling and chaos-related localization study quantum transport of a single particle held in an amplitude-modulated and tilted optical lattice. We find some near-resonant regions crossing chaotic and regular regions in the parameter space, and demonstrate that chaos can heighten velocity of delocalization in the chaos-resonance overlapping regions, while chaos may aid localization in the other chaotic regions. The degree of localization enhances with increasing the distance between parameter points and near-resonant regions. The results could be useful for experimentally manipulating chaos-assisted C 2014 AIP Publishing LLC. transport of single particles in optical or solid-state lattices. V [http://dx.doi.org/10.1063/1.4898332] A quantum particle in a static and tilted lattice is expected to undergo the celebrated Bloch oscillation rather than the uniformly accelerated motion, whereas under a periodic modulation, the dynamical localization (DL) and delocalization may occur for different parameter regions. In classical treatment of the system, the presence of periodic modulation also leads to chaos for the chaotic parameter region. We define the chaos-assisted localization (delocalization) as the localization (delocalization) appearing in the chaotic region. By investigating quantum transport of such a typical classically chaotic systems, we reveal that chaos may assist localization or delocalization, depending on whether quantum resonance happens in the chaotic region. The results clarify a longstanding contradiction between chaos-assisted tunneling and chaos-related localization.
I. INTRODUCTION
Understanding quantum tunneling and transport in classically chaotic systems is a significant problem, both fundamentally and practically. Driven double-well and optical lattice systems are two typical kinds of chaotic systems,1–4 in which some unusual transport or tunneling phenomena have been exhibited.5–9 Localization is a representative phenomenon, including DL,6,7 Anderson localization (AL)8 and coherent destruction of tunneling (CDT).9 One of the routes from localization to delocalization is the quantum resonance,10,11 which can be achieved by adjusting the modulation frequency to fit the static tilt (or bias).11 The contradiction between chaos-assisted tunneling and chaos-related localization has been found before some years. On the one hand, it is known that AL was related to disorder, and “chaos can substitute for disorder in Anderson’s scenario, leading to dynamical localization of quantum a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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probability transport” for a periodically kicked rotator which is classically chaotic.12 As the quantum suppression of classically chaotic diffusion, such a localization has been directly observed.5 Recently, we further found that the highchaoticity could replace higher disorder to aid AL in a driven optical lattice.13 On the other hand, Lin and Ballentine have proved that the tunneling rate could be highly enhanced due to the periodic modulation associated with classical chaos.4 Subsequently, many theoretical and experimental works demonstrated that classical chaos enhanced quantum tunneling rate drastically2,3,14 even without a barrier to tunnel through.2,3 Particularly, in the absence of collisions and dephasing effects, a particle in a static and tilted lattice is localized in a certain range, due to the celebrated Bloch oscillations,15 while the presence of a periodically varying ramp results in chaos16–18 and delocalization.7,19 In addition, there exist other situations, where both the quantum localization and delocalization may be independent of the classical chaos.20 Why can chaos promote localization and/or delocalization such a pair of totally contrary phenomena? In this letter, we will give an answer to the abovementioned interesting question by investigating quantum transport of a single particle held in a one-dimensional (1D) amplitude-modulated and tilted optical lattice.11,21,22 It is well known that delocalization and localization in a lattice occur in different parameter regions,7,22,23 and the chaotic and regular regions also coexist in a chaotic system.17,18 When localization (delocalization) occurs in the chaotic region, we call it the chaos-assisted localization (delocalization). We will analytically and numerically illustrate some near-resonant regions, where the driving frequency equates or approaches the lattice tilt,11 and reveal that chaos can assist delocalization in the chaos-resonance overlapping regions, while chaos may aid localization in the other chaotic regions. The degree of localization is controlled by adjusting the ramp of lattice and the distance to the near-resonant regions. The results could be useful for experimentally manipulating chaos-assisted quantum transport of single
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particles in the shaken and tilted lattices, and can be extended to studying chaotic electron transport in semiconductor superlattice induced by a combination of ac and dc electric field.24
II. CLASSICALLY CHAOTIC AND REGULAR REGIONS
