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Nonlinearly-enhanced energy transport in many dimensional quantum chaos D. S. Brambila1,2 and A. Fratalocchi1∗ 1PRIMALIGHT, Faculty of Electrical Engineering; Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia 2Max-Born Institute, Max-Born-Straβe 2 A, 12489, Berlin, Germany (Dated: January 23, 2013) By employinga nonlinearquantumkickedrotormodel, we investigatethetransport of energy in multidimensional quantum chaos. Parallel numerical simulations and analytic theory demonstrate that the interplay between nonlinearity and Anderson localization establishes a perfectly classi- 3 cal correspondence in the system, neglecting any quantum time reversal. The resulting dynamics 1 exhibitsa nonlinearly-induced,enhanced transport of energy through soliton waveparticles. 0 2 PACSnumbers: 05.45.Mt,05.45.Yv,03.75.-b n a J Anderson localization is a fundamental concept that, tion, is only possible at the quantum level [18, 19] and 2 originally introduced in solid-state physics to describe sustained by Anderson localization, which breaks diffu- 2 conduction-insulator transitions in disordered crystals, sivetransportandsuppressesthe mixing ability of chaos has permeated several research areas and has become [21]. However, when more dimensions are considered, ] n the subject of great research interest [1–11]. Theories numerical simulations predict that ergodicity is fully re- n and subsequent experiments demonstrated that disorder stored and diffusive transport settles is again, thus re- - s favors the formation of spatially localized states, which establishing the classical features of chaos and prevent- di sustain diffusion breakdown and exponentially attenu- ingquantumtime reversal[19]. Nevertheless,theoretical . ated transmission in random media [1]. Although many workreportedtodateconsideredonlynoninteractingsys- t a properties of wave localization are now well understood, tems, characterized by linear equations of motion. The m several fundamental questions remains. Perhaps one of Loschmidtparadox,conversely,involvedthe useofinter- - themostintriguingproblemisrelatedtothetransportof acting atoms, whose interplay in the mean field regime d energy. Intuitively,onecanexpectthatdisorder—byfa- is accounted by short and/or long ranged nonlinear re- n o voringexponentiallylocalizedstated—arrestsingeneral sponses [22–24]. Besides that, as pointed out in the lit- c any propagation inside a noncrystalline medium. How- erature[18],atomsinteractionsareofcrucialimportance [ ever, the interplay between localization and disorder is in quantum localization and diffusion. A key question 1 nontrivial [5, 12] and under specific conditions random- thereforelies inunderstandingthe roleofnonlinearityin v ness can significantly enhance energy transport. In par- transportingenergyinmultidimensionalquantumchaos. 8 ticular,ithasbeenobservedthatquasi-crystalswithmul- In this Letter, we theoretically investigate this problem 9 tifractaleigenstatesand/ormaterialsystemswithtempo- by employing both numerical simulations and analytic 0 ralfluctuationsofthepotential(orrefractiveindex),lead techniques. Topursue a generaltheory,we hereconsider 5 . to anomalous diffusion in the phase space [13–16]. This the following two dimensional model: 1 originates counterintuitive dynamics including ultralow 0 ∂ψ 3 conductivities [13], as well as the formation of mobility i + 2ψ+ drR(r′ r)ψ(r′)+UψδT(t)=0, (1) 1 edges even in one dimensional systems [16]. All these ∂t ∇ Z − : studies focused on specific