Table Of Content

Theory of chaotic atomic transport in an optical lattice V.Yu. Argonov and S.V. Prants Laboratory of Nonlinear Dynamical Systems, V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok, Russia A semiclassical theory of chaotic atomic transport in a one-dimensional nondissipative optical lattice is developed. Using the basic equations of motion for the Bloch and translational atomic variables, wederiveastochasticmap forthesynchronizedcomponentoftheatomicdipolemoment that determines the center-of-mass motion. We find the analytical relations between the atomic and lattice parameters under which atoms typically alternate between flying through the lattice and being trapped in the wells of the optical potential. We use the stochastic map to derive 7 formulas fortheprobability densityfunctions(PDFs) fortheflightandtrappingevents. Statistical 0 properties of chaotic atomic transport strongly depend on the relations between the atomic and 0 lattice parameters. We show that there is a good quantitative agreement between the analytical 2 PDFs and those computed with thestochastic map and thebasic equations of motion for different n ranges of the parameters. Typical flight and trapping PDFs are shown to be broad distributions a with power law “heads” with the slope −1.5 and exponential “tails”. The lengths of the power J law and exponential parts of the PDFs depend on the values of the parameters and can be varied 9 continuously. Wefindanalyticalconditions,underwhichdeterministicatomictransport hasfractal properties, and explain a hierarchical structureof thedynamical fractals. 1 v PACSnumbers: 42.50.Vk,05.45.Mt,05.45.Xt 8 3 0 I. INTRODUCTION trap is infinite. 1 0 The anomalous atomic transport of cold atoms in 7 The transport properties of cold atoms in optical opticallatticesisofadifferentnature. Acoldatomin 0 lattices depend on the lattice and atomic parameters anopticallatticecanbetrappedinthepotentialwells / h and can be very diverse. The optical lattice is a peri- anditcanmoveovermanywavelengthsindependence p odic structureofmicron-sizedpotentialwellswhichis on whether its energy is below or above the potential - createdbyalaserstandingwavemadeoftwocounter- barrier. Transitionsbetweentheseeventsarestochas- t n propagatinglaser beams. Dilute atomic samples with ticallyinducedbythecoolingforceanditsfluctuations a negligibly small atom-atom interactions are used to (heating due to randomness of spontaneous emission u probe experimentally single-atom phenomena. The and fluctuations of the atomic dipole moment). Mea- q atomic motion can take form of ballistic transport, surement of the trajectory of a single cold ion in a : v oscillations in wells of the optical potential, Brown- one-dimensionalopticallattice,bytracingitsposition i X ian motion, random walks, Lévy flights, and chaotic throughfluorescentphotons,havedemonstratedLévy transport. flights [4]. By decreasing the optical potential depth, r a Mostoftheexperimental[1,2,3,4,5,6,7]andthe- achangeofthetransportcharacteristicsfromdiffusive oretical [1, 7, 8, 9, 10, 11] works on the atomic trans- to quasiballistic was observed [4]. port in optical lattices have been done in the context The anomalous properties of atomic transport dis- oflasercoolingofatoms. One particularlyinteresting cussed briefly above are associated with random aspect of these works is the discovery of anomalous atomic recoils in spontaneous emission processes transport properties of cold atoms and Lévy flights which are generally inevitable in any cooling scheme [12,13]. ALévyflightisarandomprocessinthespace and which make the transport looks like a random (time)domainwiththedistributionofthelength(du- walk. However, the problem may be considered not ration)offlighteventsthatisgivenbyaLévylawpos- in a laser cooling but in a more general context as a sessing an algebraically decaying “tail” and infinite study of deterministic motion of atoms with compar- variance. As a consequence, the atomic trajectories atively large momentum interacting with a standing- (time series)haveself-similar(fractal)natureandthe wavelightfield. Itisafundamentalnonlinearinterac- probability to find superlong flights is not negligibly tionbetweendifferentdegreesoffreedom(thetransla- small as in a case of normal diffusion. tionalandinternalatomic degreesoffreedomandthe In the context of sub-recoil laser cooling, the Lévy field ones) that can be treated in a Hamiltonian form flights have been found experimentally [1, 2] in the when neglecting any losses. In this context sponta- distributions of trapping and escape times for ultra- neous emission may be consideredas a noise imposed coldatoms trappedin a momentum state close to the on the coherent atomic dynamics. It has been pre- dark state. It was shown in Refs. [1, 2] that not only dicted in Refs. [14, 15] that, besides the well-known thevariancebutthe meantime foratomstoleavethe transportpropertiesofatomsinopticallattices,there 2 should exist a deterministic chaotic transport with a II. HAMILTON-SCHRO¨DINGER complicatedalternationofatomic oscillationsin wells EQUATIONS OF MOTION of the optical potential and atomic flights over many potential wells when the atom may change the direc- We consider a two-level atom with mass m and a tion