Nonlinearly-enhanced energy transport in many dimensional quantum chaos
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Nonlinearly-enhanced energy transport in many dimensional quantum chaos
D. S. Brambila , and A. Fratalocchi ∗ PRIMALIGHT, Faculty of Electrical Engineering; Applied Mathematics and Computational Science,King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Max-Born Institute, Max-Born-Stra β e 2 A, 12489, Berlin, Germany (Dated: April 5, 2018)By employing a nonlinear quantum kicked rotor model, we investigate the transport of energy inmultidimensional quantum chaos. Parallel numerical simulations and analytic theory demonstratethat the interplay between nonlinearity and Anderson localization establishes a perfectly classi-cal correspondence in the system, neglecting any quantum time reversal. The resulting dynamicsexhibits a nonlinearly-induced, enhanced transport of energy through soliton wave particles. PACS numbers: 05.45.Mt, 05.45.Yv, 03.75.-b
Anderson localization is a fundamental concept that,originally introduced in solid-state physics to describeconduction-insulator transitions in disordered crystals,has permeated several research areas and has becomethe subject of great research interest [1–11]. Theoriesand subsequent experiments demonstrated that disorderfavors the formation of spatially localized states, whichsustain diffusion breakdown and exponentially attenu-ated transmission in random media [1]. Although manyproperties of wave localization are now well understood,several fundamental questions remains. Perhaps one ofthe most intriguing problem is related to the transport ofenergy. Intuitively, one can expect that disorder —by fa-voring exponentially localized stated— arrests in generalany propagation inside a noncrystalline medium. How-ever, the interplay between localization and disorder isnontrivial [5, 12] and under specific conditions random-ness can significantly enhance energy transport. In par-ticular, it has been observed that quasi-crystals with mul-tifractal eigenstates and/or material systems with tempo-ral fluctuations of the potential (or refractive index), leadto anomalous diffusion in the phase space [13–16]. Thisoriginates counterintuitive dynamics including ultralowconductivities [13], as well as the formation of mobilityedges even in one dimensional systems [16]. All thesestudies focused on specific geometries and linear mate-rials, while nothing is practically known about the roleof nonlinearity in enhancing (or depleting) the transportof energy in disordered media. This problem acquires astrong fundamental character when refereed to the fieldof quantum localization. In this area, quantum-classicalcorrespondences mediated by Anderson localization pos-sess many implications in the irreversible behavior of timereversible systems, which are at the basis of a long stand-ing physical dispute —i.e., the Loschmidt paradox [17]—as well as many fascinating quantum phenomena suchas the time reversal of classical irreversible systems andthe quantum echo effect [18, 19]. It has been argued,in particular, that microscopic chaos is at the basis ofthe irreversible entropy growth observed in classical sys-tems [20]. Time reversal, according to this interpreta- tion, is only possible at the quantum level [18, 19] andsustained by Anderson localization, which breaks diffu-sive transport and suppresses the mixing ability of chaos[21]. However, when more dimensions are considered,numerical simulations predict that ergodicity is fully re-stored and diffusive transport settles is again, thus re-establishing the classical features of chaos and prevent-ing quantum time reversal [19]. Nevertheless, theoreticalwork reported to date considered only noninteracting sys-tems, characterized by linear equations of motion. TheLoschmidt paradox, conversely, involved the use of inter-acting atoms, whose interplay in the mean field regimeis accounted by short and/or long ranged nonlinear re-sponses [22–24]. Besides that, as pointed out in the lit-erature [18], atoms interactions are of crucial importancein quantum localization and diffusion. A key questiontherefore lies in understanding the role of nonlinearity intransporting energy in multidimensional quantum chaos.In this Letter, we theoretically investigate this problemby employing both numerical simulations and analytictechniques. To pursue a general theory, we here considerthe following two dimensional model: i ∂ψ∂t + ∇ ψ + Z d r R ( r ′ − r ) ψ ( r ′ ) + U ψδ T ( t ) = 0 , (1)with r = ( x, y ), ∇ = ∂ /∂x + ∂ /∂y , δ T = P n δ ( t − nT ) a periodic delta-function of period T , R a gen-eral nonlinear response and U ( x, y ) = γ (cos x + cos y ) + ǫ cos( x + y ) a two dimensional periodic potential withstrength defined by ǫ and γ . Equation (1) defines atwo dimensional, nonlinear quantum kicked rotor: for R = 0 it reduces to the linear quantum kicked rotator[19] while for U = 0 it corresponds to the 2D nonlinearSchr¨odinger equation (NLS), which represents a universalmodel of nonlinear waves in dispersive media [24]. In onedimension, conversely, Eq. (1) generalizes the nonlinearmodel investigated in [21] with classical chaos parame-ter K = 2 γT . Despite its deterministic nature, Eq. (1)can be precisely mapped to the Anderson model with arandom potential [25], and therefore furnishes a funda-mental model for studying energy transport in random Time [T] < P > nonlinear linear ε=1.2ε=0.8ε=0 (b) -2π -π 0 π 2πx-2π-π0π2πy-2π-π0π2πy (a)(c) t-0t-100T FIG. 1. (Color Online). (a)-(b) spatial density | ψ | dis-tribution at (a) t = 0 and at (b) t = 100 T ; (c) momentumdiffusion h P i versus time in linear (dashed lines) and nonlin-ear (solid lines) conditions and for increasing coupling ǫ . Inthe simulations we set σ = 0 . ω = 0 . A = 4 and K = 1 . systems. The nonlinear response n = R d r R ( r ′ − r ) ψ ( r ′ )is modeled as a nonlocal term following a general diffu-sive nonlinearity (1 − σ ∇ ) n = | ψ | , with nonlocalitycontrolled by σ . When σ = 0, the system response islocal with n = | ψ | . For σ = 0, conversely, the systemnonlinearity becomes long ranged with kernel given by R ( r ) = π K ( r σ ), being K the modified Bessel functionof second kind. Diffusive nonlinearities are particularlyinteresting in the context of nonlinear optics, as they canbe easily accessed in liquids [26, 27], as well as in Bose-Einstein Condensates (BEC), where they generalize pre-viously investigated models [28, 29].We begin our theoretical analysis by calculating the mo-mentum diffusion h P i = h ψ | ˆ p | ψ i versus time, with ˆ p = ∇ /i the momentum operator and h ψ | f | ψ i = R d r f | ψ | the quantum average. Parallel numerical simulations areperformed by a direct solution of (1) with an uncondi-tionally stable algorithm. In order for the field ψ to ex-plore the periodic potential U , we here consider wavepackets whose spatial extension ∆ r ≪ π . Figure 1 sum-marizes our results obtained for σ = 0 .
2, by launchingat the input a gaussian beam ψ = Ae − x /ω with waist ω = 0 . A = 4 (Fig. 1a). The stochas-tic parameter K has been set to K = 1 . > K ∗ , abovethe stochastization threshold K ∗ ≈ .
97 where the lin-ear classical uncoupled rotor exhibits diffusive transportin momentum space [19]. For comparison, we also cal-culated the linear dynamics resulting from R = 0 (Fig.1b dotted line). As seen from Fig. 1b, the 2D nonlinearrotor behaves dramatically different with respect to itslinear counterpart, demonstrating the strong role playedby nonlinearity in the process. In particular, the linearsystem exhibits Anderson localization and diffusion sup-pression for ǫ = 0 (uncoupled condition), while for grow-ing ǫ it shows a monotonically increasing sub-diffusion(Fig. 1b). In the nonlinear regime, conversely, Andersonlocalization is suppressed even for ǫ = 0, and the dynam- ε L y a punov λ ε(a) (b) FIG. 2. (Color Online). Positive Lyapunov exponent λ versus coupling ǫ , calculated for (a) Eqs. (4) and (b) Eq. (5).In the simulation we set K = 5. ics shows an erratic, random-like behavior that does notmanifest any simple monotonic increase for growing val-ues of ǫ . These results are also significantly different fromthe nonlinear kicked rotor in one dimension [21], wherenonlinearity was observed to induce localization effects.To theoretically understand this dynamics, we reduce thesystem to a nonlinear map modeling the nonlinear evo-lution of the ground state of Eq. (1). This analysis isjustified by the observation that the spatial field profile,despite the chaotic motion, is not significantly altered intime (Fig. 1a,c). Due to the nonintegrability of the 2DNLS equation, we found analytical expressions by a vari-ational analysis [28, 29]. In particular, we begin from theLagrangian density L of Eq. (1): L = i (cid:18) ψ ∗ ∂ψ∂t − ψ ∂ψ ∗ ∂t (cid:19) − | ψ | × (cid:20) U δ T + 12 Z d r ′ K ( r − r ′ ) | ψ ( r ′ ) | (cid:21) + |∇ ψ | , (2)and study the