We consider a single particle held in a periodically modulated and tilted 1D optical lattice potential, which is governed by the Hamiltonian p2 q 2 q H ð p; q; tÞ ¼ þ V0 ½1 A sinðxtÞ sin þ n : (1) 2 2 2 p ffiffi ffi Hereafter, H ¼ H 0 =Er ; p ¼ 2p0 =ðhkÞ; q ¼ 2x ¼ 2kx0 ; t ¼ xr t0 , and x ¼ x0 =xr are the rescaled quantities of the original Hamiltonian H 0 , momentum p0 , position x0 , time t0 , and driving frequency x0 with wave vector k, mass m, recoil 2 2 energy Er ¼ h2mk , and recoil frequency xr ¼ Er =h. The dimensionless constant A denotes the driving strength, V0 and np (with p being the dimensionless lattice length in x coordinate) are the lattice depth and tilt11,21 in units of hxr and k hxr , respectively. Such a system has been realized experimentally by applying an amplitude-modulated laser standing wave and a linear potential produced by a magnetic field gradient.11,21 Note that our rescaled coordinate and momentum obey pffiffiffi the quantum-mechanical commutation relation ½q; p ¼ 2 2 i which is different from the corresponding commutator in the previous works,2,3 because of the different scaled units. The effective Planck’s constant is a dimensionhef f ¼ 8xr =x ¼ h=I0 with I0 being the action of less ratio2,3 a free particle over the distance k/2 ¼ p/k in the time T ¼ 2p/ x, which describes the importance of quantum treatment for a system with the action I0. For our system, it can be defined as the commutator only for the particular units of the rescaled position and momentum.2,3 We are interested in the quantum regime with I0 in the order of h, and aim at the effect of classical chaos on quantum transport. Let us at first seek the classically chaotic andpregular paffiffiffiffiffi rameter regions. After rescaling time as s ¼ 2 V0 t, from Eq. (1) the Newton’s equation m€ x 0 ¼ H 0x0 ðp0 ; x0 ; t0 Þ is transformed into xs p ffiffiffiffiffi sin q n=V0 : (2) qss þ sin q ¼ eðq; sÞ ¼ A sin 2 V0 For small values of A and n /V0, the e(q, s) can be treated as perturbation such that the perturbed solution reads q(s) ¼ q0(s) þ q1(s) with the zero-order heteroclinic solution q0(s) and the first-order “chaotic solution” q1(s) in the forms18,25 q0 ¼ 2arctan½sinhðs þ t0 Þ; ðs ðs eðq0 ; sÞq0;s ds q0;s eðq0 ; sÞgðsÞds; q1 ¼ gðsÞ C1
(3)
C2
where constants t0, C1, and C2 relate to initial conditions, q0,s(s) and g(s) are the bounded and unbounded functions, respectively
q0;s ðsÞ ¼ 2sechðs þ t0 Þ; ð gðsÞ ¼ q0;s q2 0;s ds 1 ¼ ½sinhðs þ t0 Þ þ ðs þ t0 Þsechðs þ t0 Þ: 4
(4)
Inserting the solution q(s) ¼ q0(s) þ q1(s) into Eq. (2), one can immediately prove the assertion. Generally, the corrected solution q1(s) is unbounded at s ! 6 1, because of the unboundedness of the function g(s). But using l’Hospital rule, we can prove that if and only if the condition ðs eðq0 ; sÞq0;s ds ¼ 0 (5) I6 ¼ lim t!61 C 1
is satisfied, the first correction q1(s) will maintain bounded. On the other hand, the corresponding Melnikov function reads17,18 ð1 Mðt0 Þ ¼ eðq0 ; sÞq0;s ds 1 px2 xt0 px 2pn cos pffiffiffiffiffi csch pffiffiffiffiffi : (6) ¼A 2V0 V0 2 V0 4 V0 The condition (5) implies the onset criterion of chaos17,18 M(t0) ¼ Iþ I ¼ 0, so we call q1(s) the chaotic solution. From the chaotic criterion and Eq. (6), one derives the inequality18,25 A
4n px p ffiffiffiffiffi ; sinh x2 4 V0
(7)
where the sign “>” indicates the chaotic region of parameter space, and “¼” means boundary between the chaotic and regular regions. Fixing the lattice depth V0 ¼ 3, we plot three boundary curves in Fig. 1(a) for the tilts pn ¼ 0.8 (thin dashed curve), pn ¼ 1.8 (solid curve), and pn ¼ 3 (dotted curve), respectively. For a given tilt, chaos may occur in the chaotic region above the corresponding boundary curve. Note that for 87Rb atom the frequency unit xr is in order of 103 Hz so that the chaotic regions up to x ¼ 12(xr) is experimentally realizable.2,3,21 A special curve is displayed by the thick dashed curve in Figs. 1(a) and 1(b), which corresponds to the resonant parameters11 pn ¼ x. Therefore, in the region above this curve and labeled “U,” chaos and resonance may coexist. This assertion is illustrated by plotting the four nearresonant shadow regions fi (i ¼ 1, 2, 3, and 4) crossing the different chaotic and regular regions for pn ¼ x ¼ 0.8, 1.5, 1.8, and 3, respectively. The width of any near-resonant region fi depends on the system parameters, and for pn ¼ x ¼ 1.5 the maximal width shown in the inset of Fig. 1(b) is estimated as Dx 0.07 from the inset of Fig. 4(c). The chaos-resonance overlapping regions “fi \ U” are described by the gray shaded areas embedded in U. In fact, for A < 1 and pn < 4.5 one can obtain many different chaotic regions and chaos-resonance coexistence regions associated with different pn values, and these regions are shrunk with increasing the tilt pn. The classical Poincare section for the
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FIG. 1. (a) Chaotic and regular parameter regions with different boundary curves for V0 ¼ 3 and pn ¼ 0.8 (thin dashed curve), pn ¼ 1.8 (solid curve), and pn ¼ 3 (dotted curve). The corresponding near-resonant regions (shadows) are labeled by fi for i ¼ 1, 2, and 3. The chaos-resonance overlapping regions “fi \ U” exist above the thick dashed curve with pn ¼ x. (b) Typical parameter points Qi(x, A) Q1(1.46, 0.63), Q2(1.5, 0.63), Q3(1.5, 0.4), and Q4(4, 0.4) on the chaotic and regular regions for pn ¼ 1.5 (solid boundary curve). The approximate width of near-resonant region f4 and the distance between Q1 and Q2 are exhibited in the inset. (c) Poincare section for the rescaled position q and momentum p of a classical particle for the parameters near point Q2 of (b). Hereafter, all the quantities plotted in the figures are dimensionless.