geometries and linear mate- v with r = (x,y), 2 = ∂2/∂x2+∂2/∂y2, δ = δ(t i rials, while nothing is practically known about the role ∇ T n − X nT) a periodic delta-function of period T, RPa gen- of nonlinearity in enhancing (or depleting) the transport eral nonlinear response and U(x,y) = γ(cosx+cosy)+ r of energy in disordered media. This problem acquires a a ǫcos(x + y) a two dimensional periodic potential with strong fundamental character when refereed to the field strength defined by ǫ and γ. Equation (1) defines a of quantum localization. In this area, quantum-classical two dimensional, nonlinear quantum kicked rotor: for correspondencesmediated by Andersonlocalizationpos- R = 0 it reduces to the linear quantum kicked rotator sessmanyimplicationsintheirreversiblebehavioroftime [19] while for U = 0 it corresponds to the 2D nonlinear reversiblesystems,whichareatthebasisofalongstand- Schrödingerequation(NLS),whichrepresentsauniversal ingphysicaldispute—i.e.,the Loschmidtparadox[17]— modelofnonlinearwavesindispersivemedia[24]. Inone as well as many fascinating quantum phenomena such dimension, conversely, Eq. (1) generalizes the nonlinear as the time reversal of classical irreversible systems and model investigated in [21] with classical chaos parame- the quantum echo effect [18, 19]. It has been argued, ter K = 2γT. Despite its deterministic nature, Eq. (1) in particular, that microscopic chaos is at the basis of can be precisely mapped to the Anderson model with a the irreversible entropy growth observedin classicalsys- random potential [25], and therefore furnishes a funda- tems [20]. Time reversal, according to this interpreta- mental model for studying energy transport in random 2 2π (a) 12000 (a) 0.14(b) π nonlinear linearε=1.2 (b) 1 0.12 y 0 10000 εε==00.8 λ 0.1 v 0.8 -2-ππ t-0 P> 68000000 Lyapuno0.6 000...000468 2π (c)< 0.4 0.02 π 4000 0 1 2 3 4 0 0.5 1 1.5 2 2.5 y 0 2000 ε ε -π -2π t-100T 0 FIG. 2. (Color Online). Positive Lyapunov exponent λ -2π -π x0 π 2π 0 20 T40ime [T60] 80 100 versuscouplingǫ,calculatedfor(a)Eqs. (4)and(b)Eq. (5). In thesimulation we set K =5. FIG. 1. (Color Online). (a)-(b) spatial density |ψ|2 dis- tribution at (a) t = 0 and at (b) t = 100T; (c) momentum ics shows an erratic, random-like behavior that does not diffusion hPi versus time in linear (dashed lines) and nonlinear (solid lines) conditions and for increasing coupling ǫ. In manifest any simple monotonic increase for growing val- thesimulationswesetσ=0.2,ω0 =0.3,A=4andK =1.8. uesofǫ. Theseresultsarealsosignificantlydifferentfrom the nonlinear kicked rotor in one dimension [21], where nonlinearity was observed to induce localization effects. Totheoreticallyunderstandthisdynamics,wereducethe systems. The nonlinear response n= drR(r′ r)ψ(r′) system to a nonlinear map modeling the nonlinear evo- − is modeled as a nonlocal term followinRg a general diffu- lution of the ground state of Eq. (1). This analysis is sive nonlinearity (1 σ2 2)n = ψ 2, with nonlocality justified by the observation that the spatial field profile, − ∇ | | controlled by σ. When σ = 0, the system response is despite the chaotic motion, is not significantly altered in local with n = ψ 2. For σ = 0, conversely, the system time (Fig. 1a,c). Due to the nonintegrability of the 2D | | 6 nonlinearity becomes long ranged with kernel given by NLSequation,wefoundanalyticalexpressionsbyavari- R(r)= 21πK0(σr), being K0 the modified Bessel function ationalanalysis[28,29]. Inparticular,webeginfromthe of second kind. Diffusive nonlinearities are particularly Lagrangiandensity of Eq. (1): interestinginthecontextofnonlinearoptics,astheycan L ∗ be easily accessed in liquids [26, 27], as well as in Bose- =i ψ∗∂ψ ψ∂ψ ψ 2 Einstein Condensates (BEC), where they generalize pre- L 2(cid:18) ∂t − ∂t (cid:19)−| | viously investigated models [28, 29]. 