ground state for U = 0 by the followingGaussian ansatz: ψ = q Pπ e − r /a a , defined by the power P = h ψ | ψ i and waist a ( t ). By substituting the ansatz in(2), after performing a variational derivative over a , weobtain a classical dynamics described by the followingHamiltonian H : H = 12 (cid:18) ∂a∂t (cid:19) + V , V = 8 a − P πσ Z ( a ) , (3)with Z ( x ) = e x Γ(0 , − x ) and V acting as a potential forthe one dimensional motion of a . The potential V has abell shape profile that possesses a unique absolute mini-mum V ( a ∗ ) for every combination of P and σ . The fixedpoint a (0) = a ∗ corresponds to a soliton wave of thesystem, which propagates in a translational fashion withfixed waist a ( t ) = a ∗ , while different initial values lead toa breather [30] characterized by a periodic oscillation of a in time. When the kicks are turned on, for U = 0, the dy-namics of the ground state is perturbed by an addition ofmomentum p = ( p x , p y ), with a consequent translation ofits center of mass. In order to model this dynamics, weconsidered the following general ansatz for the ground ε quadratic fitcomputation from Eq. (1) < P > Time [T]
Eq. (1) linearε=1.2ε=0.8ε=0Map d i ff u s i on D (a) (b) FIG. 3. (Color Online). (a) Momentum diffusion h ¯ P i versustime calculated from Eq. (1) (solid lines), Eqs. (4)-(5) (dia-mond markers) and Eq. (1) in linear regime (dashed lines);(b) diffusion coefficient D versus coupling ǫ . In the simula-tions we set σ = 0 . K = 5. state evolution: ψ = q Pπ e − ( r − r0 )2 /a i p ( r − r0 ) / T a , with p ( t ), a ( t ) and r ( t ) = [ x ( t ) , y ( t )] Lagrangian variableswhose equations of motion, after an integration from nT to ( n + 1) T , are found to be: p n +1 = p n − [ γ n g + ǫ n u sin( x + y )] , r n +1 = r n + p n +1 , (4)with γ n = Ke − a n , ǫ n = 2 ǫT e − a n , g = [sin x , sin y ], u = [1 , f n ≡ f ( nT ) and a n +1 calculated from theintegration of the following equation: ∂ a∂t = − ∂ V ∂a + δ T e − a × [ γ (cos x + cos y ) + 2 e − a ǫ cos( x + y )] . (5)Equations (4) can be regarded as a variant of the four di-mensional standard map, which is randomized by timedependent coupling parameters γ n and ǫ n . The lat-ter depend on Eq. (5), which represents the motionof a two dimensional nonlinear kicked rotor. The sys-tem possesses an a dependent chaos parameter, given by K a = e − a / γT . For K > K ∗ , Eq. (5) is fully chaoticand can be regarded as an external noise source to Eqs.(4), increasing the mixing of the overall system [31]. Tohighlight such a dynamics, we plot in Fig. 2a and Fig. 2bthe positive Lyapunov exponent λ [32] calculated for Eqs.(4) and Eq. (5), respectively. As seen in Fig. 2a, Eqs. (4)show a strong hyperchaotic behavior, with two positiveLyapunov exponents whose largest value grows linearlywith ǫ . Fig. 2b, conversely, displays the chaotic natureof wave packet extension a , whose Lyapunov coefficient λ increases significantly fast (quadratically) with coupling.We investigate the diffusion in momentum p by observ-ing that above the stochastization threshold K > K ∗ ,the change in momentum ∆ p = p n +1 − p n ∝ K becomeslarge compared to π . The classic position r n , which istaken modulo 2 π , can be treated a random process, sta-tistically uncorrelated in time and uniformly distributedin [ − π, π ]. The diffusion coefficient D is therefore evalu- ated as follows: D = (cid:28) ∆ p n (cid:29) = K h sin x ih e − a n / i + 2 ǫ T h e − a n / i× h sin( x + y ) i = K h e − a n / i + ǫ T h e − a n / i , (6)To evaluate the averages h e − a n / i and h e − a n / i , we canconsider a as a random variable (due to its chaotic motionin the phase-space), uniformly distributed between itsoscillation extrema a min and a max : h e − a n /τ i = √ πτ × (cid:20) erf (cid:18) a max τ (cid:19) − erf (cid:18) a min τ (cid:19)(cid:21) = 1 + O ( a max /τ ) (7)being ∆ = a max − a min and having expanded the errorfunctions up to second order, due to the smallness oftheir arguments a/τ <