rescaled position q and momentum p and the parameters near point Q2 of Fig. 1(b) is shown in Fig. 1(c), where the central regions of the potential wells near q ¼ 62np, n ¼ 0, 1, 2… consist of some small-amplitude “islands” of stability which are separated by the “chaotic sea.”2,3 Clearly, in the case n ¼ 0, system (1) is reduced to the previously investigated one in which the chaos-assisted quantum dynamical tunneling between separated momentum regions in phase space was observed.2,3 Adding the static tilt, we here aim at clarifying the contradiction between chaos-assisted localization and chaos-assisted delocalization associated with tunneling finite and infinite barriers between the spatially separated islands of stability, respectively. Does chaos aid delocalization or localization in a given chaotic region? We will take the typical parameter points Qi for i ¼ 1, 2, 3, 4 in Fig. 1(b) with pn ¼ 1.5 as examples to investigate such a question. III. MANIPULATING CHAOS-ASSISTED LOCALIZATION AND DELOCALIZATION
In the nearest-neighbor tight binding approximation, the Hamiltonian governing quantum dynamics of the system (1) reads7,21 HðtÞ ¼ JðtÞ
ðjmihm þ 1j þ jm þ 1ihmjÞ
pn
mðjmihmjÞ:
þ
Here, jmi represents the Wannier state localized on lattice site m, the coupling parameter between adjacent sites i, j is given as26
J ðsÞcosðpnsÞds
0
2 1 ) 2
J ðsÞsinðpnsÞds
(11)
for the initial conditions P0(0) ¼ 1, Pn6¼0(0) ¼ 0, where J n is n-th bessel function of the first kind. The corresponding mean-square displacement reads7 hm2 i ¼
X
Pn ðtÞn2
n
dxhij r2 þ ð1 A sinðxtÞÞV0 sin2 xjji
1
¼ J0 þ dJ sin xt
ðt
2
t
0
(8)
1
JðtÞ ¼
which forms an orthogonal set approximately. In 1986, Dunlap and Kenkre have investigated an analogous system with constant coupling and periodic tilt, and obtained an analytical solution of the probability amplitude an(t) in the P quantum state7 jwðtÞi ¼ n an ðtÞjni. Such a periodically tilted lattice system is a typical classical chaotic system.16 It is worth noting that the analytical expressions for quantum treatment of the chaotic system are based on the nearest-neighbor tight binding approximation, however, the quantum particle is generally delocalized except for the case when the ac field parameters are taken some specific values,7 because of the quantum tunneling. Directly applying our periodic coupling and static tilt to replace the constant coupling and periodic tilt in Dunlap’s analytical solution, respectively, produces the exact probability of the particle at site n as ( " ð Pn ðtÞ ¼ jan ðtÞj2 ¼ J 2n 2
1 X
1 1 X
ð1
with constant J0 in units of recoil energy Er ¼ hxr and dJ proportional to AV0. The used Wannier state vector reads27 pffiffiffiffi pffiffiffiffiffi 2 jji ¼ wðx xj Þ ¼ ð V0 =pÞ1=4 e V0 ðxjpÞ =2 ; (10)
(9)
ð t 2 ð t 2 ¼2 JðsÞ cosðpnsÞds þ 2 JðsÞ sinðpnsÞds (12) 0
0
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which characterizes the degree of localization and a quite large hm2 i value means the delocalization. For V0 ¼ 3, we numerically get the coupling function in Eqs. (11) and (12) as JðtÞ ¼ 0:058 0:033A sinðxtÞ:
(13)
On the basis of Eqs. (11) and (12), we plot time evolutions of the probability P0(t) of initially occupied state and the mean-square displacement hm2 i in Fig. 2 for the tilt pn ¼ 1.5 and the driving frequencies and strengths of the four points Qi(x, A) in Fig. 1(b). In Figs. 2(a) and 2(b), we show that P0(t) (hm2 i) returns to unity (zero) with different periods and amplitudes. This means, the occurrence of dynamical localizations are of different degrees.6 The larger amplitude corresponds to the lower degree of localization, and is associated with the smaller distance between Q1 and the near-resonant region. While in Figs. 2(c) and 2(d), the P0(t) (hm2 i) cannot return to unity (zero), and the meansquare displacement tends to infinity as t ! 1, which means delocalization caused by the resonance. The tunneling rate4 is inverse proportional to the time from P0(0) ¼ 1 to P0(t) ¼ 0, and the velocity of delocalization can be defined as the gradient of the hm2 i envelope. In Figs. 2(c) and 2(d), we observe that chaos related to the solid curves enhances the tunneling rate and the velocity of delocalization. We are interested in manipulating the chaos-assisted localization and delocalization by modulating parameters between the regular and chaotic regions. Fixing the driving amplitude A ¼ 0.63 and adjusting frequency from Q1(x ¼ 1.46) of regular region to Q2(x ¼ 1.5) of chaotic region, the dashed curve of Fig. 2(a) and solid curve of Fig. 2(c) display that chaos results in the transition from localization to delocalization. On the other hand, fixing the driving amplitude A ¼ 0.4 and increasing frequency from Q3(x ¼ 1.5) of regular region to Q4(x ¼ 4) of chaotic region, the dashed curve of Fig. 2(c) and solid curve of Fig. 2(a) indicates that chaos leads the initial delocalization to the