of motion many times. This phenomenon looks transition frequency ω , moving with the momentum a like a random walk but it should be stressed that it P along the axis X in a one-dimensional classical mayoccurwithoutanyrandomfluctuationsofthelat- standing laser wave with the frequency ω and the f tice parameters and any noise like spontaneous emis- wave vector k . In the frame, rotating with the fre- f sion. The deterministic chaotic transport is a result quency ω , the Hamiltonian is the following: f ofchaotic atomicdynamics in the standing-wavefield P2 1 that means an exponential sensitivity of the inter- Hˆ = + ~(ωa ωf)σˆz ~Ω(σˆ−+σˆ+)coskfX. nal and translational atomic variables to small vari- 2ma 2 − − (1) ations in the lattice parameters and/or initial conditions. The chaotic transport and its manifestations Here σˆ±,z are the Pauli operators which describe the transitions between lower, 1 , and upper, 2 , atomic likeatomicdynamicalfractalsmayoccurbothinclas- | i | i states, Ω is the Rabi frequency which is proportional sical[16,17,18,19]andquantizedlightfields[20,21]. to the square root of the number of photons in the Spontaneous emission events interrupt the coherent wave √n. The laser wave is assumed to be strong atomic dynamics in randominstants of time andmay enough (n 1), so we can treat the field classically. give rise anomalous statistical properties of atomic ≫ The simple wavefunction for the electronic degree of transport [22]. freedom is Ψ(t) =a(t)2 +b(t)1 , (2) | i | i | i whereaandbarethecomplex-valuedprobabilityam- plitudes to find the atom in the states 2 and 1 , | i | i respectively. Using the Hamiltonian (1), we get the In our previous papers [17, 18, 21, 23] we have Schrödinger equation found chaotic atomic transport and dynamical frac- da ω ω a f tals in numerical experiments and studied its proper- i = − a Ωbcosk X, f dt 2 − ties andmanifestations. The rangesofthe lattice and (3) atomic parameters and initial conditions, for which db ω ω f a the center-of-mass motion may be chaotic, have been i = − b ΩacoskfX. dt 2 − established. In this paper we develop a semiclassical Let us introduce instead of the complex-valued prob- Hamiltoniantheoryofthechaoticatomictransportin ability amplitudes a and b the following real-valued aone-dimensionalopticallatticeandconfirmtheana- variables: lytical results by the numerical simulation. In Sec. II ∗ ∗ we derive the basic equations of motion and give the u 2Re(ab ), v 2Im(ab ), ≡ ≡− (4) result of computation of the maximum Lyapunov ex- z a2 b2, ponent whose positive values determine the ranges of ≡| | −| | the atom-field detuning ∆ and initial atomic momen- whereuandvareasynchronized(withthelaserfield) tum p for which chaotic transport occurs. Sec. III and a quadrature components of the atomic electric 0 briefly reviews distinct regimes of motion. Using ap- dipole moment,respectively,andz is the atomic pop- proximate solutions of the basic equations, we con- ulation inversion. struct in Sec. IV a stochastic map for the synchro- In the process of emitting and absorbing photons, nized component of the atomic dipole moment u that atoms not only change their internal electronic states determines the chaotic transport. In Sec. V we intro- but their external translational states change as well duce a simple illustrativemodelofrandomwalkingof due to the photon recoil. If the atomic mean mo- the quantity arcsinu on a circle. Depending on the mentum is large as compared to the photon momen- relations between the lattice and atomic parameters, tum ~kf, one can describe the translational degree of the transport properties may be diverse. We find in freedomclassically. The positionand momentumofa Sec.Vtheconditionunderwhichtheprobabilityden- point-likeatomsatisfyclassicalHamiltonequationsof sity functions (PDFs) for the flights and trappings of motion. Full dynamics in the absence of any losses is the atoms either purely exponential or have promi- nowgovernedby the Hamilton-Schrödingerequations nent power law slopes. We derive the PDFs analyti- for the real-valued atomic variables cally and compare the results with simulation of the x˙ =ω p, r stochastic map and the basic equations. The results p˙ = usinx, obtainedinSec.IVareusedinSec.VItofindthecon- − u˙ =∆v, (5) ditionsforappearingdynamicalatomicfractalsandto explain their structure. Finally, Sec. VIIgives conclu- v˙ = ∆u+2zcosx, − sions. z˙ = 2vcosx, − 3 where x k X and p P/~k are classical atomic the plot marks the value of the maximum Lyapunov f f ≡ ≡ center-of-mass position and momentum, respectively. exponent λ. In white regions the values of λ are al- Dotdenotesdifferentiationwithrespecttothedimen- sionlesstimeτ Ωt. Thenormalizedrecoilfrequency, ω ~k2/m Ω≡ 1, and the atom-field detuning, r ≡ f a ≪ ∆ (ω ω )/Ω, are the control parameters. The f a ≡ − system has two integrals of motion, namely the total energy ω ∆ H rp2 ucosx z, (6) ≡ 2 − − 2 and the Bloch vector u2+v2+z2 =1. (7) The conservation of the Bloch vector length immedi- ately follows from Eqs. (4). Equations of motion similar to the set (5) were obtained in our previous papers [14, 15, 17] in order to describetheinteractionbetweenatwo-levelatomand the cavity radiationfield in the strong-couplinglimit. FIG. 1: Maximum Lyapunov exponent