1 Webeginourtheoreticalanalysisbycalculatingthemo- UδT + dr′K(r r′)ψ(r′)2 + ψ 2, (2) ×(cid:20) 2Z − | | (cid:21) |∇ | mentum diffusion P = ψ pˆ2 ψ versus time, with pˆ= /i the momentuhmioperhat|or2 |anid ψ f ψ = drf ψ 2 and study the ground state for U = 0 by the following ∇the quantumaverage. Parallelnumehric|a|lsiimulaRtions|ar|e Gaussianansatz: ψ = 2P e−r2/a2,definedbythepower π a performed by a direct solution of (1) with an uncondi- q P = ψ ψ and waist a(t). By substituting the ansatz in tionally stable algorithm. In order for the field ψ to ex- h | i (2), after performing a variational derivative over a, we plore the periodic potential U, we here consider wave obtain a classical dynamics described by the following packetswhosespatialextension∆r 2π. Figure1sum- ≪ Hamiltonian : marizes our results obtained for σ = 0.2, by launching H at the input a gaussian beam ψ = Ae−x2/ω02 with waist 1 ∂a 2 8 P = + , = Z(a), (3) ω0 = 0.3 and amplitude A = 4 (Fig. 1a). The stochas- H 2(cid:18)∂t(cid:19) V V a2 − 2πσ2 ∗ tic parameter K has been set to K = 1.8 > K , above the stochastization threshold K∗ 0.97 where the lin- with Z(x) = exΓ(0, x) and acting as a potential for ≈ − V ear classical uncoupled rotor exhibits diffusive transport the one dimensional motion of a. The potential has a V in momentum space [19]. For comparison, we also cal- bell shape profile that possesses a unique absolute mini- ∗ culated the linear dynamics resulting from R = 0 (Fig. mumV(a )foreverycombinationofP andσ. Thefixed ∗ 1b dotted line). As seen from Fig. 1b, the 2D nonlinear point a(0) = a corresponds to a soliton wave of the rotor behaves dramatically different with respect to its system, which propagatesin a translationalfashion with ∗ linear counterpart, demonstrating the strong role played fixedwaista(t)=a ,whiledifferentinitialvaluesleadto by nonlinearity in the process. In particular, the linear abreather[30]characterizedbyaperiodicoscillationofa system exhibits Anderson localizationand diffusion sup- intime. Whenthekicksareturnedon,forU =0,thedy- 6 pressionfor ǫ=0 (uncoupled condition), while for grow- namicsofthegroundstateisperturbedbyanadditionof ing ǫ it shows a monotonically increasing sub-diffusion momentump=(p ,p ),withaconsequenttranslationof x y (Fig. 1b). In the nonlinear regime, conversely,Anderson its center of mass. In order to model this dynamics, we localizationissuppressedevenforǫ=0,andthedynam- considered the following general ansatz for the ground 3 10000Eq. (1)Maplinearε=1.2 (a) quadratic fit (b) ated as follows: <P>468000000000 εε==00.8 diffusion D345 computation from Eq. (1) D =si(cid:28)n(∆x02p+2n(cid:29)y0=)2K2=2hKsin2xe20−iahe2n−/8a2n+/8iǫ2+T22ǫe2T−a22nhe/4−a,2n/4(i6) 2000 2 ×h i 4 h i h i 00 20 40 60 80 100 10 0.5 1 To evaluate the averages e−a2n/8 and e−a2n/4 , we can Time [T] ε consideraasarandomvarhiable(diuetoithschaotiicmotion in the phase-space), uniformly distributed between its FIG.3. (Color Online). (a)MomentumdiffusionhP¯iversus oscillation extrema a and a : time calculated from Eq. (1) (solid lines), Eqs. (4)-(5) (dia- min max mond markers) and Eq. (1) in linear regime (dashed lines); √πτ (b) diffusion coefficient D versus coupling ǫ. In the simula- e−a2n/τ2 = h i 2∆ tions we set σ=0.2 and K =5. a