final localization. Taking different parameter points in the chaotic region of Fig. 1(b) to plot the mean-square displacements which are not shown here, we find that chaos aids delocalization only for the chaos-resonance overlapping region “f4 \ U,” and for the other chaotic region chaos may assist localization. Further, we take different tilt values to make the similar plots and find that chaos-assisted delocalization may occur only in some sub-regions of the “U” region. Hence, for any lattice tilt this phenomenon cannot happen for the parameters A < 0.6 or x > 4.5 in the region of Fig. 1. Adopting Eq. (11) with pn ¼ 1.5, the spatiotemporal evolutions of the exact probability Pn(t) are shown in Fig. 3. In Fig. 3(a), we show that the degree of localization is higher for the top image at point Q4 compared with the bottom image related to Q1, because of the larger distance between Q4 and the near-resonant region. The shorter tunneling time and higher velocity of delocalization are shown in bottom image of Fig. 3(b) for the parameters of Q2 in chaotic region compared with the top image for Q3 in regular region. We have also numerically confirmed the above results by applying Eq. (4). IV. NEAR-RESONANT REGIONS AND DEGREE OF LOCALIZATION
To intuitively observe the effect of parameters on the localization and delocalization, we numerically take timeaverage ð 1 Np Pm ¼ Pm ðtÞdt; Np 0 (14) ð Np 1 2 2 hm i ¼ hm idt Np 0 as the functions of parameters x, A, and n for a sufficiently large cumulative time Np. Fixing x ¼ 1.5 and N ¼ 100, we make the plots of the average probability P0 in the initially occupied state and the time-average hm2 i of mean-square
FIG. 2. Time evolutions of the probability P0(t) of initially occupied state [(a), (c)] and the mean-square displacement [(b), (d)] with pn ¼ 1.5. The dynamical localization is exhibited in (a) and (b) for the parameters of Q1 (dashed line) and Q4 (solid line) in the off-resonant region. The delocalization is shown in (c), (d) for the parameters of Q2 (solid line) and Q3 (dashed line) in the near-resonant region. The horizontal lines in (a) and (c) label the reference lines P0 ¼ 1 and P0 ¼ 0, respectively.
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FIG. 3. Spatiotemporal evolutions of the probability Pn(t) for pn ¼ 1.5, where the lighter areas correspond to the larger values of probability. The spatial distributions of probability indicate the different degrees of localization in (a) for the parameters of Q1 (bottom) and Q4 (top). The different tunneling times and velocities of delocalization are shown in (b) for the parameters of Q2 (bottom) and Q3 (top).
displacement versus pn for A ¼ 0.63 (solid curves) and A ¼ 0.4 (dashed curves), as shown in Figs. 4(a) and 4(b). In the two figures, we can see that P0 ðhm2 iÞ rapidly climbs (descends) as increasing pn from 0 to 0.4. After pn > 0.4 and except for a small near-resonant region, 0:8 < P0 1 and hm2 i < 1 are kept. At the resonant point pn ¼ 1.5, the meansquare displacement tends to infinity for an infinite cumulative time Np. The results mean that delocalization occurs only in the near-resonant region or for a weak tilt, while localization can occur in all the other parameter regions. Taking P0 < 0:8 as a criterion of near-resonance, from the inset of Fig. 4(a), we estimate the maximal width of the near-resonant region as pDn 0.07. Similarly, setting pn ¼ 1.5 and N ¼ 100, we plot P0 and hm2 i as functions of x for A ¼ 0.63 (solid curves) and A ¼ 0.4 (dashed curves), as in Figs. 4(c) and 4(d). Here, we find that P0 and hm2 i keep values 1 and 0, respectively, except for a small near-resonant region. At the resonant point x ¼ 1.5, the infinite meansquare displacement means delocalization. From the inset of Fig. 4(c), we estimate the maximal width of the nearresonant region as Dx 0.07. Comparison between the solid curves and dashed curves of Fig. 4 shows that the different amplitude values bring only minor differences of the results. The degree of localization can be characterized by the magnitude of hm2 i, and the smaller magnitude corresponds
to higher degree. Figures 4(b) and 4(d) exhibit that the degree of localization increases as enhancing the lattice tilt or the distance between parameter points and the nearresonant region for a sufficiently large pn value. Thus, we can control such a degree by adjusting both the lattice tilt and the distance of parameters to the near-resonant regions.
V. CONCLUSION
In summary, we have defined the chaos-assisted localization and delocalization appeared in the classically chaotic regions. For a single particle held in an amplitudemodulated and tilted optical lattice, we analytically and numerically illustrated the near-resonant regions crossing chaotic and regular regions, and revealed that chaosassisted delocalization can occur in the chaos-resonance overlapping regions, while chaos-assisted localization may appear in the other chaotic regions. The degree of localization is controlled by tuning the lattice tilt and the distance between parameter points and the near-resonant regions. The results clarify the long-standing contradiction “does chaos assist localization or delocalization,” and could be useful for experimentally manipulating chaos-assisted
FIG. 4. Plots of P0 and hm2 i as the functions of pn for x ¼ 1.5 and A ¼ 0.63 (solid line), A ¼ 0.4 (dashed line) [(a), (b)], and as the functions of x for pn ¼ 1.5 and A ¼ 0.63 (solid line), A ¼ 0.4 (dashed line) [(c), (d)]. The insets in (a) and (c) denote the near-resonant regions.