λ vs atom-field Taking into account a back reaction of the atom on detuning ∆ (in units of the laser Rabi frequency Ω) and the radiation field and within the semiclassical ap- initial atomic momentum p0 (in units of the photon mo- proximation, we were able to get the corresponding mentum~kf): ωr =10−5, u0 =z0 =0.7071, v0 =0. versionofthe Hamilton-Schrödingerequationsforthe atomicpositionx,momentump,populationinversion most zero, and the atomic motion is regular in the z, and two combined atom-field variables which were corresponding ranges of ∆ and p . In shadowed re- 0 denoted by the same letters u and v as the atomic gions positive values of λ imply unstable motion. dipole-moment components in Eqs. (5) Those equa- Inallnumericalsimulationsweshallusethenormal- tions has been shown in Refs. [14, 15, 17] to have ized value of the recoil frequency equal to ω =10−5. r a positive Lyapunov exponent in a wide range of the The initial atomic position is taken to be x = 0. 0 control parameters and initial atomic momentum p . 0 The detuning ∆ will be varied in a wide range, and It implies dynamical chaos in the usual sense of ex- the Blochvariablesarerestrictedby the length ofthe ponential sensitivity to smallchanges in initial condi- Bloch vector (7). It should be noted that we use in tions and/orcontrolparameters. The sameshouldbe this paper the normalization to the laser Rabi fre- valid with the set of equations (5) describing the dif- quency Ω, not to the vacuum (or single-photon) Rabi ferent physical situation — a two level atom in open frequency as it has been done in our previous papers space with a strong standing-wave field. [14, 15, 16, 17]. So the ranges of the normalized con- Equations (5) constitute a nonlinear Hamiltonian trolparameters,takeninthis paper,differ fromthose autonomoussystemwithtwoandhalfdegreesoffree- in the cited papers. Figure 1 demonstrates that the domwhich,owingtotwointegralsofmotion,moveon center-of-massmotionbecomesunstableifthedimen- a three-dimensionalhypersurface with a given energy sionless momentum exceeds the value p 300 that 0 value H. In general, motion in a three-dimensional ≈ corresponds (with our normalization) to the atomic phase space in characterized by a positive Lyapunov velocity v 1 m/s of a cesium atom in the field a exponent λ, a negative exponent equal in magnitude ≈ withthewavelengthclosetothetransitionwavelength to the positive one, and zero exponent. The sum of λ 852 nm. a all Lyapunov exponents of a Hamiltonian system is ≃ zero [24]. The maximum Lyapunov exponent charac- terizes the meanrate ofthe exponentialdivergenceof initially close trajectories, III. REGIMES OF MOTION 1 d(τ) λ lim λ(τ), λ(τ) lim ln , (8) A. Regular atomic motion at exact atom-field ≡τ→∞ ≡d(0)→0τ d(0) resonance and serves as a quantitative measure of dynamical chaos in the system. Here, d(τ) is a distance (in the Thecaseofexactresonance,∆=0,wasconsidered Euclidean sense) at time τ between two trajectories in detail in Ref. [15, 23]. Now we briefly repeat the closeto eachotheratinitialtime momentτ =0. The simple results for the sake of self-consistency. At zero result of computation of the maximum Lyapunov ex- detuning, the variable u becomes a constant, u =u , 0 ponent in dependence on the detuning ∆ and the ini- andthefast(u,v,z)andslow(x,p)variablesaresep- tialatomicmomentump is showninFig.1. Colorin arated allowing one to integrate exactly the reduced 0 4 equations of motion. The total energy is equal to maximumLyapunovexponentλdepends bothonthe parametersω and∆, and oninitial conditions ofthe ω r H = rp2 ucosx, (9) system (5). It is naturally to expect that off the res- 0 2 − onance atoms with comparatively small values of the where u = u0. The center-of-mass atomic motion in initial momentum p0 will be at once trapped in the this spatially periodic potential of the standing wave first well of the optical potential, whereas those with isdescribedbythesimplenonlinearequationforafree large values of p0 will fly through. The question is physical pendulum what will happen with atoms, if their initial kinetic energy will be close to the maximum of the optical x¨+ωru0sinx=0, (10) potential. Numerical experiments demonstrate that such atoms will wander in the optical lattice with al- and does not depend on evolution of the internal de- ternatingtrappingsinthewellsoftheopticalpotential grees of freedom. and flights over its hills. The direction of the center- The translationalmotion is trivial when u is zero. 0 of-mass motion of wandering atoms may change in a The atom moves in one direction with a constant ve- chaotic way (in the sense of exponential sensitivity locity,andthe Rabioscillationsaremodulatedby the to small variations in initial conditions). A typical standing wave. Equations (9) and (10) describe the chaotic atomic trajectory is shown in Fig. 2. atomic motion in the simple cosine potential u cosx 0 with three types of trajectories which are possible in 20 dependence on the value of the energy H: oscillator- like motion in a potential well if H0 < u0 (atoms are 0 trappedbythestanding-wavefield[25]),motionalong -20 the separatrix if H = u , and ballistic-like motion if 0 0 H0 >u0. ExactsolutionsofEq. (10) areeasilyfound x -40 in terms of elliptic functions (see [15, 23]). -60 As to internal atomic evolution, it depends on the translational degree of freedom since the strength of -80 the atom-field coupling depends on the position of -100 atom in a periodic standing wave. At ∆ = 0, it is 0 5*104 105 1.5*105 2*105 2.5*105 easy to find the exact solutions of Eqs. (5) τ τ v(τ)= 1 u2 cos 2 cosxdτ′+χ , FIG.2: Typicalatomictrajectoryintheregimeofchaotic ± −  0 transport: x0 = 0, p0 = 300, z0 = −1, u0 = v0 = 0, p Z0 (11) ωr =10−5,∆=−0.05. Atomicposition is shown in units  τ  of k−1, time τ in units of Ω−1. f z(τ)= 1 u2 sin 2 cosxdτ′+χ , 0 ∓ −   p Z0 It follows from (5) that the translational motion   of the atom at ∆ = 0 is described by the equation where u=u0, and cos[x(τ)] is a given function of the of a nonlinear phys6ical pendulum with the frequency translational variables only which can be found with modulation the help of the exact solution for x [15, 23]. The sign of v is equal to that for the initial value z and 0 x¨+ω u(τ)sinx=0, (13) r z 0 χ arcsin (12) 0 where u is a function of all the other dynamical vari- ≡∓ 1 u2 − 0 ables. is an integrationconstant. Thpe internal energy of the atom, z, and its quadrature dipole-moment component v could be considered as frequency-modulated IV. STOCHASTIC MAP FOR CHAOTIC signals with the instant frequency 2cos[x(τ)] and the ATOMIC TRANSPORT modulation frequency ω p(τ), but it is correct only r if the maximum value of the first frequency is much Chaoticatomictransportoccursevenifthenormal- greater than the value of the second one, i. e., for ized detuning is very small, ∆ 1 (Fig. 1). Under | | ≪ ωrp0 2. this condition, we will derive in this section approx- | |≪ imate equations for the center-of-mass motion. The atomic energy at ∆ 1 is given with a good accu- B. Chaotic atomic transport off the resonance racybythesimple|re|so≪nantexpression(9). Returning tothebasicsetoftheequationsofmotion(5),wemay In Fig. 1 we show the λ-map in the space of the neglect the first term in the fourth equation since it initial momentum p and detuning ∆ values. The is very small as compared with the second one there. 0 5 However, we cannot now exclude the third equation 0.8 from the consideration. Using the solution (11) for v, we can transform this equation as u˙ = ∆ 1 u2 cosχ, (14) u 0.6 ± − p where τ 0.4 χ 2 cosxdτ′+χ . (15) 0 1000 2000 3000 4000 0 ≡ Z0 τ Far from the nodes of the standing wave, Eq. (14) canbe approximatelyintegratedunder the additional FIG. 3: Typical evolution of the atomic dipole-moment condition, ω p 1, which is valid for the ranges | r | ≪ componentuforacomparativelyslowandslightlydetuned of the parameters and the initial atomic momentum atom: x0 = 0, p0 = 550, v0 = 0, u0 = z0 = 0.7071, where chaotic transport occurs. Assuming cosx to ωr =10−5, ∆=−0.01. be a slowly-varying function in comparison with the function cosχ, we obtain far from the nodes the ap- proximatesolutionfortheu-componentofthe atomic nodes (so small that the atomic momentum has no dipole moment time to change its value significantly) we may use the Raman-Nath approximation of the constant velocity. ∆ In this approximation, we have u sin sinχ+C , (16) ≈ ±2cosx (cid:18) (cid:19) 2 χ sinx+χ . (17) where C is an integration constant. Therefore, the ≃ ω p 0 r node amplitude of oscillations of the quantity u for com- Substituting this expression into Eq. (14), we can paratively slow atoms (ω p 1) is small and of the r | |≪ integrateitintheinterval06x6π(whichcomprises order of ∆ far from the nodes. | | only one node) and obtain the value of u just after It follows from the third and forth equations in the crossing the first node set (5) that u satisfies to the equationof motionfor a drivenharmonicoscillatorwith the naturalfrequency π ∆ and the driving force 2zcosx whose frequency is ∆ | | u1 sin arcsinu0 cosχdx = space and time dependent. At ∆ = 0, the synchro- ≈  ± ω p  nized component of the atomic |dip|ole moment u= is r node Z0   a constant whereas the other Bloch variables z and ∆π (18) =sin arcsinu v∆os=cill0ataenidnfaacrcforrodmantcheewniothdetsh,ethseoluvatiroianb(le11u).peArt- 0± ωrpnode 1−u20 × f|or|m6s shallow oscillations for the natural frequency 2 p 2 v J + z E , 0 0 0 0 |∆| is small as compared with the Rabi frequency. ×(cid:20) (cid:18)ωrpnode(cid:19) (cid:18)ωrpnode(cid:19)(cid:21)(cid:19) However, the behavior of u is expected to be very where special when an atom approaches to any node of the standing wave since near the node the oscillations of 2H p (19) the atomic population inversion z slow down and the node ≡ ω corresponding driving frequency becomes close to the r r resonance with the natural frequency. As a result, is the value of the atomic momentum at the instant sudden “jumps” of the variable u are expected to oc- whentheatomcrossesanode(whichisthesamewith cur near the nodes. This conjecture is supported by agivenvalueoftheenergyH forallthenodes),J0and the numericalsimulation. In Fig. 3 we show a typical E0 arezero-orderBesselandWeberfunctions,respec- behaviorofthevariableuforacomparativelyslowand tively. Inthe limit ofthe largeargument2/(ωrpnode), slightly detuned atom. The plot clearly demonstrates both the functions have a harmonic asymptotics [27], sudden “jumps” of u near the nodes of the standing and the expression (18) reduces to the form wave and small oscillations between the nodes. Approximatingthevariableubetweenthenodesby ∆ π u sin 1 constant values, we can construct a discrete mapping ≈ ± 1−u20 (cid:20)rωrpnode × um =f(um−1),whereum isavalueofujustafterthe 2 pπ 2 π m-thnodecrossing(includingmultiplecrossingsofthe v cos z sin 0 0 × ω p − 4 − ω p − 4 − same node). In order to estimate the values of u just (cid:18) (cid:18) r node (cid:19) (cid:18) r node (cid:19)(cid:19) afterthecrossing,oneneedstointegrateEq. (14)near z ]+arcsinu ). 