a erf max erf min =1+O(a3 /τ3) (7) ×(cid:20) (cid:18) τ (cid:19)− (cid:18) τ (cid:19)(cid:21) max state evolution: ψ = 2P e−(r−r0)2/a2+ip(r−r0)/2T, with p(t), a(t) and r0(t) = [qx0(πt),y0(t)] Lagarangian variables bfuenincgtio∆ns=upamtaoxs−ecaomnidn oarndderh,advuinegteoxpthanedsemdatlhlneeessrroorf whose equations of motion, after anintegrationfromnT their arguments a/τ < 1. By substituting (7) into (6), to (n+1)T, are found to be: weobtainthediffusioncoefficient,whichreadsasfollows: pn+1 =pn−[γng+ǫnusin(x0+y0)], D = K2 +ǫ2T2 (8) rn+1 =rn+pn+1, (4) 4 Equation (8) allows to derive interesting properties for with γn = Ke−a82n, ǫn = 2ǫTe−a42n, g = [sinx0,siny0], the nonlinear dynamics of Eq. (1). In particular, the u = [1,1], fn ≡ f(nT) and an+1 calculated from the quantum average hPi results from an hyperchaotic sys- integration of the following equation: tem described by a four dimensional standard map with random coefficients, and each realization manifests itself ∂2a ∂ −a2 as a random walk in Fig. 1b. The map diffusion rate ∂t2 =− ∂Va +δTe 8 is identical to the momentum diffusion of the classical −a2 linear rotor [19], hence, an additional average (in time [γ(cosx0+cosy0)+2e 8 ǫcos(x0+y0)]. (5) × or over an ensemble of input conditions) re-establishes a Equations(4)canberegardedasavariantofthefourdi- perfectclassicalcorrespondenceforeverycouplingǫ 0. ≥ mensional standard map, which is randomized by time It is worthwhile observing that the classical correspon- dependent coupling parameters γ and ǫ . The lat- dence in the multidimensional linear quantum rotor is n n ter depend on Eq. (5), which represents the motion manifested only for very high coupling ǫ, and in general of a two dimensional nonlinear kicked rotor. The sys- the quantum diffusion P follows a fractional behavior tem possessesana dependent chaosparameter,givenby with P tβ<1 (see e.hg.,i[19] or Fig. 1b dashed lines). h i ∝ K = e−a2/8γT. For K > K∗, Eq. (5) is fully chaotic As a result, the linear quantum rotor sub-diffuses at a a and can be regarded as an external noise source to Eqs. slower rate than its classical counterpart. Conversely, (4), increasing the mixing of the overall system [31]. To Eq. (8) predicts a perfect classical correspondence for highlightsuchadynamics,weplotinFig. 2aandFig. 2b every coupling ǫ, which is re-established thanks to non- thepositiveLyapunovexponentλ[32]calculatedforEqs. lineareffects. In orderto demonstratethis dynamics, we (4)andEq. (5),respectively. AsseeninFig. 2a,Eqs. (4) performed extensive numerical simulations from Eq. (1) show a strong hyperchaotic behavior, with two positive and calculated the average diffusion through a quantum Lyapunov exponents whose largest value grows linearly average followed by an average over different input con- with ǫ. Fig. 2b, conversely, displays the chaotic nature ditions P¯ = Dψ ψ pˆ2 ψ . Figure 3a summarizes our h i 2 ofwavepacketextensiona,whoseLyapunovcoefficientλ resultsobtaineRdforK(cid:10) (cid:12)=5(cid:12),σ(cid:11)=0.2andbyconsideringan (cid:12) (cid:12) increasessignificantly fast(quadratically)with coupling. initial wave packet composed by a Gaussian beam with We investigate the diffusion in momentum p by observ- waist ω0 =0.3 and amplitude A=4. In complete agree- ∗ ing that above the stochastization threshold K > K , ment with Eqs. (4)-(8), we observe a diffusive behavior thechangeinmomentum∆p=pn+1 pn K becomes P¯ t for every ǫ 0 (Fig. 3a solid lines and diamond − ∝ h i∝ ≥ large compared to π. 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