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quantum transport of single particles in the shaken and tilted optical or solid-state lattices. ACKNOWLEDGMENTS
This work was supported by the NNSF of China under Grant Nos. 11175064, 11205021, and 11475060 and the Construct Program of the National Key Discipline of China. 1
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CHAOS 24, 043114 (2014)
Does chaos assist localization or delocalization? Jintao Tan,1 Gengbiao Lu,2 Yunrong Luo,1 and Wenhua Hai1,a) 1
Department of Physics and Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081, China 2 Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004, China
(Received 12 August 2014; accepted 6 October 2014; published online 20 October 2014) We aim at a long-standing contradiction between chaos-assisted tunneling and chaos-related localization study quantum transport of a single particle held in an amplitude-modulated and tilted optical lattice. We find some near-resonant regions crossing chaotic and regular regions in the parameter space, and demonstrate that chaos can heighten velocity of delocalization in the chaos-resonance overlapping regions, while chaos may aid localization in the other chaotic regions. The degree of localization enhances with increasing the distance between parameter points and near-resonant regions. The results could be useful for experimentally manipulating chaos-assisted C 2014 AIP Publishing LLC. transport of single particles in optical or solid-state lattices. V [http://dx.doi.org/10.1063/1.4898332] A quantum particle in a static and tilted lattice is expected to undergo the celebrated Bloch oscillation rather than the uniformly accelerated motion, whereas under a periodic modulation, the dynamical localization (DL) and delocalization may occur for different parameter regions. In classical treatment of the system, the presence of periodic modulation also leads to chaos for the chaotic parameter region. We define the chaos-assisted localization (delocalization) as the localization (delocalization) appearing in the chaotic region. By investigating quantum transport of such a typical classically chaotic systems, we reveal that chaos may assist localization or delocalization, depending on whether quantum resonance happens in the chaotic region. The results clarify a longstanding contradiction between chaos-assisted tunneling and chaos-related localization.
I. INTRODUCTION
Understanding quantum tunneling and transport in classically chaotic systems is a significant problem, both fundamentally and practically. Driven double-well and optical lattice systems are two typical kinds of chaotic systems,1–4 in which some unusual transport or tunneling phenomena have been exhibited.5–9 Localization is a representative phenomenon, including DL,6,7 Anderson localization (AL)8 and coherent destruction of tunneling (CDT).9 One of the routes from localization to delocalization is the quantum resonance,10,11 which can be achieved by adjusting the modulation frequency to fit the static tilt (or bias).11 The contradiction between chaos-assisted tunneling and chaos-related localization has been found before some years. On the one hand, it is known that AL was related to disorder, and “chaos can substitute for disorder in Anderson’s scenario, leading to dynamical localization of quantum a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
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probability transport” for a periodically kicked rotator which is classically chaotic.12 As the quantum suppression of classically chaotic diffusion, such a localization has been directly observed.5 Recently, we further found that the highchaoticity could replace higher disorder to aid AL in a driven optical lattice.13 On the other hand, Lin and Ballentine have proved that the tunneling rate could be highly enhanced due to the periodic modulation associated with classical chaos.4 Subsequently, many theoretical and experimental works demonstrated that classical chaos enhanced quantum tunneling rate drastically2,3,14 even without a barrier to tunnel through.2,3 Particularly, in the absence of collisions and dephasing effects, a particle in a static and tilted lattice is localized in a certain range, due to the celebrated Bloch oscillations,15 while the presence of a periodically varying ramp results in chaos16–18 and delocalization.7,19 In addition, there exist other situations, where both the quantum localization and delocalization may be independent of the classical chaos.20 Why can chaos promote localization and/or delocalization such a pair of totally contrary phenomena? In this letter, we will give an answer to the abovementioned interesting question by investigating quantum transport of a single particle held in a one-dimensional (1D) amplitude-modulated and tilted optical lattice.11,21,22 It is well known that delocalization and localization in a lattice occur in different parameter regions,7,22,23 and the chaotic and regular regions also coexist in a chaotic system.17,18 When localization (delocalization) occurs in the chaotic region, we call it the chaos-assisted localization (delocalization). We will analytically and numerically illustrate some near-resonant regions, where the driving frequency equates or approaches the lattice tilt,11 and reveal that chaos can assist delocalization in the chaos-resonance overlapping regions, while chaos may aid localization in the other chaotic regions. The degree of localization is controlled by adjusting the ramp of lattice and the distance to the near-resonant regions. The results could be useful for experimentally manipulating chaos-assisted quantum transport of single
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C 2014 AIP Publishing LLC V
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particles in the shaken and tilted lattices, and can be extended to studying chaotic electron transport in semiconductor superlattice induced by a combination of ac and dc electric field.24