0 0 − anode. Since uchangeslargelyina smallregionnear (20) 6 It is the deterministic solution obtained with the ap- V. STATISTICAL PROPERTIES OF proximations mentioned above. It should be stressed CHAOTIC TRANSPORT that the solution contains trigonometric functions with large values of the arguments which are in- A. Model for chaotic atomic transport verselyproportionaltotheatomicmomentum. Inthe Raman-Nath approximation, we take p p . In node Withgivenvaluesofthecontrolparametersandthe ≃ fact, even small deviations from this mean value may energyH,thecenter-of-massmotionisdeterminedby resultinlargechangesinthemagnitudeofthetrigono- the values of u (see Eq. (13)). One can obtain from m metric functions. Therefore, they can be treated as, theexpressionfortheenergy(9)theconditionsunder practically random variables in the range [ 1,1]. Be- which atoms continue to move in the same direction − yondtheRaman-Nathapproximation,thevalueofthe aftercrossinganodeorchangethedirectionofmotion atomicmomentumpdependsonthevalueofu which m notreachingthenearestantinode. Moreover,asinthe changeseverytimewhentheatomcrossesanode. So, resonance case, there exist atomic trajectories along we can replace arguments of the trigonometric func- whichatomsmovetoantinodeswiththevelocitygoing tions by random variables. Finally, we introduce the asymptotically to zero. It is a kind of separatrix-like stochastic map motionwithaninfinitetimeofreachingthestationary points. π The conditions for different regimes of motion de- um ≡sin ∆ ω p sinφm+arcsinum−1 = pend on whether the crossing number m is even (cid:18) r r node (cid:19) or odd. Motion in the same direction occurs at m π ( 1)m+1u < H, separatrix-like motion — at =sin∆rωrpnode j=1sinφj +arcsinu0, (−−1)m+1umm = H, and turns — at (−1)m+1um > H. X Itissobecauseevenvaluesofmcorrespondtocosx>   (21) 0,whereasoddvalues—tocosx<0. Thequantityu whereφ arerandomphasestobechosenintherange m duringthe motionchangesits valuesin a random-like [0,2π]. When deriving this map, we neglected the manner (see Fig. 3) taking the values which provide term z in Eq. (20) which is small as compared with 0 the atomeither toprolongthe motioninthe samedi- the factor π/ω p . r node rectionortoturn. Therefore,atomsmaymovechaot- With givpen values of ∆, ωr, and pnode, the map ically in the optical lattice. The chaotic transport (21)hasbeenshownnumericallytogiveasatisfactory occursiftheatomicenergyisintherange0<H <1. probabilistic distribution of magnitudes of changes in At H <0, atoms cannot reach even the nearest node the variable u just after crossing the nodes. The and oscillate in the first potential well in a regular stochastic map (21) is valid under the assumptions manner(see Fig.1). AtH >1,the values ofuare al- of small detunings (∆ 1) and comparatively slow ways satisfy to the flight condition. Since the atomic | |≪ atoms (ωrp 1). Furthermore, it is valid only for energy is positive in the regime of chaotic transport, | | ≪ thoserangesofthecontrolparametersandinitialcon- the corresponding conditions can be summarized as ditionswherethemotionofthebasicsystem(5)isun- follows: at u < H, atom always moves in the same | | stable. Forexample,inthoserangeswherealltheLya- direction, whereas at u > H, atom either moves in | | punovexponentsarezero,ubecomesaquasi-periodic the same direction, or turns depending on the sign of function and cannot be approximated by the map. cosx in a given interval of motion. In particular, if The stochastic map (21) allows to reduce the basic the modulus of u is larger for a long time then the setofequationsofmotion(5)tothefollowingeffective energy value, then the atom oscillates in a potential equations of motion: well crossingtwo times eachof two neighbor nodes in the cycle. The conditions stated above allow to find a di- x˙ = ω p, r rectcorrespondencebetweenchaoticatomictransport p˙ = u sinx, (22) − m in the optical lattice and stochastic dynamics of the m˙ = ωrpnode δ(cosx), Bloch variable u. It follows from Eq. (21) that | | the jump magnitude um um−1 just after crossing − where u is found fromEq. (21). The third equation the m-th node depends nonlinearly on the previous m in the set (22) gives a correspondence between the value um−1. For analyzing statistical properties of continuous evolution of the atomic motion and a dis- the chaoticatomic transport,it is more convenientto cretecrossingnumberm. Theintegrationofthedelta introduce the map for arcsinum function δ(cosx) over time at points with cosx = 0 θ arcsinu = gives 1/(ω p ) in dependence on the direction of m m r node ≡ ± motion and whether the serial number of the node is π (23) evenorodd. Since we calculateabsolutevaluesofthe =∆ ω p sinφm+arcsinum−1, r r node delta function, it is easyto