1. By substituting (7) into (6),we obtain the diffusion coefficient, which reads as follows: D = K ǫ T (8)Equation (8) allows to derive interesting properties forthe nonlinear dynamics of Eq. (1). In particular, thequantum average h P i results from an hyperchaotic sys-tem described by a four dimensional standard map withrandom coefficients, and each realization manifests itselfas a random walk in Fig. 1b. The map diffusion rateis identical to the momentum diffusion of the classicallinear rotor [19], hence, an additional average (in timeor over an ensemble of input conditions) re-establishes aperfect classical correspondence for every coupling ǫ ≥ ǫ , and in generalthe quantum diffusion h P i follows a fractional behaviorwith h P i ∝ t β< (see e.g., [19] or Fig. 1b dashed lines).As a result, the linear quantum rotor sub-diffuses at aslower rate than its classical counterpart. Conversely,Eq. (8) predicts a perfect classical correspondence forevery coupling ǫ , which is re-established thanks to non-linear effects. In order to demonstrate this dynamics, weperformed extensive numerical simulations from Eq. (1)and calculated the average diffusion through a quantumaverage followed by an average over different input con-ditions h ¯ P i = R Dψ (cid:10) ψ (cid:12)(cid:12) ˆ p (cid:12)(cid:12) ψ (cid:11) . Figure 3a summarizes ourresults obtained for K = 5, σ = 0 . ω = 0 . A = 4. In complete agree-ment with Eqs. (4)-(8), we observe a diffusive behavior h ¯ P i ∝ t for every ǫ ≥ D of the non-linear system for increasing ǫ (Fig. 3b). In perfect agree-ment with our theory, we observe a quadratic behaviorversus the coupling parameter ǫ .In conclusion, motivated by the large interest in thestudy of energy transport in complex media, we inves-tigated the quantum-classical correspondences in many-dimensional quantum chaos. In particular, we em-ployed a two-dimensional, nonlinear quantum kicked ro-tor (NQKR) and study the role of nonlinearity in enhanc-ing or depleting energy diffusion and quantum-classicalcorrespondences. We analytically tackled the problem bya variational analysis, reducing the dynamics to a four-dimensional standard map with random coefficients. Insuch an hyperchaotic system, a perfect classical corre-spondence is established by nonlinearity and an enhanceddiffusion is observed due to solitons wave-particles, whichare able to diffuse energy at a faster rate with respectto linear waves. These results show that quantum timereversal of classical irreversible systems is completelyprevented in many dimensions, and demonstrate thatnonlinearity can be effectively employed to increase thetransport of energy in complex media. This work is ex-pected to stimulate further theory and experiments inthe broad area dealing with quantum chaos and energytransport phenomena.A. Fratalocchi thanks S. Trillo for fruitful discussions.We acknowledge funding from KAUST (Award No.CRG-1-2012- FRA-005). ∗ Introduction to Wave Scattering, Localization,and Mesoscopic Phenomena (Academic Press, San Diego,1995).[2] F. Scheffold et al. , Nature , 206 (1999).[3] T. Schwartz et al. , Nature , 52 (2007).[4] S. E. Skipetrov and B. A. van Tiggelen,Phys. Rev. Lett. , 043902 (2006).[5] C. Conti and A. Fratalocchi, Nature Phys. , 794 (2008).[6] P. A. Lee and T. V. Ramakrishnan,Rev. Mod. Phys. , 287 (1985). [7] Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Moran-dotti, D. N. Christodoulides, and Y. Silberberg,Phys. Rev. Lett. , 013906 (2008).[8] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht,P. Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer,and A. Aspect, Nature , 891 (2008).[9] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus-cio, Nature , 895 (2008).[10] K. Efetov, Supersymmetry in disorder and chaos (Cam-bridge, New York, 1997).[11] S. Fishman, D. R. Grempel, and R. E. Prange,Phys. Rev. Lett. , 509 (1982).[12] D. Molinari and A. Fratalocchi,Opt. Express , 18156 (2012).[13] D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, andP. Lanco, Phys. Rev. Lett. , 3915 (1993).[14] L. Levi, M. Rechtsman, B. Freedman, T. Schwartz,O. Manela, and M. Segev, Science , 1541 (2011).[15] A. Madhukar and W. Post,Phys. Rev. Lett. , 1424 (1977).[16] F. M. Izrailev and A. A. Krokhin,Phys. Rev. Lett. , 4062 (1999).[17] J. Loschmidt, Sitzungsber. der kais. Akad. d. W. Math.Naturw. II , 128 (1876).[18] J. Martin, B. Georgeot, and D. L. Shepelyansky,Phys. Rev. Lett. , 074102 (2008).[19] S. Adachi, M. Toda, and K. Ikeda,Phys. Rev. Lett. , 659 (1988).[20] P. Gaspard et al. , Nature , 865 (1998).[21] F. Benvenuto, G. Casati, A. S. Pikovsky, and D. L. She-pelyansky, Phys. Rev. A , R3423 (1991).[22] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, andT. Pfau, Phys. Rev. Lett. , 160401 (2005).[23] M. Klawunn, R. Nath, P. Pedri, and L. Santos,Phys. Rev. Lett. , 240403 (2008).[24] A. Griffin, D. W. Snoke, and S. Stringari, Bose-EinsteinCondensation (Cambridge University Press, Cambridge,1995).[25] F. Haake,