II. CLASSICALLY CHAOTIC AND REGULAR REGIONS
We consider a single particle held in a periodically modulated and tilted 1D optical lattice potential, which is governed by the Hamiltonian p2 q 2 q H ð p; q; tÞ ¼ þ V0 ½1 A sinðxtÞ sin þ n : (1) 2 2 2 p ffiffi ffi Hereafter, H ¼ H 0 =Er ; p ¼ 2p0 =ðhkÞ; q ¼ 2x ¼ 2kx0 ; t ¼ xr t0 , and x ¼ x0 =xr are the rescaled quantities of the original Hamiltonian H 0 , momentum p0 , position x0 , time t0 , and driving frequency x0 with wave vector k, mass m, recoil 2 2 energy Er ¼ h2mk , and recoil frequency xr ¼ Er =h. The dimensionless constant A denotes the driving strength, V0 and np (with p being the dimensionless lattice length in x coordinate) are the lattice depth and tilt11,21 in units of hxr and k hxr , respectively. Such a system has been realized experimentally by applying an amplitude-modulated laser standing wave and a linear potential produced by a magnetic field gradient.11,21 Note that our rescaled coordinate and momentum obey pffiffiffi the quantum-mechanical commutation relation ½q; p ¼ 2 2 i which is different from the corresponding commutator in the previous works,2,3 because of the different scaled units. The effective Planck’s constant is a dimensionhef f ¼ 8xr =x ¼ h=I0 with I0 being the action of less ratio2,3 a free particle over the distance k/2 ¼ p/k in the time T ¼ 2p/ x, which describes the importance of quantum treatment for a system with the action I0. For our system, it can be defined as the commutator only for the particular units of the rescaled position and momentum.2,3 We are interested in the quantum regime with I0 in the order of h, and aim at the effect of classical chaos on quantum transport. Let us at first seek the classically chaotic andpregular paffiffiffiffiffi rameter regions. After rescaling time as s ¼ 2 V0 t, from Eq. (1) the Newton’s equation m€ x 0 ¼ H 0x0 ðp0 ; x0 ; t0 Þ is transformed into xs p ffiffiffiffiffi sin q n=V0 : (2) qss þ sin q ¼ eðq; sÞ ¼ A sin 2 V0 For small values of A and n /V0, the e(q, s) can be treated as perturbation such that the perturbed solution reads q(s) ¼ q0(s) þ q1(s) with the zero-order heteroclinic solution q0(s) and the first-order “chaotic solution” q1(s) in the forms18,25 q0 ¼ 2arctan½sinhðs þ t0 Þ; ðs ðs eðq0 ; sÞq0;s ds q0;s eðq0 ; sÞgðsÞds; q1 ¼ gðsÞ C1
(3)
C2
where constants t0, C1, and C2 relate to initial conditions, q0,s(s) and g(s) are the bounded and unbounded functions, respectively
q0;s ðsÞ ¼ 2sechðs þ t0 Þ; ð gðsÞ ¼ q0;s q2 0;s ds 1 ¼ ½sinhðs þ t0 Þ þ ðs þ t0 Þsechðs þ t0 Þ: 4
(4)
Inserting the solution q(s) ¼ q0(s) þ q1(s) into Eq. (2), one can immediately prove the assertion. Generally, the corrected solution q1(s) is unbounded at s ! 6 1, because of the unboundedness of the function g(s). But using l’Hospital rule, we can prove that if and only if the condition ðs eðq0 ; sÞq0;s ds ¼ 0 (5) I6 ¼ lim t!61 C 1
is satisfied, the first correction q1(s) will maintain bounded. On the other hand, the corresponding Melnikov function reads17,18 ð1 Mðt0 Þ ¼ eðq0 ; sÞq0;s ds 1 px2 xt0 px 2pn cos pffiffiffiffiffi csch pffiffiffiffiffi : (6) ¼A 2V0 V0 2 V0 4 V0 The condition (5) implies the onset criterion of chaos17,18 M(t0) ¼ Iþ I ¼ 0, so we call q1(s) the chaotic solution. From the chaotic criterion and Eq. (6), one derives the inequality18,25 A
4n px p ffiffiffiffiffi ; sinh x2 4 V0
(7)
where the sign “>” indicates the chaotic region of parameter space, and “¼” means boundary between the chaotic and regular regions. Fixing the lattice depth V0 ¼ 3, we plot three boundary curves in Fig. 1(a) for the tilts pn ¼ 0.8 (thin dashed curve), pn ¼ 1.8 (solid curve), and pn ¼ 3 (dotted curve), respectively. For a given tilt, chaos may occur in the chaotic region above the corresponding boundary curve. Note that for 87Rb atom the frequency unit xr is in order of 103 Hz so that the chaotic regions up to x ¼ 12(xr) is experimentally realizable.2,3,21 A special curve is displayed by the thick dashed curve in Figs. 1(a) and 1(b), which corresponds to the resonant parameters11 pn ¼ x. Therefore, in the region above this curve and labeled “U,” chaos and resonance may coexist. This assertion is illustrated by plotting the four nearresonant shadow regions fi (i ¼ 1, 2, 3, and 4) crossing the different chaotic and regular regions for pn ¼ x ¼ 0.8, 1.5, 1.8, and 3, respectively. The width of any near-resonant region fi depends on the system parameters, and for pn ¼ x ¼ 1.5 the maximal width shown in the inset of Fig. 1(b) is estimated as Dx 0.07 from the inset of Fig. 4(c). The chaos-resonance overlapping regions “fi \ U” are described by the gray shaded areas embedded in U. In fact, for A < 1 and pn < 4.5 one can obtain many different chaotic regions and chaos-resonance coexistence regions associated with different pn values, and these regions are shrunk with increasing the tilt pn. The classical Poincare section for the
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FIG. 1. (a) Chaotic and regular parameter regions with different boundary curves for V0 ¼ 3 and pn ¼ 0.8 (thin dashed curve), pn ¼ 1.8 (solid curve), and pn ¼ 3 (dotted curve). The corresponding near-resonant regions (shadows) are labeled by fi for i ¼ 1, 2, and 3. The chaos-resonance overlapping regions “fi \ U” exist above the thick dashed curve with pn ¼ x. (b) Typical parameter points Qi(x, A) Q1(1.46, 0.63), Q2(1.5, 0.63), Q3(1.5, 0.4), and Q4(4, 0.4) on the chaotic and regular regions for pn ¼ 1.5 (solid boundary curve). The approximate width of near-resonant region f4 and the distance between Q1 and Q2 are exhibited in the inset. (c) Poincare section for the rescaled position q and momentum p of a classical particle for the parameters near point Q2 of (b). Hereafter, all the quantities plotted in the figures are dimensionless.