show that m is a constant if cosx=0 and increases by one if cosx=0. wherethe jump magnitudedoes notdepend ona cur- 6 7 rentvalueofthevariable. Themap(23)visuallylooks then the internal atomic variable θ arcsinu m m ≡ asa randommotionofthe point alonga circle ofunit just after crossing the m-th node may take with the radius(Fig.4). Theverticalprojectionofthispointis sameprobabilitypracticallyany value fromthe range [ π/2,π/2] (see Fig. 4). With given values of the r−ecoil frequency ω = 10−5 and the energy in the u r trapping range0<H <1correspondingto chaoticallymoving +1 atoms, large jumps take place at medium detunings H n H j|∆us|t∼aft0e.r1.crToshseinpgraobnaobdileitiysPeq−uafolrtoanthaetopmrobtoabtiulirtny flight arcsi um to get to one of the trapping regions in Fig. 4 (which onedependsonwhetherthecrossingnumbermiseven θ m 0 or odd) and is given by flight arccosH 1 P− = < , P+ =1 P−, (25) π 2 − where P is the probability to prolong the motion in -H + the samedirectionaftercrossingthenode. Itis easily -1 trapping to get from Eq. (25) the probability for an atom to cross l successive nodes before turning FIG. 4: Graphic representation for the maps of um and Pfl(l)=P+lP− = θm ≡arcsinum. H is a given value of the atomic energy. = arccosH exp l ln 1 arccosH . (26) Atomseitheroscillateinopticalpotentialwells(trapping) π − π (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) or fly through theoptical lattice (flight). It is a flight probability density function (PDF) in u . The value of the energy H specifies four regions, terms of the discrete flight lengths. The exponential m two of which correspond to atomic oscillations in a decay means that the atomic transport is normal for well, and two other ones — to ballistic motion in the sufficientlylargevaluesofthejumpmagnitudesofthe optical lattice. variable u. We will call “a flight” such an event when atom The statistics of the center-of-mass oscillations in passes, at least, two successive antinodes (and three thepotentialwellscanbeobtainedanalogously. With nodes). The continuous flight length L>2π is a dis- large values of the jump magnitudes (24), trapping tance between two successive turning points at which occurs, largely, in the π-wide wells. The probability the atom changes the sign of its velocity, and the dis- for a trapped atom to cross the corresponding well crete flight length is a number of nodes l the atom node l times before escaping from the well is crossed. They arerelatedin asimple way,L πl, for ≃ P (l)=P lP = sufficiently long flight. tr − + Center-of-mass oscillations in a well of the optical arccosH arccosH (27) = 1 exp l ln . potential will be called “a trapping”. At extremely − π π small values of the detuning (the exact criterion will (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) be given below), the jump magnitudes are small and thetrappingoccurs,largely,inthe2π-widewells,i. e., C. Statistics of chaotic atomic transport at inthespaceintervalofthelength2π. Atintermediate small jump magnitudes of u valuesofthedetuning,itoccurs,largely,intheπ-wide wells, i. e. in the space interval of the length π. Far Inthis subsectionweconsiderthe caseofsmallval- from the resonance, ∆ & 1, trapping occurs only in ues of the jump magnitudes of the variable u | | the π-wide wells. Just like to the case of flights, the number of nodes l, atom crossed being trapped in a π π ∆ . (28) well, is a discrete measure of trapping. | | ω p ≪ 2 r r node Now,itmaytakealongtimeforanatomtoexitfrom B. Statistics of chaotic atomic transport at large one of the trapping or flight regions in Fig. 4. So, jump magnitudes of u we need to calculate the time of exit of the random variable θ arcsinu from one of these regions. m m ≡ If the jump magnitudes of the variable u are suffi- The result will depend on what is the length of the ciently large correspondingcirculararc in Fig.4 as comparedwith the jump lengths. π π Firstly, let us consider the case if the jump lengths ∆ & , (24) | | ω p 2 are small as compared with the lengths both of the r r node 8 flight and trapping arcs, i. e., if expressions (33), we replaced θ by arcsinH and max arccosH,respectively. Itiseasilytorealizethatwhen π j exceeds the value θ /(π√Dl), the corresponding ∆ min arcsinH,arccosH . (29) max | |rωrpnode ≪ { } terms of the series (33) decrease rapidly. ThePDFs(33)canbewritteninamuchmoresim- Motionofθ alongthe circlecanbe now treatedas a m ple form for two conditions imposed on the number one-dimensional diffusion process for a fictitious par- of jumps (node crossings) l a particle needs to quit ticle described by the equation the interval θ θ . If l & θ2 /D, then all the c ± max max ∂ (θ,m) ∂2 (θ,m) terms in the sums (33) are small as compared with P =D P , (30) the first one. Both the flight and trapping statistics ∂m ∂θ2 are exponential in this case. where is a probability density to find the particle To the contrary, if l θ2 /D, then one should P ≪ max at the angular position θ just after crossing the m-th takeintoaccountalargenumberoftermsinthesums node and D is the corresponding diffusion coefficient (33), and eachsum canbe replacedapproximatelyby of the particle the integral. In this case the result does not depend onthelengthoftheregion2θ andwegetthepower (θm θm−1)2 θm θm−1 2 ∆2π law decay max D h − i−h − i = , ≡ 2 4ω p r node ∞ (31) Q (j+1/2)2π2Dl which was calculated for the particle jumping ran- P(l) (j+1/2)2exp− dj ≃ θ3 θ2 ≃ domly with the mean magnitude θm θm−1 (equal max Z0 max h − i to zero in our case of symmetric jump distribution) Qπ−2.5D−1.5 θ2 and the variance h(θm−θm−1)2i. Thus, we reduced ≃ 4 l−1.5, l ≪ mDax the task to the firstpassagetime probability problem (34) for a continuous Markov process. Using the results both for the flight and trapping PDFs. The power- of this theory for a Wiener diffusion process [28] de- law statistics (34) implies anomalous atomic trans- scribedby Eq. (31), wecalculate the probabilityden- port. Dividing the length 2θ by the jump magni- max sity for a particle to exit from the interval θc θmax tude (28), we obtain the quantity ± after crossing l nodes ∞ lcr θmax/√D (35) 2πD ≡ P(l)= ( 1)j(j+1/2) θ2 − × that is a minimum number of jumps (node crossings) max j=0 X a particle needs to pass the regionthrough. With the cos (j+1/2)π(θ0−θc) exp−(j+1/2)2π2Dl, number of jumps l <lcr, a particle, randomly moving × (cid:20) θmax (cid:21) θm2ax on the circle, may get out of the interval θc ±θmax (32) only through the same border where it got in. The whereθ0isaninitialangularpositionoftheparticlein statistics of exit times is known to be a power-law the region under consideration and θc is the center of one with the transport exponent equal to 1.5. The − the region(with four possible values 0,π/2,π, 3π/2). length of the interval does not matter in this case. Since ina case ofsmalljumps a particle gets to the If l & l , then particles may exit through both the cr region near its limit point, it is possible to replace θ0 borders, and the corresponding statistics cannot be bythequantityθ0 =θc θmax+ǫwithasmallpositive approximated by a single transport exponent. If l & − valueofǫ. Expandingthecosinein(32)inthevicinity l2 θ2 /D, then we expect an exponential decay. cr ≡ max of the nodes and taking into account only the terms The size of the trapping and flight regions depends of the first order of smallness, we get the flight and on the value of the atomic energy H (see Fig. 4). At trapping PDFs H > √2/2 (arcsinH > π/4), the flight PDF has a ∞ longerdecaythanthetrappingPDF.Onthecontrary, P (l) Q (j+1/2)2 at H <√2/2, the Ptr’s decay is longer than the Pfl’s fl ≃ arcsin3H × one. If the jump magnitude (28)is ofthe orderofthe j=0 X size of the flight or trapping regions (j+1/2)2π2Dl exp− , × arcsin2H π π ∞ (33) ∆ arcsinH , P (l) Q (j+1/2)2 | |rωrpnode ∼ ≪ 2 tr ≃ arccos3H × or (36) j=0 X (j+1/2)2π2Dl π π ∆ arccosH , ×exp− arccos2H , | |rωrpnode ∼ ≪ 2 where Q is a normalization constant that is the same then a particle may pass through the region making inboththe cases. Whenderivingthe firstandsecond a small number of jumps l. So, the approximation of 9 the diffusionprocess(29) fails, andthe corresponding trapping events. They are normalized histograms for PDF is exponential. a number of standing-wave nodes l which the atom crossedinarun. EachPDFiscomputedwithasingle but very long (up to τ 108 for the basic equations 000 ∼ of motion) atomic trajectory. ---111 ---222 000 ---333 ---111 flflfl PPP ---444 ---222 000 111 ---555 flflfl ggg PPP ---333 ooo ---666 000 lll 111 ---444 ---777 ggg ooo ---555 ---888 lll ---666 ---999 ---777 ---111000 000 111 222 333 444 ---888 000 111 222 333 444 000 000 ---111 ---111 ---222 ---222 ---333 rrr ---333 PPPttt ---444 PPPtrtrtr ---444 000 111 ---555 000 ggg 111 ---555 lololo ---666 ogogog ---666 ---777 lll ---777 ---888 ---888 ---999 ---999 ---111000 000 111 222 333 444 ---111000 000 111 222 333 444 llloooggg lll 111000 llloooggg lll 111000 FIG.5: TheflightPfl and trappingPtr PDFsfor achaot- FIG. 6: The same as in Fig. 5 but with the energy ically moving atom with comparatively small values of the jump magnitudes of the variable u. The detuning H = 0.101 (p0 = 402) providing a narrow flight region ∆ = −0.001 is small, and the energy value H = 0.724 as compared with thetrapping one. (p0=535)providesapproximatelyequalsizesoftheflight InFig.5 we comparethe results(in a log-logscale) andtrappingregionsinFig.4. Whiteandblackcirclesrep- resent results of integration of the basic (5) and reduced in the case of small jump magnitudes of the variable (22) equations of motion, respectively, and the solid lines u (∆ = 0.001) and approximately equal sizes of represent the analytical PDFs (33). x0 = 0, z0 = −1, the flight−and trapping regions (H = 0.724 √2/2). u0 =v0 =0, ωr =10−5. Whiteandblackcirclesrepresenttheresultso∼fnumer- icalintegrationofthebasic(5)andreduced(22)equa- In order to check the analytical results obtained in tions of motion, respectively, and the solid line repre- this section, we compare them with numerical sim- sents the analytical predictions (33). The agreement ulation of the reduced (22) and basic (5) equations betweenthe PDFs computedwithEqs. (5), (22), and of motion. The reduced equations (22) describe the (33) is rather good. All the flight and trapping PDFs center-of-mass motion modulated by a jump-like be- exhibit in their dependence on the crossing number havior of the internal variableu which satisfies to the l three kinds of decay which are fairly approximated stochastic map (21). To explore different regimes of by the formulas (33). The critical value of the cross- the chaotic atomic transport we integrate Eqs. (22) ing number l is estimated to be l 55 with chosen cr ≃ and (5) numerically with two values of the detuning values of the parameters and initial conditions. In ∆ = 0.001 and ∆ = 0.01 (the cases of small and the range l . 