rescaled position q and momentum p and the parameters near point Q2 of Fig. 1(b) is shown in Fig. 1(c), where the central regions of the potential wells near q ¼ 62np, n ¼ 0, 1, 2… consist of some small-amplitude “islands” of stability which are separated by the “chaotic sea.”2,3 Clearly, in the case n ¼ 0, system (1) is reduced to the previously investigated one in which the chaos-assisted quantum dynamical tunneling between separated momentum regions in phase space was observed.2,3 Adding the static tilt, we here aim at clarifying the contradiction between chaos-assisted localization and chaos-assisted delocalization associated with tunneling finite and infinite barriers between the spatially separated islands of stability, respectively. Does chaos aid delocalization or localization in a given chaotic region? We will take the typical parameter points Qi for i ¼ 1, 2, 3, 4 in Fig. 1(b) with pn ¼ 1.5 as examples to investigate such a question. III. MANIPULATING CHAOS-ASSISTED LOCALIZATION AND DELOCALIZATION
In the nearest-neighbor tight binding approximation, the Hamiltonian governing quantum dynamics of the system (1) reads7,21 HðtÞ ¼ JðtÞ
ðjmihm þ 1j þ jm þ 1ihmjÞ
pn
mðjmihmjÞ:
þ
Here, jmi represents the Wannier state localized on lattice site m, the coupling parameter between adjacent sites i, j is given as26
J ðsÞcosðpnsÞds
0
2 1 ) 2
J ðsÞsinðpnsÞds
(11)
for the initial conditions P0(0) ¼ 1, Pn6¼0(0) ¼ 0, where J n is n-th bessel function of the first kind. The corresponding mean-square displacement reads7 hm2 i ¼
X
Pn ðtÞn2
n
dxhij r2 þ ð1 A sinðxtÞÞV0 sin2 xjji
1
¼ J0 þ dJ sin xt
ðt
2
t
0
(8)
1
JðtÞ ¼
which forms an orthogonal set approximately. In 1986, Dunlap and Kenkre have investigated an analogous system with constant coupling and periodic tilt, and obtained an analytical solution of the probability amplitude an(t) in the P quantum state7 jwðtÞi ¼ n an ðtÞjni. Such a periodically tilted lattice system is a typical classical chaotic system.16 It is worth noting that the analytical expressions for quantum treatment of the chaotic system are based on the nearest-neighbor tight binding approximation, however, the quantum particle is generally delocalized except for the case when the ac field parameters are taken some specific values,7 because of the quantum tunneling. Directly applying our periodic coupling and static tilt to replace the constant coupling and periodic tilt in Dunlap’s analytical solution, respectively, produces the exact probability of the particle at site n as ( " ð Pn ðtÞ ¼ jan ðtÞj2 ¼ J 2n 2
1 X
1 1 X
ð1
with constant J0 in units of recoil energy Er ¼ hxr and dJ proportional to AV0. The used Wannier state vector reads27 pffiffiffiffi pffiffiffiffiffi 2 jji ¼ wðx xj Þ ¼ ð V0 =pÞ1=4 e V0 ðxjpÞ =2 ; (10)
(9)
ð t 2 ð t 2 ¼2 JðsÞ cosðpnsÞds þ 2 JðsÞ sinðpnsÞds (12) 0
0
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which characterizes the degree of localization and a quite large hm2 i value means the delocalization. For V0 ¼ 3, we numerically get the coupling function in Eqs. (11) and (12) as JðtÞ ¼ 0:058 0:033A sinðxtÞ:
(13)
On the basis of Eqs. (11) and (12), we plot time evolutions of the probability P0(t) of initially occupied state and the mean-square displacement hm2 i in Fig. 2 for the tilt pn ¼ 1.5 and the driving frequencies and strengths of the four points Qi(x, A) in Fig. 1(b). In Figs. 2(a) and 2(b), we show that P0(t) (hm2 i) returns to unity (zero) with different periods and amplitudes. This means, the occurrence of dynamical localizations are of different degrees.6 The larger amplitude corresponds to the lower degree of localization, and is associated with the smaller distance between Q1 and the near-resonant region. While in Figs. 2(c) and 2(d), the P0(t) (hm2 i) cannot return to unity (zero), and the meansquare displacement tends to infinity as t ! 1, which means delocalization caused by the resonance. The tunneling rate4 is inverse proportional to the time from P0(0) ¼ 1 to P0(t) ¼ 0, and the velocity of delocalization can be defined as the gradient of the hm2 i envelope. In Figs. 2(c) and 2(d), we observe that chaos related to the solid curves enhances the tunneling rate and the velocity of delocalization. We are interested in manipulating the chaos-assisted localization and delocalization by modulating parameters between the regular and chaotic regions. Fixing the driving amplitude A ¼ 0.63 and adjusting frequency from Q1(x ¼ 1.46) of regular region to Q2(x ¼ 1.5) of chaotic region, the dashed curve of Fig. 2(a) and solid curve of Fig. 2(c) display that chaos results in the transition from localization to delocalization. On the other hand, fixing the driving amplitude A ¼ 0.4 and increasing frequency from Q3(x ¼ 1.5) of regular region to Q4(x ¼ 4) of chaotic region, the dashed curve of Fig. 2(c) and solid curve of Fig. 2(a) indicates that chaos leads the initial delocalization to the