55 the PDFs are expected to demon- − − medium jump magnitudes of the variable u, respec- strate the power law decay with the slope 1.5 given tively) and different values of the atomic energy and bytheformula(34). Intherange55.l .−l2 3000, cr ≃ momentum and compute the PDFs of the flight and there should exist a number of different transportex- 10 ponents. The initial partof this rangeis fitted by the 000 same power-law function. However, the other part of ---111 the range cannot be fitted by a simple function. In the range l & lc2r ≃ 3000, the decay is expected to PPPflflfl ---222 be purely exponentialin accordancewith the the first 000 ---333 expression in Eqs. (33). 111 ggg ---444 In Fig. 6 we show the PDFs computed at the same ooo detuning ∆ = 0.001 as in Fig. 5 but with the value lll ---555 − of the atomic energy H = 0.101 for which the flight ---666 region in Fig. 4 is narrower than the trapping one. ---777 Thejumpmagnitudeisnowofthe orderofthelength 000 111 222 333 444 000 of the flight region (see (36)) and the flight PDFs are expected to be exponential (l 9). The crit- ---111 cr ≃ ical value of the crossing number l for trapping is rrr ---222 lcr 58 with given values of the energy and the de- PPPttt ≃ ---333 tuning. Therefore, in the range l . 58 the trapping 000 111 PDFs are expected to vary as the power law decay ggg ---444 ooo with the slope −1.5 given by the formula (34). In lll ---555 the range l & l2 3400, the decay is expected to cr ≃ ---666 be purely exponential in accordancewith the the second expression in Eqs. (33). The trapping PDFs in ---777 000 111 222 333 444 the range 58 . l . l2 3400 are neither power law cr ≃ nor exponential demonstrating a complicated behav- llloooggg lll ior. This speculation is confirmed by the numerical 111000 computation shown in Fig. 6. In order to demonstrate what happens with larger values of the jump magnitudes, we take the detun- FIG.7: Theflight Pfl andtrappingPtr PDFsfor achaot- ing to be ∆ = 0.01 increasing the jump magnitude ically moving atom with comparatively large values of in ten times as−compared with the preceding cases. the jump magnitudes of the variable u. The detuning ∆ = −0.01 is medium, and the energy value H = 0.8055 With the taken value of the energy H = 0.8055 we provide a slight domination of flights over trappings. (p0=550) providesadomination oftheflight eventsover the trapping ones. White and black circles represent re- The jump magnitude is now so large that particles sultsofintegrationofthebasic(5)andreduced(22)equa- may pass both through the flight and trapping re- tions of motion, respectively,and thesolid lines represent gionsmakingasmallnumberofjumps. Itisexpected the analytical PDFs (33). x0 =0, z0 = −1, u0 =v0 =0, that all the PDFs, both the flight and trapping ones, ωr =10−5. should be practically exponential in the whole range of the crossing number l. It is really the case (see Fig. 7). cated structures that cannot be resolved in principle, no matter how large the magnification factor. The secondandthirdpanels in Fig.8demonstrate succes- VI. DYNAMICAL FRACTALS sive magnifications of the detuning intervals shown in the upper panel. Further magnifications reveal Various fractal-like structures may arise in chaotic a self-similar fractal-like structure that is typical for Hamiltonian systems [26, 29, 30]. In our previous pa- Hamiltonian systems with chaotic scattering [26, 30]. pers[17,23]wehavefoundnumericallyfractalproper- The exit time T, corresponding to both smooth and tiesofchaoticatomictransportincavitiesandoptical unresolved ∆ intervals, increases with increasing the lattices. In this section we apply the analytical re- magnification factor. Theoretically, there exist atoms sults of the theory of chaotic transport, developed in never crossing the border nodes at x = π/2 or − the preceding sections, to find the conditions under x=3π/2inspiteofthefactthattheyhavenoobvious which the dynamical fractals may arise. energyrestrictionstodothat. Tinyinterplaybetween We place atoms one by one at the point x = 0 chaotic external and internal atomic dynamics pre- 0 with a fixed positive value of the momentum p and vents those atoms from leaving the small space. The 0 compute the time T whenthey crossoneofthe nodes similar phenomenon in chaotic scattering is knownas at x= π/2 or x=3π/2. In these numerical experi- dynamical trapping. In Ref. [17] we have computed − ments we change the value of the atom-field detuning the Hausdorff dimension for the similar fractal and ∆ only. All the initial conditions p = 200, z = 1, shown that it is not an integer. We note that the 0 0 u = v = 0 and the recoil frequency ω = 10−5−are fractal-like structures similar to that shown in Fig. 8 0 0 r fixed. The exit time function T(∆) in Fig. 8 demon- arise in numerical experiments with longer distances stratesanintermittencyofsmoothcurvesandcompli- betweenthebordernodesbutthecorrespondingcom-