final localization. Taking different parameter points in the chaotic region of Fig. 1(b) to plot the mean-square displacements which are not shown here, we find that chaos aids delocalization only for the chaos-resonance overlapping region “f4 \ U,” and for the other chaotic region chaos may assist localization. Further, we take different tilt values to make the similar plots and find that chaos-assisted delocalization may occur only in some sub-regions of the “U” region. Hence, for any lattice tilt this phenomenon cannot happen for the parameters A < 0.6 or x > 4.5 in the region of Fig. 1. Adopting Eq. (11) with pn ¼ 1.5, the spatiotemporal evolutions of the exact probability Pn(t) are shown in Fig. 3. In Fig. 3(a), we show that the degree of localization is higher for the top image at point Q4 compared with the bottom image related to Q1, because of the larger distance between Q4 and the near-resonant region. The shorter tunneling time and higher velocity of delocalization are shown in bottom image of Fig. 3(b) for the parameters of Q2 in chaotic region compared with the top image for Q3 in regular region. We have also numerically confirmed the above results by applying Eq. (4). IV. NEAR-RESONANT REGIONS AND DEGREE OF LOCALIZATION
To intuitively observe the effect of parameters on the localization and delocalization, we numerically take timeaverage ð 1 Np Pm ¼ Pm ðtÞdt; Np 0 (14) ð Np 1 2 2 hm i ¼ hm idt Np 0 as the functions of parameters x, A, and n for a sufficiently large cumulative time Np. Fixing x ¼ 1.5 and N ¼ 100, we make the plots of the average probability P0 in the initially occupied state and the time-average hm2 i of mean-square
FIG. 2. Time evolutions of the probability P0(t) of initially occupied state [(a), (c)] and the mean-square displacement [(b), (d)] with pn ¼ 1.5. The dynamical localization is exhibited in (a) and (b) for the parameters of Q1 (dashed line) and Q4 (solid line) in the off-resonant region. The delocalization is shown in (c), (d) for the parameters of Q2 (solid line) and Q3 (dashed line) in the near-resonant region. The horizontal lines in (a) and (c) label the reference lines P0 ¼ 1 and P0 ¼ 0, respectively.
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FIG. 3. Spatiotemporal evolutions of the probability Pn(t) for pn ¼ 1.5, where the lighter areas correspond to the larger values of probability. The spatial distributions of probability indicate the different degrees of localization in (a) for the parameters of Q1 (bottom) and Q4 (top). The different tunneling times and velocities of delocalization are shown in (b) for the parameters of Q2 (bottom) and Q3 (top).
displacement versus pn for A ¼ 0.63 (solid curves) and A ¼ 0.4 (dashed curves), as shown in Figs. 4(a) and 4(b). In the two figures, we can see that P0 ðhm2 iÞ rapidly climbs (descends) as increasing pn from 0 to 0.4. After pn > 0.4 and except for a small near-resonant region, 0:8 < P0 1 and hm2 i < 1 are kept. At the resonant point pn ¼ 1.5, the meansquare displacement tends to infinity for an infinite cumulative time Np. The results mean that delocalization occurs only in the near-resonant region or for a weak tilt, while localization can occur in all the other parameter regions. Taking P0 < 0:8 as a criterion of near-resonance, from the inset of Fig. 4(a), we estimate the maximal width of the near-resonant region as pDn 0.07. Similarly, setting pn ¼ 1.5 and N ¼ 100, we plot P0 and hm2 i as functions of x for A ¼ 0.63 (solid curves) and A ¼ 0.4 (dashed curves), as in Figs. 4(c) and 4(d). Here, we find that P0 and hm2 i keep values 1 and 0, respectively, except for a small near-resonant region. At the resonant point x ¼ 1.5, the infinite meansquare displacement means delocalization. From the inset of Fig. 4(c), we estimate the maximal width of the nearresonant region as Dx 0.07. Comparison between the solid curves and dashed curves of Fig. 4 shows that the different amplitude values bring only minor differences of the results. The degree of localization can be characterized by the magnitude of hm2 i, and the smaller magnitude corresponds
to higher degree. Figures 4(b) and 4(d) exhibit that the degree of localization increases as enhancing the lattice tilt or the distance between parameter points and the nearresonant region for a sufficiently large pn value. Thus, we can control such a degree by adjusting both the lattice tilt and the distance of parameters to the near-resonant regions.
V. CONCLUSION
In summary, we have defined the chaos-assisted localization and delocalization appeared in the classically chaotic regions. For a single particle held in an amplitudemodulated and tilted optical lattice, we analytically and numerically illustrated the near-resonant regions crossing chaotic and regular regions, and revealed that chaosassisted delocalization can occur in the chaos-resonance overlapping regions, while chaos-assisted localization may appear in the other chaotic regions. The degree of localization is controlled by tuning the lattice tilt and the distance between parameter points and the near-resonant regions. The results clarify the long-standing contradiction “does chaos assist localization or delocalization,” and could be useful for experimentally manipulating chaos-assisted
FIG. 4. Plots of P0 and hm2 i as the functions of pn for x ¼ 1.5 and A ¼ 0.63 (solid line), A ¼ 0.4 (dashed line) [(a), (b)], and as the functions of x for pn ¼ 1.5 and A ¼ 0.63 (solid line), A ¼ 0.4 (dashed line) [(c), (d)]. The insets in (a) and (c) denote the near-resonant regions.
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quantum transport of single particles in the shaken and tilted optical or solid-state lattices. ACKNOWLEDGMENTS
This work was supported by the NNSF of China under Grant Nos. 11175064, 11205021, and 11475060 and the Construct Program of the National Key Discipline of China. 1
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